Magnetic Skyrmions and Skyrmion Clusters in the Helical Phase of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

Skyrmions are nanometric spin whirls that can be stabilized in magnets lacking inversion symmetry. The properties of isolated Skyrmions embedded in a ferromagnetic background have been intensively studied. We show that single Skyrmions and clusters of Skyrmions can also form in the helical phase and investigate theoretically their energetics and dynamics. The helical background provides natural one-dimensional channels along which a Skyrmion can move rapidly. In contrast to Skyrmions in ferromagnets, the Skyrmion-Skyrmion interaction has a strong attractive component and thus Skyrmions tend to form clusters with characteristic shapes. These clusters are directly observed in transmission electron microscopy measurements in thin films of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. Topological quantization, high mobility, and the confinement of Skyrmions in channels provided by the helical background may be useful for future spintronics devices.

Figure 1

Skyrmions in the helical phase ($B=0.2B_0$). The color code denotes the out-of-plane component of the magnetization as shown in (a). The in-plane component is sketched by white arrows. (a) and (b) Topologically equivalent single Skyrmion states with winding number $-1$ that can be smoothly transformed one into the other. The energy of the $H$-shaped Skyrmion (b) is, however, much lower compared to the interstitial Skyrmion of panel (a). In (c) an anti-Skyrmion with opposite winding number is shown. Because of attractive interactions, two Skyrmions form a dimer bound state with winding number $-2$, shown in panel (d). (e)–(h) Multi-Skyrmion bound states, see text.

Figure 2

Energetics of the Skyrmion configurations and effective interaction potentials. Left: The $H$-shaped Skyrmion of Fig. 1 has always a much lower energy (red) than the interstitial Skyrmion of Fig. 1 (blue). The energy (light green) of the dimer (2-Skyrmion bound state), Fig. 1, is slightly lower than the energy of two separate Skyrmions due to attractive interactions. The thermodynamic phase transition to the Skyrmion lattice is indicated with vertical lines (gray). Dashed lines show the energies of the corresponding anti-Skyrmion states. Right: Interaction potentials at $B=0.2B_0$ of two Skyrmions on the same track ($n=0$) and on neighboring tracks ($n=1$) as a function of their distance ($\lambda$ is the wavelength of the helix at $B=0$). For comparison, also the purely repulsive potential of two Skyrmions in a polarized background at $B=1B_0$ is shown ($FM$). Skyrmion-dimer and dimer-dimer interactions can to a good approximation be obtained by adding monomer-monomer potentials; see Supplemental Material [47].

Figure 3

Real-space cryo-Lorentz images reveal the coexistence of Skyrmion and helical phases. Several types of Skyrmion clusters, I–IV, can be identified corresponding to the simulated structures shown in Figs. 1.