Reservoir Computing with Spin Waves
Propagating via a Continuous Magnetic Film on a Chip
Ryosho Nakane1 and Gouhei Tanaka1, 2, 3
1. Department of Electronic Engineering and Information Systems, Graduate School of Engineering
2. International Research Center for Neurointelligence (IRCN)
3. Department of Mathematical Informatics, Graduate School of Information Technology and Science
The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113-8656, Japan

Summary. Numerical experiments on spin-wave-based reservoir computing (RC) are studied using
our on-chip device with a continuous ferrimagnetic yttrium iron garnet (YIG) film. RC is performed
with a reservoir output vector extracted from multiple device output signals in response to an input
signal that is a time-series 1-GHz cosine carrier wave with 6- or 4-quantized amplitude modulation.
In the 10th-order nonlinear autoregressive moving average task and channel equalization task, weachieve high prediction performance comparable to those obtained by RC with optical systems.

Toward drastic advances in Internet-of-Things (IoT) technology, increasing demand is to create
on-chip machine-learning devices that allow energy-efficient information processing of time-series datanear edges of information networks. Physical reservoir computing (RC) has a high potential for tackling
this challenge because it is suitable for temporal data processing and feasible devices are applicable [1]. A
successful RC can be achieved through a mechanism that a reservoir with rich physical dynamics
nonlinearly transforms a time-series input data to a temporal higher-dimensional output state vector, i.e., areservoir device generates diverse output signals in response to an input signal. Nonetheless, simple
solutions for achieving such function can be hardly found for on-chip devices, since there are manydifficulties, such as intractable technical problems in fabricating enormous internal wirings andunestablished device design for multiple output nodes up to a few hundreds. Using virtual nodes and a
feedback loop can mitigate strict conditions, but it imposes large overheads, such as high-speed circuits for
output signal sampling, as well as possibly causes an unstable operation under a high feedback gain. In
addition, since robust, reproducible, and tolerant device characteristics are strongly required for long-term
use in practical applications, it is desirable to use a fatigue-tolerant property, e.g., an electron-relatedquantity. Motivated by these backgrounds, we have studied spin-wave-based RC device and system [2–4].
A magnetic material has spontaneous magnetic moments originating from its electronic structure.
Here, such a magnetic moment is called “spin” for simplicity. Spin waves indicate transmission of spin
motions, and they have rich dynamical properties including emergent phenomena, such as linear / nonlinear
propagation and interference, parametric oscillation, bifurcation, and chaotic behavior, which are causedby dynamical competitions between short-range parallel or antiparallel interactions (spin exchangeinteractions), common forces acting on each spin (magnetocrystalline anisotropic and external magneticfields), and long-range antiparallel interactions (magnetic dipole interactions). We proposed an RC systemwith our spin-wave-based device, as shown in Fig. 1, where we eliminate a top magneto-electric couplinglayer and topmost exciter input / detector output electrodes for a conceptual illustration. A notable featureis that a continuous ferrimagnetic yttrium iron garnet (YIG) used as an excitable media allows nointerconnections, leading to a feasible structure with many output electrodes. The device operation is asfollows. Spin waves are excited through an application of a time-series input voltage to the input electrodeat the center of the plane and they propagate toward various directions. The resulting spin motions spatiallydistributed in the YIG are obtained as time-domain waveforms at output detectors outside the exciter. A
key to high computational performance is how to realize diverse output signals through the spatial dynamics.

Figure 1 . Spin-wave-based reservoir computing system

Numerical experiments on spin waves are performed using a micromagnetic simulator, where a
YIG film is 12 µm × 12 µm × 320 nm in volume, material parameters estimated from actual experiments
in literatures are adopted, an input exciter at the center of the x-y plane is 800 nm × 4 µm × 40 nm involume, and 120 output detectors regularly arranged on the x-y plane are 200 nm × 200 nm × 40 nm involume (Fig. 2(a)). To realize spatially-distributed rich dynamics, we used two approaches. The first
approach is to use small external magnetic fields µ0Hex = 2, 5, and 8 mT along the y direction, under which
unstable spin motions leading to nonlinear phenomena are expected. In addition, such µ0Hex values realize
stripe magnetic domain structures, as shown in Fig. 2(b), which can enhance multiple reflections duringspin-wave propagations. Dynamical properties of the entire YIG film are characterized by the ferromagnetic
resonant frequency fFMR and transient relaxation time : (in increasing µ0Hex order) fFMR = 0.98, 1.17, and1.34 GHz and  = 5.4, 5.3, and 10.7 ns. The second approach is to collect output signals under all the µ0Hexvalues to construct a temporal reservoir state vector y with a Ny dimension (Ny = 120, 240, 360). From theµ0Hex-dependent properties, it is expected that computational performance is enhanced with increasing Ny.
We perform two time-series benchmark tasks: the 10th-order nonlinear autoregressive moving
average (NARMA-10) task and nonlinear channel equalization (NCE) task with signal-to-noise ratio (SNR)12 – 32 dB. In the preprocessing part, an input signal K(t) corresponding to a time-series magnetic
anisotropy in the exciter is prepared, where a sequential 6- or 4-quantized random values u(n) with a time
step n (= 1 – 8100) are linearly converted to the amplitude of a 1-GHz cosine carrier wave with the nduration of 2 ns. While K(t) is applied, xi(t) at the ith detector (i = 1 – 120) is estimated from the averaged
vertical z component of spins in each detector (Fig. 2(c)). Then, y(n) of dimension Ny is generated fromxi(t) through postprocessing with envelope processing, low-pass filtering, and averaging over each time stepduration. In a training phase, y(n) with n = 101 – 5098 are used to form a reservoir state collection matrix
Y. Next, an optimum weight matrix Ŵ is calculated with a pseudo-inverse matrix of Y and a teacher
collection matrix D for system outputs. In a testing phase, a system output vector z*(n) = Ŵy(n) iscalculated with unknown u(n) in n = 5099 – 8098, and the computational performance is evaluated withnormalized mean square error (NMSE) for NARMA-10 and symbol error rate (SER) for NCE.
We find that NMSE and SER decrease as Ny is increased. The results are clear evidence that thecollection of diverse output signals efficiently increases the dimensionality of y(n) (i.e. reservoir-staterichness) and thereby contributes to high computational performance. The lowest NMSE = 0.042 at Ny =360 and SER = ~6 × 10-4 (under SNR = 24 dB) at Ny = 120 (Fig. 2(d)) are comparable to those obtainedwith optical RC systems and the computational performance is highest among emergent RC discrete devices
on a chip. Our spin-wave-based RC system is promising for future practical applications in IoT technology.

Figure 2. (a) Top-view schematic illustration of a 320-nm-thick YIG film with an input exciter and 120 detectors. (b)Domain structure (the z component of normalized spin). (c) Input signal and output signals at three different detectors.
(d) Symbol error rate (SER) plotted against signal to noise ratio (SNR) in the nonlinear channel equalization task.
(Figs. 2(a) and (b) are reproduced from [4], licensed under CC-BY 4.0.)

References[1] G. Tanaka, et al., “Recent advances in physical reservoir computing: A review”, Neural Netw. 115, 100 (2019).
[2] R. Nakane, G. Tanaka, and A. Hirose, “Reservoir computing with spin waves excited in a garnet film”, IEEE
Access 6, 4462 (2018).
[3] R. Nakane, A. Hirose, and G. Tanaka, “Spin waves propagating through a stripe magnetic domain structure andtheir applications to reservoir computing”, Phys. Rev. Res. 3, 033243 (2021).
[4] R. Nakane, A. Hirose, and G. Tanaka, “Performance enhancement of a spin-wave-based reservoir computing
system utilizing different physical conditions”, Phys. Rev. Appl. 19, 034047 (2023).