Real-space observation of skyrmion clusters with mutually orthogonal skyrmion tubes

We report the discovery and direct visualization of skyrmion clusters with mutually orthogonal orientations of constituent isolated skyrmions in chiral liquid crystals and ferromagnets. We show that the nascent conical state underlies the attracting inter-skyrmion potential, whereas an encompassing homogeneous state leads to the repulsive skyrmion-skyrmion interaction. The crossover between different regimes of skyrmion interaction could be identified upon changing layer thickness and/or the surface anchoring. We develop a phenomenological theory describing two types of skyrmions and the underlying mechanisms of their interaction. We show that isolated horizontal skyrmions with the same polarity may approach a vertical isolated skyrmion from both sides and thus constitute two energetically different configurations which are also observed experimentally. In an extreme regime of mutual attraction, the skyrmions wind around each other forming compact superstructures with undulations. We also indicate that our numerical simulations on skyrmion clusters are valid in a parameter range corresponding to the A-phase region of cubic helimagnets.

Figure 1

Placed into the conical phase, each vertical IS marked by V develops a transitional region (named shell) with positive energy density [radial region that stretches between radii $R_1$ and $R_2$ in (a)]. Then, an attractive V-V interaction and skyrmion cluster formation (b) can be explained by the overlap of shells and ensuing reduction of total energy. The color plots in (a) and (b) indicate the energy density averaged over $z$ coordinates and plotted on the $xy$ plane. The same “energetic” arguments can be applied to also address an attraction between vertical and horizontal ISs in (c) and (d). The color plots in (c) and (d) indicate the energy density counted from the conical state and plotted on the $xz$ plane for a cross section along the skyrmion centers. The black arrows are projections of the magnetization onto the $xz$ plane. $H/H_D=0.5$ for both cases. The attracting nature of IS-IS interaction can alternatively be explained based on the principles of compact cluster formation in (e)–(h) (see text for further details). Then, the cluster formed out of two vertical ISs in (f) recreates the fragment of a metastable SkL with the homogeneous state between two skyrmions. Such a homogeneous state is formed if the coils of two skyrmions are getting “zipped” together. We extract the spins corresponding to the conical phase from the obtained spin configurations which facilitates such way of reasoning. The magnetic field (green arrow) is applied along $z$. Pink arrows in (g) and (h) show the magnetization direction along $y$, i.e., the polarity of horizontal skyrmions. Red and blue arrows indicate the magnetization direction in the cores and transitional regions of horizontal skyrmions.

Figure 2

Reduced energy for the interaction between two ISs in the configuration $(\downarrow \uparrow V)$ (a) and $(V\downarrow \uparrow )$ (b) calculated as a function of the distance $r$ between the skyrmion centers for different values of the applied magnetic field. Numerical solutions for $(\downarrow \uparrow V)$ configurations $[h=0.7$ (d), $h=0.3$ (f)] show that for lower values of the fields a horizontal skyrmion is strongly bent by the vertical one (f). In the configuration $(V\downarrow \uparrow )$ in (c) and (e), no pronounced bending is observed. Thus, in the mixed configuration that includes both $(V\downarrow \uparrow )$ and $(\downarrow \uparrow V)$, the undulated structure of a horizontal skyrmion is only stipulated by the vertical skyrmions from the right side, whereas vertical skyrmions from the left already adjust to such a curved structure (g).

Figure 3

Experimental demonstration of orthogonal skyrmion interactions in a chiral nematic LC, visualized via polarizing optical microscopy images. Computer-simulated visualization of the confined chiral nematic liquid crystal director-field configuration (a) is depicted using cylinders with colored ends. The horizontal black bars represent the confining cell substrates. Vertical skyrmions within the colored ovals characterize different states according to the interaction potential in Fig. 2 (see text for details). In polarizing optical microscopy, the crossed polarizer orientation is marked with white double arrows and the scale bars corresponding to 100 $\mu \mathrm{m}$ are marked separately for (b)–(c) and (d)–(e) in (b) and (d), respectively.