Fig. 1. Illustrations of a series of 1D, 2D and 3D topological spin textures in magnetic materials^[1,13]: (a) Néel-type skyrmion (w = –1); (b) Bloch-type skyrmion (w = –1); (c) antiskyrmion (w = +1); (d) biskyrmion (w = –2); (e) vortex (w = –0.5); (f) meron (w = –0.5); (g) bimeron (w = –1); (h) skyrmionium (w = 0); (i) skyrmion tube, and (j) magnetic bobber. The arrow represents the spin direction and the out-of-plane spin component (m_z) is represented by the color: Red is out of the plane, white is in-plane, and blue is into the plane; (k) left-handed helimagnetic structures and soliton lattices; (l) right-handed helimagnetic structures and soliton lattices © (2016) The Physical Society of Japan; (m), (n) illustrations of soliton lattices under a small magnetic field perpendicular to c-axis: the solitons number remains unchanged, while the chiral period becomes longer.

Fig. 2. (a) Crystal structure of Cr_1/3NbS₂, Cr, Nb and S atoms are denoted by blue, green and yellow spheres, respectively; (b) schematic illustration of Cr_1/3NbS₂ single crystal growth^[22]; (c)–(g) schematic diagrams illustrating different spin configurations of Cr_1/3NbS₂: (c) Chiral helimagnetic order (CHM), in which the yellow box denotes a complete period of a chiral soliton; (d) chiral conical phase (CCP); (e) chiral soliton lattice (CSL); (f) tilted chiral soliton lattice (TCSL); (g) forced ferromagnetic (FFM) state.

Fig. 3. Phase diagram of Cr_1/3NbS₂ in presence of a magnetic field perpendicular to c-axis below curie temperature T_C. Here HM is equivalent to the CHM mentioned in this review^[15].

Fig. 4. (a) Temperature dependence of resistivity of Cr_1/3NbS₂. The inset shows the temperature deriative of resistivity^[33]; (b) ΔS_M-T curves for ΔH = 100–1000 Oe, a temperature gap exists between T_C and order-disorder temperature T^*. The inset is the ΔS_M-T curves near the T_C^[44]; (c) the Lorentz TEM micrographs of CHM in Cr_1/3NbS₂ at 0 T and 110 K, with a line profile of the contrast intensity integrated in a dotted square region. Vertical grid spacing corresponds to 15 nm. The period is estimated to be 46 nm^[12]; temperature dependent period (d) and wave number (e) of CHM are divided into three regions. Blue squares represent the data obtained by the Lorentz Fresnel and DPC methods. Green circle and red triangles represent the SAES data with increasing and decreasing T, respectively. The inset in (d) shows the 3D mean-field theory and the measurement data^[45].

Fig. 5. (a)−(c)The underfocused Lorentz micrographs in 0 T (a), magnetic fields are 0.208 T (b) and 0.224 T (c) which are perpendicular to thec-axis^[12]; (d) line profile of the contrast intensity integrated in the dotted area in (c) ^[12]; (e) the measurement data and theoretical calculation results of [L(H)-L(0)]/L(0)^[12]; (f) experimental results and theoretical calculation results of L(0)/L(H)^[12]; (g) magnetic chirality determines the characteristic magnetic patterns in Lorentz micrographs of CSL^[12]; (h) M-H curves of vertical magnetic field at 2 K^[33]; (i) magnetoresistance with a vertical magnetic field at 2 K, The upper inset is the variation in the slope of the magnetoresistance, and the lower inset is the derivative of the resistivity with respect to the magnetic field^[33]; (j) temperature dependent total intensity of A₀^[34]; (k) temperature dependent the muon spin precession frequency f under different magnetic field^[34].

Fig. 6. (a) Schematics of CSL-1 and CSL-2^[42]; (b)–(d) temperature dependence of the AC magnetic response M_3ω under H_ac = 3.9 Oe and H_dc = 0.2 kOe (b), H_dc = 1.0 kOe (c) and H_dc = 1.6 kOe (d). The frequency of all the AC magnetic field in the range of 1 to 300 Hz^[42]; (e) ESR signal at T = 3.5 K for H⊥c. The H_c1 and H_c2 indicate the appearing and disappearing fields of the anomalous signal^[53]; (f) H-T phase diagram of Cr_1/3NbS₂, The red solid and open circles indicate the H_c1 and H_c2, respectively^[53].

Fig. 7. (a) M-T curves under different fields for Cr_1/3NbS₂ with H⊥c, the inset shows the zero-field-cooling (ZFC) and field-cooling (FC) curves under 10 Oe with H⊥c; (b) the relationship between normalized resistivity and magnetic field with H⊥c at various temperatures^[33]; (c) the normalized interlayer magnetoresistance (MR) curves which is applied 5 mA in a temperature range from 10 to 110 K^[57]; (d)–(e) DPC-STEM images of dislocations (rotated red “⊥” symbols) at low (left column, 104 Oe) and high (right column, 2348 Oe) fields at 102 K in presence of a magnetic field perpendicular to c axis^[43]; (f) ferromagnetic resonance measurements on a sample with a 50-μm-long chiral axis at 50 K in the range of + 0.2 to – 0.2 T with 1 mT steps, showing the effect of magnetic properties with magnetic phase transitions ^[43]; (g) a simplified sketch of (f), the field region has mixed characteristics which marked as “*” ^[43]; (h)–(j) the Lorentz Fresnel images of solitons running vertically in a flake sample, showing the unidirectional guided movement of soliton dislocations (rotated red “⊥” symbols) as the applied (h)–(j) magnetic field decreases during the FFM to CSL phase transition^[43].

Fig. 8. (a) The upper diagram shows the magnetization component (M_∥) parallel to magnetic fields as a function of H with different angles at 5 K. The inset is the measuring configuration. The lower diagram shows the in-plane magnetoresistance measured under corresponding magnetic fields. Vertical lines indicate the polarization field with different direction^[60]; (b)–(c) scheme of Hall effect (b), anomalous Hall effect (c); (d) the relationship between magnetic field and ρ_xy at various temperature with H∥c^[60]; (e) the relationship between ordinary Hall coefficient (R_H), anomalous Hall coefficient (S_H) and temperature, which are determined from ρ_yx^[60]; (f) the relationship between magnetic field and ρ_yx besides ordinary Hall effect measured at temperature in the range of 2 to 120 K^[60]; (g) scheme of the electrical magnetochiral (EMC) effect; (h) magnetic field dependent R_EMC measured under a large range of temperature^[63].

Fig. 9. (a) Scanning ion micrograph of the device based on Cr_1/3NbS₂ crystals, the size of the flake is ^[69]; (b), (c) the MR curves at 20 (b) and 10 K (c), the inset is the MR curves under a higher magnetic field in (c) ^[69]; (d)–(f) Lorentz Fresnel images of CSL taken under the underfocused condition around the crystal grain of right-handed chirality at 1560 (d), 1771 (e) and 1781 Oe (f) at 100 K. Red arrows represent chiral boundary and blue arrows represent right-handed chiral boundary ^[69]; (g), (h) the relationship between soliton period and magnetic field. The inset shows the sample dimensions^[69]; (i) it shows the corresponding soliton density, The initial number of confined solitons is 20^[69].

Fig. 10. (a) Optical microscope images of Cr_1/3NbS₂ crystals with various thickness^[14]; (b) the magnetoresistance of Cr_1/3NbS₂ with different thicknesses measured at 250 mK^[14]; (c)–(e) the relationship between the x component of spins on the Cr atoms (S_x) and the thickness at zero magnetic field. Calculating results using the model to with t = 1.5L₀ (c), 2.5L₀ (d), and 5.5L₀ (e) (L₀ = 48 nm) ^[14]; (f) the lowest energy distribution under different magnetic fields when t = 2.5 L₀^[14]; (g) and (h) show the corresponding lowest energy configurations for single and zero soliton states, respectively^[14].