Skyrmions and multisublattice helical states in a frustrated chiral magnet

We show that frustrated exchange interactions could stabilize chiral states in a chiral ferromagnet by considering the competition between ferromagnetic (FM) nearest-neighbor (NN) interaction and antiferromagnetic (AFM) next-nearest-neighbor (NNN) interaction. In the low-field regime, a multisublattice helical state is found, which is different from the normal one-sublattice helical states in a FM and the two-sublattice helical states in a two-sublattice AFM. As the field increases further, the skyrmion lattice is even stable with a much larger energy-preferable window than a system without frustration. We argue that the enlargement of the stability window of skyrmions is a consequence of the reduced effective exchange interaction caused by the frustration. As a byproduct, the hysteresis loop of the frustrated chiral system shrinks as the magnetization goes to zero and then opens up again, which is known as the wasp-waist hysteresis loop. The critical field that separates the narrow and wide part of the wasp-waist loop depends exponentially on the strength of NNN coupling. Our results provide physical insight into the chiral states in the magnetic systems with the coexistence of frustrated exchange and chiral interaction and might even prove important for novel devices, where the stability window of skyrmions is a key asset.

Figure 1

(a),(b) Ground states of the system for $J_2/J_1=0.8$ and 0, respectively, under zero field. The pink arrows indicate the direction of period. The color codes $S_z$ from $-1$ (green), 0 (white), to +1 (orange). (c) $S^z$ as a function of the spin position ($y$). (d) Replot of $S^z-y$ when the sites are classified into three classes, $i=3n-2,3n-1,3n$, where $n$ is a positive integer. $y=ia$, where $a$ is the lattice constant. Black, red, and blue squares represent $3n-2,3n-1,3n$, respectively. (e) $S^z-y$ for $3n-2$ sites at various fields in the units of $J_1$. (f) The average magnetization of the $3n-2,3n-1$, and $3n$ sites $\overline{S^z}$ as a function of unit-cell coordinates ($y$) for various fields. $\overline{S^z}=(S_{3n-2}^z+S_{3n-1}^z+S_{3n}^z)/3$.

Figure 2

The calculated phase diagram in the $H{J}_1/D^2-J_2/J_1$ plane. $T=6.6\times {10}^{-3}{J}_1/{\mathrm{k}}_{\mathrm{B}}$. The color represents various phases including FM (yellow), skyrmion (orange), H (light blue), transition (dark blue), and MSH (pink). The dashed line refers to $J_2=0$.

Figure 3

(a) Scheme of a site $A$ and its neighbors $A_{\mathrm{nn}},A_{\mathrm{nnn}}$ on a square lattice. (b) Effective NN exchange $J_{\mathrm{eff}}$ as a function of NNN exchange strength. The red line is the fitting curve using the formula $J_{\mathrm{eff}}/J_1=\mathrm{Exp}(-1.22J_2/J_1)$. The blue line is the theoretical prediction. (c) Calculated phase boundary using effective NN coupling $J_{\mathrm{eff}}$ (red line) for H/skyrmion ($0.24H{J}_{\mathrm{eff}}/D^2$) and skyrmion/FM ($0.73H{J}_{\mathrm{eff}}/D^2$). (d) Energy density difference between a helical state and a FM state calculated from a 1D frustrated model under fields $H/J_1=0$ (black square), 0.6 (red dots), and 1.2 (blue up-triangles), respectively. The light blue/yellow region refers to the regions with FM and helix as the ground states, respectively.

Figure 4

First magnetization curves for (a) $J_2=0$ and (b) $J_2=0.4$, and hysteresis loops for (c) $J_2/J_1=0.0$ and (d) 0.4. $\langle S_z\rangle$ is defined as $\langle S_z\rangle =1/L^2{\sum }_i{S}_i^z$. Blue: helix; orange: skyrmion; yellow: FM state; gray: mixing of helix and skyrmion. (e) is the enlarged figure of the circled part in (d). $T=6.6\times {10}^{-3}{\mathrm{J}}_1/{\mathrm{k}}_{\mathrm{B}}$, $L=32$. (e) $\langle S_z\rangle$ as a function of sample size for the descending branch (blue circles) and ascending branch (red squares), respectively. The left inset is the critical field $H_c$ as a function of $J_2/J_1$. The right inset is the size effect of the hysteresis loop at small fields for $L=32$ (black circles), 96 (blue triangles), and 192 (red hexagons), respectively.