Abstract
The dynamic motion of topological defects in magnets induces an emergent electric field, as exemplified by the continuous flow of skyrmion vortices. However, the electrodynamics underlying this emergent field remains poorly understood. In this context, magnetic domain walls—one-dimensional topological defects with two collective modes, sliding and spin-tilt—offer a promising platform for exploration. Here we demonstrate that the dissipative motion of domain walls under oscillatory current excitation generates an emergent electric field. We image domain patterns and quantify the domain-wall length under applied magnetic fields in mesoscopic devices based on the magnetic Weyl semimetal NdAlSi. These devices exhibit exceptionally strong domain-wall scattering and a pronounced emergent electric field, as observed in the imaginary component of the complex impedance. Spin dynamics simulations reveal that domain-wall sliding dominates over spin-tilting, in which the phase delay of the domain-wall motion with respect to the driving force impacts the emergent electric field. Our findings establish domain-wall dynamics as a platform for studying emergent electromagnetic fields and motivate further investigations of the coupled motion of magnetic solitons and conduction electrons.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$32.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to the full article PDF.
USD 39.95
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
All experimental data needed to reproduce the figures are available via Zenodo at https://doi.org/10.5281/zenodo.17188494 (ref. 50). Source data are provided with this paper.
Code availability
The source code used to perform the calculations described in this paper is available from the corresponding authors upon request.
References
Volovik, G. E. Linear momentum in ferromagnets. J. Phys. C 20, L83 (1987).
Tatara, G. & Kohno, H. Theory of current-driven domain wall motion: spin transfer versus momentum transfer. Phys. Rev. Lett. 92, 086601 (2004).
Barnes, S. E. & Maekawa, S. Generalization of Faraday’s law to include nonconservative spin forces. Phys. Rev. Lett. 98, 246601 (2007).
Kishine, J.-i., Ovchinnikov, A. S. & Proskurin, I. V. Sliding conductivity of a magnetic kink crystal in a chiral helimagnet. Phys. Rev. B 82, 064407 (2010).
Nagaosa, N. Emergent inductor by spiral magnets. Jpn. J. Appl. Phys. 58, 120909 (2019).
Neubauer, A. et al. Topological Hall effect in the A phase of MnSi. Phys. Rev. Lett. 102, 186602 (2009).
Schulz, T. et al. Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys. 8, 301–304 (2012).
Nagaosa, N. & Tokura, Y. Emergent electromagnetism in solids. Phys. Scr. T146, 014020 (2012).
Yokouchi, T. et al. Emergent electromagnetic induction in a helical-spin magnet. Nature 586, 232–236 (2020).
Yang, S. A. et al. Universal electromotive force induced by domain wall motion. Phys. Rev. Lett. 102, 067201 (2009).
Shoka, Y. et al. Observation of anisotropic magneto-inductance effect. Appl. Phys. Express 16, 053006 (2023).
Matsushima, Y. et al. Emergent magneto-inductance effect in permalloy thin films on flexible polycarbonate substrates at room temperature. Appl. Phys. Lett. 124, 022404 (2024).
Zhang, Z. et al. Emergent magneto-inductance effect in NiFe thin films on glass substrates at room temperature. J. Magn. Magn. Mater. 610, 172500 (2024).
Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain-wall racetrack memory. Science 320, 190–194 (2008).
Finocchio, G., Büttner, F., Tomasello, R., Carpentieri, M. & Kläui, M. Magnetic skyrmions: from fundamental to applications. J. Phys. D 49, 423001 (2016).
Fert, A., Reyren, N. & Cros, V. Magnetic skyrmions: advances in physics and potential applications. Nat. Rev. Mater. 2, 17031 (2017).
Catalan, G., Seidel, J., Ramesh, R. & Scott, J. F. Domain wall nanoelectronics. Rev. Mod. Phys. 84, 119 (2012).
Shibata, J., Tatara, G. & Kohno, H. A brief review of field- and current-driven domain-wall motion. J. Phys. D 44, 384004 (2011).
Birch, M. T. et al. Dynamic transition and Galilean relativity of current-driven skyrmions. Nature 633, 554–559 (2024).
Litzius, K. et al. Skyrmion Hall effect revealed by direct time-resolved X-ray microscopy. Nat. Phys. 13, 170–175 (2017).
Furuta, S., Moody, S. H., Kado, K., Koshibae, W. & Kagawa, F. Energetic perspective on emergent inductance exhibited by magnetic textures in the pinned regime. npj Spintronics 1, 1 (2023).
Kurebayashi, D. & Nomura, K. Theory for spin torque in Weyl semimetal with magnetic texture. Sci. Rep. 9, 5365 (2019).
Kurebayashi, D. & Nagaosa, N. Electromagnetic response in spiral magnets and emergent inductance. Commun. Phys. 4, 260 (2021).
Yamane, Y., Fukami, S. & Ieda, J. Theory of emergent inductance with spin-orbit coupling effects. Phys. Rev. Lett. 128, 147201 (2022).
Araki, Y. & Ieda, J. Emergence of inductance and capacitance from topological electromagnetism. J. Phys. Soc. Jpn. 92, 074705 (2023).
Oh, T. & Nagaosa, N. Emergent inductance from spin fluctuations in strongly correlated magnets. Phys. Rev. Lett. 132, 116501 (2024).
Gaudet, J. et al. Weyl-mediated helical magnetism in NdAlSi. Nat. Mater. 20, 1650–1656 (2021).
Bouaziz, J., Bihlmayer, G., Patrick, C. E., Staunton, J. B. & Blügel, S. Origin of incommensurate magnetic order in the RAlSi magnetic Weyl semimetals (R = Pr, Nd, Sm). Phys. Rev. B 109, L201108 (2024).
Xu, S.-Y. et al. Discovery of Lorentz-violating type II Weyl fermions in LaAlGe. Sci. Adv. 3, e1603266 (2017).
Yamada, R. et al. Nernst effect of high-mobility Weyl electrons in NdAlSi enhanced by a Fermi surface nesting instability. Phys. Rev. X 14, 021012 (2024).
Kobayashi, K., Ominato, Y. & Nomura, K. Helicity-protected domain-wall magnetoresistance in ferromagnetic Weyl semimetal. J. Phys. Soc. Jpn. 87, 073707 (2018).
Ominato, Y., Kobayashi, K. & Nomura, K. Anisotropic magnetotransport in Dirac-Weyl magnetic junctions. Phys. Rev. B 95, 085308 (2017).
Nguyen, A. K., Shchelushkin, R. V. & Brataas, A. Intrinsic domain-wall resistance in ferromagnetic semiconductors. Phys. Rev. Lett. 97, 136603 (2006).
Xuan Mei, X., Chen, M. & Li, H. Magnetotransport in magnetic junctions based on tilted Weyl semimetals. J. Appl. Phys. 130, 203901 (2021).
Araki, Y. Magnetic textures and dynamics in magnetic Weyl semimetals. Ann. Phys. 532, 1900287 (2020).
Li, D.-X., Shao, X.-Q. & Yi, X.-X. Effect of environment on the scattering of electrons by a junction of different topological materials. Ann. Phys. 532, 1900399 (2020).
Ieda, J. & Yamane, Y. Intrinsic and extrinsic tunability of Rashba spin-orbit coupled emergent inductors. Phys. Rev. B 103, L100402 (2021).
Seib, J. & Fähnle, M. Calculation of the Gilbert damping matrix at low scattering rates in Gd. Phys. Rev. B 82, 064401 (2010).
Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First-principles calculation of the Gilbert damping parameter via the linear response formalism with application to magnetic transition metals and alloys. Phys. Rev. B 87, 014430 (2013).
Bouzidi, D. & Suhl, H. Motion of a Bloch domain wall. Phys. Rev. Lett. 65, 2587 (1990).
Stamp, P. C. E. Quantum dynamics and tunneling of domain walls in ferromagnetic insulators. Phys. Rev. Lett. 66, 2802 (1991).
Braun, H.-B. & Loss, D. Berry’s phase and quantum dynamics of ferromagnetic solitons. Phys. Rev. B 53, 3237 (1996).
Le Maho, Y., Kim, J.-V. & Tatara, G. Spin-wave contributions to current-induced domain wall dynamics. Phys. Rev. B 79, 174404 (2009).
Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Magnetization dissipation in ferromagnets from scattering theory. Phys. Rev. B 84, 054416 (2011).
Kim, S. K., Tchernyshyov, O., Galitski, V. & Tserkovnyak, Y. Magnon-induced non-Markovian friction of a domain wall in a ferromagnet. Phys. Rev. B 97, 174433 (2018).
Chudnovsky, E. M., Iglesias, O. & Stamp, P. C. E. Quantum tunneling of domain walls in ferromagnets. Phys. Rev. B 46, 5392 (1992).
Braun, H.-B., Kyriakidis, J. & Loss, D. Macroscopic quantum tunneling of ferromagnetic domain walls. Phys. Rev. B 56, 8129 (1997).
Wendl, A. et al. Emergence of mesoscale quantum phase transitions in a ferromagnet. Nature 609, 65–70 (2022).
Aharoni, A. Demagnetizing factors for rectangular ferromagnetic prisms. J. Appl. Phys. 83, 3432 (1998).
Yamada, R. Dataset for: Emergent electric field induced by dissipative sliding dynamics of domain walls in a Weyl magnet. Zenodo https://doi.org/10.5281/zenodo.17188494 (2025).
Kurumaji, T. Note on the interpretation of magnetic diffraction in NdAlSi: helical or fan? Preprint at https://arxiv.org/abs/2506.04000 (2025).
Acknowledgements
We acknowledge M. Birch, I. Belopolski, P. R. Baral, G. Chang, S. Sen, A. Ozawa, N. Nagaosa and T.-h. Arima for fruitful discussions, and the RIKEN CEMS Semiconductor Science Research Support Team for technical assistance. R.Y. was supported by the JSPS (KAKENHI Grant Nos. 22K20348, 23K13057, 24H01604 and 25K17336), the Japan Science and Technology Agency (JST) (PRESTO Grant No. JPMJPR259A), the Foundation for the Promotion of Material Science and Technology of Japan, the Yashima Environment Technology Foundation, the Yazaki Memorial Foundation for Science and Technology, and the ENEOS Tonengeneral Research/Academic Foundation. Y.F. was supported by the JST (PRESTO Grant No. JPMJPR2597). S.O. was supported by the JSPS (KAKENHI Grant Nos. 22K13998 and 23K25816) and the JST (PRESTO Grant No. JPMJPR2595). T.Y. was supported by the JSPS (KAKENHI Grant No. 24K00566) and the JST (PRESTO Grant No. JPMJPR235B). Y. Taguchi was supported by the JST (CREST Grant No. JPMJCR20T1) and the RIKEN TRIP initiative (Many-body Electron Systems and Advanced General Intelligence for Science Program). Y. Tokura was supported by the JSPS (KAKENHI Grant No. 23H05431) and the JST (CREST Grant No. JPMJCR1874). M.H. was supported by the JSPS (KAKENHI Grant Nos. 21K13877, 22H04463, 23H05431 and 24H01607), the JST (CREST Grant No. JPMJCR20T1 and FOREST Grant No. JPMJFR2238), the Fujimori Science and Technology Foundation, the New Materials and Information Foundation, the Murata Science Foundation, the Mizuho Foundation for the Promotion of Sciences, the Yamada Science Foundation, the Hattori Hokokai Foundation, the Iketani Science and Technology Foundation, the Mazda Foundation, the Casio Science Promotion Foundation, the Takayanagi Foundation, and Inamori Foundation. M.H. is also supported by the JST as part of Adopting Sustainable Partnerships for Innovative Research Ecosystem (ASPIRE; Grant No. JPMJAP2426). M.H. is supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) through Transregio (Grant No. TRR 360 - 492547816). O.A.T. acknowledges support from the Australian Research Council (Grant Nos. DP200101027 and DP240101062) and an NCMAS grant. This work is based on experiments performed at the Japan Research Reactor 3 (proposal no. 23515). This research was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy. This Article has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a non-exclusive, paid-up, irrevocable, worldwide licence to publish or reproduce the published form of this Article, or allow others to do so, for US government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (www.energy.gov/doe-public-access-plan).
Author information
Authors and Affiliations
Contributions
Y. Tokura and M.H. conceived the project. R.Y., A.K. and Y. Taguchi synthesized the bulk NdAlSi single crystal. R.Y. and Y.F. fabricated the FIB devices. R.Y. measured the MFM images and analysed the data with support from D.N. and F.S.Y. R.Y. performed the transport measurements and analysis with support from D.K., T.Y. and M.H. D.K. and O.A.T. developed the theoretical model and performed the numerical calculations. R.Y. and T.N. performed the polarized neutron scattering measurements. S.O. performed the spin model calculations. R.Y. and M.H. wrote the paper with contributions from all other co-authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data
Extended Data Fig. 1 Comparison of transport properties for bulk and device samples of NdAlSi.
a, Magnetoresistivity for a bulk single crystal and an FIB fabricated device (Device A). Clear quantum oscillations are observed in both cases, and the transition fields are similar, indicating little degradation during the fabrication process. b, Similar value of the Hall resistivity for a bulk single crystal and for an FIB fabricated device. This indicates that the carrier density is not changed by FIB fabrication. c,d, Oscillatory component in the magnetoresistance in the ferrimagnetic (Ferri) and ferromagnetic (FM) phases, respectively. e,f, A fast Fourier transformed spectrum of the oscillatory component with two frequencies (αS and \({\alpha }_{S}^{{\prime} }\)) and one frequency (βS) in the Ferri and FM phases, respectively.
Extended Data Fig. 2 Polarized neutron scattering experiments and estimation of the strength of the single-ion anisotropy K.
a, Measurement geometry for polarized neutron scattering. The intensities of the non-spin-flip (NSF) and spin-flip (SF) channels are related to the c-axis moment and to the moment in the ab-plane perpendicular to the Q-vector, respectively: \({I}_{{{\rm{NSF}}}}\propto {{M}^{2}}_{{{\rm{c}}}}\) and \({I}_{{{\rm{SF}}}}\propto {\mid{M}^{\perp }}_{{{\rm{in}}}}( Q)\mid^{2}\). b,c, Magnetic reflections observed at Q1/3 = (1/3 + δ1, 1/3 + δ1, 0) and Q2/3 = (2/3 + δ2, 2/3 + δ2, 0) in the incommensurate phase at 5 K, where δ1 ≠ 0 and δ2 ≠ 0. d,e, Magnetic reflections observed at Q1/3 and Q2/3 in the commensurate phase at 2.5 K, where δ1 and δ2 are nearly zero. The large ISF at Q1/3 indicates that the dominant contribution of the magnetic moment related to Q1/3 is in the ab-plane, while Mc is dominant for the Q2/3 modulation. f, Noncollinear ferrimagnetic order of NdAlSi with the in-plane magnetization component perpendicular to the Q-vector27,51. g, Definition of the spin tilt angle η of the noncollinear ferrimagnetic order (Supplementary Notes 7 and 8 and Ref. 51). h, Noncollinear ferrimagnetic order of NdAlSi in our one-dimensional spin model. i, Definition of the spin tilt angle \({\eta }^{{\prime} }\) in our spin model. j, Calculated \({\eta }^{{\prime} }\) as a function of the ratio of single-ion anisotropy K to the exchange interaction J. The experimental value of the relative angle (η ~ 20 degrees) can be reproduced by a modest value of K/J between 0.05 and 0.2.
Extended Data Fig. 3 Imaginary impedance in the device with a configuration where spin-orbit torque is forbidden.
Note that spin-orbit torque (SOT) can in principle be relevant for spin dynamics even in a bulk device of an inversion-breaking, polar material such as NdAlSi. a,b, Measurement configurations for two devices: B∥c, I∥[110] and B∥I∥c referred to as devices A and B, respectively. In device B (panel b), the current is applied along the c-axis of NdAlSi, which corresponds to the polar axis of the crystal. The spin-orbit torque must disappear in this configuration, because the spin polarization p ~ I × P vanishes. c,d, Magnetoresistance (Re(ρxx)) of the two devices. Enhanced Re(ρxx) appears around zero field and originates from the DW resistivity e,f, Imaginary impedance (Im(ρxx)) of the two devices. The similar magnitude of Im(ρxx) between device A and B indicates that the dominant driver of the imaginary impedance in NdAlSi is likely spin-transfer torque.
Extended Data Fig. 4 Negative imaginary impedance over a wide range of current density and frequency.
Imaginary impedance multiplied by the voltage terminal distance and divided by the excitation frequency and the cross-sectional area for various frequencies and current densities (Device C). The sign of the imaginary impedance is negative over the whole regime of the parameters we explored. Our theoretical calculation shows that a negative imaginary impedance appears over a wide range of frequency and current density when Ωint/Ωext is small (area below black dotted lines in Fig. 3a). Therefore, the overall negative sign of the imaginary impedance in a wide range of parameters supports our model and the dominant sliding motion of the DW with friction.
Extended Data Fig. 5 Time dependence of the excitation, potential, and dissipation energies.
a-d, Square of the excitation current density ja.c., potential energy VX for the sliding mode X, potential energy Vϕ for the tilting mode ϕ, and time-dependent change in total energy of the spin system dEtot/dt. In this column, we focus on the limit of small intrinsic pinning frequency. e-h, Same for the case of finite intrinsic pinning frequency (Ωint = 10 kHz). The following parameters are used to perform numerical calculations (in both panels): current excitation (a.c.) frequency f = 1 kHz, extrinsic pinning frequency Ωext = 1 MHz, Gilbert damping parameter α = 0.1, and nonadiabatic damping parameter β = 0.05.
Extended Data Fig. 6 Field evolution of MFM images around zero magnetic field.
a-m, Images are taken at T = 2 K after the field-cooling (FC) at 6 T followed by the field-sweep down to 0 T for an FIB-fabricated sample with a thickness of 6 μm.
Extended Data Fig. 7 Angular dependence of the magnetic field Bd where the domain wall length goes to zero.
The inset shows the configuration of the magnetic field and current in the corresponding resistivity measurements Re(ρxx) on a mesoscopic device (Device A). The scaling of Bd with 1/ cos θ indicates that the c-component of the magnetic field plays a dominant role in the magnetization reversal process between the domains of positive and negative net magnetization.
Extended Data Fig. 8 Asymmetric imaginary impedance in the field sweep in comparison with MFM images.
a, Field dependence of the imaginary impedance measured when the magnetic field is decreasing, starting from high field (‘field down’ sweep). The three black dots correspond to the magnetic field values of three MFM images shown in panels b-d. b-d, MFM images taken at − 0.04 T, 0.0 T, and + 0.04 T with the field down condition. The asymmetric profile in the imaginary impedance is related to different domain configurations observed in the MFM images at + 0.04 T and − 0.04 T.
Supplementary information
Supplementary Information (download PDF )
Supplementary Notes 1–11, Figs. 1–4 and Tables 1–3.
Source data
Source Data Fig. 2 (download XLSX )
Measurement source data.
Source Data Fig. 3 (download XLSX )
Measurement source data.
Source Data Fig. 4 (download XLSX )
Measurement source data and data obtained by a fit to a theoretical model.
Source Data Extended Data Fig. 1 (download XLSX )
Measurement source data.
Source Data Extended Data Fig. 2 (download XLSX )
Measurement source data and data obtained by theoretical calculation.
Source Data Extended Data Fig. 3 (download XLSX )
Measurement source data.
Source Data Extended Data Fig. 5 (download XLSX )
Data obtained by theoretical calculation.
Source Data Extended Data Fig. 7 (download XLSX )
Measurement source data.
Source Data Extended Data Fig. 8 (download XLSX )
Measurement source data.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yamada, R., Kurebayashi, D., Fujishiro, Y. et al. Emergent electric field induced by dissipative sliding dynamics of domain walls in a Weyl magnet. Nat. Phys. 22, 239–244 (2026). https://doi.org/10.1038/s41567-025-03124-z
Received:
Accepted:
Published:
Version of record:
Issue date:
DOI: https://doi.org/10.1038/s41567-025-03124-z
