Emergence and magnetic-field variation of chiral-soliton lattice and skyrmion lattice in the strained helimagnet ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

We demonstrate emergence of both the chiral soliton lattice and skyrmion lattice and investigate their magnetic-field variation in the strained ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ thin plate by means of small-angle resonant soft x-ray scattering. The tensile strain stabilizes a helical spin structure with the modulation vector along the strain direction. Consequently, when increasing the field perpendicular to the modulation vector, it undergoes large shrinkage and higher-order diffraction peaks are also observed, which is in accord with a typical feature of the chiral soliton lattice. The skyrmion lattice also appears in the presence of tensile strain but is elongated along the strain direction in reciprocal space. The magnetic-field dependence of the modulation vector along the strain direction agrees with the theoretical model quantitatively, while that along other directions does not. The agreement, however, becomes worse at lower temperatures, which may be attributed to the large potential barrier for skyrmion annihilation.

Figure 1

(a–c) Schematic illustrations of helical state (a), chiral soliton lattice (b), and skyrmion lattice (c).

Figure 2

(a, b) Scanning electron microscope images of device structures without (a) and with tensile strain (b). ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ and tungsten (W) are highlighted in green and blue, respectively. (c–f) Typical diffraction patterns for strain-free and strained samples. The helical state at zero field (c) and the skyrmion lattice at a field of 50 mT (d) in the strain-free sample. The helical state at zero field (e) and the skyrmion lattice at a field of 70 mT (f) in the strained sample. (g, h) Magnetic phase diagrams of strain-free (g) and strained samples (h). The red, blue, and black symbols indicate observations of six, two and no diffraction spots, respectively.

Figure 3

(a) The magnetic-field variation of the diffraction patterns at 17 K in the strained sample. Diffraction spots existing at the right side are highlighted by white arrows. (b) The profile of the integrated intensity along the $q$ vector. The diffraction intensity is integrated so as to encompass the diffraction spots, as indicated by black lines in (a-2).

Figure 4

(a) The magnetic-field dependence of $\left|q\right|$, $\left|2q\right|$, and $\left|3q\right|$ and peak width of Bragg spots derived from the Gaussian fit of the CCD camera image along the $q$ vector. (b) The magnetic-field dependence of $\left|q\right|$, $\left|2q\right|$, and $\left|3q\right|$ normalized by zero-field values and the theoretical calculation of the CSL. (c) The magnetic-field dependence of the intensity of the second-order diffractions (red circles) and the theoretical calculation (blue curve). In (a) and (b), error bars represent the peak width along the $q$ vector. The gray-shaded region indicates the region in which the chiral soliton lattice is disordered. The data were taken in field-increasing runs.

Figure 5

(a) The magnetic-field variation of the diffraction patterns at 50 K in the strained sample. In (a-5), $q_1$, $q_2$, and $q_3$ vectors are defined. (b) The magnetic-field dependence of the $q_1$ vector (red and blue filled circles) and $q_{2,3}$ vectors (red open circles). (c) Magnetic-field dependence of the ellipticity $1-q_{\mathrm{S}}/q_{\mathrm{L}}$ (green circles) and $q_{\mathrm{S}}{q}_{\mathrm{L}}$ (purple circles) of the ellipse traced by diffraction spots of the SkL. The ellipse in the inset represents an ellipse traced by diffraction spots of the SkL and $q_{\mathrm{S},\mathrm{L}}$ is also defined. (d) The magnetic-field dependence of the $q_1$ vector normalized by the helical $q$ vector at zero field ($q_0$). Square and circle symbols represent the data for the strain-free and strained samples, respectively. Dashed blue and red lines indicate theoretical curves of the CSL and SkL, respectively. The gray-shaded region indicates the region in which the ring-shaped diffraction pattern is observed.

Figure 6

(a) The magnetic phase diagram of the strain-free sample including the metastable SkL. (b) Magnetic-field variation of diffraction patterns of the metastable SkL at 15 K after the field cooling with a field of 40 mT. (c) The magnetic-field dependence of the $q$ vector for each temperature. The magnetic field and the $q$ vector are normalized by the ferromagnetic transition field ($H_{\mathrm{c}}$) and the helical period at zero field ($q_0$), respectively. Open and filled symbols represent the data taken in thermodynamically stable and metastable SkL, respectively. Circle and square symbols represent the data taken in strain-free and strained samples, respectively. For example, filled square symbols represent the data taken in the metastable SkL in the strain-free sample. The dashed line indicates the theoretical curve for the SkL.

Figure 7

Several diffraction patterns of the skyrmion lattice at 55 K, 40 mT.

Figure 8

(a) Calculated magnetic-field dependence of magnetization ($M$) of the CSL. The magnetization and magnetic field are normalized by the saturated moment ($M_{\mathrm{ferro}}$) and ferromagnetic transition field ($H_{\mathrm{c}}$), respectively. (b) Calculated magnetic-field dependence of the normalized $q$ vector of the CSL (black) and the theoretical curve calibrated by the demagnetization factor (red). $q_0$ represents the helical $q$ vector at zero field. (c) Calculated magnetic-field dependence of $M$ of the SkL. (d) Calculated magnetic-field dependence of the normalized $q$ vectors of the SkL (black) and the theoretical curve calibrated by the demagnetization factor (red).