# Emergent electromagnetic induction beyond room temperature

Contributed by Naoto Nagaosa, July 9, 2021 (sent for review March 20, 2021; reviewed by Achim Rosch and Kang L. Wang)

## Significance

Emergent inductors that utilize emergent electric fields generated by the current-induced motion of spiral spin textures have the potential to realize dramatic miniaturization of inductance elements. By using YMn

_{6}Sn_{6}, which has been attracting attention as a kagome lattice magnetic material in recent years, we have realized the room-temperature operation of emergent inductors. This micron-scale emergent inductor device shows not only an inductance as large as a commercially available product but also a sign change of inductance, which is a previously undescribed phenomenon.## Abstract

Emergent electromagnetic induction based on electrodynamics of noncollinear spin states may enable dramatic miniaturization of inductor elements widely used in electric circuits, yet the research is still in its infancy and many issues must be resolved toward its application. One such problem is how to increase working temperature to room temperature, and possible thermal agitation effects on the quantum process of the emergent induction are unknown. We report here large emergent electromagnetic induction achieved around and above room temperature, making use of a few tens of micrometer-sized devices based on the high-temperature (up to 330 K) and short-period ($\le $ 3 nm) spin-spiral states of a metallic helimagnet. The observed inductance value

*L*and its sign are observed to vary to a large extent, depending not only on the spin-helix structure controlled by temperature and applied magnetic field but also on the applied current density. The present finding on room-temperature operation and possible sign control of*L*may provide a step toward realizing microscale quantum inductors on the basis of emergent electromagnetism in spin-helix states.### Sign up for PNAS alerts.

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Conventional inductors based on classical electromagnetism are one of the most important elements in electric circuits, as characterized by the relation

*V*=*L dI*/*dt*, where*V*,*I*, and*L*are voltage, current, and inductance, respectively. Since*L*of the inductor coil is proportional to the product of the square of the coil’s winding number and the coil’s cross-section, it is difficult to reduce the dimensions of the inductor while keeping*L*large enough. To overcome the size problem of the coil-shaped inductor, the simple scheme of the electromagnetic induction has recently been proposed to use the current-induced spin dynamics in a helical-spin system (1) and experimentally verified for the helimagnetic phases of a metallic compound (2). The idea is to utilize the time-dependent emergent electromagnetic field or dynamics of the Berry phase (3, 4) produced by the conduction electrons flowing on the helical spin texture (1).The emergent magnetic field (where where

**b**) acting on conduction electrons is realized in a noncoplanar spin texture endowed with scalar spin chirality, typically in the skyrmion-lattice phase (5, 6). For example, the current-driven motion of skyrmions accompanying**b**produces the emergent electric field (**e**), which is known to give a current-dependent correction to the topological Hall effect in the skyrmion-lattice phase (7). The generalized Faraday’s law (8) states that $\nabla \times \mathbf{e}=-\partial \mathbf{b}/\partial t$ or $\mathbf{e}=-\partial \mathbf{a}/\partial t$, where $\mathbf{a}$ is the Berry connection satisfying that $\mathbf{b}=\nabla \times \mathbf{a}$. Thus, to generate the emergent electric field on the spin helix, the originally static $\mathbf{b}$ is not necessarily required but only the time-dependent deformation of the spin helix by alternating electric current (ac) is sufficient. In consideration of spin transfer torque on spin helix in the continuum limit, the coordinate component of emergent electric field (*e*_{i}) is described as (4, 5, 8)$$\begin{array}{c}{e}_{i}=\frac{h}{2\pi e}n\cdot \left({\partial}_{i}\mathbf{n}\times {\partial}_{t}\mathbf{n}\right),\end{array}$$

[1]

**n**,*h*, and*e*are a unit vector parallel to the direction of spins, Planck’s constant, and bare electron charge, respectively. As opposed to $\mathbf{b}$, $\mathbf{e}$ is related to the dynamics of spin structures and proportional to the solid angle dynamically swept by**n**(*t*). Hence, the motion of noncollinear spin structures can induce*e*_{i}(9–14). For example, in the case of a proper-screw (Bloch-wall-like) helix (Fig. 1*A*), the emergent electric field can be described as (1)$$\begin{array}{c}{e}_{x}=\frac{Ph}{e\lambda}{\partial}_{t}\varphi ,\hspace{1em}\end{array}$$

[2]

*λ*is the period of helix,*P*is a spin polarization factor, $\varphi $ is the tilting angle of the spin from the spiral plane, and the*x*axis is taken parallel to the magnetic modulation vector (**q**). We note that*e*_{x}can be generated regardless of the direction of the helical plane, e.g., also in cycloidal-type (Néel-wall-like) spin modulations. Also, the sign of*e*_{x}is independent of spin helicity, which allows the emergent induction even in a multidomain state with different helical structures. This is a striking difference from structural chirality-driven phenomena such as chiral-induced spin selectivity (15–17) and nonreciprocal transport (18) in chiral systems.Fig. 1.

The appearance of the emergent electromagnetic inductance was recently confirmed for the helimagnet Gd

_{3}Ru_{4}Al_{12}(2)_{,}in which various noncollinear spin structures, such as proper-screw and transverse conical (see Fig. 1*D*and*E*), show up below 20 K with a short helical pitch of*λ*∼ 2.8 nm due to the magnetic frustration effect of the localized Gd moments coupled via a Ruderman–Kittel–Kasuya–Yosida interaction (19–23). Several important features of the emergent inductor were experimentally confirmed or clarified by this first experimental demonstration (2): 1) the inversed size scaling law, as anticipated, that*L*for the emergent inductor increases with the reduction of the element cross-section*S*, 2) the mostly negative sign of the emergent inductance*L*in contrast to the positive value for conventional classic inductors, 3) the highly nonlinear behavior of the emergent inductance*L*with the current density, and 4) the frequency dependence of the emergent inductance showing the Debye-type relaxation feature. Whether these features of the emergent electromagnetic inductance are generic and common to all the possible spin-helix states, in particular to the room-temperature helix states, is an important subject to be verified by experiments, in addition to the possible further enhancement of the inductance value at higher temperatures.## Results and Discussion

### Magnetic Phases and Emergent Inductance.

The main purpose of the present study is to realize the large-enough magnitude, e.g., exceeding microhenry level, of the emergent electromagnetic inductance in the micrometer-sized device around and above room temperature; this would be a primarily step toward the application of the emergent electric field to the actual electronic devices. As the natural extension of the related study (2), we sought a high-temperature helimagnetic state with nanometer-scale spin-helical pitch in metallic compounds. Among several candidate materials, we target here the helimagnet YMn

_{6}Sn_{6}, where strong exchange interactions between 3*d*-moments on Mn kagome lattices (Fig. 1*B*) and strong Hund’s-rule-like couplings to itinerant electrons are expected. The compound indeed undergoes the helimagnetic transition approximately below a temperature as high as 330 K (Fig. 1*H*) (24–28), showing the short-period helimagnetic states (28–30) such as proper-screw helix (H, Fig. 1*D*) and transverse conical state (TC, Fig. 1*E*), whose magnetic modulation vectors**q**run parallel to the*c*-axis, i.e., normal to the Mn kagome-lattice plane. As increasing the magnetic field applied parallel to the*a*-axis, i.e., normal to the**q**direction or*c*-axis, the H state turns into TC, and (at low temperatures through the fan-like state [FL, Fig. 1*F*]) finally to the forced ferromagnetic state (FF, Fig. 1*G*), as shown in the phase diagram in the temperature vs. magnetic-field plane (Fig. 1*H*). Here, the phase diagram was obtained by the magneto-transport measurements on the actual micrometer-scale device (see Fig. 1*H*, Inset) with referring to the results reported for the bulk crystal and the magnetic-phase assignments done in previous reports (29); the phase transition temperatures and critical magnetic fields show a good agreement between the micrometer-sized device and the bulk crystal of YMn_{6}Sn_{6}(see*SI Appendix*, Fig. S2).The complex helical magnetic structures, in particular for the H and TC phases, appearing in this compound at relatively low magnetic fields and relatively high temperatures should be mentioned. As partly elucidated by recent neutron scattering studies (29–31), there may be coexisting plural

*q*-modulations in the H and TC phase. The*q*-values (corresponding to*λ*values less than 2.6 nm) appear to be plural, such as two- or threefold; for example, there appear coexistent multiple magnetic Bragg satellites from incommensurate*q*-states, and the relative weight of the respective*q*-states appears to change depending on each phase, temperature, or magnetic field (29–31). Such a complex feature and variation of the spin-helix modulation may influence the variation of the emergent inductance magnitude or even its sign, yet the elaborate arguments may have to await a future study to fully clarify the detailed spin structures. Nonetheless, the very short helical pitches (large*q*values) are obviously favorable for the generation of high inductance value, as argued in Eq.**2**. Below, we examine the characteristics of the emergent electromagnetic inductance for the micrometer-scale devices made of YMn_{6}Sn_{6}on which the electric current is applied along the*c*-axis (i.e., parallel to the**q**vector) while changing temperature and magnetic field applied along the*a*-axis (normal to the*c*-axis), as shown in Fig. 1*H*, Inset.Fig. 2and plotted on the right ordinate scale for the present device. The absolute value of

*A*exemplifies the imaginary part of ac resistivity (${\rho}^{1f})$ at 270 K measured using the ac input current density $j={j}_{0}\mathrm{sin}\left(2\pi ft\right)$ (${j}_{0}=2.5\times {10}^{4}\text{A}/{\text{cm}}^{2}$ and*f*= 500 Hz); the device (#1) size are 4.8 μm $\times $ 9.3 μm in cross-section (*S*) and 25.0 μm in voltage-terminal distance (*d*). We also confirmed reproducibility (see*SI Appendix*, Fig. S1). The real part of inductance value*L*can be directly related to Im${\rho}^{1f}$ via the relation$$L=\text{Im}{\rho}^{1f}d/2\pi fS$$

[3]

*L*is nearly constant with magnetic field within the low-field H phase and increases in the TC phase, exceeding a value as large as 2 μH (microhenry). In further increasing the magnetic field above 4 T, the absolute value of*L*steeply decreases within the TC phase, and around the TC-to-FF transition it changes sign and shows a positive peak. As anticipated from the spin-helix origin of the emergent electric field generation, the*L*almost disappears when deeply entering the FF phase.Fig. 2.

On the same device structure (#1) under the same current excitation condition (${j}_{0}=2.5\times {10}^{4}\text{A}/{\text{cm}}^{2}$), the temperature dependence of Im${\rho}^{1f}$(

*H*) curve is shown in Fig. 2*B–I*together with the assignments of the magnetic phases (colored vertical bands). The variation of Im${\rho}^{1f}$ at ${j}_{0}=2.5\times {10}^{4}\text{A}/{\text{cm}}^{2}$ is also displayed in Fig. 2*N*as a color contour map on the plane of temperature (*T*) and magnetic field (*H*). With lowering*T*, typically below100 K, the absolute value of Im${\rho}^{1f}$ is rapidly suppressed, perhaps due to the increase of the*ab*-plane magnetic anisotropy which tends to suppress the current-induced spin deformation (amplitude in $\varphi $ or ${\partial}_{t}\varphi $ term in Eq.**2**). In the temperature region below 270 K, the Im${\rho}^{1f}$ takes mostly negative values in the H, TC, and FL phases, while the magnitude is different in the respective phases and also depends on temperature. Notably, around the boundary between helimagnetic TC (or FL) and FF phases, the Im${\rho}^{1f}$ once increases to take the positive-value peak (see the red-colored region in Fig. 2*N*). The positive value of Im${\rho}^{1f}$ or*L*is more clearly and broadly observed at higher temperatures above 300 K. For example, at 300 K the Im${\rho}^{1f}$ is still negative within the H phase but shows a positive value in the whole TC phase, accompanying a sharp negative dip upon the field-induced H-to-TC transition. Surprisingly, even above the magnetic transition temperature (330 K), e.g., at 350 K, the broad positive peak is observed in low-field region, followed by the negative background in higher-field (>3 T) region. As for the lower current-density (${j}_{0}=4.4\times {10}^{3}\text{A}/{\text{cm}}^{2}$) behavior of Im${\rho}^{1f}$ as shown in Fig. 2*J–M*see the discussion in the next section.### Frequency Dependence and Current Nonlinearity.

Next, we proceed to the frequency dependence and the current nonlinearity for the present emergent inductor. To investigate a wide range of frequency dependence, we performed the LCR-meter measurement on the two-terminal device (#2 with 5.5 μm$\times $1.8 μm in

*S*and 28.8 μm in*d*; see*SI Appendix*, Fig. S1) at zero field. Fig. 3,*Inset*exemplifies the frequency*(f*) dependence of the real and imaginary parts of the inductance*L*at 100 K. A prototypical Debye-type relaxation behavior is observed there with the characteristic frequency (*f*_{0}) around 1 kHz, at which Im*L*shows a peak (2, 31) (see*Materials and Methods*for the definition of Im*L*). The similar relaxation-type*f*dependence of Re*L*is observed at various temperatures, as shown in the main panel of Fig. 3. The characteristic frequency*f*_{0}∼ 1 kHz is rather insensitive to the temperature variation, while the*L*value changes from negative to positive in approaching the magnetic transition temperature (*T*_{N}∼ 330 K). This means that the deformation of spins cannot fully follow alternating currents with a frequency above 1 kHz, which is ascribed to the extrinsic pinning effect stemming from defects/impurities. As compared with the case of the low-*T*_{N}(∼20 K) helimagnet Gd_{3}Ru_{4}Al_{12}where*f*_{0}∼ 10 kHz and*λ*∼ 2.8 nm (2), one order of magnitude lower*f*_{0}in the present compound with comparable*λ*may be ascribed to the smaller extrinsic pinning effect. Such frequency dependence of the*L*would be improved to show higher*f*_{0}by enhancing the extrinsic pinning effect via, for example, artificially introducing the pinning sites.Fig. 3.

As for the current-nonlinear behavior of

*L*, this compound shows dramatic but complex magnetic-field-dependent features. At first, we show the very low current density regime of the*H-*dependent inductance in this compound at various temperatures in Fig. 2*J–M*. The Im${\rho}^{1f}$ or*L*is always positive in the linear response regime, i.e., low current density ${j}_{0}=4.4\times {10}^{3}\text{A}/{\text{cm}}^{2}$ while turning into mostly negative as*j*increases to ${j}_{0}=2.5\times {10}^{4}\text{A}/{\text{cm}}^{2}$ in the helimagnetic phases H and TC ; see and compare Fig. 2*B–I*and*J–M*for ${j}_{0}=2.5\times {10}^{4}\text{A}/{\text{cm}}^{2}$ and ${j}_{0}=4.4\times {10}^{3}\text{A}/{\text{cm}}^{2}$ (low current density), respectively. When the current density*j*is further increased, the magnetic-field variation of the inductance or Im${\rho}^{1f}$ is observed to change in a qualitative manner, even including its sign. Shown in Fig. 4*A*is the typical result at 270 K on device #3 (3.8 μm$\times $8.5 μm in*S*and 35.5 μm in*d*; see*SI Appendix*, Fig. S1); the magnetic-field dependence at the current density ${j}_{0}$ ∼$2\times {10}^{4}\text{A}/{\text{cm}}^{2}$ therein nearly reproduces the results of device #1 shown in Fig. 2*A*at ${j}_{0}=2.5\times {10}^{4}\text{A}/{\text{cm}}^{2}$. In all the helimagnetic phases multiple sign changes of Im${\rho}^{1f}$ occur upon the development of nonlinearity. To see this more clearly, we plot the evolution of Im${\rho}^{1f}$ with the current density as monitored at 0 T (H phase), 2.5 T (lower-field side of TC phase), and 4.2 T (higher-field side of TC phase) in Fig. 4*B*together with the negative-maximal and positive-maximal magnitudes in this magnetic field range eye-guided by the envelope curves. Although the magnitude of Im${\rho}^{1f}$ and the frequency at the sign change of Im${\rho}^{1f}$ are different with the magnetic states, the overall profile of its*j*dependence is generic. Positive Im${\rho}^{1f}$ appears and generally shows a moderate increase in the low-*j*region below $2\times {10}^{4}\text{A}/{\text{cm}}^{2}$. With increasing*j*, Im${\rho}^{1f}$ sharply decreases in combination with a sign reversal then subsequently exhibits accelerated rise with another sign change (see*SI Appendix*, Fig. S3). One possible interpretation of this nontrivial*j*dependence is that the positive inductance linear in*j*is buried in the nonlinear component pronounced in the high-*j*region. Indeed, a recent theoretical study provides a consistent argument (32, 33). The complex conductivity $\sigma \left(\omega \right)$ comprises multiple components with each distinct response to*j*due to rich dynamical characteristics of the spin helix. While the dominant term is the Drude conductivity ${\sigma}_{\text{d}}$ of the metallic electrons, the contributions associated with the collective modes of spin spirals ${\sigma}_{\text{c}}\left(\omega \right)$ are a source of emergent inductance. The inductance of Im$\rho $ is therefore approximated as $-\frac{\text{Im}{\sigma}_{\text{c}}\left(\omega \right)}{{\sigma}_{\text{d}}^{2}}$, which is subdivided into two contributions, one from the tilt $\varphi $ and the other from the phason, both of which constitute the canonical conjugate pair. The former is equivalent to the emergent electric field giving the positive inductance, while the latter depends sensitively on the impurity pinning of the phason. We now understand that the phason contribution is the major origin of the nonlinearity and sign change of the inductance, since the pinning frequency depends on the temperature and current density, and also the depinning beyond the threshold current density gives even more dramatic change in the phason dynamics. Nevertheless, other mechanisms may be also relevant to the nonlinearity. For example, the current-induced change of the magnetic structure itself, such as the weight change of the plural*q*-value components even within the TC phase region, should be taken into account for the highly*j*-nonlinear. This is to be confirmed by the in situ neutron or magnetic resonant X-ray scattering studies while changing the magnetic field and electric current density. In turn, the possible control of the emergent inductance sign and magnitude in terms of the current excitation would be an important function for this class of quantum inductor.Fig. 4.

## Conclusion

We have demonstrated the potential of the above-room-temperature quantum inductor as derived from the emergent electric field generated in the short-pitch (∼3 nm) helimagnetic states, such as proper-screw helix and transverse conical states, of the YMn

_{6}Sn_{6}crystal plate. The imaginary part of ac resistivity Im${\rho}^{1f}$ as the representation of the intrinsic material parameter for the electromagnetic induction can show large absolute values of approximately microohm centimeter level, ensuring the large inductance value for the micrometer-sized or thin-film device, in which the current density can be increased. The emergent inductance is highly nonlinear with the current density exceeding 10^{4}A/cm^{2}, mostly negative but turns into a large positive value near the phase boundary to the forced ferromagnetic region or with increasing the current density in the transverse conical spin state. Apart from the case of the low-current-density (<10^{4}A/cm^{2}) excitation region, the positive sign of the emergent inductance is likely due to the gapless nature of collective spin excitation. The strong nonlinear behavior of emergent inductance can also be related to the collective phason modes sensitively responding to the impurity pinning; however, the detailed mechanism remains to be clarified. The possibility of the varying magnitude and sign of the emergent inductance with current density may lead to a useful functionality of the quantum inductor as well as the advantage of several orders-of-magnitude miniaturization as compared with the classical coil inductor. Since the low Debye frequency (less than a few kilohertz) and hence the low*Q*-factor (2π*fL*/*R*∼0.6%; also see*SI Appendix*, Fig. S2 for resistivity value) critically limit the performance of emergent inductance in YMn_{6}Sn_{6}, the improvement of operating frequency is a next important challenge. Further investigations by tuning magnetic period, anisotropy, and pinning strength in this class of compounds are needed.## Materials and Methods

### Crystal Growth and Device Fabrication.

The single crystals of YMn

_{6}Sn_{6}were synthesized by a Sn-flux method (26). A mixture of ingredient elements with atomic ratio of Y:Mn:Sn = 1:6:30 was put in an evacuated quartz tube and heated to 1,050 °C, subsequently cooled slowly to 600 °C, and then quenched to room temperature. Any remaining flux was centrifuged, followed by soaking in hydrochloric acid solution. The single crystallinity was indicated by the well-developed facet structures and was also confirmed by Laue X-ray diffraction. No impurity phase of the single crystal was detected by powder X-ray diffraction. We cut thin plates out of the single crystals by using the focused ion beam (FIB) technique (NB-5000; Hitachi). The thin plates were mounted on silicon substrates with patterned electrodes. We fixed the thin plates to the substrates and electrically connected them to the electrodes by using FIB-assisted tungsten deposition. We made Au/Ti-bilayer electrode patterns by an electron-beam deposition method.### Transport and Magnetization Measurements.

Magnetic-field dependence of complex resistivity was measured with use of lock-in amplifiers (SR-830; Stanford Research Systems). We input a sine-wave current and recorded both in-phase (Re${V}^{1f}$) and out-of-phase (Im${V}^{1f}$) voltage with a standard four-terminal configuration. Background signals were estimated by measuring a short circuit where the terminal pads were connected by Au/Ti-bilayer electrode patterns. We subtracted the background signals from the measurement data. The possible temperature increase Δ

*T*upon current excitation was checked by monitoring the temperature-dependent resistance value of the sample by passing the direct current (dc); note that the average joule heating by the dc density*j*_{dc}is twice as large as that by the ac density (*j*_{0}) of the same amplitude. In the current excitation corresponding to the case of*j*_{0}∼$3.6\times {10}^{4}\text{A}/{\text{cm}}^{2}$, close to the maximal value used to obtain the result of Fig. 4, the estimated temperature increase is Δ*T*= +2.5 K at the base temperature of 270 K, indicating little influence of the heating on the current-induced effects discussed in this work.Frequency dependence of complex inductance was measured with use of an LCR meter (E4980A; Agilent Technologies). We employed the two-terminal method to reduce parasitic impedance (the device #2 shown in

*SI Appendix*, Fig. S1). We corrected the contributions from the cables and the electrodes with a standard open/short correction procedure. We also subtracted the contributions from electrical contacts between the sample and the electrodes, which were estimated by measurements with low current density (${j}_{0}=1.0\times {10}^{3}\text{A}/{\text{cm}}^{2}$). Here, the observed complex impedance $\stackrel{\sim}{Z}\left(\omega \right)$ is the sum of frequency-independent resistance ($R$) and frequency-dependent reactance of complex inductance [$\omega \stackrel{\sim}{L}\left(\omega \right)=\omega \text{Re}L\left(\omega \right)+i\omega \text{Im}L\left(\omega \right)$]. The real and imaginary components of inductance can be estimated as $\text{Re}L\left(\omega \right)=\text{Im}\stackrel{\sim}{Z}\left(\omega \right)/\omega ,\text{Im}L\left(\omega \right)=(\text{Re}\stackrel{\sim}{Z}\left(\omega \right)-R)/\omega $.## Data Availability

All study data are included in the article and/or

*SI Appendix*.## Acknowledgments

We thank M. Kawasaki, Y. Kaneko, and Y. Onishi for enlightening discussions. This work was supported by Core Research for Evolutional Science and Technology, Japan Science and Technology Agency grants JPMJCR1874 and JPMJCR16F1 and Japan Society for the Promotion of Science KAKENHI grants JP18H03676, JP20H01859, and JP20H05155.

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### Information

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#### Copyright

© 2021. Published under the PNAS license.

#### Data Availability

All study data are included in the article and/or

*SI Appendix*.#### Submission history

**Published online**: August 13, 2021

**Published in issue**: August 17, 2021

#### Keywords

#### Acknowledgments

We thank M. Kawasaki, Y. Kaneko, and Y. Onishi for enlightening discussions. This work was supported by Core Research for Evolutional Science and Technology, Japan Science and Technology Agency grants JPMJCR1874 and JPMJCR16F1 and Japan Society for the Promotion of Science KAKENHI grants JP18H03676, JP20H01859, and JP20H05155.

### Authors

#### Competing Interests

Competing interest statement: Y.T., N.K., and reviewer A.R. are coauthors on a research article [P. Schoenherr

*et al.*,*Nature Phys.***14**, 465–468 (2018)]. Y.T., N.N., and reviewer A.R. are coauthors on a roadmap article with a collection of independent subarticles [C. Back*et al.*,*J. Phys. D***53**, 363001 (2020)]. However, these papers are not directly related to the present work, and the latter is an article by many authors.## Metrics & Citations

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