Spin waves propagating through a stripe magnetic domain structure and their applications to reservoir computing

Spin waves propagating through a stripe domain structure and reservoir computing with their spin dynamics have been numerically studied, focusing on the relation between physical phenomena and computing capabilities. Our system utilizes a spin-wave-based device that has a continuous magnetic garnet film and one-input/72-output electrodes on top. To control spatially distributed spin dynamics, a stripe magnetic domain structure and amplitude-modulated triangular input waves were used. The spatially arranged electrodes detected spin vector outputs with various nonlinear characteristics that were leveraged for reservoir computing. By moderately suppressing nonlinear phenomena, our system achieves $100\%$ prediction accuracy in temporal exclusive-OR problems with a delay step up to 5. At the same time, it shows perfect inference in delay tasks with a delay step more than 7 and its memory capacity has a maximum value of 21. This study demonstrated that our spin-wave-based reservoir computing has a high potential for edge-computing applications and also can offer a rich opportunity for further understanding the underlying nonlinear physics.

Figure 1

Spin-wave-based reservoir computing system that is composed of an input preprocessing, a reservoir part, and a readout part. The computing is executed following the order of (1)–(8) in the training, whereas it is executed following the order of (3)–(6) and then (9)–(10) in the testing. In a spin-wave-based reservoir device in the reservoir part, a red area is the input exciter, a zebra pattern with pale and deep greens represents a stripe magnetic domain structure, and bended black arrows schematically illustrate the expected propagation of spin waves with multiple reflections by the magnetic domains. Blue curves schematically illustrate time-series output wave forms obtained at 72 detectors in the device

Figure 2

Schematic illustration of a spin-wave-based reservoir computing device that is composed of (from the bottom to top) a conductive substrate, a magnetic garnet film, a magnetoelectric coupling layer, and input (a red cuboid)/output (gray cylinders) electrodes. The stripes in the magnetic garnet film denote a stripe magnetic domain structure. In the operation of an actual device, spin waves in the magnetic garnet film are excited by an input voltage, they propagate through the magnetic garnet film, and their dynamics beneath the output electrodes are detected by output voltages.

Figure 3

(a) Top and (b) cross-sectional views of the spin-wave-based reservoir computing device utilizing a magnetic garnet film, where a Cartesian coordinate is defined in each figure, pale-green and deep-green regions are the transmission and damping regions of spin waves in the garnet film, respectively, and red and blue regions are the input exciter and detectors, respectively. The damping constant ${\alpha }_0$ in the transmission region is 0.001, whereas the damping constant $\alpha$ in the damping regions is varied ($\alpha$ = 0.001, 0.01, 0.1, and 1) to examine the change in spin dynamics, as described in Sec. 3b. The $x\text{−}y$ plane and thickness along the $z$ axis of the magnetic garnet film is 12 $\times$ 12 $\mu {\text{m}}^2$ and 320 nm, respectively. The depths of the input exciter and detectors are 40 nm from the surface, as shown in (b). (c) Detailed top-view structure of the input exciter and detectors on the top of the garnet film, where detectors with 200 $\times$ 200 ${\text{nm}}^2$ are arranged with 400 nm and 1000 nm in pitch along the $x$ and $y$ axes, respectively, and the number is defined for each detector. All the detectors are divited into four groups: areas 1, 2, 3, and 4, each of which has 18 detectors.

Figure 4

(a) Top and (b) cross-sectional views of the distributions of $s_z$ in the magnetic garnet film after the relaxation under a magnetic field ${\mu }_0{H}^{\text{EX}}$ = 0.005 T along the $+y$ direction, where the magnitude is defined by a color bar. The cross-sectional view is along the $x$ axis between two open arrows and it is extended along the $z$ axis for visibility.

Figure 5

(a) Example of a reservoir input signal $K_{\text{u}}\left(n\right)$ with the time step length $X$ = 4 ns: 0 and 1 in $u\left(n\right)$ are expressed by $\mathrm{\Delta }{K}_{\text{u,}\text{0}}$ = $K_{\text{u}}^{\text{H}}-K_{\text{u}}^{\text{L}}$ and $\mathrm{\Delta }{K}_{\text{u,}\text{1}}$ = $K_{\text{u}}^{\text{H}}-K_{\text{u}}^{\text{M}}$, respectively, with a use of $K_{\text{u}}^{\text{H}}$ as the baseline, where $K_{\text{u}}^{\text{H}}$, $K_{\text{u}}^{\text{M}}$, and $K_{\text{u}}^{\text{L}}$ are 5.0, 4.5, and 4.0 $\text{kJ}/{\text{m}}^3$, respectively. To analyze the effect of $X$ on the properties of spin waves as well as on the capability of the reservoir computing, reservoir input signals with various $X$ values were prepared; $X$ = 1, 2, 4, 6, and 8 ns. The repetition number of triangular pulses for one bit can be calculated such that $X$ ns is divided by 1 ns. (b) Schematic illustration of a DFT spectrum, where $f_{\text{IN}}$, odd, and even denote the pulse repetion frequency (1 GHz), the odd-number harmonic frequencies (3 and 5 GHz), and the even-number harmonic frequencies (2 and 4 GHz), respectively, and sub-$f_{\text{IN}}$ and IH denote interhamonics below $f_{\text{IN}}$ and interharmonics between the interger-number harmonic frequencies, respectively.

Figure 6

Red lines in the top two figures: A wave form and a DFT spectrum of input triangular pulses with two modulated amplitudes corresponding to bit sequence data, for which the transformation rule from the bit to amplitude of $K_{\text{U}}$ is shown in Fig. 5. On the left-hand side, the unit of the horizontal axis is time step $n$, i.e., the time normalized by $X$ ns. On the right-hand side, the DFT spectrum calculated for $X$ = 4 ns is shown. Blue lines: Wave forms and DFT spectra of the resultant spin waves detected at detector 1 when $\alpha$ = 0.1 and various time steps $X$ (= 1, 2, 4, 6, and 8 ns). In the wave forms, the green ranges correspond to the bit value 0 ($\mathrm{\Delta }{K}_{\text{u,}\text{0}}$), whereas the rest white ranges correspond to the bit value 1 ($\mathrm{\Delta }{K}_{\text{u,}\text{1}}$). Note that the widths of the lines for $X$ = 1 and 2 ns are broadened compared to those of the others to clearly see the outlines.

Figure 7

Wave forms and DFT spectra of reservoir output signals obtained at detector 1 when various $\alpha$ values (= 0.001, 0.01, 0.1, and 1) and $X$ = 4 ns.

Figure 8

Wave forms and DFT spectra of reservoir output signals obtained at detectors 37, 13, and 1 when $\alpha$ = 0.1 and $X$ = 4 ns.

Figure 9

Wave forms and DFT spectra of reservoir output signals obtained at detectors 1, 7, and 12 when $\alpha$ = 0.1 and $X$ = 4 ns.

Figure 10

Postprocessing of a reservoir output signal, which is executed in the order of (a) reservoir output signal, (b) an envelope processing, (c) a low-pass filtering, and (d) averaging over the time step $X$. Blue lines in (a)$-$(d) are the same signal obtained at detector 1 when $\alpha$ = 0.1 and $X$ = 4 ns. Colored lines in (b)$-$(d) are the signals after the processing. The $x_i\left(n\right)$ value in (d) is assigned to $y_i\left(n\right)(i=1,\cdots ,72)$ in $\mathbf{y}\left(n\right)$.

Figure 11

Singals in solving a XOR problem using 72-dimensional $\mathbf{y}\left(n\right)$ with the target output $d_3\left(n\right)=\mathrm{XOR}(u(n-1),u(n-3))$ (corresponding to delay $k$ = 3) when $\alpha$ = 0.1 and $X$ = 4 ns. Red open circles are the input binary data and blue open circles represent the calculated values in the readout part. (a) Input $u\left(n\right)$ that has a bit value (0 or 1) changing randomly with $n$. (b) Target signal $d_3\left(n\right)=\mathrm{XOR}(u(n-1),u(n-3))$. (c) Model output $z_3^{*}\left(n\right)$. (d) Binary output $b_3^{*}\left(n\right)$ calculated from $z_3^{*}\left(n\right)$ with a threshold of 0.5.

Figure 12

Prediction accuracy plotted against delay $k$, which were estimated by solving XOR problems using 72-dimensional $\mathbf{y}\left(n\right)$ with $d_k\left(n\right)=\mathrm{XOR}(u(n-1),u(n-k))$ for delay $k=2,\cdots ,24$: $\alpha$ = (a) 0.001, (b) 0.01, (c) 0.1, and (d) 1, where black, red, blue, brown, and green dots represent the results for $X$ = 1, 2, 4, 6, and 8 ns, respectively, and colored lines connecting nearest-neighbor dots are guides.

Figure 13

Prediction accuracy plotted against delay $k$, which were estimated by solving XOR problems using 18-dimensional $\mathbf{y}\left(n\right)$ with $d_k\left(n\right)=\mathrm{XOR}(u(n-1),u(n-k))$ for delay $k=2,\cdots ,24$: $\alpha$ = (a) 0.001, (b) 0.01, (c) 0.1, and (d) 1, where grey, orange, pale-blue, and pale-green dots represent the results with $X$ = 4 ns for areas 1, 2, 3, and 4, respectively, and colored lines connecting nearest-neighbor dots are guides.

Figure 14

Coefficient of determination ${\gamma }^2\left(k\right)$ plotted against delay $k$, which were estimated by solving delay tasks using 72-dimensional $\mathbf{y}\left(n\right)$ with the target output $d\left(n\right)=u(n-k)$ for delay $k=1,\cdots ,72$: $\alpha$ = (a) 0.001, (b) 0.01, (c) 0.1, and (d) 1, where black, red, blue, brown, and green dots represent the results for $X$ = 1, 2, 4, 6, and 8 ns, respectively, and colored lines connecting nearest-neighbor dots are guides.

Figure 15

Time-domain wave form at detector 1 (the upper side) and instantaneous top-view sz distributions in space (the bottom side), when $\alpha$ = 0.1 and $X$ = 4 ns. The $s_{\text{z}}$ distributions with 12 $\times$ 12 $\mu {\text{m}}^2$ in area are the differences from the reference $s_z$ distribution at the end of $n=100$ (after [1, 0, 1, 0, 0, 0] input sequence), where bright and dense-gray areas are the detector and exciter areas, respectively, light-gray areas are the other areas, and the direction and strength of $s_z$ is defined by a color bar.

Figure 16

(a) Detector numbers and group names, where center, inside, middle, and outside groups are referred to as C, I, M, and O, respectively, in (b) and (c). (b) Two types of methods for changing detector numbers): Type I increases the groups toward the outside boundaries (from I to O), whereas type II increases the groups toward the inside exciter (from O to I). (c) Computing capabilities for temporal XOR problems and delay tasks plotted against delay step $k$, where each color of line/dots shown in the inset corresponds to the combination of the groups in (b).