Multifarious skyrmion phases on a trilayer triangular lattice

The instability toward a magnetic skyrmion crystal in centrosymmetric trilayer magnets is investigated based on a spin model with layer-dependent Dzayloshinskii-Moriya interaction. We find various types of skyrmion crystal phases with different skyrmion numbers in a low-temperature phase diagram by performing the simulated annealing. In addition to the Néel skyrmion crystal phase that is expected to emerge in the presence of the polar-type Dzayloshinskii-Moriya interaction, we obtain the skyrmion crystal phases characteristics of the layered system: the twisted surface skyrmion crystal, antiskyrmion crystal, and high-topological-number skyrmion crystal phases. The rich magnetic phases are brought about by the synergy among the layer-dependent Dzayloshinskii-Moriya interaction, interlayer exchange interaction, and an external magnetic field. Our results indicate that the layer degree of freedom at the surface and heterostructures is a good platform to engineer and design the topological spin textures.

Figure 1

(a) The trilayer triangular-lattice system consisting of layers A, B, and C. The lattice constants, $a$ and $c$, and the interlayer exchange coupling $J_{\parallel }$ are also shown. (b) Three layers viewed from the $z$ axis. The green arrows denote the DM vectors in each layer, where $\mathit{E}$ represents the local crystalline electric field.

Figure 2

(a) Phase diagram of the model in Eq. (6) obtained by the simulated annealing while changing $J_{\parallel }$ and $H$. The contour shows the total scalar chirality ${\chi }^{\mathrm{sc}}$. The layer-resolved scalar chiralities, ${\chi }_{\mathrm{A}}^{\mathrm{sc}}$ and ${\chi }_{\mathrm{B}}^{\mathrm{sc}}$, are plotted by the contour in (b) and (c), respectively, on the same phase diagram as (a).

Figure 3

Real-space spin (left) and scalar chirality (right) configurations of (a) phase I [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,0,-1)$] at $J_{\parallel }=0.1$ and $H=1.5$ and (b) phase II [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,0,-1)$] at $J_{\parallel }=-0.5$ and $H=1.2$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component.

Figure 4

Real-space spin (left) and scalar chiraity (right) configurations of (a) phase III [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,-1,-1)$] at $J_{\parallel }=-0.2$ and $H=1$ and (b) phase IV [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,1,-1)$] at $J_{\parallel }=0.2$ and $H=1$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component. The three types of vortex cores are represented by green circles, triangles, and squares.

Figure 5

Real-space spin (left) and scalar chirality (right) configurations of (a) phase V [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,-1,-1)$] at $J_{\parallel }=-1$ and $H=0.9$ and (b) phase VI [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,2,-1)$] at $J_{\parallel }=0.4$ and $H=1$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component.

Figure 6

Real-space spin (left) and scalar chirality (right) configurations of (a) phase VII [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,-2,-1)$] at $J_{\parallel }=-0.6$ and $H=0.65$ and (b) phase VIII [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(-1,x,-1)$] at $J_{\parallel }=-0.6$ and $H=0.75$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component. The green circles stand for the core positions.

Figure 7

Real-space spin (left) and scalar chirality (right) configurations of (a) phase IX [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(0,1,0)$] at $J_{\parallel }=-0.3$ and $H=0.4$ and (b) phase IX [SkX with $(n_{\mathrm{sk}}^{\left(\mathrm{A}\right)},n_{\mathrm{sk}}^{\left(\mathrm{B}\right)},n_{\mathrm{sk}}^{\left(\mathrm{C}\right)})=(0,1,0)$] at $J_{\parallel }=0.3$ and $H=0.4$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component.

Figure 8

Real-space spin (left) and scalar chirality (right) configurations of (a) phase i at $J_{\parallel }=-0.1$ and $H=0.4$ and (b) phase ii at $J_{\parallel }=-0.4$ and $H=0.2$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component. The green circles represent the core positions discussed in the main text.

Figure 9

Real-space spin (left) and scalar chirality (right) configurations of (a) phase iii at $J_{\parallel }=-0.5$ and $H=1.5$ and (b) phase iv at $J_{\parallel }=-1$ and $H=1.05$ on layer A (top), layer B (middle), and layer C (bottom). In the left panels, the arrows represent the $xy$ components of the spin moment and the color shows the $z$ component.

Figure 10

(Left and second left) The square root of the $xy$ and $z$ components of the spin structure factor for layer A, respectively, in (a) phase I at $J_{\parallel }=0.1$ and $H=1.5$, (b) phase II at $J_{\parallel }=-0.5$ and $H=1.2$, (c) phase III at $J_{\parallel }=-0.2$ and $H=1$, (d) phase IV at $J_{\parallel }=0.2$ and $H=1$, (e) phase V at $J_{\parallel }=-1$ and $H=0.9$, (f) phase VI at $J_{\parallel }=0.4$ and $H=1$, and (g) phase VII state at $J_{\parallel }=-0.6$ and $H=0.65$. Black hexagons represent the first Brillouin zone. The middle and right two panels represent the data for layers B and C corresponding to the left two ones, respectively.

Figure 11

(Left and second left) The square root of the $xy$ and $z$ components of the spin structure factor for layer A, respectively, in (a) phase VIII at $J_{\parallel }=-0.6$ and $H=0.75$, (b) phase IX at $J_{\parallel }=-0.3$ and $H=0.4$, (c) phase IX at $J_{\parallel }=0.3$ and $H=0.4$, (d) phase i at $J_{\parallel }=-0.1$ and $H=0.4$, (e) phase ii at $J_{\parallel }=-0.4$ and $H=0.2$, (f) phase iii at $J_{\parallel }=-0.5$ and $H=1.5$, and (g) phase iv state at $J_{\parallel }=-1$ and $H=1.05$. Black hexagons represent the first Brillouin zone. The middle and right two panels represent the data for layers B and C corresponding to the left two ones, respectively.

Figure 12

$H$ dependences of the magnetization $M_{\eta }^z$ (left) and the scalar chirality ${\chi }_{\eta }^{\mathrm{sc}}$ (right) for layers $\eta =$ A, B, and C at (a) $J_{\parallel }=-0.1$, (b) $-0.4$, (c) $-0.7$, and (d) $-1$. The vertical dashed lines represent the phase boundaries between different phases, where the corresponding phases are presented above in each figure.

Figure 13

$H$ dependences of the magnetization $M_{\eta }^z$ (left panel) and the scalar chirality ${\chi }_{\eta }^{\mathrm{sc}}$ (right panel) for layers $\eta =$ A, B, and C at (a) $J_{\parallel }=0.1$, (b) 0.2, (c) 0.3, and (d) 0.5. The vertical dashed lines represent the phase boundaries between different phases, where the corresponding phases are presented above in each figure.

Figure 14

$H$ dependences of the magnetization $M_{\eta }^z$ (left panel) and the scalar chirality ${\chi }_{\eta }^{\mathrm{sc}}$ (right panel) for layers $\eta =$ A, B, and C at (a) $J_{\parallel }=-0.1$ and $D=0.1$, (b) $J_{\parallel }=-0.4$ and $D=0.4$, (c) $J_{\parallel }=0.1$ and $D=0.1$, and (d) $J_{\parallel }=0.2$ and $D=0.4$. The vertical dashed lines represent the phase boundaries between different phases, where the corresponding phases are presented above in each figure.