Geometrical protection of topological magnetic solitons in microprocessed chiral magnets

A chiral soliton lattice stabilized in a monoaxial chiral magnet ${\mathrm{CrNb}}_3{\mathrm{S}}_6$ is a magnetic superlattice consisting of magnetic kinks with a ferromagnetic background. The magnetic kinks are considered to be topological magnetic solitons (TMSs). Changes in the TMS number yield discretized responses in magnetization and electrical conductivity, and this effect is more prominent in smaller crystals. We demonstrate that, in microprocessed ${\mathrm{CrNb}}_3{\mathrm{S}}_6$ crystals, TMSs are geometrically protected through element-selected micromagnetometry using soft x-ray magnetic circular dichroism (MCD). A series of x-ray MCD data is supported by mean-field and micromagnetic analyses. By designing the microcrystal geometry, TMS numbers can be successfully changed and fixed over a wide range of magnetic fields.

Figure 1

Images of (a) a helimagnet, (b) a chiral soliton lattice, and (c) a forced ferromagnet. In (b), the dc magnetic field $H$ applied to the chiral helical axis stabilizes CSL.

Figure 2

Images and schematics of micrometer-sized ${\mathrm{CrNb}}_3{\mathrm{S}}_6$ specimens (a) A, (b) B, and (c) C, each attached to a Ta plate pierced by a single 5-$\mu \mathrm{m}$-diameter pinhole (green dotted circle). (d) Overview of the experimental setup of soft x-ray MCD measurement. Both the x-ray beam axis and magnetic field $H$ were perpendicular to the chiral helical axis ($c$ axis; see the text). (e) Hysteresis curves of element-specific magnetization for Cr $M_{\mathrm{Cr}}$ obtained by the soft x-ray MCD measurements at $T=10$ K. The MCD signal was observed over only the areas marked with green dots in (a)–(c). (f) Hysteresis curves of $M_{\mathrm{Cr}}$ after calibrating the demagnetization effect. The black solid squares and purple curve represent bulk millimeter-sized crystals from [24] and the mean-field (MF) theoretical results tracing the minimum-energy state for an infinite system, respectively [5]. In both (e) and (f), $M_{\mathrm{Cr}}$ is normalized to the saturated magnetization $M_{\mathrm{s}}$.

Figure 3

Minor loop (ML) magnetization curves of specimen C of ${\mathrm{CrNb}}_3{\mathrm{S}}_6$ for ML-1 and ML-2. Magnetization profiles were obtained with the soft x-ray MCD measurements at $T=10$ K. (a) and (b) ML-1 starting from zero field. (c) and (d) ML-2 starting from above $H_{\mathrm{c}}$. For reference, the full loop and MF theory data, shown in Fig. 2, are also presented in (a)–(d). The original $M_{\mathrm{Cr}}$ vs $H$ data corresponding to (a)–(d) are shown in Fig. S5 [27]. (e) and (f) Hysteresis areas of ML-1 and ML-2 as a function of the return field $H_{\mathrm{RE}}$ evaluated by the levels of (e) $H_{\mathrm{eff}}$ and (f) $M_{\mathrm{Cr}}$/$M_{\mathrm{s}}$. Two hysteresis thresholds, $H^{*}$ and $H^{**}$, appear for the $H$-increasing and $H$-decreasing processes, respectively. In (a)–(d), reversible and irreversible regions are hatched with orange and light blue, respectively.

Figure 4

Minor-loop (ML) magnetization curves of specimen C of ${\mathrm{CrNb}}_3{\mathrm{S}}_6$ for ML-3 and ML-4. Magnetization profiles were obtained with the soft x-ray MCD measurements. (a) ML-3 at $T=36$ K and (b) ML-4 at $T=10$ K. The original $M_{\mathrm{Cr}}$ vs $H$ data are shown in Fig. S6 [27]. Here, the data after the field calibration are shown. The ML-3 process is $H=0.00\to 2.53\to 0.90\to 4.70$ kOe and $H=0.00\to 2.43\to 0.60\to 4.70$ kOe; the ML-4 process is $H=4.00\to 0.21\to 2.40\to 0.00$ kOe. The reversible $H_{\mathrm{eff}}$ regions are hatched with orange.

Figure 5

(a) Theoretical calculation of the magnetization curve with fixed soliton numbers by finite-temperature three-dimensional mean-field theory under a periodic boundary condition. The soliton number $w$ was fixed at values between 0 and 105. Points 1–5 have the same TMS number of $w=50$, and for point 6, corresponding to the forced-ferromagnetic state, $w=0$. (b) Spatial distributions of magnetic moments viewed from the direction perpendicular to the chiral helical axis are displayed for each point. (c) Spatial distributions for each point are displayed in three-dimensional perspectives.

Figure 6

Explanation of $M_{\mathrm{Cr}}$ for specimens A, B, and C with the help of the 3D-MF theory. In (a), guide curves and lines are shown for both specimens A (${\mathrm{A}}_0\to {\mathrm{A}}_1\to {\mathrm{A}}_2\to {\mathrm{A}}_0$) and B (${\mathrm{B}}_0\to {\mathrm{B}}_1\to {\mathrm{B}}_2\to {\mathrm{B}}_0$). The arrows in (a) indicate the directions of change in $H$. In (b), the full loop and four minor loops (ML-1, ML-2, ML-3, ML-4) for specimen C are indicated by solid and dashed curves, respectively. In (b), some points are labeled as ${\mathrm{C}}_n$ ($n=0\text{–}11$), and the full and minor loops are expressed as follows: full loop, point ${\mathrm{C}}_0\to {\mathrm{C}}_1\to {\mathrm{C}}_2\to {\mathrm{C}}_6\to {\mathrm{C}}_7\to {\mathrm{C}}_0$; ML-1, ${\mathrm{C}}_0\to {\mathrm{C}}_1\to {\mathrm{C}}_0$ or ${\mathrm{C}}_0\to {\mathrm{C}}_2\to {\mathrm{C}}_3\to {\mathrm{C}}_0$; ML-2, ${\mathrm{C}}_6\to {\mathrm{C}}_7\to {\mathrm{C}}_6$ or ${\mathrm{C}}_6\to {\mathrm{C}}_8\to {\mathrm{C}}_5\to {\mathrm{C}}_6$; ML-3, ${\mathrm{C}}_0\to {\mathrm{C}}_2\to {\mathrm{C}}_4\to {\mathrm{C}}_{10}\to {\mathrm{C}}_4\to {\mathrm{C}}_6$; and ML-4, ${\mathrm{C}}_6\to {\mathrm{C}}_8\to {\mathrm{C}}_9\to {\mathrm{C}}_{11}\to {\mathrm{C}}_9\to {\mathrm{C}}_0$. Both ML-1 and ML-2 have one $H_{\mathrm{RE}}$ point, whereas both ML-3 and ML-4 have two $H_{\mathrm{RE}}$ points. In both (a) and (b), the magnetization $M$ in the vertical axis is normalized by the spin quantum number $S=3/2$, and the magnetic field $H$ is normalized by a multiple of the ferromagnetic exchange interaction on a plane perpendicular to the chiral helical axis $J^{\left|\right|}$ and $S$.

Figure 7

Magnetization curve calculated by time evolution of the Landau-Lifshitz-Gilbert equation. (a) A microprocessed simulation was performed for the system landscape containing 500 $\times 20$ unit cells. The magnetic field $H$ was applied perpendicular to the chiral helical axis. (b) Dependence of the magnetization $M$ on $H$ for $H_{\mathrm{RE}}$ values of 1.0, 2.6, and 4.0 kOe. (c) The spin mapping shown at representative field points identified as A–J in (b) (A–F for $H_{\mathrm{RE}}=4.0$ kOe and A–C and G–J for $H_{\mathrm{RE}}=2.6$ kOe), where red and blue represent the positive and negative moments along $H$, respectively. (d) and (e) Images of the soliton's release from and insertion into the chiral soliton lattice state presented for the $H$-increasing and $H$-decreasing processes, respectively. The magnetic moments are colored according to the coloring in (c).

Figure 8

Annihilation of TMSs in a flat, thin system modeling specimen C. The present system is schematized in Fig. 7. (a)–(f) Snapshots of the magnetization configuration (200 nm $<X<$ 300 nm) as the number of magnetic solitons drastically changes in the hysteresis curve (as shown in Fig. 7). A series of snapshots shows how the soliton is released from the sample above $H^{*}$ in the $H$-increasing process [from (a) to (f)].

Figure 9

Creation of TMSs in a flat, thin system modeling specimen C. The present system is schematized in Fig. 7. (a)–(f) show how the soliton penetrates into the sample below $H^{**}$ in the $H$-decreasing process [from (a) to (f)]. The created soliton propagates to the center of the sample, while another soliton is created at the edge ($H=580$ Oe).