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

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 as emergent inductors. The time evolution of a magnetic texture creates 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, creating 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 ; Furuta2 , 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. Here, we raise fundamental questions associated with the interpretation.

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 . In contrast, for $j_{0}\sim 10^{8}$ A m^-2, 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 . 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, contradicting the experimental observations Textbook (for details, see Supplemental 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$1–10 kHz for micrometer-sized fabricated Gd₃Ru₄Al₁₂ and YMn₆Sn₆ YokouchiNature ; KitaoriPNAS and $\sim$0.1 kHz for needle-like bulk FeSn₂ YokouchiArxiv . These results were interpreted as indicating that the magnetic texture under consideration has slow dynamics, but the experimentally obtained 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. In Gd₃Ru₄Al₁₂ YokouchiNature and YMn₆Sn₆ KitaoriPNAS , $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$ exhibits complicated magnetic-field dependence in response to the successive magnetic phase transitions. Although these observations reveal a considerable correlation between $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})/\omega$ and the magnetic phases, the complicated behavior has not yet been well substantiated in terms of EEF.

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. 1(a) and (b), respectively, were reported YokouchiNature . A static temperature increase 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. 1(c)]. $\operatorname{Re}\rho\gg\operatorname{Im}\rho$ at low $\omega$ is a characteristic of dissipative responses of a resistor, and therefore, the nonlinear impedance $\Delta\rho^{1\omega}(j_{0})$ is resistor-like. This perspective is in contrast to the nonlinear inductance mechanism, which assumes nondissipative characteristics, i.e., $\operatorname{Im}\Delta\rho^{1\omega}\gg\operatorname{Re}\Delta\rho^{1\omega}$ at low $\omega$. The observation of $\operatorname{Re}\rho\gg\operatorname{Im}\rho$ at low $\omega$ implies that the nonlinear impedance observed in Gd₃Ru₄Al₁₂ represent not a manifestation of an nonlinear emergent inductor but a manifestation of a nonlinear resistor. Thus, it appears more reasonable to regard the $\operatorname{Im}\Delta\rho^{1\omega}(j_{0})$ under an AC current as a delayed response of the much larger $\operatorname{Re}\Delta\rho^{1\omega}(j_{0})$, rather than the emergent inductance.

This notice led us to reconsider the unresolved issues regarding nonlinear $\operatorname{Im}\Delta\rho^{1\omega}$ from a perspective of 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. 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 cause 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 finding motivated us to scrutinize whether the nonlinear $\operatorname{Im}\rho^{1\omega}$ caused by AC Joule heating is truly negligible in the reported results YokouchiNature ; KitaoriPNAS ; YokouchiArxiv .

The paper is organized as follows: In Sec. II, we construct a Joule-heating model up to the power-linear order and derive the expressions of the Joule-heating-induced AC electrical response. In Sec. III, we show the experimental observations of the Joule-heating-induced AC electrical response in CuIr₂S₄, which contains no magnetic order, and verify our Joule-heating model. Section IV is devoted to reconsidering the nonlinear emergent inductance previously reported and showing that the Joule-heating-induced AC electrical response approximately reproduces the temperature and magnetic field dependences, cutoff frequency, and magnitude of the $\operatorname{Im}\rho/\omega$. In Sec. V, we consider the coexistence of dissipative and nondissipative mechanisms in impedance and argue that a nonlinear low-frequency regime should be avoided when exploring nondissipative signals in conducting materials. Section VI summarizes and concludes the paper.

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 (approximately, 1.3$\times$0.7$\times$0.14 mm³); images of the two samples are shown in the APPENDIX [Figs. 9(a) and (b)]. 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 results 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. The results for the bulk MoTe₂ are shown in the APPENDIX (Figs. 10 and 11).

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., ${\rm d}\rho_{0}/{\rm d}T>0$ with $\rho_{0}$ representing a linear-response DC value) 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 4(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 AC 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 the APPENDIX [Fig. 9(c)]. In contrast, for $\rho^{3\omega}$, we discover that the background signal is not significant, so we display the raw data.

In the $\operatorname{Re}\rho^{1\omega}$–$T$ profile [Fig. 4(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 assumed to be generated mainly in the bulk, rather than at the contacts of the current electrodes.

Figures 4(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 density increases [Figs. 4(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. 4(e)], which is 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. 4(f)].

The frequency dependences and Cole-Cole representations of $\Delta\rho^{1\omega}$ and $\rho^{3\omega}$ at 210 K are shown in Figs. 5(a)–(f). The Cole-Cole representations [Figs. 5(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. 3(a) and (c)]. The lengths of the arc strings are approximately the same for both cases (200 $\mu\Omega$ cm). These observations confirm that the observed $\Delta\rho^{1\omega}$ and $\rho^{3\omega}$ have dissipative characteristics and that the origin of their imaginary parts lies in the delay of the nonlinear real parts. 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 the APPENDIX, Fig. 11), indicating that the sample dimensions are a crucial factor determining the cutoff frequency of the Joule-heating-induced AC electrical response. The cutoff frequency in the microfabricated CuIr₂S₄ 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.

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 $\operatorname{Im}\Delta\rho^{1\omega}$–$j_{0}$ profile up to a higher current density, and the results are shown in Fig. 6. 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.

Having experimentally verified the Joule-heating-induced AC electrical response, we examine whether the nonlinear $\operatorname{Im}\rho^{1\omega}(\omega,j_{0})$ previously reported YokouchiNature ; KitaoriPNAS ; YokouchiArxiv has the characteristics of the Joule-heating-induced AC electrical response. In particular, we discuss the temperature and magnetic field dependences, cutoff frequency, and magnitude of $\operatorname{Im}\rho^{1\omega}$, and show that the Joule-heating-induced AC electrical response gives answers to questions I–III. We begin by discussing the temperature and magnetic field dependences and answering question III.

In general, dissipative and nondissipative mechanisms can coexist in impedance. Phenomenologically, the nonlinear $\Delta\rho(j_{0})$ is thus likely described by the sum of the two mechanisms: $\Delta\rho(j_{0})=\Delta\rho_{\rm diss}(j_{0})+\Delta\rho_{\rm nondiss}(j_{0})$, where $\Delta\rho_{\rm diss}$ and $\Delta\rho_{\rm nondiss}$ represent the nonlinear impedances due to dissipative mechanisms and nondissipative mechanisms, respectively. Note that by definition, at low $\omega$ close to DC, $\operatorname{Re}\Delta\rho_{\rm diss}\gg\operatorname{Im}\Delta\rho_{\rm diss}$, and $\operatorname{Im}\Delta\rho_{\rm nondiss}\gg\operatorname{Re}\Delta\rho_{\rm nondiss}$. As mentioned in Sec. II C, the inductor mechanism due to the EEF caused by a pinned magnetic texture assumes $\operatorname{Im}\Delta\rho^{1\omega}\gg\operatorname{Re}\Delta\rho^{1\omega}$ at low $\omega$ and therefore belongs to $\Delta\rho_{\rm nondiss}$, whereas the Joule-heating-induced AC electrical response satisfies $\operatorname{Re}\Delta\rho^{1\omega}\gg\operatorname{Im}\Delta\rho^{1\omega}$ at low $\omega$ and therefore belongs to $\Delta\rho_{\rm diss}$. As shown in Sec. IV, we have discovered that regardless of whether the material under consideration exhibits a magnetic order (as in YokouchiNature ; KitaoriPNAS ; YokouchiArxiv ) or not (as in this study), the observed nonlinear $\Delta\rho(j_{0})$ is categorized into $\Delta\rho_{\rm diss}$ and has the characteristics of the Joule-heating-induced AC electrical response [Eqs. (7)–(10)]. This finding implies that unless the time-varying Joule heating is negligibly small, any nondissipative signals that may coexist are easily masked by the Joule-heating-induced AC electrical response. Note that the nonlinear $\operatorname{Im}\Delta\rho^{1\omega}/\omega$ becomes pronounced when large AC currents are used at low frequencies for impedance measurements, as indicated by Eq. (8). Therefore, when exploring nondissipative signals in conducting materials, it is important to avoid a nonlinear low-frequency regime and to suppress the time-varying Joule heating as much as possible. Since an order of magnitude estimate of the Joule-heating-induced $\operatorname{Im}\Delta\rho^{1\omega}(j_{0})$ can be obtained if the amount of the temperature increase due to the Joule heating and the thermal-response time are determined from experimental results, it would also be important to double-check that the nondissipative signals thus obtained are greater than this estimated value.

To clarify fundamental questions that have remained unanswered in the previous experiments on 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 within a power-linear regime have been clarified. To further examine the Joule-heating-induced AC electrical response, we have performed experiments on two materials that exhibit no magnetic order, CuIr₂S₄ and 1$T$’-MoTe₂, and verified the characteristics of the Joule heating model. We have reconsidered the nonlinear emergent inductance previously reported and determined that the temperature and magnetic field dependences, cutoff frequency, and magnitude of $\operatorname{Im}\rho^{1\omega}/\omega$ can be naturally explained in terms of the Joule-heating-induced AC electrical response.

The Joule-heating-induced AC electrical response inevitably yields finite $\operatorname{Im}\rho^{1\omega}/\omega$ and $\operatorname{Im}\rho^{3\omega}/\omega$, 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 may cause a $\operatorname{Im}\Delta\rho^{1\omega}/\omega$ of considerable magnitude, which is much larger than that expected from the linear-response EEF. In previous experiments on emergent inductors, the nonlinear $\operatorname{Im}\rho^{1\omega}/\omega$ below the cutoff frequency was discussed, and the nonlinearity more pronouncedly occurred in real impedance than in imaginary impedance. Our results thus suggest that the reported data regarding emergent inductors need to be reconsidered by taking into account the impact of the Joule-heating-induced AC electrical response.

Note added in proof. During the review process, our manuscript was commented by Yokouchi et al. from Tokura group Condmat . After careful consideration of their comment and new data, we found no need to change our conclusion. Our response to the comment is provided in Ref. Reply .