Discretized topological Hall effect emerging from skyrmions in constricted geometry

We investigate the skyrmion formation process in nanostructured FeGe Hall-bar devices by measurements of the topological Hall effect, which extracts the winding number of a spin texture as an emergent magnetic field. Stepwise profiles of the topological Hall resistivity are observed in the course of varying the applied magnetic field, which arise from instantaneous changes in the magnetic nanostructure such as the creation, annihilation, and jittering motion of skyrmions. The discrete changes in the topological Hall resistivity demonstrate the quantized nature of an emergent magnetic flux inherent in each skyrmion, which had been indistinguishable in many-skyrmion systems on a macroscopic scale.

Figure 1

(a) Scanning electron microscope image of a Hall-effect measurement circuit with a linewidth of $w=50$ nm. (b) Schematic illustrations of the skyrmion-lattice state in a 40-nm-thick FeGe thin film. The magnetic modulation period in the FeGe thin film is assumed to be the same length of 70 nm as the bulk sample. Magnetic-field dependence of (c) Hall resistivity ${\rho }_{yx}$ and (d) $\Delta {\rho }_{yx}$ at various inclination angles at 2 K in a 10-$\mu \mathrm{m}$-wide circuit. (See text for the definition of $\Delta {\rho }_{yx}$.) A sudden shrinkage of hysteresis loops in (c) and (d) represents declines in skyrmion numbers.

Figure 2

(a)–(f) Magnetic-field dependence of Hall resistivity for FeGe nanoscale circuits with various widths ranging from 250 to 32 nm. We subtracted the $H$-symmetric component using the equation ${\rho }_{yx}(\pm H)=\pm \frac{[V_y\left(H\right)-V_y(-H)]t}{2I}$.

Figure 3

(a)–(f) Magnetic-field dependence of the difference between Hall resistivities in increasing and decreasing field processes, which is denoted by $\Delta {\rho }_{yx}^{\mathrm{T}}$. Normalized $\Delta {\rho }_{yx}^{\mathrm{T}}$ by its maximum value is presented. Insets are magnified images of each ${\rho }_{yx}$ at low fields.

Figure 4

Conceptual diagrams for skyrmion formation and corresponding Hall resistivities in a 250-nm-wide nanoscale circuit. Variations of numbers of core-up and core-down skyrmions as functions of magnetic field in (a) increasing and (b) decreasing field processes. Corresponding (c) Hall resistivity and (d) its decomposition into ${\rho }_{yx}^{\mathrm{N}}$ (NHE), ${\rho }_{yx}^{\mathrm{A}}$ (AHE), and ${\rho }_{yx}^{\mathrm{T}}$ (THE). (e) Series of schematic illustrations representing an expected development of magnetic structure as a function of magnetic field.