Multiplet of Skyrmion States on a Curvilinear Defect: Reconfigurable Skyrmion Lattices

Typically, the chiral magnetic Skyrmion is a single-state excitation. Here we propose a system, where multiplet of Skyrmion states appears and one of these states can be the ground one. We show that the presence of a localized curvilinear defect drastically changes the magnetic properties of a thin perpendicularly magnetized ferromagnetic film. For a large enough defect amplitude a discrete set of equilibrium magnetization states appears forming a ladder of energy levels. Each equilibrium state has either a zero or a unit topological charge; i.e., topologically trivial and Skyrmion multiplets generally appear. Transitions between the levels with the same topological charge are allowed and can be utilized to encode and switch a bit of information. There is a wide range of geometrical and material parameters, where the Skyrmion level has the lowest energy. Thus, periodically arranged curvilinear defects can result in a Skyrmion lattice as the ground state.

Figure 1

Individual Skyrmion profiles and Skyrmion lattices. (a): Equilibrium magnetization states of a single Gaussian concave bump ($\mathcal{A}=-3$, $r_0=1$ and $d=1$) are shown in a vertical cross section view. Arrows indicate the magnetization distribution and the normal component $m_n=\cos \mathrm{\Theta }$ is color coded. The corresponding solutions $\mathrm{\Theta }(s)$ of Eq. (3) are shown in the insets I, II, ${\mathrm{I}}^{\prime }$, and ${\mathrm{II}}^{\prime }$. Vertical axis $\mathcal{E}=E/E_{\mathrm{BP}}$ shows the distribution of the corresponding energy levels obtained from (S8) with $E_{\mathrm{BP}}=8\pi AL$ being energy of the Belavin-Polyakov soliton [66]. (b) Two Skyrmion states with big (I) and small (II) radii are shown on the same bumps arranged in a square lattice. These Skyrmion solutions can be considered as logical states “1” and “0” of an information bit. (c) Skyrmion lattice as a ground state. (d) Sketch of the geometry with the Cartesian $\{{\mathit{e}}_x,{\mathit{e}}_y,{\mathit{e}}_z\}$ and curvilinear $\{{\mathit{e}}_s,{\mathit{e}}_{\chi },\mathit{n}\}$ frames of reference used throughout the manuscript.

Figure 2

Energies of different solutions. Solid lines I–IV and dashed lines ${\mathrm{I}}^{\prime }$, ${\mathrm{II}}^{\prime }$ show energies (S8) of topological nontrivial (Skyrmion) and trivial states, respectively, for the bump with $\mathcal{A}=2$ and $r_0=1$. Energy of the planar Skyrmion is shown by the thin line P. States I, II, ${\mathrm{I}}^{\prime }$, ${\mathrm{II}}^{\prime }$ are similar to the same name states in Fig. 1. States III and IV correspond to Skyrmions whose radius much exceeds the lateral bump size, see Figs. S5 and S8. The background filling corresponds to the number of stable Skyrmion states, see also Fig. 3. For the case $d<d_2$, the Skyrmion state I has the lowest energy in the system; this corresponds to the dashed area in the Fig. 3.

Figure 3

Diagram of Skyrmion states for Gaussian bump with $r_0=1$. In the white area, the Skyrmion solutions do not exist. The number of any other area (see legend) coincides with the number of stable Skyrmion solutions. At least one Skyrmion solution exists within the gray area “0”; however, the bump center is a position of unstable equilibrium for it. Within the other areas the corresponding number of Skyrmion are pinned at the bump center. The horizontal dashing shows areas where the lowest energy level is Skyrmion one. The star marker shows the parameters of Fig. 1. The solutions spectra for points a–f are presented in Sec. S.VI. The dotted horizontal line $\mathcal{A}=2$ corresponds to Fig. 2.