# Feedback-voltage driven chaos in three-terminal spin-torque oscillator

Tomohiro Taniguchi<EMAIL_ADDRESS>National Institute of
Advanced Industrial Science and Technology (AIST), Research Center for
Emerging Computing Technologies, Tsukuba 305-8568, Japan

###### Abstract

Recent observations of chaos in nanomagnet suggest a possibility of new
spintronics applications such as random-number generator and neuromorphic
computing. However, large amount of electric current and/or magnetic field are
necessary for the excitation of chaos, which are unsuitable for energy-saving
applications. Here, we propose an excitation of chaos in three-terminal spin-
torque oscillator (STO). The driving force of the chaos is voltage-controlled
magnetic anisotropy (VCMA) effect, which enables us to manipulate
magnetization dynamics without spending electric current or magnetic field,
and thus, energy efficient. In particular, we focus on the VCMA effect
generated by feedback signal from the STO since feedback effect is known to be
effective in exciting chaos in dynamical system. Solving the Landau-Lifshitz-
Gilbert (LLG) equation numerically and applying temporal and statistical
analyses to its solution, the existence of the chaotic and transient-chaotic
magnetization dynamics driven by the feedback VCMA effect is identified.

Chaos is a kind of nonlinear dynamics which is sensitive to the initial state
and thus, is difficult to predict its time evolution. These characteristics
have attracted attention not only from fundamental aspect Strogatz (2001) but
also from practical applications such as random-number generator Stojanovski
and Kocarev (2001), chaotic communication Cuomo and Oppenheim (1993), and
brain-inspired computing Verstraeten  _et al._ (2007). Recent predictions and
observations of chaos in nanostructured ferromagnets suggest a possibility to
realize these applications by spintronics devices Kudo, Sato, and Mizushima
(2006); Li, Li, and Zhang (2006); Yang, Zhang, and Li (2007); Petit-Watelot
_et al._ (2012); Devolder  _et al._ (2019); Bondarenko  _et al._ (2019);
Montoya  _et al._ (2019); Taniguchi (2019); Williame  _et al._ (2019);
Taniguchi  _et al._ (2019); Yamaguchi  _et al._ (2019); Kamimaki  _et al._
(2021); Taniguchi (2024a, b). Since chaos can appear only in a high-
dimensional phase space Strogatz (2001); Alligood, Sauer, and Yorke (1997);
Ott (2002), various methods to increase the number of the dynamical degree of
freedom have been used in the previous works, such as injecting periodic
signal Li, Li, and Zhang (2006); Yang, Zhang, and Li (2007); Yamaguchi  _et
al._ (2019) and increasing the number of the ferromagnets Kudo, Sato, and
Mizushima (2006); Taniguchi (2019). Among them, adding feedback signal is an
efficient approach Biswas and Banerjee (2018) because feedback effect
generally increases the number of the dynamical degree of freedom
significantly and thus, makes it easy to excite chaos. A large number of the
dynamical degree of freedom is also preferable for practical applications. For
example, the number of the linearly independent dynamical degree of freedom
bounds the computational capability of physical reservoir computing Kubota,
Takahashi, and Nakajima (2021). Motivated by these reasons, chaos in spin-
torque oscillators (STOs) driven by feedback electric current or magnetic
field has intensively been investigated both theoretically Williame  _et al._
(2019); Taniguchi  _et al._ (2019); Taniguchi (2024b) and experimentally
Kamimaki  _et al._ (2021); Tsunegi  _et al._ (2023). However, using the
feedback electric current causes considerable the Joule heating, and using the
feedback magnetic field prevents all-electrical control of devices.

Another candidate as the feedback signal is the voltage-controlled magnetic
anisotropy (VCMA) effect Weisheit  _et al._ (2007); Maruyama  _et al._ (2009);
Shiota  _et al._ (2009), where an application of voltage to magnetic tunnel
junctions modifies magnetic anisotropy energy by a modulation of electron
states Duan  _et al._ (2008); Nakamura  _et al._ (2009); Tsujikawa and Oda
(2009) and/or induction of magnetic moment Miwa  _et al._ (2017) near
ferromagnet metal/insulating barrier interface. The VCMA effect does not, in
principle, increase Joule heating and enables the system to work by all-
electrical manipulation. Therefore, it is of great value to examine whether
chaos can be excited by the feedback VCMA effect.

In this work, we study an excitation of chaos in a three-terminal STO with
feedback VCMA effect by numerical simulations of the Landau-Lifshitz-Gilbert
(LLG) equation. An STO is placed on a bottom electrode, which generates spin
current through the spin Hall effect Dyakonov and Perel (1971); Hirsch (1999);
Zhang (2000); Kato  _et al._ (2004) and drives magnetization oscillation. The
electric current flowing in the bottom electrode also generates electric
signal reflecting magnetization oscillation through anisotropic
magnetoresistance (AMR) effect. The AMR signal is converted to voltage which
is then applied to the STO from another terminal. As can be seen, using the
three-terminal structure allows us to use the spin-transfer (spin-orbit)
torque Slonczewski (1996); Berger (1996) and VCMA effects simultaneously,
which enables us to control the magnetization excitation in the STO and the
bifurcation from simple to chaotic dynamics independently. This was impossible
in two-terminal STOs studied previously. The existence of chaos, as well as
transient chaos Lai and Tél (2011), caused by the feedback VCMA effect in the
three-terminal STO is identified from the temporal solution of the LLG
equation and its statistical analyses.

Figure 1:  Schematic illustration of three-terminal STO with feedback voltage.
Electric current density $J$ flowing in the $x$ direction generates spin
current polarized in the $y$ direction and is injected in the STO. The spin
current excites spin-transfer (spin-orbit) torque on the magnetization
$\mathbf{m}$ of a (bottom) free layer in the STO and induces magnetization
oscillation around the $y$ axis. The electric current density in the bottom
electrode also generates AMR signal ($\propto m_{x}^{2}$), which passes
through feedback circuit with delay time $\tau$ and is converted to feedback
voltage applied to the STO from another terminal.

Figure 1 is a schematic illustration of a three-terminal STO, where electric
current density $J$ flowing in a bottom electrode (along the $x$ direcction)
generates spin current polarized in the $y$ direction and injected into the
STO. The spin current excites spin-transfer (spin-orbit) torque on
magnetization in a free layer. The magnetization oscillation can be detected
by AMR effect, as in the case of spin-torque ferromagnetic resonance
experiments in three-terminal structures Liu  _et al._ (2011). Then, this AMR
signal is sent to the other terminal through a feedback circuit and is used to
determine the feedback voltage, as shown in Fig. 1. Therefore, both the spin
Hall and VCMA effects coexist in this structure. We should mention here that
the possibility of utilizing both of them simultaneously was experimentally
confirmed in three-terminal magnetoresistive memory Shirotori  _et al._
(2017). In the present work, the magnitude of the voltage for this VCMA effect
is determined by the feedback signal. The advantage and/or necessity to use
three-terminal structures, rather than two-terminal STOs, in the present study
will also be discussed at the end of this paper. Note also that the feedback
circuit can amplify or attenuate the input signal and yield a delay time
$\tau$ Kamimaki  _et al._ (2021); Tsunegi  _et al._ (2023), where a typical
value of $\tau$ available in experiments is on the order of $1$-$10$ ns.

Table 1:  Parameters used in the numerical calculations: $M$, saturation magnetization; $H_{\rm K}$, in-plane magnetic anisotropy field; $\gamma$, gyromagnetic ratio; $\alpha$, Gilbert damping constant; $d$, thickness of the free layer; $\vartheta$, spin Hall angle; $J$, current density; $\tau$, delay time.  Quantity | Value
---|---
$M$ | 1000 emu/cm3
$H_{\rm K}$ | 200 Oe
$\gamma$ |  $1.764\times 10^{7}$ rad/(Oe s)
$\alpha$ | $0.010$
$d$ |  $2$ nm
$\vartheta$ | 0.34
$J$ |  $15$ MA/cm2
$\tau$ |  $20$ ns

The magnetization dynamics in the proposed STO is described by the LLG
equation given by

$\frac{d\mathbf{m}}{dt}=-\gamma\mathbf{m}\times\mathbf{H}-\frac{\gamma\hbar\vartheta
J}{2eMd}\mathbf{m}\times\left(\mathbf{e}_{y}\times\mathbf{m}\right)+\alpha\mathbf{m}\times\frac{d\mathbf{m}}{dt},$
(1)

where $\mathbf{m}=(m_{x},m_{y},m_{z})$ is the unit vector pointing in the
direction of the magnetization in the free layer, while $\mathbf{e}_{k}$
($k=x,y,z$) is the unit vector in the $k$ direction. The gyromagnetic ratio,
saturation magnetization, thickness along the $z$ direcction, and the Gilbert
damping constant of the free layer are denoted as $\gamma$, $M$, $d$, and
$\alpha$, respectively. The spin Hall angle is $\vartheta$. We assume that the
free layer in the STO is in-plane magnetized and its magnetic field
$\mathbf{H}$ is given by

$\mathbf{H}=H_{\rm K}m_{y}\mathbf{e}_{y}-4\pi
M\left\\{1+\nu[m_{x}(t-\tau)]^{2}\right\\}m_{z}\mathbf{e}_{z},$ (2)

where $H_{\rm K}$ is the in-plane magnetic anisotropy field along the $y$
direction while $4\pi M$ is the demagnetization field along the $z$ direction.
The term $\nu[m_{x}(t-\tau)]^{2}$ represents the modulation of the
perpendicular magnetic anisotropy field by the VCMA effect, where the
magnitude of the modulation is determined by the AMR feedback signal
proportional to $m_{x}^{2}$. The dimensionless coefficient $\nu$ is defined as
$H_{{\rm K},V}/(4\pi M)$, where $H_{{\rm K},V}=2\eta V/(Mdd_{\rm I})$
represents the magnitude of the VCMA effect in terms of magnetic field with
the VCMA efficiency $\eta$, voltage $V$, and the thickness $d_{\rm I}$ of the
insulating layer in the STO. The VCMA efficiency recently obtained in the
experiment reaches about $300$ fJ/(V m) Nozaki  _et al._ (2020), which makes
$H_{{\rm K},V}$ be on the order of kilo Oersted for typical device structures.
The magnitude of the voltage, $|V|$, which might include an amplification or
attenuation by the feedback circuit, is limited to be about $0.5$ V at maximum
to avoid electrostatic breakdown of the STO. The saturation magnetization $M$
in typical STOs and/or VCMA devices are about $1000$ emu/cm3. Regarding these
values, the magnitude of $\nu$ can be on the order of $0.1$. The values of the
parameters used in the following calculations, except $\nu$, are summarized in
Table 1. We solved Eq. (1) with an initial condition $\mathbf{m}(0)=(\sin
90^{\circ}\cos 89^{\circ},\sin 90^{\circ}\sin 89^{\circ},\cos 90^{\circ})$
from $t=0$ to $t=t_{\rm max}$, where $t_{\rm max}=1.0$ $\mu$s throughout this
work. As a general consequence for chaos and transient chaos Lai and Tél
(2011), the results shown below, such as their boundary, depend on the choice
of $t_{\rm max}$, which will be discussed later. The fourth-order Runge-Kutta
method extended to feedback system Hairer and Wanner (1993); Taniguchi  _et
al._ (2019) was applied for solving Eq. (1), where the time increment of the
LLG equation is defined from the delay time $\tau$ and an integer $N$ as
$\Delta t=\tau/N$ ($N=2\times 10^{4}$, or equivalently, $\Delta t=1$ ps, in
this work). For the latter discussion, we note that the number of the
dynamical degree of the freedom in this numerical scheme is $2(N+1)$ Hairer
and Wanner (1993); Taniguchi  _et al._ (2019).

Figure 2:  Temporal dynamics of $m_{x}$ and $m_{y}$ for $\nu$ of (a) $0$, (b)
$0.1$, and (c) $0.5$. The insets in (a) and (b) show their dynamics in a time
range of $1$ ns. Fourier spectra, $|m_{x}(f)|$, of $m_{x}(t)$ for those $\nu$
are shown in (d)-(f).

Figures 2(a)-2(c) summarize typical dynamics of $m_{x}$ (red) and $m_{y}$
(blue) near $t=0$, where $\nu$ is (a) $0$, (b) $0.1$, and (c) $0.5$, while the
Fourier spectra, $|m_{x}(f)|$, of $m_{x}(t)$ for these $\nu$ are shown in
Figs. 2(d)-2(f). The Fourier spectra for various $\nu$ are summarized in Fig.
3. In the absence of the feedback effect ($\nu=0$), an auto-oscillation with a
single main frequency is excited, as in the case of the conventional three-
terminal STOs Liu  _et al._ (2012); see Figs. 2(a) and 2(d), as well as the
inset in Fig. 2(a). In the presence of the feedback effect, however, the
dynamics is largely modulated, although the dynamics during a short time scale
looks similar to a simple auto-oscillation; see Fig. 2(b) and compare the
insets in Fig. 2(a) and 2(b). The complexity of the dynamics is also found
from the Fourier spectrum in Fig. 2(e), where the spectra shows broadened
structure. The magnetization switching is also observed for some values of
$\nu$, as shown in Fig. 2(c). Note that the oscillation amplitude repeats
amplification and reduction before the switching, which is also reflected in
the broadened structure of the Fourier spectrum in Fig. 2(f). This switching
behavior is quite different from the conventional spin-transfer (spin-orbit)
torque switching Taniguchi  _et al._ (2022a), where the work done by the spin-
transfer torque monotonically supply energy into the ferromagnet and thus, the
oscillation amplitude also increases monotonically. The complex behavior
before the magnetization switching found in Fig. 2(c) suggest that the
dynamics is classified as transient chaos Lai and Tél (2011), where nonlinear
dynamical system shows chaotic behavior before saturating to a fixed point.

Figure 3:  Power spectrum density, $|m_{x}(f)|$, for various $\nu$, where
$m_{x}(f)$ is the Fourier transformation of $m_{x}(t)$.

Summarizing the results, it is shown that the feedback VCMA effect causes the
modulation of the oscillation amplitude in the STO, and the dynamics becomes
complex. The complexity of the dynamics results in the broadened structure of
the Fourier spectra, as summarized in Fig. 3. These results are similar to the
characteristics found in chaotic spintronics devices with feedback current or
magnetic field Taniguchi  _et al._ (2019); Kamimaki  _et al._ (2021);
Taniguchi (2024b). We should, however, recall that an appearance of complex
dynamics is not a direct evidence of the existence of chaos. Although the
complexity of dynamics is a typical characteristic of chaos, not all complex
dynamics is classified as chaos. Therefore, to identify chaos, we evaluate the
Lyapunov exponent Strogatz (2001).

The Lyapunov exponent is an expansion rate of a distance between two solutions
of the LLG equation having an infinitesimally different initial conditions.
Let us denote the solution of the LLG equation with an initial condition
$\mathbf{m}(t)$. Note that we need to store the value of $N+1$ $\mathbf{m}$
[$\mathbf{m}(t)$, $\mathbf{m}(t-\Delta t)$, $\cdots$, $\mathbf{m}(t-\tau)$]
for performing the numerical simulation, where $\mathbf{m}(t-\Delta t)$,
$\cdots$, $\mathbf{m}(t-\tau)$ are the past values of $\mathbf{m}(t)$ and are
used as feedback signal. To evaluate the Lyapunov exponent, for each
$\mathbf{m}(t-k\Delta t)$ ($k=0,1,\cdots,N$), we introduce the zenith and
azimuth angles, $\theta_{k}=\cos^{-1}m_{z}(t-k\Delta t)$ and
$\varphi_{k}=\tan^{-1}[m_{y}(t-k\Delta t)/m_{x}(t-k\Delta t)]$. Then, we
introduce $\tilde{\mathbf{m}}(t-k\Delta t)$, whose zenith and azimuth angles,
$\tilde{\theta}_{k}=\cos^{-1}\tilde{m}_{z}(t-k\Delta t)$ and
$\tilde{\varphi}_{k}=\tan^{-1}[\tilde{m}_{y}(t-k\Delta
t)/\tilde{m}_{x}(t-k\Delta t)]$, are defined by shifting $\theta_{k}$ and
$\varphi_{k}$ with infinitesimal differences
($|\epsilon_{k,\theta}|,|\epsilon_{k,\varphi}|\ll 1$) as
$\tilde{\theta}_{k}=\theta_{k}+\epsilon_{k,\theta}$ and
$\tilde{\varphi}_{k}=\varphi_{k}+\epsilon_{k,\varphi}$. Therefore, the
distance between $\mathbf{m}(t-k\Delta t)$ and $\tilde{\mathbf{m}}(t-k\Delta
t)$ at time $t$ is given by
$\epsilon_{k}=\sqrt{\epsilon_{k,\theta}^{2}+\epsilon_{k,\varphi}^{2}}$, and
their sum is $\epsilon=\sum_{k=0}^{N}\epsilon_{k}$. Then, we evaluate the time
evolution of $\mathbf{m}(t)$ and $\tilde{\mathbf{m}}(t)$ from time $t$ to
$t+\Delta t$ by solving the LLG equation and obtain $\mathbf{m}(t+\Delta t)$
and $\tilde{\mathbf{m}}(t+\Delta t)$. Note that the distance
$\epsilon_{0}^{\prime}$ between these $\mathbf{m}(t+\Delta t)$ and
$\tilde{\mathbf{m}}(t+\Delta t)$, in terms of
$\theta_{0}^{\prime}=\cos^{-1}m_{z}(t+\Delta t)$,
$\varphi_{0}^{\prime}=\tan^{-1}[m_{y}(t+\Delta t)/m_{x}(t+\Delta t)]$,
$\tilde{\theta}_{0}^{\prime}=\cos^{-1}\tilde{m}_{z}(t+\Delta t)$, and
$\tilde{\varphi}_{0}^{\prime}=\tan^{-1}[\tilde{m}_{y}(t+\Delta
t)/\tilde{m}_{x}(t+\Delta t)]$, is generally different from that,
$\epsilon_{0}$, between $\mathbf{m}(t)$ and $\tilde{\mathbf{m}}(t)$. Then,
defining $\epsilon^{\prime}=\epsilon-\epsilon_{0}+\epsilon_{0}^{\prime}$, the
expansion rate of the solution of the LLG equation during $[t,t+\Delta t]$ is
given by $\epsilon^{\prime}/\epsilon$. Therefore, a temporal Lyapunov exponent
at time $t+\Delta t$ is defined as

$\lambda=\frac{1}{\Delta t}\ln\frac{\epsilon^{\prime}}{\epsilon}.$ (3)

We repeat this procedure and evaluate the temporal Lyapunov exponent at every
time from $t=\tau$. Finally, the Lyapunov exponent $\varLambda$ is defined as
a statistical average of the temporal Lyapunov exponents as
$\varLambda=(1/\mathscr{N})\sum_{i=1}^{\mathscr{N}}\lambda_{i}$, where
$\mathscr{N}$ is the number of the averaging and $\lambda_{i}$ is the $i$th
temporal Lyapunov exponent; see also Ref. Taniguchi  _et al._ (2019), which
refers to how to define the Lyapunov exponent in feedback system. The sign of
the Lyapunov exponent indicates the dynamical state of the system, i.e.,
negative exponent corresponds to dynamics saturating to a fixed point, zero
exponent corresponds to a periodic dynamics, and positive exponent corresponds
to chaos. Figure 4 summarizes the Lyapunov exponent for various $\nu$. The
value is zero at $\nu=0$, i.e., in the absence of the feedback effect, because
the magnetization dynamics in this case is a periodic oscillation (auto-
oscillation), as shown in Fig. 2(a). For finite feedback effect ($\nu\neq 0$),
such as the dynamics in Fig. 2(b), the Lyapunov exponent becomes positive,
which can be an evidence of the existence of chaos in the present system. The
Lyapunov exponent can also be negative for some $\nu$ when the magnetization
switches its direction, as shown in Fig. 2(c). Summarizing them, the sign of
the Lyapunov exponent reasonably reflects the dynamical behavior of the
magnetization.

Figure 4:  Lyapunov exponent for various $\nu$. A horizontal dotted line is
added as a guide for eyes to clarify a boundary of zero exponent.

Note that analyses on both temporal dynamics in Fig. 2 and statistical
property (Lyapunov exponent) in Fig. 4 are necessary to identify dynamical
state of the magnetization. The temporal analysis clarifies the existence of
complex dynamics, at least as temporary, which might be missed from the
Lyapunov exponent. For example, for the dynamics shown in Fig. 2(c), chaotic
behavior appeared initially might be missed when only the Lyapunov exponent is
evaluated. On the other hand, the evaluation of the Lyapunov exponent can be
used as a figure of merit to distinguish between chaos and complex but non-
chaotic dynamics, which is hard to determine from just observing temporal
dynamics. While the temporal dynamics in STOs has been frequently measured and
used in practical applications Tsunegi  _et al._ (2018), one may be interested
in how to estimate the Lyapunov exponent experimentally. In experiments, it is
difficult to control the initial state of the magnetization, and thus, the
evaluation method of the Lyapunov exponent used in the numerical simulation is
hardly applicable. There are, however, various methods to estimate the
Lyapunov exponent from experimental data by reproducing the dynamical
trajectory in an embedded space Alligood, Sauer, and Yorke (1997) or identify
chaos from different quantities, such as noise limit Barahona and Poon (1996);
Poon and Barahona (2001); Devolder  _et al._ (2019); Kamimaki  _et al._
(2021). Therefore, it will be possible to examine the validity of the present
results by experiments in future.

Recall that the identification of chaos sometimes depends on the measurement
time $t_{\rm max}$, especially when transient chaos possibly occurs Lai and
Tél (2011). For example, we did not observe magnetization switching until
$t_{\rm max}=1$ $\mu$s for $\nu=0.1$ shown in Fig. 2(b); therefore, the
corresponding Lyapunov exponent for this $\nu$ is positive, as shown in Fig.
4. If we use, however, $t_{\rm max}$ longer than $1$ $\mu$s, a switching might
occur; in this case, the Lyapunov exponent will be negative and the dynamics
will be classified as transient chaos. On the other hand, if we, for example,
use $t_{\rm max}=100$ ns, the dynamics for $\nu=0.5$ shown in Fig. 2(c) might
be classified as chaos and the Lyapunov exponent becomes positive because at
$t=100$ ns, the switching does not occur. As shown in these examples, the
identification of chaos depends on the measurement time. This is not a
particular property for spintronics devices; rather, this is a general
consequence in chaotic system Lai and Tél (2011). It is generally difficult to
estimate the transient time, if transient chaos exists, because it depends not
only on the values of parameters but also on the initial conditions and so on
Lai and Tél (2011). The time scale during which chaos can be kept determines
the manipulation time of chaos applications, and thus, is of great interest.
We keep this issue as a future work.

At the end of this work, let us mention advantage and/or necessity using the
three-terminal structure, instead of the two-terminal one studied previously
Williame  _et al._ (2019); Taniguchi  _et al._ (2019); Kamimaki  _et al._
(2021); Taniguchi (2024b); Tsunegi  _et al._ (2023). The three-terminal
structure allows us to utilize both the spin-transfer torque and VCMA effects
simultaneously Shirotori  _et al._ (2017), as mentioned earlier. In the two-
terminal device, these effects hardly coexist because their source current and
voltage are not independent from each other, and they require contradicting
conditions for resistance of the device. The spin-transfer torque effect
arises from a flow of electrons Slonczewski (1996); Berger (1996), while the
VCMA effect arises from an accumulation of electrons Nakamura  _et al._
(2009); Tsujikawa and Oda (2009). Thus, the resistances of the spin-transfer
torque and VCMA devices are greatly different. For example, the resistance of
an STO in Ref. Kubota  _et al._ (2013) is at least one or two orders of
magnitude smaller than that of the VCMA devices Shiota  _et al._ (2017).
Accordingly, it is difficult to fabricate devices in which two effects
coexist. Although small current can flow even in VCMA devices for reading the
magnetization direction, it is not sufficient for the current required in STOs
applications Taniguchi  _et al._ (2022b). The three-terminal structure solves
this contradiction by separating the paths of the electric current generating
the spin-transfer torque and the voltage for the VCMA effect.

In summary, we proposed a three-terminal STO with feedback VCMA effect. The
spin Hall effect in the bottom electrode results in an auto-oscillation of the
magnetization in the STO. The magnetization oscillation can be detected by the
AMR effect, whose signal is sent to another terminal and is converted to the
feedback voltage. The feedback VCMA effect induces chaos in the STO without
generating Joule heating or without applying magnetic field. Therefore, the
proposed structure will be an attractive candidate for applying spintronics
technology to practical applications utilizing chaos. The three terminal
structure is necessary for allowing the coexistence of the spin-transfer
(spin-orbit) torque and VCMA effects, where the former drives the
magnetization oscillation while the latter makes it chaotic. The existence of
chaos, as well as transient chaos, in this STO was confirmed by solving the
LLG equation numerically and applying temporal and statistical analyses to its
solution.

The work is supported by JSPS KAKENHI Grant Number 20H05655 and 24K01336.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## Author Contributions

Tomohiro Taniguchi: Conceptualization, Investigation, Visualization, Writing

## Data Availability

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

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