Research ArticlePHYSICS

Current-induced dynamics of skyrmion strings

See allHide authors and affiliations

Science Advances  10 Aug 2018:
Vol. 4, no. 8, eaat1115
DOI: 10.1126/sciadv.aat1115
  • 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 (Embedded Image) 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 Embedded Image) 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 Embedded Image 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 Embedded Image) 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 jth and the crossover point jCO at B = 0.15 T. The values of jth and jCO 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 Embedded Image 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 Embedded Image 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 (qz). (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 jth and jCO, 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 [(vevSk) × b]z (C) and the emergent electric field Embedded Image (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 (vevSk) × b (D) and Embedded Image (G) at several time points. (E and H) Time dependence of the average z components of both (vevSk) × b (E) and Embedded Image (H) over the skyrmion string.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/8/eaat1115/DC1

    Section S1. Determination of the magnetic phase diagram for MnSi thin plates

    Section S2. Relationship between the nonreciprocal nonlinear Hall effect and second-harmonic resistivity

    Section S3. Current dependence of the nonlinear Hall effect measured by using square-wave current

    Section S4. Frequency dependence of skyrmion velocity

    Section S5. Calculation of nonreciprocal nonlinear Hall responses to AC

    Section S6. Calculation of current-induced dynamics of a single skyrmion string

    Fig. S1. Functional forms of the dimensionless functions fb(λ) and fe(λ).

    Fig. S2. Temperature dependence of longitudinal resistivity and magnetic field dependence of planar Hall resistivity.

    Fig. S3. Current dependence of the nonreciprocal nonlinear Hall effect measured by using square-wave current.

    Fig. S4. Frequency dependence of skyrmion velocity.

    References (3234)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Determination of the magnetic phase diagram for MnSi thin plates
    • Section S2. Relationship between the nonreciprocal nonlinear Hall effect and second-harmonic resistivity
    • Section S3. Current dependence of the nonlinear Hall effect measured by using square-wave current
    • Section S4. Frequency dependence of skyrmion velocity
    • Section S5. Calculation of nonreciprocal nonlinear Hall responses to AC
    • Section S6. Calculation of current-induced dynamics of a single skyrmion string
    • Fig. S1. Functional forms of the dimensionless functions fb(λ) and fe(λ).
    • Fig. S2. Temperature dependence of longitudinal resistivity and magnetic field dependence of planar Hall resistivity.
    • Fig. S3. Current dependence of the nonreciprocal nonlinear Hall effect measured by using square-wave current.
    • Fig. S4. Frequency dependence of skyrmion velocity.
    • References (3234)

    Download PDF

    Files in this Data Supplement:

Navigate This Article