Effect of magnetic anisotropy on skyrmions with a high topological number in itinerant magnets

We report our numerical results for the effect of magnetic anisotropy on a skyrmion crystal with a high topological number of two, which was recently discovered in an itinerant electron model [R. Ozawa, S. Hayami, and Y. Motome, Phys. Rev. Lett. 118, 147205 (2017)]. By performing numerical simulations based on the kernel polynomial method and the Langevin dynamics for the Kondo lattice model on a triangular lattice, we find that the topological property remains robust against single-ion anisotropy, while the magnetic texture is deformed continuously. The resultant spin structure is characterized by three wave numbers (triple-$Q$ state), in which the $xy$ component of the spins forms a magnetic vortex crystal and the $z$ component of the spins behaves as a sinusoidal wave. For a larger anisotropy, we show that the system exhibits a phase transition from a skyrmion crystal to topologically trivial phases with vanishing scalar chirality: a single-$Q$ collinear state and double-$Q$ noncoplanar states for the easy-axis and easy-plane anisotropy, respectively. We also examine the effect of single-ion anisotropy in an external magnetic field, and find that the field range of the skyrmion crystal is rather insensitive to the anisotropy, in contrast to another skyrmion crystal with a topological number of one whose field range is considerably extended (reduced) by the easy-axis (easy-plane) anisotropy.

Figure 1

KPM-LD results for the model in Eq. (1) at zero field: single-ion anisotropy $A$ dependences of the ${\mathbf{Q}}_{\nu }$ components of the magnetization [Eq. (5)] and the net spin scalar chirality [Eq. (6)]. The vertical dashed lines show the phase boundaries.

Figure 2

Leftmost: Snapshots of the spin configurations in (a) the $3Q$ SkX with $n_{\mathrm{sk}}=2$ for $A=0.0025$ and (b) the $2Q$ noncoplanar state for $A=-0.0025$ at $H=0$. The contour shows the $z$ component of the spin moment. Middle left: Snapshots of the spin scalar chirality. Middle right and rightmost: The square root of the $xy$ and $z$ components of the spin structure factor, respectively. In the right two columns, the hexagons represent the first Brillouin zone.

Figure 3

Left: Snapshots of the spin configurations in (a) the $1Q$ collinear state for $A=0.008$ and (b) the $2Q$ coplanar state for $A=-0.006$ at $H=0$. The contour shows the $z$ component of the spin moment. Right: The square root of the $z$ ($xy$) component of the spin structure factor in the upper panel (lower). The $xy$ ($z$) component of the spin structure factor is negligible in (a) [(b)]. In the right column, the hexagons represent the first Brillouin zone.

Figure 4

(a) Magnetization curves and (b) the net scalar chirality for $A=0.002$ and $-0.001$ under the magnetic field $H$. The data at $A=0$ are also shown for comparison. (c), (d) $H$ dependences of $m_{{\mathbf{Q}}_{\nu }}$ ($\nu =1–3$) for (c) $A=0.002$ and (d) $A=-0.001$.

Figure 5

Leftmost: Snapshots of the spin configurations in (a) the $3Q$ SkX with $n_{\mathrm{sk}}=2$ for $H=0.002$ and $A=0.002$, (b) the $3Q$ SkX with $n_{\mathrm{sk}}=1$ for $H=0.004$ and $A=0.002$, (c) the $2Q$ state with $n_{\mathrm{sk}}=0$ for $H=0.018$ and $A=0.002$, and (d) the $3Q$ state with $n_{\mathrm{sk}}=0$ for $H=0.006$ and $A=-0.001$. The contour shows the $z$ component of the spin moment. Middle left: Snapshots of the spin scalar chirality. Middle right and rightmost: The square root of the $xy$ and $z$ components of the spin structure factor, respectively. In the right two columns, the hexagons represent the first Brillouin zone. In (c), the field-induced $\mathbf{q}=\mathbf{0}$ component is subtracted for clarity.

Figure 6

Snapshots of (a) the spin configurations and (b) the spin scalar chirality in the $3Q$ SkX with $n_{\mathrm{sk}}=2$. The data are $A=-0.0015$, 0.0015, 0.0035, and 0.0055 from the left panel.