We have proposed a spin-wave transducer structure named film-penetrating transducers (FPTs). FPTs penetrate an on-chip magnetic film for a spin-wave transmission medium and allow flexible spatial arrangements of many exciters/detectors due to their zero-dimensional feature. We constructed four device models with different spatial arrangements of FPT/conventional exciters using a 10-nm-thick ferrimagnetic garnet film with a central FPT detector. We performed numerical experiments that combine electromagnetics with micromagnetics including thermal noise at 300 K. We evaluated important device features of FPTs, such as the signal-to-noise ratios (SNRs), input/output signal transmission efficiencies, and nonlinear phenomena of spin waves. We applied in-phase sinusoidal input currents with various amplitudes and frequencies and altered the damping strengths near the film boundaries. We obtained sufficient SNRs for the practical use of FPTs and revealed that FPTs have both higher transmission efficiencies and nonlinear strengths than conventional antennas, as the input frequency approaches the ferromagnetic resonance frequency of the film. Moreover, we observed and analyzed various nonlinear phenomena of spin waves, including beats in the time-domain waveform, components of integer harmonic frequencies, wide-range scatterings of inter-harmonic frequencies, and frequency doubling in spin precession. These characteristics probably originate from various device effects: FPTs effectively excite dipolar spin waves with large-angle precession, propagating spin waves reflect from the film boundaries, and spin waves dynamically and nonlinearly interfere with each other. This study demonstrated that FPTs have promising features for both their applications to reservoir computing and the studies on the physics of nonlinear and space-varying spin waves.

## I. INTRODUCTION

Spin-wave-based computation systems have been extensively explored in recent years due to their nature of low intrinsic power and high operation frequency.^{1,2} Spin-wave transduction methods, hence, are becoming crucial for investigations to suitably satisfy the requirements of various system designs. One possible approach that is actively being researched is the utilization of magnetoelectric effects,^{3–8} which usually relies on the physical properties of piezoelectric or magnetostrictive multi-layers that generate stress or strain to alter magnetic anisotropies. Excitation of spin waves based on magnetoelectric effects has been studied and examined at the interfaces of various compounding structures such as Ta/CoFeB/MgO,^{3} Fe/MgO(001),^{4} and other multiferroic heterostructures.^{6} Voltage control of magnetic anisotropy (VCMA) is another strategy that is similar to magnetoelectric effects, where the perpendicular magnetic anisotropy is modulated near the interface between an ultrathin magnetic metal and an insulator by a voltage-controlled electric field.^{7}

Spin-wave antenna is another typical transduction method that directly manipulates magnetic fields for transduction, especially in radio frequency operation. This type of antenna is usually made of microstrip lines or coplanar waveguides. Spin-wave antenna designs have been studied and experimented involving their mutual inductance, power transmission, and dispersion characteristics.^{9–15} A brief introduction of conventional spin-wave antenna designs is provided in Ref. 9. Studies and designs of conventional antennas with respect to their impedance matching and radiation efficiency under a linear spin-wave response are provided in Refs. 14 and 15. Thus far, most established spin-wave antennas share the same structural feature where they are flatly positioned on the top of a magnetic film. Coplanar waveguides based on this structure can excite and detect planar spin waves that have fundamental and harmonic wavelengths determined by the electrode period, the gaps between electrodes, and the wavelength-frequency dispersion of spin waves. Therefore, this conventional planar structure is beneficial and effective when spin waves are served as information carriers for data processing in binary logic and majority logic manners, where manipulation of wavelength/phase and reduction of nonlinear strength are required. For example, a spin-wave-based three-terminal logic gate design proposed in Refs. 16 and 17 incorporates spin-wave antennas using such a structure for signal transduction.

As an emerging machine learning framework, reservoir computing receives increasing attention from researchers.^{18,19} A reservoir computing system generally consists of an input part, a reservoir part, and a readout part. The parameters of the reservoir part are fixed and only the weights in the readout part are trained with an algorithm. Hence, the training process is fast and the training cost of such a system is significantly suppressed. The reservoir part of this system is not restricted to software recurrent neural networks but can also be achieved by certain physical phenomena that possess the following three key properties including (1) high-dimensional mapping, (2) nonlinearity, and (3) short-term memory. Such physical reservoir computing benefits from avoiding the implementation of complicated and massive network connections between neurons and has been deeply studied in a variety of research fields, such as electronics, photonics, and spintronics.^{20}

Spin waves can meet the requirements for a physical reservoir because of their input-history dependencies, high-dimensional mapping, and nonlinearity in propagation and interference.^{21,22} Recently, we have proposed an on-chip reservoir computing device based on spin-wave propagation and interference.^{23–26} We have conducted computation tasks including temporal XOR logic training, waveform classification, and delay tasks to verify its performance and properties in terms of reservoir computing.^{24,25} We have also numerically explored the arrangements of signal detectors in the designing perspectives of a platform reservoir chip.^{26} However, the spin-wave transduction method in our studies so far is not practically concerned, where spin waves are excited by modulation of the magnetic anisotropy of a ferrimagnetic film in micromagnetic simulations and the local magnetic moments (or in other words, spins) are straightly treated as the detected signals for readout computations. A more realistic spin-wave transduction process, thus, needs to be incorporated for the near-future realization of our system. In the context of reservoir computing, both exciting nonlinear spin waves with high-dimensional mappings and detecting such intricate information at numerous positions over the film are necessary to enhance the computational performance.^{26} However, the conventional planar antenna structure strictly limits the possible numbers of transducers that can be implemented on the film and are, thus, considered impractical for the application to a spin-wave reservoir computing device.

In this paper, we propose a new spin-wave antenna structure that we name film-penetrating transducers (FPTs) that penetrate an on-chip magnetic film for a spin-wave transmission medium. FPTs allow flexible and multiple arrangements of spin-wave exciters and detectors in the film and can be fabricated by the present technology such as reactive ion etching^{27} and focused ion beam.^{28} To examine the transmission of FPTs, we construct four device models with different spatial arrangements of exciters. The exciters in two of the models have FPT structure and those in the other two have conventional antenna structure. We numerically explored these device models with both input frequency and amplitude sweeps to investigate and compare the transmission efficiency between our proposed FPTs and the conventional antennas.

Although the influence of thermal gradient on spin-wave propagation is physically and experimentally studied in Ref. 29, to our best knowledge, it is often difficult to numerically evaluate the influences of thermal noise regarding spin-wave propagation due to the limitations of the micromagnetic simulation environment. We successfully construct a module for numerical analysis that integrates electromagnetics as well as thermal noise field into spin-wave micromagnetics. Hence, we also provide a thorough evaluation of our FPT structure in respect of power and signal-to-noise ratio (SNR).

We also investigate the decline in transmission efficiency that is observed in the amplitude sweep at the ferromagnetic resonance (FMR) frequency, where the FMR refers to a phenomenon in which the energy of applied electromagnetic waves is maximally absorbed by the magnetization of a magnetic medium. The FMR occurs when the frequency of the applied electromagnetic waves matches the precession frequency of the magnetization.^{30} To explore the physical mechanism behind the decline at the FMR frequency, we analyze the damping effects on the transmission, the frequency spectra of output signals, and the precession of spin waves. The results reveal the effective enhancement of nonlinear strength in output signals with respect to the increasing input power. These observations provide deep insight into the nonlinear phenomena of spin waves that are excited and detected by our proposed FPT structure.

This paper is organized as follows. Section II introduces the fundamental physics and structure of our proposed FPTs. Section III explains and compares transmission efficiency between the FPTs and the conventional spin-wave antennas. Section IV evaluates the practicality of the FPT structure in terms of power and SNR. Section V explores the decline in transmission efficiency and interprets the observed nonlinear phenomena in spin waves. Section VI discusses the applicability of our proposed FPTs to reservoir computing. Section VII concludes the paper.

## II. PROPOSAL: FILM-PENETRATING TRANSDUCERS

Spin-wave antennas excite torque on the local magnetic moments through the external magnetic field created by an alternating input electrical current. They are also able to detect fluctuations in the magnetic moments by collecting the induced current from their neighboring dynamic dipolar fields.^{1} Spin-wave antennas are, therefore, usually made of microstrip lines or coplanar waveguides.

Conventional spin-wave antennas are placed on the top of a magnetic film and are in parallel with the film plane. Figure 1(a) illustrates the fundamental structure of the conventional antennas. This structure is convenient when spin waves are applied to carrier waves for information transfer. For instance, coplanar waveguides based on this structure can control the fundamental wavelengths of spin waves to realize effective transmission while suppressing nonlinear phenomena in the propagation. However, the one-dimensional structure of the conventional antennas is considered very space-limited for the use of reservoir computing. This is because a large number of transducers at various positions over the film are required to create and capture high-dimensional spin-wave information for a high-performance reservoir computing device.^{25,26}

We propose a zero-dimensional FPT structure as shown in Fig. 1(b), where the transducers penetrate the magnetic film instead of being placed on the top. In terms of excitation, an alternating input electrical current in an FPT exciter induces magnetic fields that circulate the transducers, exert torque on the surrounding magnetic moments, and excite dynamical spin-wave motions toward every direction of the film plane. Moreover, an FPT detector picks up information on magnetic moments more locally in the form of the induced current for computations.

One of the potential benefits of FPTs is their efficient utilization of space. Depending on the specific design and size, FPTs can be implemented at diverse locations over the film. Hence, the FPT structure not only enables multiple detector arrangements to extract sufficient information for readout computation but also inspires different possible temporal and high-dimensional patterns of spin-wave propagation due to the unbounded choices of exciter placements.

## III. NUMERICAL ANALYSIS ON TRANSMISSION EFFICIENCY

In this section, we explain the details of our simulation modeling and numerical analysis to compare the transmission efficiency between conventional antennas and FPTs. Specifically, we explain the general modeling including the structure, initialization of magnetic moments, and material parameters of our spin-wave reservoir chip in Sec. III A. We then explain device models and experimental procedures for transmission analysis in Sec. III B. Next, we explain transmission coefficient per area as our evaluation criterion in Sec. III C. Finally, we present our results on the numerical analysis in Sec. III D.

### A. General modeling of spin-wave reservoir chip

We perform our modeling and simulation using COMSOL Multiphysics (COMSOL Inc.), where a combination of electromagnetics and micromagnetics is possible. Figure 2 presents the planar view of our spin-wave reservoir chip. The material parameters and simulation methods are mostly consistent with our previous works.^{23,25,26} The reservoir chip is a 2.2 $\xd7$ 2.2 $\mu $m$2$ square that consists of a 2 $\xd7$ 2 $\mu $m$2$ inner propagation area and a 0.1-$\mu $m-wide outer damper area. The damper area in Fig. 2 is shaded in gray. The Cartesian coordinates are defined as shown in Fig. 2, where the plane of the platform chip is defined as the $xy$ plane, and we define the $+z$ direction to be pointing out of the page. The film thickness is chosen as 10 nm, which is small enough to avoid formations of magnetic domains along the $z$-axis. Free tetrahedral meshes are adaptively generated with a maximum element length of 0.08 $\mu $m. A 0.1-$\mu $m-diameter output electrode (detector) is placed at the center of the film to detect and gather spin-wave information. The detector is assumed to be an FPT structure. All exciters and detectors in the modeling are assumed to be made of copper with an electrical conductivity $\sigma =6.0\xd7107$ S/m.

In our preliminary experiments, we put the ground of the electric circuit at the bottom side of the chip along the $z$-axis. We find that simulation results are similar to the situations when the ground is neglected and only sections of current-carrying wires with a uniform current density are considered, with only slight enhancement of the transmission efficiency. In addition, horizontal leads that connect the vertical FPTs to the power source are necessary for a complete circuit for the operation of FPTs, but these horizontal leads can be placed at more than a few tens or hundreds of nm above the magnetic film so that the magnetic fields created by these leads are small enough to be negligible. Thus, for the conciseness of our study, we assume all the exciters and detectors in our modeling to have connecting wires away from the chip and we neglect the horizontal leads since they do not affect the results. In the physical designing aspect of the reservoir chip, the placement of the ground can be thoroughly examined and carefully adjusted to suit the circumstances and improve the overall efficiency in the future.

Other properties of the reservoir chip are listed in Table I. These material parameters are taken from typical ferrimagnetic garnet films such as Tm$3$Fe$5$O$12$^{31} and Y$3$Fe$5$O$12$.^{32} Since we aim to develop forward volume spin waves (FVSWs),^{33} we initialize the magnetic moments of the film as uniform along the $+z$ direction with an external magnetic field of $\mu 0$H^{EX} = 0.03 T. The same $\mu 0$H^{EX} is continuously applied to the film during the simulation. Furthermore, we define an inclination angle $\theta $ of a magnetic moment to be the angle in absolute value between the direction of a magnetic moment and the $+z$ direction. This inclination angle $\theta $ will be used in Sec. V C.

Parameter name . | Value . |
---|---|

Damping constant α | 0.001 (film) or 1 (damper) |

Exchange stiffness constant A_{EX} | 3.7 × 10^{−12} J/m |

Saturation magnetization M_{s} | 100 kA/m |

Uniaxial magnetic anisotropy K_{U} | 5 kJ/m^{3} along z-axis |

Cubic magnetic anisotropy K_{C} | 0 kJ/m^{3} |

Environmental temperature T | 300 K |

Parameter name . | Value . |
---|---|

Damping constant α | 0.001 (film) or 1 (damper) |

Exchange stiffness constant A_{EX} | 3.7 × 10^{−12} J/m |

Saturation magnetization M_{s} | 100 kA/m |

Uniaxial magnetic anisotropy K_{U} | 5 kJ/m^{3} along z-axis |

Cubic magnetic anisotropy K_{C} | 0 kJ/m^{3} |

Environmental temperature T | 300 K |

### B. Device models, comparisons, and experiments

In this paper, we primarily study the efficiency evaluation of the proposed FPTs. We apply in-phase sinusoidal electrical currents as the input signals for the excitation of spin waves. Figures 3(a)–3(d) demonstrate different device models we constructed to compare the efficiencies of the spin-wave transmission between conventional antennas and FPTs. Blue cylinders represent 0.1-$\mu $m-diameter exciters with the same geometric and physical properties as the central output detector. The exciters in models 1 and 3 are our proposed FPTs and those in models 2 and 4 are conventional antennas with a 0.05-$\mu $m-spacing between the antenna bottom and the film top. Exciters in model 3 (6 FPTs on each side) can be further developed into a loop or coil structure for one wire realization so that the reuse of the input current is possible to efficiently utilize the input power.

To thoroughly evaluate each device model, we perform both frequency and amplitude sweeps of the input electrical current. For the frequency sweep, while we fix the amplitude of the input current at 5 mA, we sweep the input frequency from 5 GHz down to 0.5 GHz in steps of 0.5 GHz. We also estimate the FMR frequency of the ferrimagnetic garnet film to be 0.1675 GHz, which is determined by an FMR simulation method in Ref. 30. Because the spin-wave power and the transmission efficiency is expected to have the maximum at the FMR frequency, we include this FMR frequency in the frequency sweep as well. Sometimes, as the input frequency approaches the FMR value, spin-wave excitation becomes stronger and magnetic moments in certain locations can flip into the $\u2212z$ direction. In such instances, we employ a smaller input current accordingly.

We select three frequencies for the amplitude sweep, including 0.1675, 0.5, and 1 GHz. For each frequency, we start with the input amplitude at a reasonably small level and gradually increase its scale until it is large enough to flip the magnetic moments at some locations of the film. The total simulation time for a single experiment in both frequency and amplitude sweeps is 15 ns with a time interval of 0.01 ns and we start to supply an input electrical current to each exciter at 0 ns. We directly obtain the induced output current from the central FPT detector over the entire simulation time for evaluations and analyses.

From Subsection III C, we compare properties among models 1–4 and we particularly consider the following two comparisons:

- $\u2219$
Comparison 1: Model 1 (FPTs) vs model 2 (conventional antennas), where spin-wave excitation occurs at the opposite corners of the film.

- $\u2219$
Comparison 2: Model 3 (FPTs) vs model 4 (conventional antennas), where spin-wave excitation occurs at the opposite boundaries of the film.

### C. Transmission coefficient per area

The transmission coefficient is a standard evaluation criterion for a transmission system. However, it is intuitive that the larger space the transducers take up in the film, the greater the transmission coefficient the system will achieve. How much space of the film that transducers occupy is also one of the crucial factors that we should take into account. We, thus, consider transmission coefficient per area $T$ in $\mu $m^{−2} as our evaluation standard, where we divide the transmission coefficient by the contact area $Acontact$ of the exciters in $\mu $m^{2}. That is,

where $Iin2\xaf$ and $Iout2\xaf$ in $\mu $A denote the average root mean square values of the input signal $Iin$ and output signal $Iout$ over the entire 15 ns, respectively. The contact area $Acontact$ in $\mu $m^{2} is defined by the overlapping area between the exciters and propagation area of the film. Since the exciter arrangements are symmetric around the film center, we only take into account the contact area on one side of the film. Figures 4(a)–4(d) show the contact areas of models 1–4, which are represented by the green shaded areas. The contact areas of each model are also summarized in Table II.

### D. Simulation results

Figures 5(a) and 5(b), respectively, present the relationships between $T$ and the input frequency in comparisons 1 and 2. From Fig. 5(a), model 2 is area-wise efficient when the input frequency is at 2.5 GHz or beyond. On the other hand, model 1 becomes more efficient as the input frequency becomes lower than 2 GHz. The relationship in comparison 2 in Fig. 5(b) is complicated, which is likely due to the intricate spin-wave propagation dynamics of model 3. A similar rise is still found at lower frequencies, where model 3 performs better in respect of the transmission and area usage at 2.5 GHz or below. Additionally, we observe the maximum $T$ at the FMR frequency (0.1675 GHz) for every model as expected.^{30}

Next, we present the results on input amplitude sweep, where we explore three input frequencies including 0.1675, 0.5, and 1 GHz. Figures 6(a) and 6(b) present the three-dimensional plots of the input amplitude, frequency, and $T$ in both comparisons 1 and 2. For most of the cases, $T$ is nearly constant while the input amplitude changes, which suggests that the transmission efficiency is independent of the amplitude of the input current. On the other hand, we observe that $T$ decreases with increasing input amplitude at 0.1675 GHz, and this decrease is especially noticeable in model 3. We further analyze this efficiency decline at 0.1675 GHz in Sec. V.

## IV. EVALUATION OF POWER EFFICIENCY AND SIGNAL-TO-NOISE RATIO

In this section, we evaluate the practicality of the FPT structure by estimating the power efficiency and SNRs of each device model. The power efficiency $\eta $ in decibel (dB) is used to evaluate a transmission system and is calculated via

We straightforwardly estimate input and output powers according to the average levels of input and output currents. Even though in our circumstances, a higher $\eta $ is correlated with a larger contact area between the exciters and ferrimagnetic garnet film, we should confirm whether $\eta $ is reasonable even when the contact area is small. Figure 7 presents $\eta $ of each device model at respective input frequencies. The maximum $\eta $ ($\u2212$44.07 dB) is achieved in model 4 at the FMR frequency due to both the FMR and its large contact area. On the other hand, the minimum $\eta $ ($\u2212$92.58 dB) is found in model 1 at 5 GHz. For most of the cases, $\eta $ lies between $\u2212$55 and $\u2212$80 dB. Nonetheless, even for the worst-case scenarios, we consider this range of $\eta $ reasonably enough and physically feasible due to its sufficient SNR, as explained in the following part of the section.

To create a more realistic simulation environment, we include thermal noise in spin-wave simulations. Thermal noise is introduced in the form of a magnetic field known as the thermal noise field, which fluctuates in time and differs among magnetic moments.^{34} (Details are explained in the Appendix.) Since the magnitude of this thermal noise field strongly depends on the scale of $\alpha $, we experiment with three different $\alpha $ values, including $\alpha =10\u22123$, $10\u22124$, and $10\u22125$. Under each $\alpha $ value, we perform simulations for all the device models at each input frequency with the thermal noise field included.

Figure 7 shows the average current levels for signal $Isignal2\xaf$ of all device models (solid lines with different colors) and noise $Inoise2\xaf$ of all the experimented $\alpha $ values (gray dotted lines). The signal current $Isignal$ is obtained under a noiseless spin-wave propagation environment and the noise current $Inoise$ is determined by subtracting the noiseless signal current $Isignal$ from the overall output current $Iout$ where the thermal noise field is included. That is,

As shown in Fig. 7, $Inoise2\xaf$ for a given $\alpha $ value gradually grows as the frequency of the signal increases correspondingly, whereas $Inoise2\xaf$ of all the $\alpha $ values are generally below $Isignal2\xaf$ of all the models.

Next, SNRs in dB of various scenarios are calculated by

Figures 8(a)–8(d) three-dimensionally present the SNRs against both the input frequency and $\alpha $ for (a) model 1, (b) model 2, (c) model 3, and (d) model 4. These plots illustrate that SNRs mostly decrease with larger $\alpha $ and higher input frequency. Nonetheless, SNRs for most of the cases are beyond 10 dB, with only a few exceptions for model 1 as shown in Fig. 8(a). Even in the worst case (model 1 with $\alpha =10\u22123$ at 5 GHz), SNR is 4.53 dB, meaning that the signal level is at least comparable to the noise level of the output current. Additionally, Figs. 8(a)–8(d) can be a guideline for the selections of device models, $\alpha $ values, and input frequencies when a certain SNR is desired for our reservoir computing device.

## V. EXPLORATIONS OF EFFICIENCY DECLINE IN THE AMPLITUDE SWEEP EXPERIMENT

From the amplitude sweep explained in Sec. III, we observe an independent relationship between $T$ and the amplitude of the input current in most instances. At the same time, we notice a significant decline in $T$ with increasing input amplitude, specifically occurring in model 3 at 0.1675 GHz (estimated FMR frequency). Although the analytical frequency-to-wavenumber dispersion relationship allows us to easily gain insights into the spin-wave phenomena, there are difficulties to estimate it in our circumstances because of spatially and temporarily varied non-planar spin-wave patterns originating from nonlinear interferences and reflections of spin waves from multiple excitation sources. Hence, we study this efficiency decline using various other techniques in this section. We examine the influence of the damper area in Sec. V A. We analyze the nonlinearity of output signals from their frequency spectra in Sec. V B. We also analyze the average precession of the magnetic moments by interpreting the inclination angle to further study the nonlinear phenomena in Sec. V C. Finally, we experiment with another input frequency in Sec. V D.

### A. Damper area experiment

One possible explanation for the efficiency decline is a loss of energy due to the damping effects. An exploration of the damping effects and energy dissipation on spin-wave amplification for a two-layer multiferroic spin-wave amplifier is described in Ref. 35. Even though spin-wave modes and excitation strategies are different, we consider it worth examining the influence of the damper area in our models. In addition, more complicated spin-wave dynamics is expected to occur by adjusting the damping effects, such as reflections from the boundaries, which is potentially beneficial to certain reservoir computing tasks.

We specifically select model 3 at 0.1675 GHz input frequency for experiments, where the efficiency decline is observed. We ignore the thermal noise field in the explorations to reduce the complexity. We consider the following two experimental cases to comparatively investigate the damping effects:

- $\u2219$
With-damper cases: $\alpha =0.001$ for the propagation area and $\alpha =1$ for the damper area, with input amplitude of 0.1, 0.2, 0.3, 0.4, and 0.5 mA. (This is consistent with the amplitude sweep in Sec. III.)

- $\u2219$
No-damper cases: $\alpha =0.001$ for both the propagation area and the damper area with input amplitude of 0.1, 0.2, 0.3, 0.4, and 0.45 mA.

Since spin waves in the no-damper cases retain more energy from the excitation and some magnetic moments flip into the $\u2212z$ direction in the case of 0.5 mA input amplitude, we adopt a smaller amplitude value of 0.45 mA for the no-damper cases.

Figure 9 presents the transmission coefficient per area $T$ for both the with-damper and no-damper cases, represented by the solid and dashed lines separately. With the reduction of the damping effects, more energy is retained in spin waves and $T$ gets enhanced overall. Nevertheless, a downward trend is still observed. This suggests that the damping effects are not the main reason that causes the decline in $T$.

### B. Frequency spectrum analysis

Spin waves possess nonlinear propagation and interference that are useful and required for reservoir computing. Since we expect that the observed decline of $T$ is strongly related to nonlinear phenomena in spin waves, we analyze the frequency spectra of output signals at various input amplitudes. We extend the individual simulation time to 250 ns in order to enhance the frequency resolution. In addition, we include the no-damper cases into our analysis to comparatively interpret damping effects on the nonlinearity of spin-wave dynamics.

Figure 10 presents the time-domain current output signals at various input amplitudes for the entire 250 ns. The left column shows the output signals for the with-damper cases, whereas the right column is those for the no-damper ones. Beats are observed in the output signals and they are particularly visible in the no-damper cases. The slight differences in beat frequency are also observed with respect to the input amplitude. Figure 11 presents the frequency spectra of the output signals in Fig. 10. Since an analysis of steady-state output signal is desired for the application to reservoir computing, the first 100 ns of the output signals are removed to minimize the influence of transient response for the frequency analysis. The zero-padding and Hann windowing techniques are used to enhance the frequency resolution and to smooth out the padded signals, respectively.

We observe the main peak at 0.168 GHz in all of the spectra, which corresponds to the frequency of the input signal. (The values are slightly off due to the frequency resolution.) Moreover, we note two major observations from the frequency spectra. First, in the with-damper cases, we observe clear spectral peaks at 0.504 GHz for every input amplitude and at 0.336 and 0.840 GHz for the 0.4 and 0.5 mA amplitude cases. These peaks correspond, respectively, to the third, second, and fifth harmonics of the input frequency. As the input amplitude increases, the third harmonic component grows accordingly and the second and fifth harmonics start to appear as well. The dominance of odd harmonics (especially the third harmonic) as shown in Fig. 11 originates from the half-wave symmetry of the output signal, which is reasonable since the in-phase sinusoidal input signal is also half-wave symmetric. On the other hand, the appearance of the second harmonic suggests that the output signals become less symmetric in their waveforms as the input power increases.^{36} More importantly, this elevation of harmonics indicates that the input power partially contributes to the nonlinear strength in spin-wave dynamics.

Meanwhile, in the no-damper cases, the integer harmonics of the input frequency are not clearly visible, but wide-range scatterings of inter-harmonic frequency components emerge. One possible origin of such scattered frequency components is the magnon–magnon scattering process under the energy and momentum conservation. Although this nonlinear scattering process probably also occurs in the with-damper cases, this process is more obviously observed in the no-damper cases possibly because the removal of the damper area promotes stronger reflections of spin waves near the film boundaries. This phenomenon is experimentally studied in Ref. 37, where a larger excitation power leads to more energy transferred to the scattered spin waves, as a result of the four-magnon scattering process from spin-wave reflections in a lineshape ferromagnetic structure. Moreover, these significant variations in the nonlinear responses between the with-damper and no-damper cases infer that a variety of nonlinear spin-wave characteristics can be engineered with manipulation of the damping effects.

Second, we notice another dominant spectral peak next to the 0.168 GHz input frequency peak. We find that this peak has a larger magnitude in the no-damper cases than in the with-damper cases. We interpret this peak as the fundamental frequency of spin waves in the model. We also find that this fundamental frequency marginally grows with the increasing input power. That is, we sequentially observe this peak at 0.188 GHz for 0.1 mA, 0.192 GHz for 0.2 mA, 0.196 GHz for 0.3 mA, 0.2 GHz for 0.4 mA, and 0.204 GHz for 0.45 mA (no-damper) and 0.5 mA (with-damper). We consider this upshift of the fundamental frequency to result from a positive nonlinear frequency shift of spin waves in the out-of-plane magnetized film under the FMR response that potentially leads to the well-known effects called FMR foldover, where the line shapes of FMR absorption profiles become distorted with upshift/downshift of the maximum absorption peak under strong excitation powers.^{38,39} A similar input power dependence of the fundamental frequency in a yttrium iron garnet (YIG) is observed in Ref. 40. Furthermore, this peak may also explain the occurrence of beats in Fig. 10, where the estimated FMR frequency (0.1675 GHz) and the observed fundamental frequency (varying with the input amplitude) are nearly equal to each other so that the time-domain beats are created while these two frequencies are combined.^{41} Since these fundamental frequency peaks in the no-damper cases have larger magnitudes than those in the with-damper cases, the beats in the no-damper cases are more evident in Fig. 10 as the mixing of frequencies strongly distorts the output waveforms.

Thus far, our explorations mainly focus on the current output signals that only reflect partial information of the spin waves. Therefore, we further interpret the inclination angle $\theta $ of the magnetic moments to comprehend their precession and related nonlinear phenomena in Subsection V C.

### C. Precession interpretation

To understand the intrinsic properties behind the efficiency decline three-dimensionally, we also interpret the precession of the magnetic moments in this subsection. The precession angle, which refers to the orientation angle of the temporal rotational axis for a magnetic moment, is normally examined for the interpretation of precession.^{35} In our circumstances, determining and effectively representing the temporal rotational axis for each magnetic moment in the film at each time step is challenging. Since the external magnetic field $\mu 0$H^{EX} = 0.03 T is continuously applied toward the $+z$ direction of the film, we instead consider the inclination angle $\theta $, which we define as the angle of inclination between the direction of a magnetic moment and the $+z$ direction. We calculate the spatially averaged inclination angle $\theta \xaf(t)$ over the entire film as

where $\theta i,j,k(t)$ is the inclination angle in the magnetic moment at time $t$ and the ($x=i$, $y=j$, $z=k$) position in the film.

Figure 12 shows $\theta \xaf(t)$ in time domain at various input amplitudes. Figure 13 shows the frequency spectra of the time-domain $\theta \xaf(t)$. As shown in Fig. 12, we observe both short-period and long-period oscillations. First, the short-period oscillation corresponds to the major spectral peak that we observe at 0.336 GHz in all the cases in Fig. 13. We note that this peak has a frequency that is twice the input frequency (0.1675 GHz). Generally, “frequency doubling” occurs in the longitudinal magnetization component when the precession lacks a cylindrical symmetry or the external magnetic field is not circularly polarized.^{42} This frequency doubling in Fig. 13 probably arises from various intricate spin-wave dynamics, such as the unpolarized alternating magnetic fields from FPTs, nonlinear spin-wave scatterings induced by the spin-wave reflections from film boundaries with/without the damper area, time-dependent interaction among incoherent spin waves, and time-dependent dipolar fields among the magnetic moments with large $\theta i,j,k(t)$. All these factors possibly contribute to an elliptical trajectory of spin motions and result in the frequency doubling in the precession. Since the frequency doubling is clear evidence for nonlinear phenomena of spin waves, this observation also supports the enhancement of nonlinear strength in model 3, as discussed previously in Sec. V B.

Next, we also observe a long-period oscillation in the time-domain $\theta \xaf(t)$ in Fig. 12. This long-period oscillation corresponds to the low-frequency component in Fig. 13, as observed at 0.02 GHz for 0.1 mA, 0.024 GHz for 0.2 mA, 0.028 GHz for 0.3 mA, 0.032 GHz for 0.4 mA, and 0.036 GHz for 0.5 mA (with-damper) and 0.45 mA (no-damper). This low-frequency component appears possibly due to the frequency mixing of the two major spectral peaks that are observed in Fig. 11. In specific, we find that the values of these low-frequency components in Fig. 13 match the differences in the frequencies between the two major peaks in Fig. 11. Furthermore, a third major peak is observed in each no-damper case in Fig. 13 at 0.356 GHz for 0.1 mA, 0.360 GHz for 0.2 mA, 0.364 GHz for 0.3 mA, 0.368 GHz for 0.4 mA, and 0.372 GHz for 0.45 mA. These frequency values match the summations of the frequencies of the two spectral peaks in Fig. 11. This third peak is not apparent in the with-damper cases in Fig. 13 since the power of the fundamental frequency component is weaker according to Fig. 11. This can also explain why the long-period oscillation in Fig. 12 is rather significant in the no-damper cases but is less noticeable in the with-damper cases. Moreover, we note that this long-period oscillation is mainly observed in the transient state and we expect it to vanish in the steady state of the system, as seen from the decay in the envelope of the waveform in Fig. 12.

Next, we interpret $\theta \xaf(t)$ in relation to the spin-wave transmission. A larger $\theta \xaf(t)$ refers to a greater deviation of magnetic moments from the $+z$ direction. In our circumstances, we consider it appropriate to link $\theta \xaf(t)$ to the strength of excitation of spin waves. In other words, when an averagely larger $\theta \xaf(t)$ is observed, we expect the excitation of spin waves to be stronger as well. We define the term “excitation efficiency” $E$ in radian per mA as

where $\theta \xaft$ refers to averaged $\theta \xaf(t)$ over the entire 250 ns and $Iin2\xaf$ is the average input current level in mA. The excitation efficiency $E$ is defined in this way to find its relation to the spin-wave transmission efficiency $T$ as defined in Eq. (1).

Figure 14 presents $E$ for all the experimental cases of model 3 at the 0.1675 GHz input frequency, where solid and dashed lines, respectively, represent $E$ for the with-damper and no-damper cases. Although the units are different, we observe similar trends between $E$ in Fig. 14 and $T$ in Fig. 9. This similarity not only suggests a strong correlation between the spin-wave excitation and transmission of our system but also infers that the decrease in $E$ substantially contributes to the decline in $T$. Moreover, according to the discussion in Sec. V B, the decrease in $E$ is also accompanied by the shift of the input power to higher-frequency harmonic components. This suggests that when the input power for the spin-wave excitation is large enough, the precessions of the excited magnetic moments become more elliptical and their complicated trajectories contribute to the nonlinear spin motions and interactions.

On the other hand, we do not observe a decline in $T$ of model 4 (the conventional antenna structure) but rather the nearly constant relationship in Fig. 6(b). This suggests that $E$ of model 4 is also nearly constant in the amplitude sweep and the enhancement of the nonlinear strength in model 4 is much weaker than that in model 3, under the same input power levels. Even though it is theoretically possible to enhance the nonlinear phenomena to the same degree by the conventional antennas with large enough input power, we find that the FPT structure can achieve the same strength of nonlinearity with much less input power needed. This power-effective development of nonlinearity by FPTs is likely due to the penetrating structure that allows direct excitation of spin waves inside the body of the ferrimagnetic film. On the contrary, the conventional transducers are placed on top of the film with a small gap in between. Therefore, their induced magnetic fields are expected to decay exponentially with the distance from the gap and their excitation strength of spin waves in the film body is, thus, significantly smaller.

### D. Experiment with a different frequency

For further study, experiments with an input frequency of 1 GHz are examined using input currents 0.1 and 0.5 mA. In both cases, we apply model 3 with the thermal noise field ignored and $\alpha =1$ in the damper area.

Figure 15(a) presents the output signals in time and frequency domains for both cases. The steady-state frequency spectra are determined consistently as those in Fig. 11. We observe a main spectral peak at 1 GHz together with a peak of the fundamental frequency at 0.188 GHz in the frequency spectra. Unlike the preceding experiments in Sec. V B, the fundamental frequency does not shift rightwards as the input power increases from 0.1 to 0.5 mA. This is probably because the excited spin waves at 1 GHz are much weaker than those at 0.1675 GHz and, thus, the intricate and nonlinear spin-wave dynamics also occurs differently. We also notice a small appearance of the third harmonic peak (3 GHz) in the 0.5 mA input case in Fig. 15(a), but this nonlinear strength is much less significant as compared to those in the 0.1675 GHz input frequency cases, as confirmed in Fig. 11.

Figure 15(b) shows the time-domain $\theta \xaf(t)$ and their respective frequency spectra for both 1 GHz cases. From their frequency spectra, we observe similar phenomena including frequency doubling and mixing as we do in the 0.1675 GHz input frequency cases in Fig. 13. This observation indicates that such nonlinear phenomena originate from a similar mechanism even when the input frequency is changed. On the other hand, we apply Eq. (6) and estimate $E$ for both cases to be 0.0510 rad/mA (0.1 mA case) and 0.0508 rad/mA (0.5 mA case). The estimated $E$ for both cases are approximately the same, which is consistent with the observation of the amplitude-independent $T$ at 1 GHz in Fig. 6(b). Overall, $E$ at 1 GHz is much smaller than that at 0.1675 GHz, whereas nonlinear phenomena are still observed such as the frequency doubling and the harmonic frequency component. This indicates that nonlinear phenomena can be introduced and strengthened even when the spin-wave excitation is not as strong as it is under the FMR frequency.

## VI. DISCUSSION

In this section, we discuss the application of FPTs to the on-chip platform reservoir computing device. It is known that successful reservoir computing requires the reservoir part to output signals with high dimensionality, nonlinearity, and short-term memory.^{20} The following describes how such features are expected to be realized through physical phenomena developed by our spin-wave reservoir chip with FPTs.

High-dimensional mapping: This refers to the space-varying spin-wave patterns observed at different positions among the chip. Since this paper primarily focuses on the transmission efficiency $T$ of FPTs, we apply simple in-phase sinusoidal input signals. We expect the high-dimensionally mapped and space-varying outputs to be easily obtained with FPTs when informative input signals are used, such as various-phase and/or various-magnitude input currents as we previously did in Refs. 23–26.

Nonlinearity: This refers to the nonlinear phenomena of spin waves during their propagation and interference that lead to a nonlinear relationship between inputs and outputs. Our previous experiments in Ref. 24 have demonstrated that the degree of nonlinearity heavily influences the computational capability of the reservoir computing device. We expect that with the effective control of nonlinear phenomena by FPTs, the requirement of nonlinearity can be easily met by our on-chip reservoir computing device.

Short-term memory: This refers to the temporary influence of former information on the processing of latter information. Our previous works have comprehensively examined the property of short-term memory in the outputs of our spin-wave reservoir device.

^{23–26}Therefore, we expect similar short-term memory to be easily achieved in signal outputs by our reservoir device with FPTs.

Moreover, according to the results in this paper, we expect that various spin-wave characteristics can be realized by flexibly adjusting the number and position of FPT exciters and detectors, changing the input frequency and input amplitude, and modifying the damping constant $\alpha $ and the damper area. Additionally, the thickness of the garnet film can be varied and controlled to realize more complicated spin-wave dynamics that can enhance the transmission efficiency and diverse output signals. We expect that future studies on these characteristics will also contribute to the physical realization of a high-performance reservoir chip together with a deep understanding of spin-wave physics.

## VII. CONCLUSION

This paper proposed the structure of FPTs for spin-wave excitation and detection. FPTs are particularly advantageous in their zero-dimensionality, which enables a large number of exciters and detectors arranged flexibly in the ferrimagnetic garnet film. The FPT structure, thus, allows sufficient outputs for the readout of reservoir computing and provides numerous possible temporal patterns of spin-wave propagation and interference for the design of reservoir computing devices. Our simulation results clearly demonstrated that the FPT structure is more efficient than the conventional antenna structure with respect to transmission and area usage of the ferrimagnetic film, particularly at the lower input frequencies (2 GHz or below for model 1 and 2.5 GHz or below for model 3). We also found the power efficiency and SNRs of FPTs to be promising for practical use at 300 K. Moreover, we revealed and discussed some nonlinear phenomena in the complicated spin-wave dynamics of our reservoir chip, including the appearance and enhancement of harmonic frequency peaks, the input-power-dependent increase in the fundamental frequency, the spin-wave scatterings of wide-range inter-harmonic frequency components due to reflections from the film boundaries with/without the damper areas, and the frequency doubling and mixing in spin-wave precession. We found that FPTs can effectively enhance the nonlinear strength of spin waves, which is probably due to their penetrating structure that provides direct contact with the ferrimagnetic film for efficient spin-wave excitation.

Whereas our main scope was to evaluate the capabilities of FPTs for their application to reservoir computing devices, this paper also demonstrated that FPTs are very useful for studies on unclarified areas of spin-wave physics, including space-varying and instantaneous nonlinear phenomena.

## ACKNOWLEDGMENTS

This work was supported in part by the Cooperative Research Project Program of Research Institute of Electrical Communication (RIEC), Tohoku University, and in part by the New Energy and Industrial Technology Development Organization (NEDO) under Project No. JPNP16007.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jiaxuan Chen:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Ryosho Nakane:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). **Gouhei Tanaka:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). **Akira Hirose:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: NUMERICAL METHODS

In this appendix, we explain our numerical methods for the study of our proposed FPTs. The simulations are performed by COMSOL Multiphysics (COMSOL Inc.), where a finite element method is employed for numerical computations and a combination of multiple physical fields of study is possible. We apply the inherently constructed AC/DC Module in COMSOL Multiphysics to study and analyze the electromagnetic systems and processes. We created a custom mathematical module to numerically solve the partial differential equation known as the Landau–Lifshitz–Gilbert (LLG) equation for spin-wave micromagnetics.

LLG equation is expressed as

with the following constraints:

where $M$ is the magnetization over space and time in A/m, $\gamma $ is the gyromagnetic ratio, $\mu 0$ is the permeability constant, $Heff$ is the overall effective magnetic field in A/m, $\alpha $ is the damping constant, $Ms$ is the saturation magnetization in A/m, and $m$ is the magnetic moment, in the form of a unit vector, over space and time.^{1} There, $Heff$ consists of various types of magnetic fields, including exchange field, external field, magnetic anisotropy field, and demagnetization field. We adopt the conventional weak formulation of the LLG equation as provided in Refs. 43 and 44. The weak formulation of the LLG equation is expressed as

where $\varphi $ is the test vector that belongs to the same vector space as $m$ and $\Omega $ represents the entire magnetic domain. One challenge to seek a solution to the LLG equation using weak formulation is the unit vector constraint of $m$, as illustrated in Eq. (A2). In this work, since FVSWs are studied with initialization of the $+z$ direction for every magnetic moment, we assume that the z-component of $m$ to be always positive and is computed from the x-component and y-component of $m$ to handle this unit vector constraint. We are considering employing Lagrange multipliers for equality constraints to deal with situations when the z-component of $m$ needs to be negative in our future work.

Furthermore, we make the following assumptions about $Heff$ as

where $Hexchange$ is the magnetic field due to the exchange interaction/coupling between neighboring magnetic moments, $Hexternal$ is the externally applied magnetic field (which is associated with input signals), $Hanisotropy$ is the magnetic field due to anisotropic behavior of magnetization from the crystal structure of the material, $Hdemag$ is the demagnetization field that is created by the magnetization itself, and $Hthermal$ is the thermal noise field.

The analytical expression of $Hexchange$ is shown as

Since $Hexchange$ is dependent on $\Delta M$, it is important that we apply the test vector for weak formulation to this expression. As a result, we introduce $Hexchange$ to Eq. (A3) by

where $AEX$ is the exchange stiffness constant in J/m, $Heff\u2217$ is the overall effective field excluding $Hexchange$, and $x1$, $x2$, $x3$ are numerical forms of the $x$, $y$, $z$ directions, respectively.

Next, $Hanisotropy$ is expressed as

where $KU$ is the uniaxial magnetic anisotropy in kJ/m^{3}, $KC$ is the cubic magnetic anisotropy in kJ/m^{3}, and $u$ is the direction vector of the easy axis, which depends on the crystal structure.

Moreover, $Hdemag$ is determined by using finite element methods to solve the following Maxwell’s equations:

where $B$ is the magnetic flux density in T under magneto-static assumptions.^{45}

Finally, $Hthermal$ is expressed by

where $\eta $ is a three-dimensional vector, with each entry containing a normally distributed random variable, $kB$ is the Boltzmann constant, $T$ is the temperature of the chip, which we assumed to be constantly 300 K, and $\Delta V$ is the unit cell volume of the simulation.^{34} Since our meshes are adaptively generated, we estimated this volume from the average mesh size as $6\xd710\u22126\mu m3$. Finally, $\Delta t$ is the unit time step of the calculation. Since the bandwidth of the signal grows as the frequency of input current signals increases, we accordingly adjusted the unit time step of each simulation. That is,

where $fin$ is the frequency of input signals. This relation is also reflected in the gradual growth of noise with respect to the increasing input frequency, as shown in Fig. 7.

## REFERENCES

*2020 International Joint Conference on Neural Networks (IJCNN)*(IEEE, 2020), p. 1.