# Chaotic magnetization dynamics driven by feedback magnetic field

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

###### Abstract

An excitation of highly nonlinear, complex magnetization dynamics in a
ferromagnet, for example chaos, is a new research target in spintronics. This
technology is applied to practical applications such as random number
generator and information processing systems. One way to induce complex
dynamics is applying feedback effect to the ferromagnet. The role of the
feedback electric current on the magnetization dynamics was studied in the
past. However, there is another way to apply feedback effect to the
ferromagnet, namely feedback magnetic field. In this paper, we developed both
numerical and theoretical analyses on the role of the feedback magnetic field
causing complex magnetization dynamics. The numerical simulation indicates the
change of the dynamical behavior from a simple oscillation with a unique
frequency to complex dynamics such as amplitude modulation and chaos. The
theoretical analyses on the equation of motion qualitatively explain several
features found in the numerical simulations, exemplified as an appearance of
multipeak structure in the Fourier spectra. The difference of the role of the
feedback electric current and magnetic field is also revealed from the
theoretical analyses.

##  I Introduction

Magnetization dynamics studied in magnetism and spintronics have mainly been
focused on simple dynamics such as magnetization switching, ferromagnetic
resonance, and microwave oscillation driven by magnetic field and/or electric
current [1, 2, 3, 4, 5, 6, 7, 8, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17].
Recently, however, an excitation of complex dynamics, for example chaos, is
studied from both experimental and theoretical perspectives [18, 19, 20, 21,
22, 23, 24, 25, 26, 27] because of the possibilities in developing new
practical applications, such as true-random-number generator and brain-
inspired computing [28, 29]. One way to excite complex dynamics in physical,
chemical, and biological systems is using feedback effects [30]. They are
frequently observed in natural and artificial systems, such as electric
circuit, population dynamics, and neural networks. From the mathematical
viewpoint, feedback effects significantly increase the number of the dynamical
degrees of freedom of systems, or dimensions of phase space. Recall that the
number of dynamical degrees of freedom strongly relates to dynamics possibly
excited in a system. For example, chaos cannot be excited in a system
described by a differential equation, whose dimension is less than or equal to
$2$. This fact is known as the Poincaré-Bendixson theorem [31, 32, 33] and is
one reason why it has been difficult to identify chaos in ferromagnets, where
magnetization dynamics were well described by various simple models including
only two dynamical variables, such as macrospin, vortex, spin wave, domain
wall, and so on. Thus, several approaches to increase additional dynamical
degree of freedom, such as vortex-core reversal [21, 22], have been proposed
[18, 19, 20, 23, 24, 25, 26, 27]. Adding feedback effects to ferromagnets is
one of these approaches to overcome this issue and expect to excite rich
dynamics which could appear in high-dimensional systems.

The previous works on the feedback effect on spintronics devices mainly focus
on using electric current as feedback signal [34, 35, 36, 37, 38]. There is,
however, another way to add the feedback effect to the magnetization dynamics,
namely, converting the feedback signal into magnetic field [39, 40]. Although
using magnetic field is not preferable for practical applications, there are
several motivations to investigate the role of the feedback magnetic field on
the magnetization dynamics. One is the fact that the torques due to the
magnetic field and electric current have different roles on the dynamics. The
former induces a sustainable oscillation of the magnetization while the latter
induces non-conservative torque through spin-transfer effect [41, 42]. The
role of the feedback magnetic field on the dynamics however still remains
unclear. Another motivation relates to an experimental viewpoint. The previous
works have mainly focused on applying feedback current to spin-torque
oscillators (STOs), and typical STOs consist of magnetic tunnel junctions
(MTJs). MTJs include an insulating barrier such as MgO, and thus, there is a
limitation on the magnitude of the feedback current to avoid electrostatic
breakdown of the tunnel barrier. This was one reason why the past works dealt
only in a weak feedback limit [34, 35]. If we use, on the other hand, the
feedback magnetic field, this issue will be circumvented [39, 40] because the
feedback signal is not injected into MTJs directly. However, precise studies
of theoretical analyses on the magnetization dynamics in the presence of
feedback magnetic field have not been developed yet.

In this paper, we study the role of feedback magnetic field on magnetic vortex
dynamics using numerical simulation of the Thiele equation. Here, we evaluate
both temporal dynamics of the vortex core and Fourier spectra and find an
appearance of chaos in a large feedback limit. We also develop theoretical
analyses on the feedback effect on the vortex oscillation. We derive equations
which self-consistently determine the relationship between electric current,
magnetic field strength, and delay time, and their solutions show qualitative
agreement with the results of the numerical simulations. The difference of the
role of the feedback electric current and magnetic field is also clarified by
deriving a theoretical formula of a threshold current density.

##  II System description

Figure 1:  Schematic illustration of a vortex STO with a feedback circuit. DC
bias generates direct current, which induces vortex-core dynamics via spin-
transfer torque effect. Output signal can be separated from the input direct
current by bias tee and is sent to a delayed feedback circuit with amplifier
and/or attenuator. Finally, the output signal flows in a metallic line as
electric current and generates feedback magnetic field, which affects the
vortex-core dynamics.

Figure 1 is a schematic illustration of a vortex STO with the feedback circuit
[39]. Direct current is injected into the STO and excites an auto-oscillation
of the vortex core. The dynamics modulates resistance of the STO due to the
magnetoresistance effect, and therefore, the output signal oscillates, which
depends on the vortex-core position. The input direct current and the output
signal can be separated by bias tee. The output signal is sent to a delayed-
feedback circuit and passes through a metallic line, which is placed on the
STO. The electric current flowing in the metallic line generates magnetic
fields, whose magnitude reflects the vortex-core position with delay time
$\tau$. The magnitude of the feedback magnetic field can also be tuned by an
attenuator and/or amplifier. The dynamics of the vortex core in the presence
of the feedback magnetic field is described by the following Thiele equation
[43, 44, 45, 46, 47, 48, 49],

$\begin{split}&-G\mathbf{e}_{z}\times\dot{\mathbf{X}}-|D|\left(1+\xi
s^{2}\right)\dot{\mathbf{X}}-\kappa\left(1+\zeta
s^{2}\right)\mathbf{X}+a_{J}Jp_{z}\mathbf{e}_{z}\times\mathbf{X}\\\
&+ca_{J}JR_{0}p_{x}\mathbf{e}_{x}-cb_{J}JRp_{x}\mathbf{e}_{y}+c\mu^{*}\mathbf{e}_{z}\times\mathbf{H}=\bm{0},\end{split}$
(1)

where $\mathbf{e}_{i}$ is the unit vector representing the $i$ direction,
$\mathbf{X}=(X,Y,0)$ is the position vector of the vortex core, while $G=2\pi
pML/\gamma$ and $D=-(2\pi\alpha ML/\gamma)[1-(1/2)\ln(R_{0}/R)]$ [45] consist
of the saturation magnetization $M$, the gyromagnetic ratio $\gamma$, the
Gilbert damping constant $\alpha$, the thickness $L$, the disk radius $R$, and
the core radius $R_{0}$. The dimensionless parameter $\xi$ was introduced to
describe the nonlinear damping in a highly excited state [48]. The magnetic
potential energy $W$ is characterized by $\kappa$ and $\zeta$ via

$W=\frac{\kappa}{2}\left(1+\frac{\zeta}{2}s^{2}\right)|\mathbf{X}|^{2},$ (2)

and $\kappa=(10/9)4\pi M^{2}L^{2}/R$ [48]. The dimensionless parameter $\zeta$
was introduced to explain the linear dependence of the oscillation frequency
on the current [48]. We also introduce
$s=|\mathbf{X}|/R=\sqrt{X^{2}+Y^{2}}/R$, which corresponds to the distance of
the vortex-core position measured from the disk center. The spin-transfer
torque strength with spin polarization $P$ is $a_{J}=\pi\hbar P/(2e)$ [46,
47], while the strength of the field-like torque is given by $b_{J}$. The
electric current density is denoted by $J$, where positive current corresponds
to that flowing from the reference to the free layer. The unit vector pointing
in the magnetization direction in the reference layer is assumed to lie in the
$xz$-plane as $\mathbf{p}=(p_{x},0,p_{z})$. Accordingly, the output signal
depends on the $y$ component $Y$ of the vortex-core position. The external
magnetic field is denoted as $\mathbf{H}$, while we define $\mu^{*}$ as
$\mu^{*}=\pi MLR$. Recall that the magnetic field is generated as the feedback
effect and the output signal depends on $Y$. Therefore, $\mathbf{H}$ can be
expressed as

$\mathbf{H}(t)=h_{y}\frac{Y(t-\tau)}{R}\mathbf{e}_{y},$ (3)

where $h_{y}$ determines the magnitude of the feedback magnetic field, while
$Y(t-\tau)/R$ is the normalized $y$ component of the vortex core position in
the past with a delay time $\tau$. Here, we assume that the feedback magnetic
field points to the $y$ direction. In the following, we express the magnitude
of $h_{y}$ in terms of gain in the feedback circuit, which is frequently used
in experiments [39, 50] (see also Appendix A). The values of the parameters
are derived from typical experiments and summarized in Appendix A. We,
however, inform here that the delay time $\tau$ is set to be $29$ ns [40],
which is sufficiently longer than the typical oscillation period of the vortex
STO (about $3$ ns). This is because complex dynamics can be easily excited
when the delay time is longer than typical time scale of systems [30], while
dynamics excited by a short-time delay effect were, for example, self-
injection locking [34, 35].

##  III Numerical simulation

Figure 2:  Fourier spectra of $Y(t)/R$ as a function of feedback gain.

Figure 2 shows the Fourier spectra of $Y(t)/R$ obtained by solving Eq. (1).
When the feedback gain is small, a single peak around $290$ MHz is observed.
Multiple peak structure appears when the feedback gain exceeds roughly over
$-20$ dB, which corresponds to feedback magnetic field of $2.2$ Oe. As the
feedback gain further increases, it becomes difficult to identify the main
peak.

Figure 3:  (a) Time evolution of $Y(t)/R$ and $s(t)$ in a steady state, and
the Fourier transformation $|Y(f)/R|$ of $Y(t)/R$. Feedback gain is $g=-60$
dB, corresponding to $h_{y}=0.1$ Oe. The feedback gain is changed as (b)
$g=-20$ dB ($h_{y}=2.2$ Oe), (c) $g=-10$ dB ($h_{y}=4.6$ Oe), and (d) $g=+20$
dB ($h_{y}=46.4$ Oe). Note that the range of time is 20 ns in (a) and (b)
while it is $200$ ns in (c) and (d).

Figure 4:  (a) Bifurcation diagram summarizing local maxima of $Y(t)/R$ and
(b) Lyapunov exponent.

The change of the peak structure reflects the temporal dynamics of the vortex
core. In Fig. 3, we show the temporal dynamics for various gains. For example,
Fig. 3(a) shows the temporal dynamics of $Y(t)/R$ and $s(t)$, as well as the
Fourier spectrum, $|Y(f)|/R$, at the feedback gain of $g=-60$ dB, which
corresponds to $h_{y}=0.1$ Oe. In this case, the feedback magnetic field is
approximately negligible. Thus, an auto-oscillation well studied previously
[48, 49] is excited, where the vortex core rotates around the disk center with
practically constant oscillating amplitude (radius). Therefore, $Y(t)/R$ is
similar to a trigonometric function, and $s(t)$ is approximately constant, as
shown in Fig. 3(a). A small-amplitude oscillation of $s(t)$ originates from
the terms related to $p_{x}$ in Eq. (1), as well as the feedback magnetic
field, which breaks the axial symmetry around the $z$ axis of Eq. (1). As can
be seen, however, the amplitude of the oscillation in $s(t)$ is very small.
The Fourier spectrum shows a sharp single peak, as mentioned above.

When the feedback gain is $-20$ dB ($h_{y}=2.2$ Oe), the vortex core still
shows a simple auto-oscillation, where the Fourier spectrum shows a single
peak; see Fig. 3(b). However, the oscillation amplitude $s(t)$ becomes small
compared with that in Fig. 3(a). The peak frequency of the Fourier spectrum
also becomes slightly low. The reason will be explained theoretically in the
next section.

When the feedback gain is further increased to $-10$ dB ($h_{y}=4.6$ Oe), the
dynamics become complex, as shown in Fig. 3(c). In a short time range, the
temporal dynamics of $Y(t)/R$ looks similar to the simple dynamics shown in
Figs. 3(a) and 3(b). However, when the temporal dynamics over a relative long
time range are shown, we find an amplitude modulation of the oscillation. This
tendency is apparently shown by $s(t)$. The Fourier spectrum shows multiple
peak structure around a main peak.

Chaotic dynamics appear clearly when the feedback gain is sufficiently large;
see Fig. 3(d), where $g=+20$ dB ($h_{y}=46.4$ Oe). The temporal dynamics
become non-periodic, and the Fourier spectrum shows a flat structure over a
wide range of the frequency. In other words, the oscillation frequency of the
vortex core is not unique. These features are typical in chaotic dynamics [32,
33].

The change of the temporal dynamical behavior can be investigated in a
different way [31]. We evaluate a bifurcation diagram, as shown in Fig. 4(a),
which summarizes local maxima of $Y(t)/R$. When the dynamics is a simple
oscillation described by a trigonometric function, for example, the values of
the local maxima are always the same, and thus, they are summarized at a
single point. On the other hand, when the dynamics becomes complex such as
chaos, the local maxima show a broadened structure. In Fig. 4(a), the local
maxima summarize at a single point when the feedback gain is smaller than $20$
dB. The local maxima show broadened and approximately symmetric structure when
the feedback gain is about in the range of $-20\lesssim g\lesssim-5$ dB, which
is typical in the amplitude-modulation dynamics. The maxima are largely
broadened and become asymmetric when the feedback gain exceeds about $-5$ dB,
which is a typical feature of chaotic dynamics [31]. We also evaluate the
Lyapunov exponent, as shown in Fig. 4(b), where the numerical method to
evaluate the Lyapunov exponent in a feedback system was developed in Ref.
[37]. The Lyapunov exponent is close to zero for the feedback gain of
$g\lesssim-5$ dB, indicating that the dynamics is periodic. The Lyapunov
exponent becomes largely positive when the feedback gain exceeds about $-5$
dB, which evidently shows the appearance of chaos. These results are
consistent with the temporal dynamics shown in Fig. 3, where the simple
oscillations in Figs. 3(a) and 3(b) and the oscillation with the amplitude
modulation in Fig. 3(c) are periodic, while the dynamics shown in Fig. 3(d)
are chaos.

Summarizing them, the vortex core dynamics changes from a simple auto-
oscillation to complex dynamics such as an oscillation with an amplitude
modulation and chaos. These changes of the dynamical behavior can be confirmed
from both temporal dynamics of the vortex core and its Fourier spectra, which
can be experimentally measured due to the magnetoresistance effect.

##  IV Theoretical analyses

Here, we discuss the bifurcation of the dynamical behavior found in the
numerical simulation from theoretical viewpoint. First, we should note that it
is difficult to obtain exact solutions of Eq. (1) analytically, even when the
feedback effect is absent. However, by applying some approximations, some of
the features, such as the appearance of multipeak structure, can be
qualitatively explained. For this purpose, we express the Thiele equation in
terms of $s=\sqrt{X^{2}+Y^{2}}/R$ and $\psi=\tan^{-1}(Y/X)$, which correspond
to the distance of the vortex-core position measured from the disk center and
the core’s phase from the $x$ axis. Neglecting small terms in the Thiele
equation but keeping terms related to the feedback magnetic field, the
approximated equations of motions for $s$ and $\psi$ become

$\dot{s}\simeq as-bs^{3}+\frac{c\mu^{*}}{GR}\sin\psi
h_{y}s_{\tau}\sin\psi_{\tau}-\frac{c\mu^{*}|D|}{G^{2}R}\cos\psi
h_{y}s_{\tau}\sin\psi_{\tau},$ (4)
$\dot{\psi}\simeq\frac{\kappa}{G}\left(1+\zeta
s^{2}\right)+\frac{c\mu^{*}}{GRs}\cos\psi
h_{y}s_{\tau}\sin\psi_{\tau}+\frac{c\mu^{*}|D|}{G^{2}Rs}\sin\psi
h_{y}s_{\tau}\sin\psi_{\tau},$ (5)

where we introduce $a=(|D|\kappa/G^{2})[(J/J_{\rm c})-1]$ and
$b=(|D|\kappa/G^{2})(\xi+\zeta)$ with $J_{\rm c}=|D|\kappa/(Ga_{J}p_{z})$, for
simplicity. We also introduce notations $s_{\tau}=s(t-\tau)$ and
$\psi_{\tau}=\psi(t-\tau)$ for delayed variables. We used the fact that
$G^{2}\gg|D|^{2}$ (or equivalently, $|D|/G\simeq\alpha\ll 1$) to neglect
several terms in Eq. (1), but, kept terms related to the feedback terms with
the coefficient $|D|/G$. The reason will be explained below. In the following,
we first review the steady state solutions of Eqs. (4) and (5) and then
discuss the effect of the feedback magnetic field

###  IV.1 Steady state solution in an unperturbed system

Note that Eqs. (4) and (5) without the feedback effect (i.e., in the limit of
$h_{y}\to 0$) are identical to the Stuart-Landau equation [51], which was
introduced by Landau to phenomenologically describe dynamics in turbulence and
derived by Stuart from hydrodynamic equations. The original Stuart-Landau
equation is known to be solved analytically [52] by introducing a generalized
phase $\theta=\psi+[\zeta\kappa/(Gb)]\ln s$. Equations (4) and (5) in the
absence of the feedback effect have the steady state solution given by

$\displaystyle\lim_{t\to\infty}s=\sqrt{\frac{a}{b}}\equiv s_{0},$
$\displaystyle\lim_{t\to\infty}\dot{\psi}=\frac{\kappa}{G}\left(1+\zeta
s_{0}^{2}\right),$ (6)

when $J/J_{\rm c}>1$, while $\lim_{t\to\infty}s=0$ when $J/J_{\rm c}\leq 1$.
These solutions for $J/J_{\rm c}>1$ correspond to the auto-oscillation of the
vortex core with a distance $s_{0}$ from the disk center and the frequency
$[\kappa/(2\pi G)](1+\zeta s_{0}^{2})$. On the other hand, when $J/J_{\rm
c}\leq 1$, the vortex core cannot move from the disk center. From this
perspective, $J_{\rm c}$ can be regarded as a threshold current density for
exciting an auto-oscillation. For the latter discussion, it is useful to
notice that the threshold current density is determined by the parameter $a$,
which is a coefficient of a linear term in Eq. (4). The coefficient $a$ can be
rewritten as

$a=\frac{a_{J}Jp_{z}}{G}-\frac{|D|\kappa}{G^{2}},$ (7)

where the first and second terms on the right-hand side correspond to the
spin-transfer and damping torques, respectively (recall that $|D|$ is
proportional to the damping constant $\alpha$). Therefore, $a$ is positive
(negative) when the spin-transfer torque is larger (smaller) than the damping
torque. We can confirm that this condition, $a>(<)0$, is equivalent to
$J/J_{\rm c}>(<)1$. These characteristics will be used in the following
section to study the modulation of the instability threshold by the feedback
effects.

###  IV.2 Threshold current and oscillation frequency in the presence of
feedback magnetic field

In the presence of the feedback effect, analytical solutions of Eqs. (4) and
(5) are hardly obtained. Recall, however, that the numerical simulation
indicated that periodic dynamics can appear even in the presence of the
feedback magnetic field when the feedback gain is relatively small. Thus, we
limit ourselves to a case of a simple auto-oscillation, where $s$ is
approximately constant and the vortex-core dynamics are still periodic. This
approximation is valid when the magnitude of the feedback effect is relatively
weak. We denote an angular frequency of this oscillation as $\varOmega$, i.e.,
$\dot{\psi}=\varOmega$. Then, we can assume that $\psi=\varOmega t$ and
$\psi_{\tau}=\varOmega(t-\tau)$. It should be emphasized here that the value
of $\varOmega$ is, at this moment, unknown. Since $s$ is assumed to be
constant, by averaging Eq. (4) over a period $2\pi/\varOmega$, the solution of
$\dot{s}=0$ after averaging is determined by an equation,

$as-bs^{3}+\beta
s\left(\cos\varOmega\tau+\frac{|D|}{G}\sin\varOmega\tau\right)=0,$ (8)

where we introduce a notation $\beta=c\mu^{*}h_{y}/(2GR)$, for simplicity. The
steady state solution of $s$ in an oscillating state can be obtained by
replacing $a$ in the unperturbed solution $s_{0}=\sqrt{a/b}$ with

$a^{\prime}=a+\beta\left(\cos\varOmega\tau+\frac{|D|}{G}\sin\varOmega\tau\right).$
(9)

Note that the solution $\sqrt{a^{\prime}/b}$ is real number only when
$J/J_{\rm
c}>1-[G\beta/(|D|\kappa)][\cos\varOmega\tau+(|D|/G)\sin\varOmega\tau]$. It
means that the feedback magnetic field modulates the threshold current density
as

$J_{\rm c}^{\prime}=J_{\rm
c}\left[1-\frac{G\beta}{|D|\kappa}\left(\cos\varOmega\tau+\frac{|D|}{G}\sin\varOmega\tau\right)\right].$
(10)

Here, the following three points should be noticed.

First, the modulation of the threshold current density comes from the fact
that the applied magnetic field originates from the feedback effect. This
argument can be confirmed from the fact that the last term in Eq. (8) includes
$s$ coming from the feedback signal, and thus, it contributes to modify $a$ to
$a^{\prime}$. If the applied magnetic field is, for example, an external
oscillating field, which is independent of $s$, such magnetic field does not
modify the coefficient $a$ and also does not change the condition of the
instability threshold. Rather, the phenomenon excited by such magnetic field
will be synchronization [52].

Second, although a modulation of the threshold current density by the feedback
effect was found in Ref. [34], where the feedback signal was injected as
electric current, the way in which the feedback effect modulates the threshold
current density is different. In the case of the feedback electric current, a
term related to the feedback effect appears in the denominator in the
definition of the threshold current; see Eq. (6) of Ref. [34]. On the other
hand, in the case of the feedback magnetic field studied here, the terms
related to the feedback effect appear as a coefficient of proportion, as shown
in Eq. (10). The difference comes from the difference of the roles for
magnetic field and electric current on an excitation of auto-oscillation. The
feedback electric current appears as a coefficient multiplied to the current
density $J$ in Eq. (7), and thus, it can be regarded as a modulation of the
spin-transfer torque [34]. On the other hand, the feedback magnetic field,
which corresponds to term related to $\beta$ in Eq. (8), does not include $J$.
Thus, the term related to the feedback effect of the magnetic field is
regarded as an additional to the second term (damping torque) on the right-
hand side of Eq. (7). Therefore, the ways the feedback effects modify the
threshold current are different for the feedback electric current and magnetic
field. Note also that the sign of this modulation can be either positive or
negative, depending on the values of $\varOmega$ and $\tau$.

Third but most importantly, at this moment, we cannot estimate the value of
Eq. (10) quantitatively because the value of $\varOmega$ is unknown. This is
the reason why we kept terms related to $|D|/G$ in Eqs. (4) and (5), although
$|D|/G\ll 1$; depending on the values of $\varOmega$ and $\tau$, for example,
$\cos\varOmega\tau$ in Eq. (8) might becomes smaller than
$(|D|/G)\sin\varOmega\tau$. Therefore, for generality, terms having the factor
$|D|/G$ in Eqs. (4) and (5) were not neglected. The value of $\varOmega$
should be determined from the following equation, which is obtained by
averaging Eq. (5) over a period $2\pi/\varOmega$ and using
$\dot{\psi}=\varOmega$;

$\begin{split}\varOmega=&\frac{\kappa}{G}\left(1+\zeta\frac{a+\beta\cos\varOmega\tau}{b}\right)\\\
&-\beta\left(\sin\varOmega\tau-\frac{|D|}{G}\cos\varOmega\tau\right).\end{split}$
(11)

Note that a general solution of Eq. (11) cannot be obtained analytically. It
becomes possible only when the delay time $\tau$ is sufficiently shorter than
$1/\varOmega$ and thus, we can use approximations, $\cos\varOmega\tau\simeq 1$
and $\sin\varOmega\tau\simeq\varOmega\tau$. It should also be noted that, to
determine the value of $\varOmega$ quantitatively from Eq. (11), the value of
the current density $J$ should be given because it is included in $a$.
Accordingly, in a strict sense, it is difficult to theoretically predict the
threshold current density and the oscillation frequency in the presence of the
feedback effect. Let us explain this point in detail in the following
sections.

###  IV.3 Difficulty of the prediction of threshold current density and
oscillation analytically

In the absence of the feedback effect, the threshold current density $J_{\rm
c}$ can be estimated just from the material parameters and the disk size, and
its estimation does not require any information on the oscillation frequency.
The auto-oscillation is excited when the magnitude of the applied current
density exceeds $J_{\rm c}$. The oscillation frequency is then given by
$[\kappa/(2\pi G)](1+\zeta s_{0}^{2})$, and its value can also be estimated
from the material parameters, the disk size, and the current magnitude.

On the other hand, in the presence of the feedback effect, the threshold
current density depends not only on the material parameters and the disk size,
in addition to the delay time $\tau$, but also on the frequency
$\varOmega/(2\pi)$, as can be seen in Eq. (10). Thus, to estimate the
threshold current density, we need to estimate the value of the frequency. The
frequency is determined by solving Eq. (11) with respect to $\varOmega$, where
an analytical expression of the frequency is hardly obtained, in general. Note
also that this frequency depends on the current density $J$ through the
parameter $a$, and thus, we need to alter some value of $J$ for the estimation
of the frequency. However, at this moment, it is not clear whether this
current density $J$ exceeds the value of $J_{\rm c}^{\prime}$ given by Eq.
(10) because $J_{\rm c}^{\prime}$ depends on $\varOmega$. Therefore, even
after determining the frequency from Eq. (11) by substituting the value of
$J$, if $J_{\rm c}^{\prime}$ estimated by using this frequency is higher than
the current density $J$, an oscillation cannot exist, and the vortex core will
be relaxed to the disk center.

As can be seen in this explanation, in the presence of the feedback effect, it
is necessary to solve Eqs. (10) [or Eq. (8)] and (11) simultaneously and self-
consistently for the estimation of the threshold current and the oscillation
frequency. Therefore, the analytical approach to estimate them is very
complex. If the strength of the feedback effect is sufficiently weak, the
frequency $\varOmega/(2\pi)$ might be approximated to the unperturbed one,
$[\kappa/(2\pi G)](1+\zeta s_{0}^{2})$, and the threshold current density, Eq.
(10), becomes a periodic function of the delay time $\tau$ [34]. In general,
however, the threshold current density and the frequency should be determined
simultaneously, as explained above.

We also explain the reason why the value of $s$ in Fig. 3(b) was smaller than
that in Fig. 3(a), although the feedback gain in the former case was stronger
than that in the latter case. The numerically estimated values of the
oscillation frequency, $\varOmega/(2\pi)$, in Figs. 3(a) and 3(b) are $289$
and $285$ MHz, respectively. Using these and $\tau=29$ ns, $\cos\varOmega\tau$
is negative for both cases, indicating that the feedback magnetic field
increases the magnitude of the threshold current density; see Eq. (10). This
increase of the threshold current density is larger for the case in Fig. 3(b)
than that in Fig. 3(a) because of the large feedback gain. Accordingly, the
excitation of the auto-oscillation is easier for the case of the small
feedback gain, compared with the case of the large one, and $s$ in Fig. 3(a)
becomes larger than that in Fig. 3(b). In addition, since the frequency
increases as $s$ increases, this difference of $s$ between Fig. 3(a) and 3(b)
is consistent with the fact that the oscillation frequency in the former is
higher than that in the latter. As can be seen in this example, an injection
of large feedback effect does not guarantee, for example, a reduction of the
threshold current. This is because various characteristics, such as the
threshold current density and the oscillation frequency, are complex functions
of the feedback strength and delay time. Nevertheless, as shown here, we can
develop the theoretical analyses which qualitatively explain the results of
the numerical simulations.

Figure 5:  Values of $f_{1}$ and $f_{2}$, given by Eqs. (12) and (13), for
feedback gains of (a) $g=-60$ dB ($h_{y}=0.1$ Oe), (b) $g=-20$ dB ($h_{y}=2.2$
Oe), (c) $g=-10$ dB ($h_{y}=4.6$ Oe), and (d) $g=+20$ dB ($h_{y}=46.4$ Oe).

###  IV.4 Theoretical viewpoint on appearance of the multipeak structure

Finally, we also show that the above theoretical analyses qualitatively
explain the appearance of the multipeak structure found in Fig. 2. The
solution of the frequency satisfying Eq. (11) can be estimated as
intersections of the following two lines;

$f_{1}=\frac{\varOmega}{2\pi},$ (12) $\begin{split}f_{2}=&\frac{\kappa}{2\pi
G}\left(1+\zeta\frac{a+\beta\cos\varOmega\tau}{b}\right)\\\
&-\frac{\beta}{2\pi}\left(\sin\varOmega\tau-\frac{|D|}{G}\cos\varOmega\tau\right).\end{split}$
(13)

Figure 5 summarizes examples of $f_{1}$ and $f_{2}$ for some values of the
feedback gain. For example, in Fig. 5(a), we show $f_{1}$ and $f_{2}$ for the
feedback gain of $-60$ dB. These two lines intersects at $311$ MHz, which is
about $8$% higher than that estimated from the numerical simulation ($289$
MHz, as mentioned above). The difference may come from terms neglected in Eqs.
(4) and (5). Note that there is only one intersection for this case, which
agrees with the fact that the Fourier spectrum in Fig. 3(a) shows a single
peak. When the feedback gain increases, we see a discrepancy between the
numerical simulation and the theoretical analyses. In Fig. 5(b) for example,
we show $f_{1}$ and $f_{2}$ for the feedback gain of $-20$ dB. While we find
multiple intersections in this figure, the Fourier spectrum in Fig. 3(b) shows
a single peak. In other words, while the theoretical analyses imply a
possibility of an appearance of multiple peak structure, the numerical
simulation does not show such behavior. The difference again may come from
approximations to simplify Eq. (1) into the forms of Eqs. (4) and (5). When
the feedback gain further increases, the number of the intersections increases
significantly, as can be seen in Figs. 5(c) and 5(d), where the feedback gains
are $-10$ and $+20$ dB, respectively. In these cases, an assumption that the
vortex core oscillates with a unique frequency, i.e., $\dot{\psi}$ is
constant, is no longer valid. In other words, the appearance of the multi-
intersections between $f_{1}$ and $f_{2}$ indicate that the theoretical
analyses developed above is no longer applicable, and complex dynamics such as
chaos can appear. From this perspective, the investigation of the number of
the intersections of $f_{1}$ and $f_{2}$ can be useful to estimate
qualitatively the feedback gain at which bifurcation from a simple oscillation
to complex dynamics occurs.

##  V Conclusion

In summary, the vortex-core dynamics in the presence of the feedback magnetic
field were studied by numerical simulation of the Thiele equation. The
bifurcation from a simple oscillation with a unique frequency to complex
dynamics, such as amplitude modulation and chaos, was found from the Fourier
spectra, the temporal dynamics, the bifurcation diagram, and the Lyapunov
exponent. Theoretical analyses on the Thiele equation were also developed to
clarify the role of the feedback magnetic field on the vortex-core dynamics.
The paper evidently shows the difficulty to obtain simple analytical solutions
of several characteristics, such as the threshold current density and the
oscillation frequency. Nevertheless, the theory was able to qualitatively
explain several features found in the numerical simulations, such as the
change of the oscillation amplitude with changing the feedback gain and the
appearance of the multipeak structure. The theory also clarifies the
difference of the role of the feedback electric current and magnetic field.

## Acknowledgements

The work is supported by Japan Society for the Promotion of Science KAKENHI
Grants No. 20H05655 and No. 24K01336.

##  Appendix A Values of parameters

The values of the parameters are derived from typical experiments using vortex
STOs and feedback circuits [39, 40] and are set as $M=1300$ emu/cm3,
$\gamma=1.764\times 10^{7}$ rad/(Oe s), $\alpha=0.010$, $L=5.0$ nm, $R=212.5$
nm, $R_{0}=10$ nm, $P=0.70$, $p_{x}=\sin 60^{\circ}$ ($p_{z}=\cos
60^{\circ}$), $\xi=3.0$, and $\tau=29$ ns. The field-like torque is assumed to
be zero. Using these values, the threshold current density $J_{\rm c}$ in the
absence of the feedback effect is estimated as $1.9$ MA/cm2. The current
density $J$ is defined from the current $I$ as $J=I/(\pi R^{2})$, where the
current is $5.5$ mA, corresponding to $J=3.9$ MA/cm2. The polarity $p$ and
chirality $c$ are each assumed to be $+1$ for convenience. The frequency
$\kappa/(2\pi G)$ of the ferromagnetic resonance is estimated to be $191$ MHz.
Some parameters, such as $\zeta$ and $\xi$, might include contributions from
Oersted field [48]. Regarding the fact that the current density is constant in
this paper, we assume that these contributions are already included in the
values shown above. In experiments, however, it is difficult to estimate the
magnitude of the applied magnetic field directly because it strongly depends
on, for example, placement of the metallic line and so on. The experimentally
controllable parameter is feedback gain (or attenuation), and we use the
feedback gain $g$ as a control parameter in the numerical simulation. The
feedback gain $g$ is defined as $h_{y}=100\times(10^{g/30}/10)$ Oe, i.e., $g$
is defined so that the feedback gain of $30$ dB corresponds to $h_{y}=100$ Oe.

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