Current-Induced Creation of Topological Vortex Rings in a Magnetic Nanocylinder

Vortex rings are ubiquitous topological structures in nature. In solid magnetic systems, their formation leads to intriguing physical phenomena and potential device applications. However, realizing these topological magnetic vortex rings and manipulating their topology on demand have still been challenging. Here, we theoretically show that topological vortex rings can be created by a current pulse in a chiral magnetic nanocylinder with a trench structure. The creation process involves the formation of a vortex ring street, i.e., a chain of magnetic vortex rings with an alternative linking manner. The created vortex rings can be bounded with monopole-antimonopole pairs and possess a rich and controllable linking topology (e.g., Hopf link and Solomon link), which is determined by the duration and amplitude of the current pulse. Our proposal paves the way for the realization and manipulation of diverse three-dimensional (3D) topological spin textures and could catalyze the development of 3D spintronic devices.

Figure 1

Dynamical vortex ring creation. (a) Schematic view of the chiral magnetic nanocylinder with a trench structure on its top surface. The cross-sectional cut shows the conical state with screw dislocation. The bottom color wheel illustrates the color scheme that encodes the spin direction in three dimensions. (b),(c) Magnetic vortex ring with positive (b) and negative (c) linking. Top row is the equispin surface for $S_z=0$. Bottom row is the linking of equispin lines with $S_x=1$ (red) and $S_x=-1$ (cyan). (d) The created vortex rings’ positions as a function of time. Termination of the line indicates the transformation and annihilation of the vortex ring. (e) Bird’s eye view of the screenshots for the creation process at $0.5B_c$. Equispin surface with $S_z=-0.4$ is plotted. The applied current density is $5\times {10}^{11}{\mathrm{A}/\mathrm{m}}^2$. (f) Cross-sectional cut of the spin texture at the $y\text{−}z$ plane corresponding to the screenshots in (e). (g) Cross-sectional cut of the emergent magnetic field’s out-of-plane component associated with the created spin texture at different times. The amplitudes of the emergent magnetic field ($B_x$) for the monopole and antimonopole are artificially reduced by a factor of 0.1 to better visualize $B_x$ for the other part of the spin texture. The black arrows in (g) indicate the rotating manner of the spin texture.

Figure 2

Created spin texture with one vortex ring and two monopole-antimonopole pairs (MAPs). (a) Cross-sectional cut of the spin texture at the $y\text{−}z$ plane. (b) Equispin surface for $S_z=0$ of the created spin texture. (c) Cross-sectional cut of (b) showing the vortex ring and MAP structure. (d) Equispin lines with $\mathbf{S}=(1,0,0)$ (red) and $\mathbf{S}=(-1,0,0)$ (cyan) associated with (b). (e) Schematic linking structures for the equispin lines in (d). (f)–(i) Corresponding plots for equispin surface with $S_z=0.4$ and equispin lines with $\mathbf{S}\approx (0.92,0,0.4)$ (light red) and $\mathbf{S}\approx (-0.92,0,0.4)$ (light cyan). Up (down) star symbol represents the monopole (antimonopole).

Figure 3

Created spin texture with two vortex rings and three MAPs. (a) Cross-sectional cut of the spin texture at the $y\text{−}z$ plane. (b) Equispin surface ($S_z=0$) of the created spin texture. (c) Cross-sectional cut of (b) showing the vortex ring and MAP structure. (d) Equispin lines with $\mathbf{S}=(1,0,0)$ (red) and $\mathbf{S}=(-1,0,0)$ (cyan) associated with (b). (e) Schematic linking structures for the equispin lines in (d). (f)–(i) The corresponding plots for equispin surface with $S_z=0.2$ and equispin lines with $\mathbf{S}\approx (0.98,0,0.2)$ (red) and $\mathbf{S}\approx (-0.98,0,0.2)$ (cyan). (j)–(m) The corresponding plots for equispin surface with $S_z=0.4$ and equispin lines with $\mathbf{S}\approx (0.92,0,0.4)$ (red) and $\mathbf{S}\approx (-0.92,0,0.4)$ (cyan). (e),(i),(m) Up (down) star symbol represents the monopole (antimonopole).

Figure 4

Created spin texture with three (a)–(d) and four (e)–(h) vortex rings. (a),(e) Equispin surfaces ($S_z=0.2$) of the created spin texture. (b),(f) Cross-sectional cuts of (a) and (e) showing the vortex ring and MAP structure. (c),(g) Equispin lines with $\mathbf{S}\approx (0.98,0,0.2)$ (red) and $\mathbf{S}\approx (-0.98,0,0.2)$ (cyan) associated with (a),(e). (d),(h) Schematic linking structures for the equispin lines shown in (c),(g). For clarity, the view point is modified based on different spin textures.

Figure 5

Phase diagram of topological vortex ring creation as a function of current pulse amplitude and duration under various conditions. (a),(b) Phase diagrams for the cylinder with varying length at $B_z=0.5B_c$ (other geometrical parameters are the same as that used in Fig. 1. (c),(d) Phase diagrams for the cylinder with varying radius at $B_z=0.5B_c$ (with fixed length $11\lambda$). The color bar indicates the number of vortex rings created.