# Reconsidering the nonlinear emergent inductance: time-varying Joule heating
and its impact on the AC electrical response

Soju Furuta  Department of Physics, Tokyo Institute of Technology, Tokyo
152-8551, Japan  Wataru Koshibae  RIKEN Center for Emergent Matter Science
(CEMS), Wako 351-0198, Japan  Keisuke Matsuura  Department of Physics, Tokyo
Institute of Technology, Tokyo 152-8551, Japan  Nobuyuki Abe  Department of
Physics, College of Humanities and Sciences, Nihon University, Tokyo 156-8550,
Japan  Fei Wang  Department of Physics, Tsinghua University, Beijing 100084,
People’s Republic of China  Shuyun Zhou  Department of Physics, Tsinghua
University, Beijing 100084, People’s Republic of China  Taka-hisa Arima  RIKEN
Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan  Department of
Advanced Materials Science, University of Tokyo, Kashiwa 277-8561, Japan
Fumitaka Kagawa<EMAIL_ADDRESS>Department of Physics, Tokyo
Institute of Technology, Tokyo 152-8551, Japan  RIKEN Center for Emergent
Matter Science (CEMS), Wako 351-0198, Japan

###### Abstract

A nonlinearly enhanced electrical reactance, $\operatorname{Im}Z$, under a
large AC current has been measured to explore emergent inductors, which
constitute a new class of inductors based on the spin-transfer torque effect.
A nonlinear $\operatorname{Im}Z$ has been observed in conducting magnets that
contain noncollinear spin textures and interpreted as the realization of an
inductance due to current-induced spin dynamics. However, curious behavior has
concomitantly been observed. For instance, the nonlinear $\operatorname{Im}Z$
always has a cutoff frequency of $10^{0}$–$10^{4}$ Hz, which is much lower
than the resonance frequency of a ferromagnetic domain wall, $\sim$107 Hz;
furthermore, the temperature and magnetic field variations in
$\operatorname{Im}Z$ appear to be considerably correlated with those in the
temperature derivative of resistance. This behavior appears to be difficult to
understand in terms of the current-induced spin dynamics, and therefore, the
earlier interpretation of the nonlinear $\operatorname{Im}Z$ should be further
verified. Here, we theoretically and experimentally show that time-varying
Joule heating and its impact on the AC electrical response can naturally
explain these observations. In the experimental approach, we study the
nonlinear AC electrical response of two conducting materials that exhibit no
magnetic order, CuIr2S4 and 1$T$’-MoTe2. Under time-varying Joule heating, a
nonlinearly enhanced $\operatorname{Im}Z$ with the curious behavior mentioned
above is observed in both systems. Our study implies that the nonlinear
$\operatorname{Im}Z$ previously observed in noncollinear magnets includes a
considerable contribution of the Joule-heating-induced apparent AC impedance.

##  I I. INTRODUCTION

The exchange of spin angular momentum between flowing conduction electrons and
an underlying magnetic texture leads to the spin-transfer-torque (STT) effect
on the magnetic texture STT1 ; STT2 . Nagaosa theoretically proposed a new
class of inductors arising from the STT-induced elastic deformation of
noncollinear magnetic textures NagaosaJJAP , and these inductors are now
referred to emergent inductors. The time evolution of a magnetic texture
brings about an emergent electric field (EEF) Volovik ; BarnesPRL2007 . Under
an AC electric current below the threshold value, the magnetic texture remains
in the pinned regime Nattermann ; ChauvePRB ; Kleemann ; IntrinsicTatara ;
ThiavilleEPL ; ExtrinsicTatara ; IntrinsicOno ; TataraReview ;
ExtrinsicNatPhys  and is periodically deformed, giving rise to a time-varying
effective U(1) gauge field and thus an oscillating EEF. From an energetic
perspective, an emergent inductor stores energy in a magnetic texture under a
current Furuta1 , in contrast to classical inductors, which store energy as
the magnetic field under a current Jackson .

Soon after the theoretical proposal of emergent inductors NagaosaJJAP ,
experimental studies were launched. The AC impedance $Z(\omega)$, also termed
complex resistivity $\rho(\omega)$ (normalized with the sample dimensions),
where $\omega$ represents the angular frequency, was extensively investigated
for materials that contain a noncollinear magnetic texture YokouchiNature ;
KitaoriPNAS ; YokouchiArxiv ; KitaoriPRB . Thus far, the imaginary part of the
complex resistivity of such magnetic materials divided by $\omega$, i.e.,
$\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$, has been commonly
found to be negligibly small for a weak AC current density, $j_{0}e^{i\omega
t}$, whereas it is significantly enhanced and becomes detectable for a
relatively large AC current density, $j_{0}\sim 10^{8}$ A m-2; here, the
complex $\rho^{1\omega}(\omega,j_{0})$ is not differential resistivity but
defined by the 1$\omega$ Fourier component of the time-varying electric field
under $j_{0}e^{i\omega t}$ divided by $j_{0}$. The nonlinearly enhanced
$\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$ has been interpreted as
emergent inductance. The present authors believe, however, that this
interpretation needs to be reconsidered; in particular, several observations
seem to be not well explained within the EEF-based inductance scenario. Below,
we raise fundamental questions associated with the interpretation.

Figure 1:  Correlation between nonlinear $\operatorname{Im}\rho(j_{0})$ and
$-{\rm d}\rho_{0}/{\rm d}T$ in the reported data. (a, b) $\rho_{0}$–$T$
profile (a) and $\operatorname{Im}\Delta\rho^{1\omega}(j_{0})$–$T$ and
$\left(-{\rm d}\rho_{0}/{\rm d}T\right)$–$T$ profiles (b) of Gd3Ru4Al12. (c,
d) $\rho_{0}$–$T$ profile (c) and $\operatorname{Im}\rho^{1\omega}(j_{0})$–$T$
and $\left(-{\rm d}\rho_{0}/{\rm d}T\right)$–$T$ profiles (d) of YMn6Sn6. (e,
f) $\rho_{0}$–$T$ profile (e) and $\operatorname{Im}\rho^{1\omega}$–$T$ and
$(-\rho_{0}$$\times$${\rm d}\rho_{0}/{\rm d}T)$–$T$ profiles (f) of FeSn2. The
$\rho_{0}$–$T$ and $\operatorname{Im}\rho^{1\omega}(j_{0})$–$T$ profiles were
taken from the literature YokouchiNature ; KitaoriPNAS ; YokouchiArxiv , and
we constructed the ($-{\rm d}\rho_{0}/{\rm d}T$)–$T$ profiles from the data.
Note that for YMn6Sn6, the $\operatorname{Im}\rho^{1\omega}(j_{0})$–$T$ and
$(-{\rm d}\rho_{0}/{\rm d}T)$–$T$ profiles were collected from different
devices.

Question I. The values of
$\operatorname{Im}\rho^{1\omega}(j_{0},\omega)/\omega$ reported in experiments
YokouchiNature ; KitaoriPNAS ; YokouchiArxiv ; KitaoriPRB  appear too large to
be ascribed to the EEF origin. Linear-response EEF theory includes $\hbar$ and
predicts only a small value of
$\operatorname{Im}\rho^{1\omega}(\omega)/\omega$, on the order of 10-11–10-13
$\mu\Omega$ cm s NagaosaJJAP ; Furuta1 . Contrastingly, the magnitudes of
experimentally observed $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$
at low $\omega$ are $\approx$$-9\times$10-7 $\mu\Omega$ cm s for Gd3Ru4Al12
YokouchiNature , $\approx$$-4\times$10-4 $\mu\Omega$ cm s for YMn6Sn6
KitaoriPNAS , and $\approx$$-3\times$10-3 $\mu\Omega$ cm s for FeSn2
YokouchiArxiv  for $j_{0}\sim 10^{8}$ A m-2. Although the experimental reports
emphasize that these values are observed in the nonlinear regime,
understanding such gigantic responses within the EEF framework is nontrivial
even if the EEF beyond the linear-response regime is considered. Furthermore,
the sign of the reported $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})$ is
negative in most cases. The negative
$\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$ was interpreted as the
inductance being negative, but the theory of dynamical systems concludes that
unstable behavior occurs when the coefficient of the time derivative of the
electric current, ${\rm d}I/{\rm d}t$, is negative Textbook  (for details, see
Supplementary Materials Supp ).

Question II. The $\omega$ dependence of the nonlinear
$\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$ reported thus far
exhibits a cutoff frequency as low as $\sim$103–104 Hz for micrometer-sized
fabricated specimens YokouchiNature ; KitaoriPNAS  and $\sim$102 Hz for
needle-like bulk crystals YokouchiArxiv . These results were interpreted as
indicating that the spin texture under consideration has slow dynamics, but
the experimentally found resonance frequency of a ferromagnetic domain wall of
$\sim$107 Hz should be noted SaitoNature . Currently, there is no
understanding of why such extremely slow dynamics are ubiquitously observed in
the study of nonlinear emergent inductance.

Question III. By inspecting the previous reports YokouchiNature ; KitaoriPNAS
; YokouchiArxiv , we find a possible correlation between the nonlinearly
enhanced $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})$ and the temperature
($T$) derivative of the linear-response DC resistivity $\rho_{0}$. As
summarized in Fig. 1, the
$\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0})$–$T$ profile appears
similar to the ($-{\rm d}\rho_{0}/{\rm d}T$)–$T$ profile for Gd3Ru4Al12 [Figs.
1(a) and (b)] YokouchiNature  and YMn6Sn6 [Figs. 1(c) and (d)] KitaoriPNAS  or
to the ($-\rho_{0}$$\times$${\rm d}\rho_{0}/{\rm d}T$)–$T$ profile for FeSn2
[Figs. 1(e) and (f)] YokouchiArxiv . In particular, regarding Gd3Ru4Al12, we
could construct the approximate ${\rm d}\rho_{0}/{\rm d}T$–$H$ profile ($H$
denotes the magnetic field) and compare this with the
$\operatorname{Im}\rho^{1\omega}$–$H$ profile (not the
$\operatorname{Im}\Delta\rho^{1\omega}$–$H$ profile because of the lack of
linear-response data), as shown in Figs. 8(a)–(c) in the APPENDIX. Remarkably,
the complicated $H$ dependence shown in $\operatorname{Im}\rho^{1\omega}$ is
reproduced by the $H$-dependent $-{\rm d}\rho_{0}/{\rm d}T$ at least
qualitatively, thereby indicating that a considerable correlation exists
between $\operatorname{Im}\rho^{1\omega}$ and $-{\rm d}\rho_{0}/{\rm d}T$.
This propensity seems difficult to understand in terms of the EEF, which is
determined by the spin dynamics and does not involve the $T$ derivative of the
scattering rate.

Figure 2:  Nonlinear $\rho^{1\omega}$ in Gd3Ru4Al12. (a, b)
$\operatorname{Re}\rho^{1\omega}$–$T$ (a) and
$\operatorname{Im}\rho^{1\omega}$–$T$ (b) profiles under various AC current
densities, $j_{0}\sin\omega t$, at $\omega/(2\pi)=10$ kHz. (c) Current-density
dependence of complex
$\Delta\rho^{1\omega}(j_{0})=\rho^{1\omega}(j_{0})-\rho^{1\omega}(0.7\times
10^{8}{\rm A~{}m^{-2}})$ at 15 K. The panels are constructed from the raw data
published in the literature YokouchiNature .

In exploring clues to answer questions I–III, we realize that the magnitude of
the nonlinear enhancement of $\operatorname{Re}\rho^{1\omega}(\omega,j_{0})$
is larger than that of $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})$; i.e.,
regarding the nonlinear part,
$\operatorname{Re}\Delta\rho^{1\omega}>\operatorname{Im}\Delta\rho^{1\omega}$
holds, where
$\Delta\rho^{1\omega}(\omega,j_{0})\equiv\rho^{1\omega}(\omega,j_{0})-\rho_{0}(\omega)$,
with $\rho_{0}(\omega)$ representing a linear-response complex value. For
Gd3Ru4Al12, for instance, the $j_{0}$-dependent
$\operatorname{Re}\rho^{1\omega}(j_{0})$–$T$ and
$\operatorname{Im}\rho^{1\omega}(j_{0})$–$T$ profiles in Figs. 2(a) and (b),
respectively, were reported YokouchiNature . The simple temperature shift
cannot explain the pronounced variations in
$\operatorname{Im}\rho^{1\omega}(j_{0})$ at 10 kHz, and on this basis, the
nonlinear $\operatorname{Im}\rho^{1\omega}(j_{0})$ was interpreted as
originating from the current-induced EEF. However, the nonlinear part, which
is here defined as
$\Delta\rho^{1\omega}(j_{0})\equiv\rho^{1\omega}(j_{0})-\rho^{1\omega}(0.7\times
10^{8}{\rm A~{}m^{-2}}$), exhibits a $\operatorname{Re}\Delta\rho^{1\omega}$
that is approximately 10 times larger than
$\operatorname{Im}\Delta\rho^{1\omega}$ for all $j_{0}$ values [Fig. 2(c)].
$\operatorname{Re}\rho\gg\operatorname{Im}\rho$ at low $\omega$ is a
characteristic of dissipative responses of a resistor, and therefore,
regarding the $\operatorname{Im}\Delta\rho^{1\omega}(j_{0})$ under an AC
current as a delayed response of
$\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$ appears more reasonable. In
contrast, the nonlinear inductance mechanism discussed in previous studies
assumes that the nonlinearity occurs mainly in the imaginary part; i.e.,
$\operatorname{Im}\Delta\rho^{1\omega}\gg\operatorname{Re}\Delta\rho^{1\omega}$
at low $\omega$, which is a characteristic of nondissipative responses of an
inductor and inconsistent with the behavior shown in Fig. 2(c).

These unresolved or hitherto overlooked issues led us to reconsider the origin
of the nonlinear $\operatorname{Im}\Delta\rho^{1\omega}$ observed in previous
experiments. In particular, on the basis of Figs. 1 and 2, we explored a
dissipative mechanism. In general, when nonlinear electrical responses under a
large current are examined, the impact of Joule heating must be considered. In
AC impedance measurements, the sample temperature is time-varying with a
$2\omega$ modulation as a result of the time-varying current, and thus, the
situation is more complicated than DC measurements, in which Joule heating
only results in a static temperature increase. Such time-varying heating may
induce a nonlinear $\operatorname{Im}\Delta\rho^{1\omega}$. While its impact
has been discussed in the context of detecting a superconducting transition
HeatingZ , to the best of the authors’ knowledge, the impact of time-varying
Joule heating on the nonlinearly enhanced $\operatorname{Im}\rho^{1\omega}$
has not been discussed in the experimental studies on emergent inductance.
This motivated us to scrutinize whether the nonlinear
$\operatorname{Im}\rho^{1\omega}$ caused by AC Joule heating is truly
negligible in the reported nonlinear emergent inductance.

##  II II. Joule heating model for the nonlinear AC electrical response

###  II.1 A. Model construction

Below, we aim to clarify $\rho(\omega,j_{0})$ characteristics when a large AC
current is applied to the sample such that the effect of time-varying Joule
heating is not negligible; that is, we consider a sample in contact with a
heat bath of temperature $T_{0}$ and derive the voltage responses under the
time-varying sample temperature due to Joule heating. For simplicity, we
neglect the temperature gradient within the sample. To analytically solve this
problem, we make the following assumptions that appear to be physically
reasonable.

Assumption $\\#$1. The instantaneous voltage drop in the time-varying self-
heated sample is given by:

$V(t)=R_{0}\bigl{(}T(t)\bigr{)}I(t),$ (1)

where $R_{0}(T)$ denotes the linear-response resistance at $T$. In this model,
only this resistive mechanism is considered for the relationship between the
voltage and current. In other words, neither inductive nor capacitive
mechanisms are considered.

Assumption $\\#$2. We treat the temperature increase of the sample, $\Delta
T(t)=T(t)-T_{0}$, with respect to the AC power input with angular frequency
$\Omega$, $P(t)=\operatorname{Re}(P_{0}e^{i\Omega t})$, as being within the
linear-response regime. This means that we consider the lowest-order nonlinear
response to the AC current input. By introducing a complex response function,
$\chi^{*}(\Omega,T_{0})=\chi^{\prime}(\Omega,T_{0})-i\chi^{\prime\prime}(\Omega,T_{0})$,
$\Delta T(t)$ is given by:

$\displaystyle\Delta T(t)$
$\displaystyle=\operatorname{Re}\bigl{[}\chi^{*}(\Omega,T_{0})P_{0}e^{i\Omega
t}\bigr{]}$ $\displaystyle=P_{0}\bigl{[}\chi^{\prime}(\Omega,T_{0})\cos\Omega
t+\chi^{\prime\prime}(\Omega,T_{0})\sin\Omega t\bigr{]},$ (2)

which is the expression for a cosine-wave power input, $P(t)=P_{0}\cos\Omega
t$. By definition, in the DC limit, $\chi^{\prime}(\Omega,T_{0})$ approaches a
finite value, $\chi_{0}(T_{0})$, and $\chi^{\prime\prime}(\Omega,T_{0})$
approaches zero; i.e., we impose $\lim_{\Omega\to
0}\chi^{\prime}(\Omega,T_{0})=\chi_{0}(T_{0})$ and $\lim_{\Omega\to
0}\chi^{\prime\prime}(\Omega,T_{0})=0$. Note that $\chi^{*}$ is determined by
the heat capacitance of the system and the heat conduction to the heat bath;
thus, $\chi^{*}$ depends on the volume and geometry of the sample, the details
of the thermal contacts, etc.

Assumption $\\#$3. A system has a nonzero thermal-response time, $\tau_{\rm
therm}$ $(>0)$, which is a phenomenon known as thermal relaxation. Thus, under
an AC power input, the thermal response of the sample is more or less delayed
(i.e., $\chi^{\prime\prime}>0$), and correspondingly, $\chi^{*}(\Omega,T_{0})$
has a $\Omega$ dependence with a cutoff frequency of $\approx 1/(2\pi\tau_{\rm
therm})$. The form of $\chi^{*}(\Omega)$ can be approximately captured by a
polydispersive Cole-Cole-type response:

$\chi^{*}(\Omega,T)=\frac{\chi_{0}(T)}{1+\bigl{(}i\Omega\tau_{\rm
therm}(T)\bigr{)}^{1-\alpha}},$ (3)

where $\alpha$ represents the polydispersivity. For the readers’ reference, we
display the functional form of Eq. (3) for the case of a monodispersive
relaxation, $\alpha=0$, in Fig. 3. The details of $\chi^{*}$ depend on the
system as mentioned above. Nevertheless, the only important feature in the
following discussion is that $\chi^{*}(\Omega,T_{0})$ has a cutoff frequency
determined by the thermal-response dynamics and satisfies $\lim_{\Omega\to
0}\chi^{\prime}(\Omega,T_{0})=\chi_{0}(T_{0})$ and $\lim_{\Omega\to
0}\chi^{\prime\prime}(\Omega,T_{0})=0$. Equation (3) is an example of a
function that satisfies these characteristics.

Figure 3:  Example of the response function,
$\chi^{*}(\Omega)=\chi^{\prime}-i\chi^{\prime\prime}$, of the temperature
change in response to the power input. The functional form of Eq. (3) with
$\alpha=0$ is displayed on a double logarithmic scale.

Assumption $\\#$4. $\Delta T(t)$ is so small that the time-dependent
resistance, $R(T(t))$, is well approximated by:

$\displaystyle R\bigl{(}T(t)\bigr{)}$ $\displaystyle=R_{0}(T_{0})+\Delta
R_{0}\bigl{(}\Delta T(t)\bigr{)}$ $\displaystyle\approx
R_{0}(T_{0})+\frac{{\rm d}R_{0}(T_{0})}{{\rm d}T}\Delta T(t),$ (4)

where ${\rm d}R_{0}(T_{0})/{\rm d}T\equiv{\rm d}R_{0}/{\rm
d}T\rvert_{T=T_{0}}$.

###  II.2 B. Derivation of the nonlinear AC electrical response

With the above assumptions, one can analytically derive $P(t)$, $\Delta T(t)$,
$R(t)$, and, finally, $V(t)$ in sequence. We consider the situation in which a
sine-wave AC current with angular frequency $\omega$, $I(t)=I_{0}\sin\omega
t$, is applied to the sample. Hence, the time-varying power input, $P(t)$, is
given by:

$\displaystyle P(t)$ $\displaystyle\approx R_{\rm
2probe}(T_{0})(I_{0}\sin\omega t)^{2}$ $\displaystyle=P_{0}\frac{1-\cos
2\omega t}{2}$
$\displaystyle=\frac{P_{0}}{2}-\frac{P_{0}}{2}\operatorname{Re}(e^{i2\omega
t}),$ (5)

where $R_{\rm 2probe}$ denotes the two-probe resistance including the contact
resistance $R_{\rm contact}$ and $P_{0}(I_{0})\equiv R_{\rm
2probe}(T_{0})I_{0}^{2}=(R_{0}(T_{0})+R_{\rm contact}(T_{0}))I_{0}^{2}$. Here,
we neglect the deviation from Eq. (II.2) due to the time-varying two-probe
resistance. By combining this result with Eq. (II.1) and further considering
Eqs. (II.1) and (1) in sequence, one obtains:

$\displaystyle\Delta T(t)$
$\displaystyle\approx\frac{\chi_{0}(T_{0})P_{0}}{2}-\frac{P_{0}}{2}\operatorname{Re}\bigl{[}\chi^{*}(2\omega,T_{0})e^{i2\omega
t}\bigr{]}$
$\displaystyle=\frac{\chi_{0}(T_{0})P_{0}}{2}-\frac{P_{0}}{2}\bigl{[}\chi^{\prime}(2\omega,T_{0})\cos
2\omega t+\chi^{\prime\prime}(2\omega,T_{0})\sin 2\omega t\bigr{]}.$
$\displaystyle\therefore R(t)$ $\displaystyle\approx R_{0}(T_{0})+\frac{{\rm
d}R_{0}(T_{0})}{{\rm
d}T}\left\\{\frac{\chi_{0}(T_{0})P_{0}}{2}-\frac{P_{0}}{2}\bigl{[}\chi^{\prime}(2\omega,T_{0})\cos
2\omega t+\chi^{\prime\prime}(2\omega,T_{0})\sin 2\omega t\bigr{]}\right\\}.$
$\displaystyle\therefore V(t)$ $\displaystyle\approx
R_{0}(T_{0})I_{0}\sin\omega t$ $\displaystyle+\frac{{\rm d}R_{0}(T_{0})}{{\rm
d}T}\left\\{\frac{\chi_{0}(T_{0})P_{0}}{2}-\frac{P_{0}}{2}\bigl{[}\chi^{\prime}(2\omega,T_{0})\cos
2\omega t+\chi^{\prime\prime}(2\omega,T_{0})\sin 2\omega
t\bigr{]}\right\\}I_{0}\sin\omega t.$
$\displaystyle\therefore\frac{V(t)}{I_{0}}$ $\displaystyle\approx
R_{0}(T_{0})\sin\omega t+\frac{{\rm d}R(T_{0})}{{\rm
d}T}\frac{\chi_{0}(T_{0})P_{0}}{2}\sin\omega t$ $\displaystyle+\frac{{\rm
d}R_{0}(T_{0})}{{\rm
d}T}\frac{P_{0}}{4}\Bigl{[}\chi^{\prime}(2\omega,T_{0})\sin\omega
t-\chi^{\prime\prime}(2\omega,T_{0})\cos\omega
t-\chi^{\prime}(2\omega,T_{0})\sin 3\omega
t+\chi^{\prime\prime}(2\omega,T_{0})\cos 3\omega t\Bigr{]}.$ (6)

Thus, for an input of $I(t)=I_{0}\sin\omega t$, the Fourier series of
$V(t)/I_{0}$ has been analytically derived up to the 3$\omega$ components.
Following previous studies YokouchiNature ; KitaoriPNAS ; YokouchiArxiv , we
introduce $Z^{n\omega}$ to describe the resulting in-phase and out-of-phase
electrical responses of the $n\omega$ components ($n=1,3$); i.e.,
$\operatorname{Re}Z^{n\omega}\equiv\operatorname{Re}\mathcal{V}^{n\omega}(\omega,I_{0})/I_{0}$
and
$\operatorname{Im}Z^{n\omega}\equiv\operatorname{Im}\mathcal{V}^{n\omega}(\omega,I_{0})/I_{0}$,
where $\mathcal{V}^{n\omega}(\omega,I_{0})$ is the $n\omega$ Fourier component
of $V(t)$ under an AC current with $\omega$. The real and imaginary parts of
$Z^{n\omega}$ are given by:

$\displaystyle\operatorname{Re}Z^{1\omega}(\omega,I_{0},T_{0})-R_{0}(T_{0})$
$\displaystyle=\frac{{\rm d}R_{0}(T_{0})}{{\rm
d}T}\frac{P_{0}(I_{0})}{4}\bigl{[}2\chi_{0}(T_{0})+\chi^{\prime}(2\omega,T_{0})\bigr{]},$
(7) $\displaystyle\operatorname{Im}Z^{1\omega}(\omega,I_{0},T_{0})$
$\displaystyle=-\frac{{\rm d}R_{0}(T_{0})}{{\rm
d}T}\frac{P_{0}(I_{0})}{4}\chi^{\prime\prime}(2\omega,T_{0}),$ (8)
$\displaystyle\operatorname{Re}Z^{3\omega}(\omega,I_{0},T_{0})$
$\displaystyle=-\frac{{\rm d}R_{0}(T_{0})}{{\rm
d}T}\frac{P_{0}(I_{0})}{4}\chi^{\prime}(2\omega,T_{0}),$ (9)
$\displaystyle\operatorname{Im}Z^{3\omega}(\omega,I_{0},T_{0})$
$\displaystyle=\frac{{\rm d}R_{0}(T_{0})}{{\rm
d}T}\frac{P_{0}(I_{0})}{4}\chi^{\prime\prime}(2\omega,T_{0}).$ (10)

Note that the right-hand sides of Eqs. (7)–(10) are $I_{0}$ dependent via
$P_{0}=R_{\rm 2probe}I_{0}^{2}$, and thus, they represent nonlinear responses.
The emergence of these nonlinear terms can be qualitatively understood as
follows: Under an AC current with angular frequency $\omega$, the sample
temperature and resistance are time-varying, with a $2\omega$ modulation; the
$2\omega$ resistance modulation couples with the AC current with $\omega$,
giving rise to additional output voltage modulations of both $\omega$ and
$3\omega$. Thus, $\operatorname{Im}Z^{1\omega}(\omega,I_{0})$ and
$\operatorname{Im}Z^{3\omega}(\omega,I_{0})$ appear due to a delay of the
thermal response (i.e., $\chi^{\prime\prime}$).

Note that regarding the nonlinear parts of the Joule-heating-induced AC
electrical response, the following relations hold at low frequencies,
$\omega\ll\omega_{c}=1/(2\tau_{\rm therm}$):

$\displaystyle\left|\frac{\operatorname{Im}Z^{1\omega}}{\operatorname{Re}Z^{1\omega}-R_{0}}\right|=\frac{\chi^{\prime\prime}(2\omega,T_{0})}{2\chi_{0}(T_{0})+\chi^{\prime}(2\omega,T_{0})}\ll
1\hskip 14.22636pt\rm for\hskip 5.69046pt\omega\ll\omega_{c}.$ (11)
$\displaystyle\left|\frac{\operatorname{Im}Z^{3\omega}}{\operatorname{Re}Z^{3\omega}}\right|=\frac{\chi^{\prime\prime}(2\omega,T_{0})}{\chi^{\prime}(2\omega,T_{0})}\ll
1\hskip 14.22636pt\rm for\hskip 5.69046pt\omega\ll\omega_{c}.$ (12)

These equations indicate that at low $\omega$, the nonlinearity is observed
mainly in the real part, rather than in the imaginary part. These
characteristics are also evident in the Cole-Cole representation [Figs. 4(a)
and (b)], in which $\operatorname{Re}Z^{1\omega}-R_{0}$ is adopted for the
real axis for clarity. In the Cole-Cole representation of $Z^{3\omega}$, the
$\omega$-evolving trajectory starts somewhere on the real axis at DC and
converges to the origin at high $\omega$ [Fig. 4(b)]. This behavior indicates
that the Joule-heating-induced AC electrical response has dissipative
characteristics. Contrastingly, when nonlinear $Z^{1\omega}$ and $Z^{3\omega}$
are caused by the nonlinear inductance, the nonlinear change should
exclusively appear in the imaginary part; that is,
$\left|\operatorname{Im}Z^{1\omega}\right|\gg\left|\operatorname{Re}Z^{1\omega}-R_{0}\right|$
and
$\left|\operatorname{Im}Z^{3\omega}\right|\gg\left|\operatorname{Re}Z^{3\omega}\right|$
should hold at low $\omega$. In the Cole-Cole representation, these
nondissipative characteristics are observed as shown in Figs. 4(c) and (d). In
particular, the $\omega$-evolving trajectory of $Z^{3\omega}$ should start
from the origin at DC [Fig. 4(d)], which is distinctly different from the
$Z^{3\omega}$ caused by Joule heating [Fig. 4(b)]. Thus, the Cole-Cole
representation provides key insight into whether the observed nonlinear AC
electrical response has dissipative or nondissipative characteristics.

Figure 4:  Schematic Cole-Cole representation of the AC electrical response.
(a, b) Cole-Cole representations of $Z^{1\omega}$ (a) and $Z^{3\omega}$ (b)
under Joule heating. $R_{0}$ represents the DC linear response impedance.
Schematics are drawn for both the cases of ${\rm d}R_{0}/{\rm d}T>0$ and ${\rm
d}R_{0}/{\rm d}T<0$. (c, d) Cole-Cole representations of $Z^{1\omega}$ (c) and
$Z^{3\omega}$ (d) of a nonlinear inductor element.

###  II.3 C. Answers to questions I–III

Below, we show that the Joule-heating-induced AC electrical response expressed
by Eqs. (7)–(10) satisfies all the types of behavior highlighted in questions
I–III. We begin by answering question III. Below we use $T$ to denote the
sample-holder temperature, which corresponds to $T_{0}$ of the previous
section.

Answer to question III. As indicated by Eqs. (8)–(10), the temperature
dependences of $\operatorname{Im}Z^{1\omega}$
$(=-\operatorname{Im}Z^{3\omega})$ and $\operatorname{Re}Z^{3\omega}$ involve
that of $P_{0}(T)$$\times$${\rm d}R_{0}/{\rm d}T$. Thus, the
$\operatorname{Im}Z^{1\omega/3\omega}$–$T$ profile is correlated with ${\rm
d}R_{0}/{\rm d}T$. Since $P_{0}(T)=\bigl{(}R_{0}(T)+R_{\rm
contact}(T)\bigr{)}I_{0}^{2}$, whether ${\rm d}R_{0}/{\rm d}T$ or
$R_{0}$$\times$${\rm d}R_{0}/{\rm d}T$ better describes the
$\operatorname{Im}Z^{1\omega/3\omega}$–$T$ profile depends on whether the
Joule heating is dominant in the bulk or at the contacts of the current
electrodes. If the Joule heat is produced exclusively in the bulk [i.e.,
$P_{0}(T)\approx R_{0}(T)I_{0}^{2}$], then
$\operatorname{Im}Z^{1\omega/3\omega}(T)$ would scale with
$R_{0}$$\times$${\rm d}R_{0}/{\rm d}T$. If the Joule heat is mainly generated
at the contacts of the current electrodes [i.e., $P_{0}(T)\approx R_{\rm
contact}(T)I_{0}^{2}$] and if $R_{\rm contact}(T)$ only weakly depends on
temperature, then $\operatorname{Im}Z^{1\omega/3\omega}(T)$ would scale with
${\rm d}R_{0}/{\rm d}T$. This consideration accounts for the fact that
$\operatorname{Im}\rho^{1\omega}(T)$ scales well with either $-{\rm
d}\rho_{0}/{\rm d}T$ or $-\rho_{0}$$\times$${\rm d}\rho_{0}/{\rm d}T$
depending on the report [Figs. 1(b), (d) and (f)]. However, note that the
agreement should only be qualitative because this $\chi^{*}(T)$, which scales
with the heat conductance and inversely scales with the heat capacitance, is
also temperature dependent. At a very low temperature, the temperature
increase for a given Joule heat can be pronounced, and thus, the scaling
between the $\operatorname{Im}Z^{1\omega/3\omega}$–$T$ and ${\rm
d}\rho_{0}/{\rm d}T$–$T$ profiles may be worse. This circumstance may be
related to the behavior below 10 K of Gd3Ru4Al12 [Fig. 1(b)].

Answer to question II. The cutoff frequency $\omega_{c}$ of the Joule-heating-
induced AC electrical response is determined by the thermal relaxation time of
the sample: $\omega_{c}\approx 1/(2\tau_{\rm therm})$. The thermal relaxation
time depends on the system details, such as the sample dimensions and thermal
contacts. For a millimeter-sized bulk sample, the typical value of $\tau_{\rm
therm}$ can be as long as $\sim$10-1–10-2 s OikeNatPhys , which leads to
$\sim$100–101 Hz for the cutoff frequency of $\chi^{*}(\omega)$. For an
exfoliated thin plate with a submicrometer thickness, the typical thermal
relaxation time is $\sim$10-6–10-3 s (depending on the thermal conductivity of
the substrate) OikeSciAdv , which leads to $\sim$102–105 Hz for the cutoff
frequency. As shown below, the microfabricated CuIr2S4 and bulk MoTe2 exhibit
cutoff frequencies of 20 kHz and 1 Hz, respectively. Thus, in general, the
cutoff frequency of the Joule-heating-induced AC electrical response can be
much lower than the resonance frequency of the magnetic texture under
consideration.

Table 1: Order of magnitude estimate of $\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0},T)/\omega$ for Gd3Ru4Al12 YokouchiNature , YMn6Sn6 KitaoriPNAS , and FeSn2 YokouchiArxiv . The parameters used for the calculations are taken from the literature YokouchiNature ; KitaoriPNAS ; YokouchiArxiv  or Fig. 2. Material |  | Gd3Ru4Al12 | YMn6Sn6 | FeSn2
---|---|---|---|---
Specimen |  | Microfabricated | Microfabricated | Bulk
Sample-holder temperature, $T_{0}$ [K] |  | 15 | 270 | 350
Applied current density, $j_{0}$ [108 A m-2] |  | 4.0 | 2.5 | 0.8
Sample volume [$\mu$m3] |  | 13 | 290 | 8.9$\times 10^{4}$
Observed cutoff frequency, $\omega_{c}$/2$\pi$ [kHz] |  | 20 | 1 | 0.1
$\operatorname{Re}\Delta\rho^{1\omega}(j_{0},T)$ at low $\omega$ [$\mu\Omega$ cm] |  | 0.7 | 1.5 | 2.9
$\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0},T)/\omega$ at low $\omega$ ($\ll\omega_{c}$) [$\mu\Omega$ cm s] | (Cal.) |  $-2\times$10-6 |  $-0.8\times$10-4 |  $-1\times$10-3
(Exp.) |  $-1.1\times$10-6 |  $-2.6\times$10-4 |  $-1.4\times$10-3

Answer to question I. For the Joule-heating-induced AC electrical response, an
order of magnitude estimate of $\operatorname{Im}\Delta\rho^{1\omega}(j_{0})$
can be obtained by referring to $\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$
and $\omega_{c}$ as follows:

$\displaystyle\frac{\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0},T)}{\operatorname{Re}\Delta\rho^{1\omega}(\omega,j_{0},T)}=\frac{\chi^{\prime\prime}(2\omega,T)}{2\chi_{0}(T)+\chi^{\prime}(2\omega,T)}.$
(13)

Note that the ratio of $\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0},T)$
to $\operatorname{Re}\Delta\rho^{1\omega}(\omega,j_{0},T)$ does not depend on
$j_{0}$. Using Eq. (3) with $\alpha=0$ for simplicity, we obtain:

$\displaystyle\frac{\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0},T)}{\omega}\approx-\frac{1}{3}\frac{\operatorname{Re}\Delta\rho^{1\omega}(j_{0},T)}{\omega_{c}}\hskip
14.22636pt\rm for\hskip 5.69046pt\omega\ll\omega_{c},$ (14)

where $\operatorname{Re}\Delta\rho^{1\omega}(j_{0},T)$ represents the low-
frequency limit value. By referring to the
$\operatorname{Re}\Delta\rho^{1\omega}(j_{0},T)$ and $\omega_{c}$ reported for
Gd3Ru4Al12 YokouchiNature , YMn6Sn6 KitaoriPNAS , and FeSn2 YokouchiArxiv , we
calculate the $\operatorname{Im}\Delta\rho^{1\omega}(\omega,j_{0},T)/\omega$
at low $\omega$ ($\ll\omega_{c}$) for each system. The results are summarized
in Table I. The calculated values are in good agreement with the reported data
for all three systems. This correspondence on the order of magnitude
demonstrates that the origin of the nonlinear
$\operatorname{Im}\Delta\rho^{1\omega}(j_{0})$ observed in the experiments
YokouchiNature ; KitaoriPNAS ; YokouchiArxiv  lies in the delay of the
nonlinear $\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$ with a cutoff
frequency $\omega_{c}$. Another question that naturally arises here is what is
the origin of the nonlinear $\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$?
The fact that the $\operatorname{Im}\rho^{1\omega}(j_{0})$–$T$ profiles scale
well with the $(-{\rm d}\rho_{0}/{\rm d}T)$–$T$ or $(-\rho_{0}$$\times$${\rm
d}\rho_{0}/{\rm d}T)$–$T$ profiles (Fig. 1) indicates that the nonlinear
$\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$ is also related to ${\rm
d}\rho_{0}/{\rm d}T$. The simplest interpretation of this observation is that
the nonlinear $\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$ is caused by
Joule heating. Thus, the similarity between the
$\operatorname{Im}\Delta\rho^{1\omega}$–$T$ and $(-{\rm d}\rho_{0}/{\rm
d}T)$–$T$ profiles implies that $\operatorname{Im}\Delta\rho^{1\omega}$
includes a considerable contribution of the Joule-heating-induced AC
electrical response.

Others. Although unrelated to questions I–III,
$\operatorname{Im}Z^{1\omega}=-\operatorname{Im}Z^{3\omega}$ is also a
characteristic consequence of the Joule heating model.

##  III IV. Experiments

To experimentally test the Joule heating model, we measured the nonlinear AC
electrical response for two conducting systems: a microfabricated CuIr2S4
crystal (sample dimensions of 20$\times$4$\times$1 $\mu$m3) and a bulk
1$T$’-MoTe2 crystal (1.3$\times$0.7$\times$0.14 mm3); images of the two
samples are shown in Figs. 9(a) and (b) in the APPENDIX. These materials
exhibit no magnetic order, and a current-induced spin dynamics contribution to
the AC electrical response can therefore be ruled out. In both materials, we
find a good agreement between the experimental and expected results for the
Joule-heating-induced AC electrical response. For readability, only results
for the microfabricated CuIr2S4 are shown in the main text. Results for the
bulk MoTe2 are shown in Figs. 10 and 11 in the APPENDIX.

CuIr2S4 shows a first-order metal-insulator transition at $T_{\rm c}\approx$
230 K CuIr2S4_Furubayashi ; CuIr2S4_Cheong . This material is paramagnetic and
metallic (i.e., ${\rm d}\rho_{0}/{\rm d}T>0$) above $T_{\rm c}$, whereas it is
nonmagnetic and semiconducting (i.e., ${\rm d}\rho_{0}/{\rm d}T<0$) below
$T_{\rm c}$. Figures 5(a), (b), (c), and (d) display the temperature
dependences of $\operatorname{Re}\rho^{1\omega}$,
$\operatorname{Im}\Delta\rho^{1\omega}$, $\operatorname{Re}\rho^{3\omega}$,
and $\operatorname{Im}\rho^{3\omega}$ at $\omega$/2$\pi=6$ kHz, respectively,
measured at various current densities. Here, we display
$\operatorname{Im}\Delta\rho^{1\omega}\equiv\operatorname{Im}\rho^{1\omega}(j_{0})-\operatorname{Im}\rho^{1\omega}(5.0\times
10^{7}$ A m-2) only for $\operatorname{Im}\rho^{1\omega}$ to subtract the
contribution of the nonnegligible linear-response background; for the raw data
of $\operatorname{Im}\rho^{1\omega}$, see Fig. 9(c) in the APPENDIX. In
contrast, for $\rho^{3\omega}$, we find that the background signal is not
significant, so we display the raw data.

Figure 5:  Temperature dependence of the AC electrical response of CuIr2S4
measured at various current amplitudes. (a) $\operatorname{Re}\rho^{1\omega}$,
(b) $\operatorname{Im}\Delta\rho^{1\omega}$, (c)
$\operatorname{Re}\rho^{3\omega}$, and (d) $\operatorname{Im}\rho^{3\omega}$.
The $\operatorname{Im}\Delta\rho^{1\omega}$ shown in (b) is defined as
$\operatorname{Im}\Delta\rho^{1\omega}\equiv\operatorname{Im}\rho^{1\omega}(j_{0})-\operatorname{Im}\rho^{1\omega}(5.0\times
10^{7}$ A m-2). (e) Comparison between $\operatorname{Im}\Delta\rho^{1\omega}$
and $-\rho_{0}$$\times$${\rm d}\rho_{0}/{\rm d}T$. The data were recorded in
the heating process. (f) Comparison between
$\operatorname{Im}\Delta\rho^{1\omega}$ and
$-\operatorname{Im}\rho^{3\omega}$.

In the $\operatorname{Re}\rho^{1\omega}$–$T$ profile [Fig. 5(a)], the apparent
transition temperature clearly decreases with increasing current, indicating
that the sample temperature is elevated from the sample-holder temperature,
$T_{\rm holder}$, by Joule heating. In the insulating phase, the four-probe
resistance is 30–80 $\Omega$ within 225–160 K, whereas the contact resistance
is $\approx$20 $\Omega$. Thus, the Joule heat is thought to be generated
mainly in the bulk, rather than at the contacts of the current electrodes.

Figures 5(b)–(f) show characteristic features consistent with the Joule
heating model. First, in the insulating phase,
$\operatorname{Im}\Delta\rho^{1\omega}$, $\operatorname{Re}\rho^{3\omega}$ and
$\operatorname{Im}\rho^{3\omega}$ nonlinearly emerge as the AC current
increases [Figs. 5(b)–(d)]. The signs of these quantities are consistent with
Eqs. (8)–(10) for the case of ${\rm d}\rho_{0}/{\rm d}T<0$. Second, the
$\operatorname{Im}\Delta\rho^{1\omega}$–$T$ profile agrees well with the
$(-\rho_{0}$$\times$${\rm d}\rho_{0}/{\rm d}T)$–$T$ profile [Fig. 5(e)],
consistent with the results expected when Joule heating occurs mainly in the
bulk.. Third,
$\operatorname{Im}\Delta\rho^{1\omega}=-\operatorname{Im}\rho^{3\omega}$ is
well satisfied [Fig. 5(f)].

The frequency dependences and Cole-Cole representations of
$\Delta\rho^{1\omega}$ and $\rho^{3\omega}$ at 210 K are shown in Figs.
6(a)–(f). Overall, the Cole-Cole representations [Figs. 6(c) and (f)] are
consistent with the predictions of the Joule heating model for the case of
${\rm d}\rho_{0}/{\rm d}T<0$ [Figs. 4(a) and (c)]. The lengths of the arc
strings are approximately the same for both cases (200 $\mu\Omega$ cm). From
the frequency dependence, the cutoff frequency in the present device is found
to be $\approx$20 kHz. Note that this value is not an intrinsic quantity of
the material but should depend on the sample volume, details of the thermal
contacts, etc. In the bulk MoTe2, for instance, the cutoff frequency is as low
as 1 Hz (see Fig. 11 in the APPENDIX), indicating that the sample dimensions
are a crucial factor determining the cutoff frequency of the Joule-heating-
induced AC electrical response. The cutoff frequency depends on temperature
only weakly, except for at the transition point, at which it decreases to 7
kHz. This decrease in the cutoff frequency is ascribed to an apparent increase
in the heat capacity at a first-order phase transition.

Figure 6:  Frequency dependence of the nonlinear AC electrical response of
CuIr2S4 at 210 K. (a) $\operatorname{Re}\Delta\rho^{1\omega}$, (b)
$\operatorname{Im}\Delta\rho^{1\omega}$, and (c) Cole-Cole representation of
$\Delta\rho^{1\omega}$. $\Delta\rho^{1\omega}$ is defined as
$\Delta\rho^{1\omega}\equiv\rho^{1\omega}(j_{0})-\rho^{1\omega}(5.0\times
10^{7}$ A m-2). (d) $\operatorname{Re}\rho^{3\omega}$, (e)
$\operatorname{Im}\rho^{3\omega}$, and (f) Cole-Cole representation of
$\rho^{3\omega}$. The data were recorded at $j_{0}=3.5\times 10^{8}$ A m-2.

For an applied current density below 5$\times$108 A m-2, the nonlinear AC
electrical response is difficult to detect in the metallic phase above 230 K.
To observe the sign of the nonlinear behavior in the metallic phase, we
measured the $j_{0}$–$\operatorname{Im}\Delta\rho^{1\omega}$ profile up to a
higher current density, and the results are shown in Fig. 7. The finite
nonlinear $\operatorname{Im}\Delta\rho^{1\omega}$ exhibits a detectable
magnitude in the metallic phase when $j_{0}$ exceeds 7$\times 10^{8}$ A m-2,
and its sign is negative. Thus, we confirm that the sign of the nonlinear
$\operatorname{Im}\Delta\rho^{1\omega}$ is negative in the metallic phase
(${\rm d}\rho_{0}/{\rm d}T>0$), whereas it is positive in the insulating phase
(${\rm d}\rho_{0}/{\rm d}T<0$). The relationship between the signs of
$\operatorname{Im}\Delta\rho^{1\omega}$ and ${\rm d}\rho_{0}/{\rm d}T$ is
consistent with the Joule heating model.

Figure 7:  Current-density dependence of $\operatorname{Im}\Delta\rho$ in the
insulating and metallic phases. $\Delta\rho^{1\omega}$ is defined as
$\Delta\rho^{1\omega}\equiv\operatorname{Im}\rho^{1\omega}(j_{0})-\rho^{1\omega}(5.0\times
10^{7}$ A m-2).

##  IV V. Discussion

By measuring two different materials that exhibit no magnetic order, i.e.,
CuIr2S4 (in the main text) and MoTe2 (in the APPENDIX), we have confirmed that
the observed nonlinear AC electrical response is consistent with the Joule-
heating-induced AC electrical response. Phenomenologically, the nonlinear
$\Delta Z(j_{0})$ is likely described by the sum of dissipative and
nondissipative mechanisms: $\Delta Z(j_{0})=\Delta Z_{\rm diss}(j_{0})+\Delta
Z_{\rm nondiss}(j_{0})$, where $\Delta Z_{\rm diss}$ and $\Delta Z_{\rm
nondiss}$ represent the nonlinear impedances due to dissipative and
nondissipative mechanisms, respectively. Note that by definition, at low
$\omega$, $\operatorname{Re}\Delta Z_{\rm diss}\gg\operatorname{Im}\Delta
Z_{\rm diss}$, and $\operatorname{Re}\Delta Z_{\rm
nondiss}\ll\operatorname{Im}\Delta Z_{\rm nondiss}$. The inductor mechanism
due to the EEF caused by a pinned magnetic texture belongs to $\Delta Z_{\rm
nondiss}$, whereas the Joule-heating-induced AC electrical response belongs to
$\Delta Z_{\rm diss}$. Since we found
$\operatorname{Re}\Delta\rho\gg\operatorname{Im}\Delta\rho$ at low $\omega$ in
previous studies YokouchiNature ; KitaoriPNAS ; YokouchiArxiv , we focused on
$\Delta Z_{\rm diss}$ and considered $\Delta Z_{\rm Joule}(j_{0})$ to be a
plausible candidate. We have thus found that regarding $\Delta Z(j_{0})$, both
the temperature dependence and magnitude are similar to those expected for
$\Delta Z_{\rm Joule}(j_{0})$, regardless of whether the material under
consideration exhibits a magnetic order (as in YokouchiNature ; KitaoriPNAS ;
YokouchiArxiv ) or not (as in this study). This finding implies that $\Delta
Z_{\rm Joule}(j_{0})$ is presumably the most prominent among several
coexisting nonlinear mechanisms, and therefore, the observed $\Delta Z(j_{0})$
cannot straightforwardly be attributed to any subdominant mechanism of the
nonlinear electrical response, such as the spin-related $\Delta Z_{\rm
nondiss}$.

In general, a comparison of the real and imaginary parts of the AC electrical
response at low frequencies would provide key insights into the fundamental
properties of the nonlinear AC impedance under consideration. In previous
experiments, the fact that a nonlinear $\operatorname{Im}\Delta\rho$ is
accompanied by an even larger nonlinear $\operatorname{Re}\Delta\rho$ was not
deeply considered. To understand the nature of the nonlinear AC electrical
response, the real and imaginary parts must be examined on an equal footing.

##  V VI. Conclusion

Aiming to clarify fundamental questions that arose in the previous experiments
on so-called emergent inductors, we have considered the impact of time-varying
Joule heating on the AC electrical responses. From a theoretical point of
view, several key characteristics of the Joule-heating-induced AC electrical
response are clarified, and they provide natural explanations for the hitherto
unexplained behavior observed in emergent-inductor experiments. To further
examine the Joule-heating-induced AC electrical response, we have performed
experiments on two materials that exhibit no magnetic order, i.e., CuIr2S4 and
1T’-MoTe2, and confirmed the characteristics of the Joule heating model. In
particular, in our microfabricated CuIr2S4, a cutoff frequency of $\approx$20
kHz is observed, which is in agreement with the value reported for the
microfabricated noncollinear magnet Gd3Ru4Al12.

The Joule-heating-induced AC electrical response inevitably gives rise to
either a positive $\operatorname{Im}Z$ (when ${\rm d}\rho_{0}/{\rm d}T<0$) or
a negative $\operatorname{Im}Z$ (when ${\rm d}\rho_{0}/{\rm d}T>0$), unless
the Joule heating is negligible and the measurement frequency is far greater
than the inverse of the thermal response time. Even temperature oscillations
as small as 0.1 K cause a $\operatorname{Im}\Delta\rho$ of considerable
magnitude, which is much larger than that expected based on the linear-
response EEF. In previous experiments on emergent inductors, the nonlinear
$\operatorname{Im}\rho$ below the cutoff frequency was discussed, and the
nonlinearity more pronouncedly occurred in $\operatorname{Re}\rho$ than in
$\operatorname{Im}\rho$. Our results thus suggest that the reported data
regarding emergent inductors need to be reconsidered by taking the impact of
the Joule heating into account.

### Acknowledgments

F.K., S.F. and W.K. thank T. Yokouchi and A. Kitaori for providing us with the
raw data from the literature YokouchiNature ; KitaoriPNAS . F.K., S.F. and
W.K. thank T. Yokouchi and S. Maekawa for fruitful discussions. We thank Y.
Tokumura for the synthesis of CuIr2S4. This work was partially supported by
JSPS KAKENHI (Grants No. 21H04442 and No. 23K03291), JST CREST (Grants No.
JPMJCR1874 and No. JPMJCR20T1).

##  VI APPENDIX

###  VI.1 1\. Magnetic field dependence of ${\rm d}\rho/{\rm d}T$ in
Gd3Ru4Al12 at 5 K

Figure 8(a) shows the magnetic-field dependence of
$\operatorname{Re}\rho^{1\omega}$ in Gd3Ru4Al12 at $T_{\rm holder}=5.1$ and
5.3 K, which was measured at a current density in the nonlinear regime,
$j_{0}=3.3\times 10^{8}$ A m-2. Although these data are not in the linear-
response regime, we estimate the ${\rm d}\rho_{0}/{\rm d}T$–$H$ profile by
simply taking the difference between the two sets of data divided by the small
$T$ increment, 0.2 K. The results are shown in Fig. 8(b). Figure 8(c) shows
the $\operatorname{Im}\rho^{1\omega}$–$H$ profile at $T_{\rm holder}=5.1$ K
under the same current density. Note that the data are not
$\operatorname{Im}\Delta\rho^{1\omega}$, so they should include the linear-
response background signal. For instance, in Fig. 2,
$\operatorname{Im}\rho^{1\omega}$ shows a $T$-dependent negative signal even
in the linear-response regime. This signal may vary with the magnetic field.
Given this uncertainty, the ${\rm d}\rho/{\rm d}T$–$H$ profile exhibits a
complicated sharp structure, which is similar to that seen in Fig. 8(c). This
similarity, at least on a qualitative level, leads us to conclude that there
is a considerable correlation between ${\rm d}\rho_{0}/{\rm d}T$ and
$\operatorname{Im}\rho$.

Figure 8:  (a) Magnetic-field dependence of $\operatorname{Re}\rho^{1\omega}$
in Gd3Ru4Al12 at $T_{\rm holder}=5.1$ and 5.3 K, measured at a relatively high
current density, $j_{0}=3.3\times 10^{8}$ A m-2. (b) Estimated $(-{\rm
d}\rho_{0}/{\rm d}T)$–$H$ profile under $j_{0}=3.3\times 10^{8}$ A m-2. (c)
$\operatorname{Im}\rho^{1\omega}$–$H$ profile under $j_{0}=3.3\times 10^{8}$ A
m-2. The data were collected from the raw data published in the literature
YokouchiNature .

###  VI.2 2\. Methods

Figure 9:  (a) Scanning electron microscopy image of the microfabricated
CuIr2S4 device used in this study. (b) Photograph of the bulk MoTe2 used in
this study. (c) Raw data of the $\operatorname{Im}\rho^{1\omega}$–$T$ profile
of the microfabricated CuIr2S4. (d) Raw data of the
$\operatorname{Im}Z^{1\omega}$–$T$ profile of the microfabricated MoTe2. (e)
$R^{2}$ vs $\operatorname{Im}Z^{1\omega}$ plot for the impedance under a weak
AC current of 5.0$\times 10^{7}$ A m-2.

The images of the microfabricated CuIr2S4 device and the bulk MoTe2 are shown
in Figs. 9(a) and (b), respectively. In the experiments on MoTe2, we used
carbon paste for the current electrodes to facilitate the Joule heating; the
resistivity of MoTe2 is too low to achieve a Joule heating in bulk with use of
$\sim$100 mA for the case of bulk crystal.

Figures 9(c) and (d) show the raw data of the
$\operatorname{Im}\rho^{1\omega}$–$T$ profile of the microfabricated CuIr2S4
device and $\operatorname{Im}Z^{1\omega}$–$T$ profile of the bulk MoTe2
crystal, respectively. The linear-response background signal is
nonmonotonically $T$ dependent in the microfabricated CuIr2S4 device, whereas
it is relatively small and weakly $T$ dependent in the bulk MoTe2 crystal. The
linear-response background in the CuIr2S4 device is likely due to the impact
of the stray capacitance $C$, which gives rise to a linear-response background
of $-i\omega R_{0}^{2}C$ Furuta1 , where $R_{0}$ is the linear-response
resistance. $\operatorname{Im}Z^{1\omega}$ is proportional to $R_{0}^{2}$ in
the present temperature range [Fig. 9(e)], indicating that the
$(-R_{0}^{2}C)$-type background dominates the imaginary part of the linear-
response signal of the microfabricated sample. Since the nonlinearly enhanced
electrical response is the main focus of this study, we present
$\Delta\operatorname{Im}Z^{1\omega}$, which represents a nonlinear change from
the linear-response value.

###  VI.3 3\. Nonlinear AC electrical response of MoTe2 bulk crystal

MoTe2 shows a clear change in ${\rm d}R_{0}/{\rm d}T$ through a first-order
structural phase transition at $T_{\rm c}\approx 250$ K ZhouNatCommun , and
thus, this feature is helpful to study the correlation between
$\operatorname{Im}Z^{1\omega}$ and $-{\rm d}R_{0}/{\rm d}T$.

Figure 10:  Temperature dependence of AC electrical response of MoTe2 measured
at various current amplitudes. (a) $\operatorname{Re}Z^{1\omega}$, (b)
$\operatorname{Im}Z^{1\omega}$, (c) $\operatorname{Re}Z^{3\omega}$, and (d)
$\operatorname{Im}Z^{3\omega}$. The data were recorded in the heating process.
(e) Comparison between $\operatorname{Im}\Delta Z^{1\omega}$ and $-{\rm
d}R_{0}/{\rm d}T$. (f) Comparison between $\operatorname{Im}\Delta
Z^{1\omega}$ and $-\operatorname{Im}Z^{3\omega}$.

Figures 10(a), (b), (c), and (d) display the temperature dependences of
$\operatorname{Re}Z^{1\omega}$, $\operatorname{Im}\Delta Z^{1\omega}$,
$\operatorname{Re}Z^{3\omega}$, and $\operatorname{Im}Z^{3\omega}$ at
$\omega$/2$\pi=1$ Hz, respectively, measured using various current amplitudes.
In the $\operatorname{Re}Z^{1\omega}$–$T$ profile [Fig. 10(a)], the apparent
transition temperature clearly decreases as the current increases, indicating
that the sample temperature is increased from the sample-holder temperature,
$T_{\rm holder}$, by Joule heating. In Figs. 10(b)–(f), characteristic
features consistent with the Joule heating model can be observed. First,
$\operatorname{Im}\Delta Z^{1\omega}$, $\operatorname{Re}Z^{3\omega}$ and
$\operatorname{Im}Z^{3\omega}$ nonlinearly emerge as the AC current increases
[Figs. 10(b)–(d)]. Second, the $\operatorname{Im}\Delta Z^{1\omega}$–$T$
profile qualitatively agrees well with the $(-{\rm d}R_{0}/{\rm d}T)$–$T$
profile [Fig. 10(e)], consistent with the results expected when Joule heating
occurs mainly at the contacts. Third, the relation of $\operatorname{Im}\Delta
Z^{1\omega}=-\operatorname{Im}Z^{3\omega}$ is well satisfied except for in the
transition region at 245 K [Fig. 10(f)]. The breakdown of
$\operatorname{Im}\Delta Z^{1\omega}=-\operatorname{Im}Z^{3\omega}$ at 245 K
may be reasonable considering that Eq. (II.1) is generally not applicable at a
first-order phase transition because of the latent heat.

Figure 11:  Frequency dependence of the nonlinear AC electrical response of
MoTe2 at 181 K. (a) $\operatorname{Re}\Delta Z^{1\omega}$, (b)
$\operatorname{Im}\Delta Z^{1\omega}$, and (c) Cole-Cole representation of
$\Delta Z^{1\omega}$. (d) $\operatorname{Re}Z^{3\omega}$, (e)
$\operatorname{Im}Z^{3\omega}$, and (f) Cole-Cole representation of
$Z^{3\omega}$. The data were recorded using an AC current of $j_{0}=179$ mA.
As the experiments progressed, the contact resistance changed from the state
shown in Fig. 10, and the sample is therefore labelled Sample A’, instead of
A. The geometries of Sample A and Sample A’ are exactly the same.

The frequency dependences and Cole-Cole representations of $\Delta
Z^{1\omega}$ and $Z^{3\omega}$ at 180 K are shown in Figs. 11(a)–(f). They are
also consistent with the predictions of the Joule heating model for the case
of ${\rm d}R_{0}/{\rm d}T>0$, although $Z^{1\omega}$ appears to be more
susceptible to the linear-response background than $Z^{3\omega}$. In the Cole-
Cole representations of $Z^{1\omega}$ and $Z^{3\omega}$, the lengths of the
arc strings are approximately the same (0.5 m$\Omega$). The cutoff frequency
of the nonlinear AC electrical response is estimated as $\approx$1 Hz. This
frequency is typical for the thermal response in a bulk crystal, as often
reported in AC-temperature calorimetry experiments ACthermal1 .

## References

  * (1)
  * (2)

## References

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##  VII Supplemental Material

Fig. S 1: Typical experimental circuit used for measurement of the complex
impedance.

In Supplemental Material, we show that the negative nonlinear reactance
reported in previous experimental studies [17–19] is not a manifestation of
negative nonlinear inductance. In previous experimental reports, the impedance
at the fundamental and higher hamonic frequencies was measured for the input
frequency $\omega/(2\pi)$ using the lock-in technique. The typical
experimental setup is displayed in Fig. S1: a magnetic material, a load
resistor with resistance, $R_{\rm L}$, and an AC voltage output of the lock-in
amplifier, $V_{0}\sin\omega t$, are connected in series. For simplicity, we
consider a sample with a two-probe configuration and do not explicitly
consider the contact resistance. The current flowing through the circuit,
$\mathcal{I}^{1\omega}(\omega)$ [the $1\omega$ Fourier component of $I(t)$],
was determined from the voltage drop at the load resistor, and the voltage
drop at the sample, $\mathcal{V}^{1\omega}_{\rm s}(\omega)$ [the $1\omega$
Fourier component of $V_{\rm s}(t)$], was measured. Then, they defined the
complex impedance $Z^{1\omega}(\omega,\mathcal{I})$ at
$\mathcal{I}^{1\omega}(\omega)$ by $\mathcal{V}^{1\omega}_{\rm
s}(\omega)/\mathcal{I}^{1\omega}(\omega)$. Note that this definition was also
used for a large $\mathcal{I}^{1\omega}(\omega)$ such that the proportionality
between $\mathcal{V}^{1\omega}_{\rm s}(\omega)$ and
$\mathcal{I}^{1\omega}(\omega)$ is no longer valid. From a theoretical
perspective, they defined an emergent inductor as an element showing the
following voltage drop:

$V_{\rm s}(t)=R_{\rm s}I(t)+L_{0}\Bigl{(}1+{\rm A}I(t)^{2}+{\rm
B}I(t)^{4}+\cdots\Bigr{)}\frac{{\rm d}I}{{\rm d}t},$ (S1)

where $R_{\rm s}$ is the resistance of the sample, $L_{0}$ denotes the
inductance in the linear response, and A and B are coefficients representing
the nonlinearity related to the inductive response. From an energetic
perspective, Eq. (S1) corresponds to the fact that under current $I$, the
inductive element with a finite resistance can store the following energy:

$E_{\rm L}\bigl{(}I(t)\bigr{)}=L_{0}\left(\frac{1}{2}I(t)^{2}+\frac{1}{4}{\rm
A}I(t)^{4}+\frac{1}{6}{\rm B}I(t)^{6}+\cdots\right).$ (S2)

Note that the inductive term in Eq. (S1), multiplied by $I$, is equal to ${\rm
d}E_{\rm L}/{\rm d}t$, indicating the relationship between the inductive
electric response and the energy stored in the inductor. Given Eq. (S1), the
circuit equation of the measurement system is given as:

$\displaystyle V_{0}\sin\omega t$ $\displaystyle=(R_{\rm L}+R_{\rm
s})I(t)+L_{0}\Bigl{(}1+{\rm A}I(t)^{2}+{\rm
B}I(t)^{4}+\cdots\Bigr{)}\frac{{\rm d}I}{{\rm d}t}$
$\displaystyle=RI(t)+L_{0}\Bigl{(}1+{\rm A}I(t)^{2}+{\rm
B}I(t)^{4}+\cdots\Bigr{)}\frac{{\rm d}I}{{\rm d}t},$ (S3)

where $V_{0}$ is the output voltage amplitude of the lock-in amplifier and $R$
is the sum of the sample and load resistances. Experimentally, they observed
$\operatorname{Im}[\mathcal{V}^{1\omega}_{\rm
s}(\omega)/\mathcal{I}^{1\omega}(\omega)]\approx-|\eta|\omega$ at low
frequencies, especially when the flowing current is large under a large
$V_{0}$. This observation was interpreted as indicating that the coefficient
of ${\rm d}I/{\rm d}t$ in Eq. (S3) became negative under large currents.
However, it has not been discussed whether Eq. (S3) truly shows
$\operatorname{Im}[\mathcal{V}^{1\omega}_{\rm
s}(\omega)/\mathcal{I}^{1\omega}(\omega)]\approx-|\eta|\omega$ when the
coefficient of ${\rm d}I/{\rm d}t$ is negative at a large $I$. Therefore, the
correspondence between the experimental observations and Eqs. (S1)–(S3)
remains unclear. In fact, as shown below, the solution of Eq. (S3) has
instability towards self-sustained oscillations when the coefficient of ${\rm
d}I/{\rm d}t$ becomes negative at a large $I$, and it does not show an
electric response such that $\operatorname{Im}[\mathcal{V}^{1\omega}_{\rm
s}(\omega)/\mathcal{I}^{1\omega}(\omega)]\approx-|\eta|\omega$; i.e., the
experimental observations are not described by Eqs. (S1)–(S3). In the
following, we numerically examine the nature of the nonlinear negative
inductance represented by Eqs. (S1)–(S3).

Equation (S3) is complicated because the coefficient of ${\rm d}I/{\rm d}t$ is
time dependent and may change the sign during the time evolution. To gain
insights into Eq. (S3), it would be instructive to begin with a simpler case.
We consider a different form of circuit equation, which is nonlinear with
respect to the input voltage amplitude, $V_{0}$, instead of the current:

$\displaystyle V_{0}\sin\omega t$ $\displaystyle=RI(t)+L_{0}(1+{\rm
A_{V}}V_{0}^{2}+{\rm B_{V}}V_{0}^{4}+\cdots)\frac{{\rm d}I}{{\rm d}t}$
$\displaystyle=RI(t)+L_{\rm V}(V_{0})\frac{{\rm d}I}{{\rm d}t}.$ (S4)

Note that in Eq. (S4), the coefficient of ${\rm d}I/{\rm d}t$, $L_{\rm
V}(V_{0})$, is constant during the time evolution of the system, and thus, the
profile of $I(t)$ can be easily deduced from the knowledge of dynamic systems
[21]. The numerical results are displayed in Figs. S2(a) and (b). For clarity,
here, we chose a simplified parameter set: $V_{0}=1,\omega=1,R=10,I(0)=0$, and
$L_{\rm V}(V_{0})=+5$ [Fig. S2(a)] or $-5$ [Fig. S2(b)]. Note that the
qualitative behavior of the solution depends only on whether the coefficient
of ${\rm d}I/{\rm d}t$ is positive or negative, and we chose parameters such
that the characteristics of $I(t)$ can be clearly observed. Figure S2(a) shows
the result for the case of $L_{\rm V}(V_{0})=+5$. When the coefficient of
${\rm d}I/{\rm d}t$ is positive, the focus, $I_{0}(t)=(V_{0}\sin\omega t)/R$
(denoted by the dotted line), is a stable focus (but time dependent due to the
AC driving force, $V_{0}\sin\omega t$); thus, $I(t)$ tracks the time-varying
focus, with a finite delay. Figure S2(b) shows the case of $L_{\rm
V}(V_{0})=-5$. When the coefficient is negative, the focus is unstable, and
thus, $I(t)$ tends to separate from it over time. As a result, once $I(t)$
deviates from the focus by an infinitesimal amount, the deviation is
unlimitedly amplified and diverges. Thus, a constant negative inductance makes
the system unstable. This divergence instability is a natural consequence of
the fact that the energy of the inductor, $\frac{1}{2}L_{V}(V_{0})I^{2}$,
diverges to negative infinity by increasing $|I|$ under constant negative
inductance $L_{V}(V_{0})$.

Fig. S 2: Time evolution of the current in the differential equation, Eq.
(S4), for the case of positive inductance, $L_{\rm v}(V_{0})=+5$ (a) and
negative inductance, $L_{\rm v}(V_{0})=-5$ (b).  Fig. S 3: Time evolution of
the current in the differential equation, Eq. (S5) with ${\rm B}=0$. (a, b)
The coefficient of ${\rm d}I/{\rm d}t$ (divided by $L_{0}$) (a) and the energy
stored in the inductor (b) as a function of current. (c, d) Time evolutions of
the coefficient of ${\rm d}I/{\rm d}t$ (c) and current (d).

We next consider the main issue, Eq. (S3), which is a more nontrivial circuit
equation in the sense that the coefficient of ${\rm d}I/{\rm d}t$ varies
during the time evolution. We set positive $L_{0}$ $(=0.3)$ so that the system
is stable in the linear response. By choosing appropriate A and B, Eq. (S3)
represents a nontrivial situation such that the coefficient of ${\rm d}I/{\rm
d}t$ changes its sign from positive to negative when the current becomes
sufficiently large. Note that Eq. (S3) can be rewritten as ${\rm d}I/{\rm
d}t=(V_{0}\sin\omega t-RI)/[L_{0}(1+{\rm A}I^{2}+{\rm B}I^{4})]$, and thus, it
shows singularity the moment $L_{0}(1+{\rm A}I^{2}+{\rm B}I^{4})$ reaches
zero. To numerically solve Eq. (S3), we therefore introduce ${\rm d}^{2}I/{\rm
d}t^{2}$ with an infinitesimally small positive coefficient, $\Gamma$
($=10^{-8}$ in the present numerical calculation), which is just for the sake
of the stability of the numerical tracking of the solution and is not
important in the following discussion. Namely, the differential equation
actually computed in this Supplemental Material is:

$\displaystyle V_{0}\sin\omega t=RI(t)+L_{0}\Bigl{(}1+{\rm A}I(t)^{2}+{\rm
B}I(t)^{4}\Bigr{)}\frac{{\rm d}I}{{\rm d}t}+\Gamma\frac{{\rm d}^{2}I}{{\rm
d}t^{2}}.$ (S5)

When considering real systems, it is also natural to assume that $I(t)$ is
always first-order differentiable (i.e., ${\rm d}I/{\rm d}t$ is always
finite), and it therefore makes sense to consider such a second-order
derivative term to ensure first-order differentiability.

First, we consider the case in which ${\rm A}=-1.25$ and ${\rm B}=0$; thus,
the coefficient of ${\rm d}I/{\rm d}t$ and $E_{\rm L}$ depend on $|I|$, as
shown in Figs. S3(a) and (b), respectively. The coefficient of ${\rm d}I/{\rm
d}t$ and ${\rm d}E_{\rm L}/{\rm d}I$ reach zero at $I\approx\pm 0.9$. Figures
S3(c) and (d) show the numerical results in the nonlinear regime such that the
coefficient of ${\rm d}I/{\rm d}t$ reaches zero: the parameter set used is:
$V_{0}=2.25,\omega=1,R=2,L_{0}=0.3,\Gamma=1\times 10^{-8},I(0)=0$, and
$I^{\prime}(0)=0$. As the current increases with time, the coefficient of
${\rm d}I/{\rm d}t$ decreases and eventually reaches zero at $t\approx 0.92$
[Fig. S3(c)]. Concomitantly, ${\rm d}I/{\rm d}t\approx(V_{0}\sin\omega
t-RI)/[L_{0}(1+{\rm A}I^{2}+{\rm B}I^{4})]$ diverges approximately positively
so that the current catches up with the time-varying focus,
$I_{0}(t)=(V_{0}\sin\omega t)/R$, from below and overshoots it [Fig. S3(d)].
This upwards overshoot causes the coefficient of ${\rm d}I/{\rm d}t$ to become
negative [Fig. S3(c)], which never reverts to a positive value. Thus, the
time-varying focus is unstable for $t>0.92$, and accordingly, the current
continues to separate from it and eventually diverges, similar to the case of
Fig. S2(b). Thus, the system is unstable for a large $V_{0}$. This behavior
reflects the fact that in the absence of the $I^{4}$ term in Eq. (S5), the
inductor energy can be unlimitedly decreased by increasing $|I|$ when the
current exceeds the threshold value, $|I|\approx 0.9$ [Fig. S3(b)].

Fig. S 4: Time evolution of the current in the differential equation, Eq. (S5)
with positive B. (a, b) The coefficient of ${\rm d}I/{\rm d}t$ (divided by
$L_{0}$) (a) and the energy stored in the inductor (b) as a function of
current. (c, d) Time evolutions of the coefficient of ${\rm d}I/{\rm d}t$ (c)
and current (d). (e, f) Enlarged view of the squared area in panel (c) and
(d), respectively.

Next, we consider the case in which ${\rm A}=-1.25$ and ${\rm B}=0.3$; i.e.,
the $I^{4}$ term with a positive coefficient is present in Eq. (S5). Thus, as
$|I|$ increases, the coefficient of ${\rm d}I/{\rm d}t$ first changes from
positive to negative and then back to positive again [Fig. S4(a)];
furthermore, for this parameter choice, the inductor energy is always positive
for arbitrarily chosen $I$ [Fig. S4(b)]. These features imply that the
simultaneous divergence of $I$ and the coefficient of ${\rm d}I/{\rm d}t$, as
observed in Figs. S3(c) and (d), does not occur. Figures S4(c) and (d) show
the numerical results in the nonlinear regime such that the coefficient of
${\rm d}I/{\rm d}t$ reaches zero: the parameter set used is:
$V_{0}=2.25,\omega=1,R=2,L_{0}=0.3,\Gamma=1\times 10^{-8},I(0)=0$, and
$I^{\prime}(0)=0$. When the coefficient of ${\rm d}I/{\rm d}t$ decreases from
positive to zero, the system starts to oscillate [see also Figs. S4(e) and
(f), which are magnified views of Figs. S4(c) and (d) during oscillation,
respectively]. This oscillating behavior can be understood by considering the
time evolution of Eq. (S5) in steps as follows. Whenever the coefficient of
${\rm d}I/{\rm d}t$ reaches zero, the current tends to quickly catch up with
the focus, $I_{0}(t)=(V_{0}\sin\omega t)/R$, overshoot it upwards from below
(or downwards from above), and diverge while $1+{\rm A}I^{2}+{\rm B}I^{4}<0$
(i.e., $1<|I|<1.75$); however, when the current falls outside this range, the
coefficient of ${\rm d}I/{\rm d}t$ becomes positive again [Fig. S4(a)]; thus,
unlike in the case of ${\rm B}=0$ [Figs. S3(c) and (d)], the divergence of the
current is halted, and the tracking towards the focus from above (or from
below) restarts, resulting in decreasing (or increasing) current; during this
tracking, the coefficient of ${\rm d}I/{\rm d}t$ decreases and reaches zero
again. In this way, a jerky oscillation of the current is achieved around the
time-varying focus [Figs. S4(d) and (f)] when the current value results in a
negative coefficient of ${\rm d}I/{\rm d}t$, exemplifying the unstable nature
inherent to a negative coefficient of ${\rm d}I/{\rm d}t$.

Fig. S 5: Oscillator function of an element that shows a negative coefficient
of ${\rm d}I/{\rm d}t$ under the application of a DC-voltage of appropriate
magnitude. (a) Experimental circuit for the demonstration of the oscillator
function. (b) The coefficient of ${\rm d}I/{\rm d}t$ (divided by $L_{0}$) as a
function of current. The broken line indicates a condition of $V_{0}$ and $R$
that show the oscillator function. (c) Time evolution of the self-sustained
current oscillation in the differential equation, Eq. (S5) with positive B.

In summary, numerical examinations have demonstrated that the hallmark of a
negative coefficient of ${\rm d}I/{\rm d}t$ is instability, such as divergence
or spontaneous oscillation, and this phenomenon is never observed as an
impedance with negative reactance. As would be best demonstrated by connecting
a DC-voltage source of appropriate magnitude, an element that has a negative
coefficient of ${\rm d}I/{\rm d}t$ in a certain current range will operate as
an oscillator [Figs. S5(a)–(c)]. Therefore, the experimental observation,
$\operatorname{Im}[\mathcal{V}^{1\omega}_{\rm
s}(\omega)/\mathcal{I}^{1\omega}(\omega)]\approx-|\eta|\omega$, in the
nonlinear regime should be considered beyond the framework of Eqs. (S1)–(S3).
As discussed in the main text, we consider this issue in terms of time-varying
Joule heating and its impact on the AC electrical response.