Electric-current-assisted nucleation of zero-field hopfion rings

Abstract

Magnetic hopfions are three-dimensional topological solitons -- knotted, vortex-like spin configurations. In chiral magnets, hopfions can appear as isolated structures or they can be linked to skyrmion strings. Previous studies employed a sophisticated protocol and a special sample geometry to nucleate such hopfions linked to one or a few skyrmion strings. Here, we introduce an electric-current-assisted nucleation protocol that is simple and independent of the sample shape and size. The resulting hopfions exhibit extraordinary stability in the presence of both positive and negative magnetic fields, in perfect agreement with micromagnetic simulations. We also present a comprehensive framework for classifying hopfions, skyrmions, and merons by deriving the corresponding homotopy group.

Figures (16)

• Figure 1: Setup for electric-current-assisted experiments in the transmission electron microscope.a, Schematic representation of an FeGe sample. b, Low-magnification TEM image of the device viewed along the $+z$ direction. Electrodes were connected to the thicker edges of the sample. An amorphous, non-conductive carbon layer was deposited on one side of the sample during fabrication with focused ion beam (FIB) milling. c, Over-focus Lorentz TEM image of a representative magnetic state that appears after a short current pulse. The image is taken at a defocus distance of 700 $\mu$m and a temperature of 95 K. d, Magnified view of the area in c, showing a cluster of magnetic skyrmions and other magnetic textures.
• Figure 1: Hopfion ring nucleation.a illustrates the configuration that has been achieved by saturating the sample in the perpendicular magnetic field, subsequently reducing the external field to zero, and finally applying a short electric current pulse. b and c illustrate the evolution of the contrast under an external magnetic field applied in the negative direction -- opposite to the saturating field. Positive and negative field directions correspond to the directions towards the viewer and away from the viewer as indicated by symbols $\otimes$ in b-d and $\odot$ in f-g. At this step, the large skyrmion clusters collapse, and we often end up with a few configurations with a hopfion ring as in c. d is a zoomed-in view of the image shown in c (dashed square). e the contrast of the hopfion ring around two skyrmion strings after the magnetic field is reduced back to zero. The images f and g show the evolution of Lorentz TEM contrast as the field increases in the positive direction but nearly the same absolute value as in the negative direction in c. h displays the contrast after reducing the external magnetic field to zero again. It indicates that the hopfion ring remains stable even after a few cycles of applying positive and negative fields. The images were taken at a defocus distance of 700 $\mu$m and temperature $95$ K. The scale bar in all images corresponds to 100 nm.
• Figure 2: Magnetic hopfion ring nucleation. Each row represents a distinct sequence of over-focus Lorentz TEM images, illustrating the protocol for the nucleation of magnetic hopfion rings around (a) one, (b) two and (c) four magnetic skyrmion strings. The first column shows the initial configuration obtained after applying a short electric current pulse in zero magnetic field. The second and third columns illustrate the evolution of the contrast under an external magnetic field applied in the negative direction (opposite to the initial direction of the saturating field). This step is required to collapse large skyrmion clusters and to obtain an isolated hopfion ring. The fourth column provides a magnified view of the images in the third column. The fifth to seventh columns show the evolution of Lorentz TEM contrast of hopfion rings as the applied field is increased in the positive direction. The rightmost column displays the contrast immediately after the hopfion ring collapses. The arrows in b mark the positions of low-intensity spots corresponding to chiral bobbers or dipolar strings (torons). The images were recorded at a defocus distance of 700 $\mu$m and a temperature of 95 K. The scale bar in all images corresponds to 100 nm.
• Figure 2: Field-driven evolution of hopfion rings. Each row of images represents the sequence of over-focus Lorentz TEM images recorded during the magnetic field reversal from a negative to a positive direction. The strength of magnetic field is labeled in the top-left of each images. Positive and negative field directions correspond to fields pointing toward and away from the viewer, as denoted by the symbols $\odot$ and $\otimes$ in a, respectively. In each row, the negative field decreases from left to right, reaches zero, and then increases in the positive direction. a and b are the extended versions of Figs. \ref{['fig3']}a and b, respectively. d the extended version of Fig. \ref{['fig2']}c. Under negative fields, bumps are observed in the hopfion rings. As the positive field increases, these bumps gradually diminish and disappear at higher positive fields. The hopfion ring remains stable within this magnetic field range. The images were taken at a defocus distance of 700 $\mu$m and temperature $95$ K. The scale bar in all images corresponds to 100 nm.
• Figure 3: Magnetic-field-driven evolution of hopfion rings. Each row shows a sequence of over-focus Lorentz TEM images recorded between zero magnetic field and the field at which the hopfion ring collapses. The strength of the magnetic field is labeled in the upper left corner of each image. In a-d, the magnetic field increases from left to right. In e, the magnetic field increases, decreases to zero and then increases again. The arrows in the images recorded in zero magnetic field indicate the position of a bump that is present in each hopfion ring. With increasing field, such bumps become less pronounced. All of the images were recorded at a sample temperature of 95 K and a defocus distance of 700 $\mu$m. The scale bar in all images is 100 nm. The three indices shown in the lower right corner in the first column represent the topological charges of the magnetic textures in the corresponding row: two 2D topological indices $q_t$ and $q_b$ and one 3D topological index $h$. The cone winding number is $v=2$ for all configurations. See Extended Data Fig. \ref{['fig:SS5']} for corresponding simulated images.