Controlling the Topological Sector of Magnetic Solitons in Exfoliated ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ Crystals

We investigate manifestations of topological order in monoaxial helimagnet ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ by performing transport measurements on ultrathin crystals. Upon sweeping the magnetic field perpendicularly to the helical axis, crystals thicker than one helix pitch (48 nm) but much thinner than the magnetic domain size ($\sim 1\mu \mathrm{m}$) are found to exhibit sharp and hysteretic resistance jumps. We show that these phenomena originate from transitions between topological sectors with a different number of magnetic solitons. This is confirmed by measurements on crystals thinner than 48 nm—in which the topological sector cannot change—that do not exhibit any jump or hysteresis. Our results show the ability to deterministically control the topological sector of finite-size ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ and to detect intersector transitions by transport measurements.

Figure 1

(a) Structure of layered ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$. The $c$-axis lattice constant is 1.21 nm. (b) Schematic illustration of the spin configurations along the $c$ axis in bulk ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ (the arrows represent the magnetization in the Cr planes). Upon increasing magnetic field ($B$) in the $ab$ plane, the $B=0$ magnetic helix (top) is gradually transformed into a chiral soliton lattice with period $L_B$ (middle), finally becoming ferromagnetic above the critical magnetic field ($B_c$; bottom). (c) In the process, the period $L_B$ increases and diverges close to $B_c$ (plot obtained from theoretical calculations based on the model discussed in the main text). (d) Magnetoresistance, $\mathrm{\Delta }R/R(0)\equiv \mathbf{(}R(B)-R(0)\mathbf{)}/R(0)$, of bulk ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ measured at 250 mK. The blue or red curve corresponds to measurements performed upon sweeping the in-plane field $B$ in the direction indicated by the blue or red arrow. An abrupt resistance drop at $B_c\sim 0.17\mathrm{T}$ with no hysteresis is observed.

Figure 2

(a) Optical microscope images of exfoliated ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ crystals with thicknesses between 28 nm and 280 nm. (b) Atomic force microscope image of the 57-nm-thick crystal, showing a flat and uniform surface, as visible in the thickness profile (c) taken along the red line. (d) Temperature dependence of the resistance $R(T)/R(280K)$ for crystals of different thicknesses (see legend). All curves merge together and exhibit a resistance drop starting around 130 K, the bulk transition temperature of the helimagnetic state. The inset in (d) shows an optical image of a device, with nanofabricated Ti/Au contacts. In all images, the scale bar is $3\mu \mathrm{m}$.

Figure 3

(a)–(d) Evolution of the magnetoresistance, $\mathrm{\Delta }R/R(0)$, measured at 250 mK on ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ crystals of different thickness (see legends; blue or red curve represents data taken while sweeping $B$ in the direction indicated by the arrows). Resistance jumps and pronounced hysteresis are seen, absent in bulk ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$ [see Fig. 1]. The insets zoom-in on a small $B$ range to make the smaller jumps visible. (e) Thickness dependence of the critical field $B_c$, corresponding to the highest value of $B$ at which a resistance jump is observed. The experimental data (dots) are in good agreement with calculations (solid line). The resistance continues to decrease for $B>B_c$, as shown in (f) by plotting $\mathbf{(}R(0.5T)-R(B_c)\mathbf{)}/R(0)$ versus $t$. The data (dots) approximately exhibit a $1/(t-L_0)$ dependence (solid line) expected as the phenomenon originates from the alignment of spins close to the crystal surfaces. The yellow regions in (e) and (f) denote the thickness range below $L_0$.

Figure 4

(a)–(c) Lowest-energy configuration of the spin helix at $B=0$, represented by the $x$ component of spins on the Cr atoms ($S^x$), calculated using the model discussed in the main text for crystals with $t=1.5$ $L_0$, 2.5 $L_0$, and 5.5 $L_0$ ($L_0=48\mathrm{nm}$). The colored areas mark complete spin windings, corresponding to the presence of one (a), two (b), and five (c) magnetic solitons. (d) The blue, red, and green lines correspond to the lowest energy of configurations containing two, one, and zero solitons, calculated as a function of $B$ for a crystal with $t=2.5$ $L_0$. At low $B$ (blue shaded region) configurations containing two solitons [panel (b)] have the lowest energy; for intermediate values of $B$ (red shaded region) configurations with one soliton [panel (e)] have the lowest energy; for the higher field, configurations with no solitons [panel (f)] are energetically favorable.

Figure 5

Magnetoresistance $\mathrm{\Delta }R/R(0)$ of crystals thinner than the helix pitch ($L_0=48\mathrm{nm}$) measured at 250 mK. The blue or red curves represent data measured while sweeping $B$ in opposite directions, as indicated by the arrows. In all cases, only a continuous decrease of the resistance is observed, and no jumps or hysteresis is present.