Rectangular and square skyrmion crystals on a centrosymmetric square lattice with easy-axis anisotropy

We report our theoretical results on the emergence of a rectangular- and square-shaped magnetic skyrmion crystals on a centrosymmetric square lattice with magnetic anisotropy. By performing the simulated annealing for a frustrated spin model with an easy-axis single-ion anisotropy on a square lattice, we find that a competition of the interactions in momentum space gives rise to the formation of two skyrmion crystals, where the skyrmion cores are aligned in a rectangle- or square-lattice way. Especially, the interactions at the higher-harmonic wave vectors play an important role in stabilizing the square/rectangular skyrmion crystals. Our results provide a reference to explore further skyrmion-hosting materials in centrosymmetric tetragonal systems.

Figure 1

(a) Real-space spin configuration of the S-SkX in Eq. (1). The arrows and the contour show $(S_i^x,S_i^y)$ and $S_i^z$, respectively. The inset of (a) represents the spin structure factor in the first Brillouin zone. The solid (dashed) circles represent ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ (${\mathit{Q}}_1^{\prime }={\mathit{Q}}_1-{\mathit{Q}}_2$ and ${\mathit{Q}}_2^{\prime }={\mathit{Q}}_1+{\mathit{Q}}_2$). (b) Contour plot of $\xi$ in the plane of $J_2$ and $J_3$. The optimal ordering vectors are also shown in each region. In the region for the ordering vectors ${\mathit{Q}}_1=(q^{*},q^{*})$ and ${\mathit{Q}}_2=(-q^{*},q^{*})$ [${\mathit{Q}}_1^{\prime }=(q^{*},0)$ and ${\mathit{Q}}_2^{\prime }=(0,q^{*})$], $\xi =J_{{\mathit{Q}}_1+{\mathit{Q}}_2}/J_{{\mathit{Q}}_1}$ ($\xi =J_{({\mathit{Q}}_1^{\prime }+{\mathit{Q}}_2^{\prime })/2}/J_{{\mathit{Q}}_1^{\prime }}$).

Figure 2

Phase diagrams while varying $\zeta =J/J^{\prime }$ and magnetic field $H$ at (a) $A=0.05$, (b) $A=0.2$, and (c) $A=0.4$ obtained by simulated annealing for $N={32}^2$. S-SkX, R-SkX, VS, CS, and FP stand for the square skyrmion crystal, rectangular skyrmion crystal, vertical spiral, conical spiral, and fully polarized phases, respectively. The lower panel shows the spin configurations, where the arrows and the contour show $(S_i^x,S_i^y)$ and $S_i^z$, respectively; see also Table 1 for the detailed spin textures in each phase.

Figure 3

$H$ dependences of (a) $M_z$ and ${\left({\chi }_0\right)}^2$, (b) ${\mathit{m}}_{{\mathit{Q}}_{\eta }^{\prime }}$, and (c) ${\mathit{m}}_{{\mathit{Q}}_{\eta }}$ ($\eta =1,2$) at $A=0.2$ and $\zeta =1$. The vertical lines represent the phase boundaries in Fig. 2.