Formation and rotation of skyrmion crystal in the chiral-lattice insulator Cu${}_2$OSeO${}_3$

Small-angle neutron scattering experiments were performed on a bulk single crystal of chiral-lattice multiferroic insulator Cu${}_2$OSeO${}_3$. In the absence of an external magnetic field, helical spin order with magnetic modulation vector $q\parallel \langle 001\rangle$ was identified. When a magnetic field is applied, a triple-$q$ magnetic structure emerges normal to the field in the A phase just below the magnetic ordering temperature $T_c$, which suggests the formation of a triangular lattice of skyrmions. Notably, the favorable $q$ direction in the A phase changes from $q\parallel \langle 110\rangle$ to $q\parallel \langle 001\rangle$ upon approaching $T_c$. Near the phase boundary between these two states, the external magnetic field induces a 30${}^{\circ }$ rotation of the skyrmion lattice. This suggests a delicate balance between the magnetic anisotropy and the spin texture near $T_c$, such that even a small perturbation significantly affects the ordering pattern of the skyrmions.

Figure 1

(a) Crystal structure of Cu${}_2$OSeO${}_3$ with two distinct Cu${}^{2+}$ sites with different oxygen coordination. (b) Proper screw spin order with a single magnetic modulation vector $q$. (c) Skyrmion crystal with triple magnetic modulation vectors $q_1,q_2,$ and $q_3$, which lies within a plane normal to the applied magnetic field ($H$). The background color represents the out-of-plane component of the local magnetic moment ($m_z$).

Figure 2

Typical SANS profiles for the (110) plane of bulk Cu${}_2$OSeO${}_3$, measured at various magnetic fields and temperatures. All the data were obtained in the $H$-increasing scans, at 56.75 [(a)–(d)], 57.25 [(e)–(g)], 58.00 [(h) and (i)], and 57.75 K (j), respectively. The inset shows the measurement setup; $k_{\mathrm{in}}$ and $k_{\mathrm{out}}$ are the incident and scattered neutron wave vectors, respectively. The magnetic field ($H$) is applied along $k_{\mathrm{in}}\parallel \langle 110\rangle$, i.e., perpendicular to the observed plane.

Figure 3

Magnetic field ($H\parallel \left[110\right]$) dependence of (a) ac magnetic susceptibility ${\chi }^{\prime }$ and (b) integrated intensity $I$ of magnetic reflections obtained from SANS profiles. Temperature dependence of ${\chi }^{\prime }$ and $I$ is also plotted in (c) and (d).

Figure 4

Temperature ($T$) and magnetic field ($H$) phase diagram obtained for the bulk Cu${}_2$OSeO${}_3$ with $H\parallel \left[110\right]$. Diamonds represent the anomalies in the ${\chi }^{\prime }$ profile, and circles indicate the data points where SANS measurements were performed. In both cases, solid and open symbols are obtained from the $H$ and $T$ scans, respectively. The background color indicates the properties of the SANS profile at each data point; color saturation represents the integrated intensity of sixfold magnetic reflections, and the hue stands for the rotation angle ($\theta$) of the magnetic Bragg position with respect to the $\langle 110\rangle$ direction (see the bottom panel). Typical SANS profiles at selected data points are shown in Figs. 2a, 2b, 2c, 2d, 2e, 2f, 2g, 2h, 2i, 2j. In the data points with cross symbols, the mixed SANS pattern as shown in Fig. 2f was observed. Here, the A phase consists of SkX(1) and SkX(2); the former corresponds to the skyrmion crystal state favoring $q\parallel \langle 110\rangle$, and the latter does the one favoring $q\parallel \langle 001\rangle$.