Emergent electric field induced by dissipative sliding dynamics of domain walls in a Weyl magnet

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 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, observed in the imaginary component of the complex impedance. Spin dynamics simulations reveal that domain wall sliding dominates over spin tilting, where 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 on the coupled motion of magnetic solitons and conduction electrons.

Figures (4)

• Figure 1: Schematics for a magnetic domain wall (DW) and its dissipative motion.a, DW motion excited by an a.c. electric current $I_\mathrm{a.c.}$ (yellow arrow). Red, blue, and white regions indicate domains with positive and negative net magnetization, as well as the DW region. Black arrows indicate the sliding motion of a magnetic DW via the time derivative of the sliding mode $\dot{X}$. b,c, Schematics of the time evolution of the sliding motion ($X$) and spin-tilting motion ($\phi$) under $I_\mathrm{a.c.}$ along the $x$ axis. We illustrate the sliding motion in the case of $\phi = 0$ in panel b, while $X$ is set to zero in panel c. d, Mass on a spring as a driven harmonic oscillator, subject to friction and an external force $F$ induced by excitation current $I_{a.c.}(\omega)$. e, Dissipative oscillator without reactive spring-force.
• Figure 2: Domain wall (DW) resistance and emergent electric field (EEF) in Weyl magnet NdAlSi.$\mathbf{a,}$ Illustration of magnetic force microscopy (MFM) measurements. The magnetic stray field from the sample is detected optically as a variation of the resonance frequency of the cantilever (Methods). $\mathbf{b,}$ MFM image of the mesoscopic device of NdAlSi at zero external field and $T=2\,\mathrm{K}$ under field-cooling conditions (Methods). In panels $\mathbf{a,b}$, the area with negative (positive) net magnetization is colored in dark blue (dark red), respectively, while the white area is the magnetic DW. $\mathbf{c,}$ False-colour image of a micrometer-sized device of NdAlSi fabricated by the focused ion beam (FIB) technique for the transport measurements (Methods). $\mathbf{d,}$ Magnetization ($M$) curves measured at $3\,\mathrm{K}$ ($<T_\mathrm{C}$) and $8\,\mathrm{K}$ ($>T_\mathrm{C}$). $\mathbf{e,}$ Magnetoresistivity ($\mathrm{Re}\rho_{xx}$) of the FIB device of NdAlSi. The dashed line is a fit to the $B^2$ term from the Lorentz force, while the area highlighted in red corresponds to DW-scattering ($\mathrm{Re} \, \Delta \rho_{xx}^\mathrm{DW}$). $\mathbf{f,}$ Imaginary impedance $\Im \rho_{xx}$ originating from DWs. The vertical dotted line in panels d-f indicates the magnetic field where the DWs disappear ($B_\mathrm{d}$). $\Im \rho_{xx}$ is enhanced below $B_\mathrm{d}$, as highlighted in red in panel f. $\mathbf{g,h,}$ Schematic time-evolution of the spin moments at a magnetic domain wall under excitation currents and the calculated scalar spin chirality (SSC) in the space-time domain (see Supplementary Fig. 4). Finite, time-alternating SSC results in an EEF detected by the imaginary impedance.
• Figure 3: Correlation of domain wall (DW) density and emergent electric field (EEF) from DW motion.$\mathbf{a,}$ Imaginary part of the complex impedance $\Im\rho_{xx}$ divided by frequency $f$ as obtained from numerical calculations of the EEF in our spin dynamics model for a single magnetic DW. $\Omega_\mathrm{int}$ and $\Omega_\mathrm{ext}$ are defined in Eq. (\ref{['EqofMotion_X_twodots']}), while $j_\mathrm{th}$ corresponds to the depinning threshold of the tilting mode in the $\Omega_\mathrm{int} \ll \Omega_\mathrm{ext}$ limit. A negative imaginary impedance, or negative EEF, appears over a wide range of $f$ when $\Omega_\mathrm{int} / \Omega_\mathrm{ext}$ is small, i.e., where spin-tilting $\phi$ has a low energy cost. This corresponds to DW motion dominated by friction, as in Fig. \ref{['Fig1']}e, where the potential term in Eq. (\ref{['EqofMotion_X_twodots']}) becomes irrelevant to the dynamics. In the regime below the black dotted line, the model predicts negative $\Im\rho_{xx}$ over a wide parameter range. $\mathbf{b,c,}$ MFM images taken during a field scan from positive to negative field (field-down sweep). When the magnetic field is close to $B_\mathrm{d}$, the size of the domain with negative net magnetization is enhanced, and accordingly the number of DWs decreases. $\mathbf{d,}$ Magnetic field dependence of the DW length. The red triangles (blue circles) show the total DW length (projection of the DW length onto the $\langle 110 \rangle$ axis) corresponding to the green lines in panels $\mathbf{b,c}$, respectively. $\mathbf{e,f,}$ The DW-resistivity $\Re\rho_{xx}^\mathrm{DW}$ and the imaginary impedance $\Im\rho_{xx}$ when the magnetic field is tilted away from the vertical axis by an angle $\theta$. The horizontal axis is the magnetic field normalised by $B_\mathrm{d}$. The blue and red shading in panel f indicates the regimes where changes in DW population and changes in DW pinning are dominant, respectively (see also Fig. 4).
• Figure 4: Scaling of domain wall (DW) resistivity and the imaginary impedance.$\Re\rho_{xx}^\mathrm{DW}$ and $\Im\rho_{xx}$, as defined in Fig. \ref{['Fig3']}, from field down-sweeps for $B>0$ and $B<0$ are shown as open and filled circles, respectively. Close to $B_\mathrm{d}$, the two quantities show near-linear correlation ("DW population dominant" regime, blue shading); but $\Im\rho_{xx}$ exhibits a plateau when $\Re \rho_{xx}^\mathrm{DW}$ is still rising, due to a $B$-dependent change in the EEF beyond the domain wall density ("DW pinning dominant" regime, red shading). The inset shows our EEF model calculation for $\Im\rho_{xx}$ with Gilbert damping $\alpha=10^{\mathchar'-3}\sim10^{\mathchar'-4}$ (see text for discussion).