Current-induced dynamics of skyrmion strings

Abstract

Dynamics of string-like objects is an important issue in a broad range of physical systems, including vortex lines in superconductors, viscoelastic polymers, and superstrings in elementary particle physics. In noncentrosymmetric magnets, string forms of magnetic skyrmions are present as topological spin objects, and their current-induced dynamics has recently attracted intense interest. We show in the chiral magnet MnSi that the current-induced deformation dynamics of skyrmion strings results in transport response associated with the real-space Berry phase. Prominent nonlinear Hall signals emerge above the threshold current only in the skyrmion phase. We clarify the mechanism for these nonlinear Hall signals by adopting spin density wave picture to describe the moving skyrmion lattice; deformation of skyrmion strings occurs in an asymmetric manner due to the Dzyaloshinskii-Moriya interaction, which leads to the nonreciprocal nonlinear Hall response originating from an emergent electromagnetic field. This finding reveals the dynamical nature of string-like objects and consequent transport outcomes in noncentrosymmetric systems.

Fig. 1. Experimental configurations and the second-harmonic Hall effect in MnSi.

(A) Schematic picture of translationally moving skyrmion strings and the experimental setup for second-harmonic Hall measurement. (B) Scanning electron microscope image of a MnSi thin-plate sample: MnSi crystal (green), gold electrodes (yellow), tungsten (light blue) to fix the MnSi and to connect the gold electrodes to MnSi, and a silicon stage (gray). (C and D) Magnetic field dependence of second-harmonic Hall resistivity (${\mathrm{\rho }}_{\mathit{z}\mathit{x}}^{2\mathit{f}}$) in right-handed (C) and left-handed (D) MnSi crystals. The blue, orange, green, and white shadows represent helical (H), SkL, conical (C), and ferromagnetic (FM) phases, respectively.

Fig. 2. The second-harmonic Hall effect near the transition temperature.

(A) Magnetic field dependence of the real part of second-harmonic Hall resistivity (Re ${\mathrm{\rho }}_{\mathit{z}\mathit{x}}^{2\mathit{f}}$) measured with current densities j = 2.1 × 10⁸ A/m² (blue lines) and j = 8.3 × 10⁸ A/m² (red lines). The blue, orange, green, and white shadows represent helical, SkL, conical, and ferromagnetic phases, respectively. (B) Contour mapping of Re ${\mathrm{\rho }}_{\mathit{z}\mathit{x}}^{2\mathit{f}}$ in the temperature–magnetic field plane. The blue, green, and red circles denote helical-to-conical, conical-to-ferromagnetic, and SkL-to-conical phase transitions, respectively, determined from kinks in the magnetic field dependence of planar linear Hall resistivity. The green squares represent helical, conical, and SKL-to-paramagnetic phase transitions determined from inflection points of the temperature dependence of longitudinal resistivity (see also the Supplementary Materials).

Fig. 3. Current density and frequency dependence of the second-harmonic Hall effect.

(A and B) Current density (j) dependence of the temperature of the MnSi thin-plate sample estimated from longitudinal resistivity (A) and the real part of second-harmonic Hall resistivity (Re ${\mathrm{\rho }}_{\mathit{z}\mathit{x}}^{2\mathit{f}}$) at B = 0.15 T measured with the frequency f = 13 Hz (B). The red solid curve is a guide to the eyes. (C) Temperature dependence of threshold current densities j_th and the crossover point j_CO at B = 0.15 T. The values of j_th and j_CO at T = 29.0 K are represented as triangles in (B). (D) Dependence of the real part (line with red dots) and the imaginary part (line with blue dots) of ${\mathrm{\rho }}_{\mathit{z}\mathit{x}}^{2\mathit{f}}$ on the input current frequency at T = 28 K and B = 0.16 T. (E) Temperature dependence of frequency (f₀), where the imaginary part of ${\mathrm{\rho }}_{\mathit{z}\mathit{x}}^{2\mathit{f}}$ peaks. The f₀ values at T = 28 K are represented by the inverse triangle in (D). The thick light blue band is a guide to the eyes.

Fig. 4. Emergent electromagnetic fields for dynamically generated deformed skyrmion strings.

(A) Dispersion relation of the low-energy excitation (ε) of SkL with a propagating vector along the external magnetic field direction (q_z). (B) Schematic for the current dependence of nonreciprocal nonlinear (second-harmonic) Hall resistivity. The solid blue line represents the theoretical calculation, which is valid for the current density between j_th and j_CO, and the broken blue line denotes the square of the current density and is simply a guide to the eyes. The red line indicates the experimentally observed current profile (see also Fig. 3B). (C and F) Schematics of emergent electromagnetic fields for the deformation of a skyrmion string when it collides with point-like impurity. u represents the displacement vector of a skyrmion string. The red arrows denote the z components of the topological Hall electric field [(v_e − v_Sk) × b]_z (C) and the emergent electric field ${\mathit{e}}_{\mathit{z}}=-{[\widehat{\mathbf{u}}\times \mathbf{b}]}_{\mathit{z}}$ (F). (D and G) Position (z) dependence of the displacement of skyrmion strings shown together with the color map of the z components of both (v_e − v_Sk) × b (D) and $\mathit{e}=-\widehat{\mathbf{u}}\times \mathbf{b}$ (G) at several time points. (E and H) Time dependence of the average z components of both (v_e − v_Sk) × b (E) and $\mathbf{e}=-\widehat{\mathbf{u}}\times \mathbf{b}$ (H) over the skyrmion string.