Surface barrier effect as evidence of chiral soliton lattice formation in chiral dichalcogenide CrTa₃S₆ crystals

Chiral helimagnets induce an antisymmetric exchange interaction strongly coupled to a chiral crystalline structure.^1,2 As a consequence of its competition with a symmetric Heisenberg exchange interaction in the absence or presence of magnetic fields, nontrivial chiral magnetic structures emerge such as chiral helimagnetic order (CHM),^1,2 chiral soliton lattice (CSL),^3–7 and chiral magnetic vortices called magnetic Skyrmions.^8,9 These chiral magnetic structures have been observed via neutron scattering or electron microscopy in the recent decade.^10–12 

CrNb₃S₆ is one of the well-established transition-metal dichalcogenides (TMDs) that exhibit chiral helimagnetism. CrNb₃S₆ forms a chiral monoaxial crystal structure with space group P6₃22,^13–15 where the symmetric and antisymmetric exchange interactions work along the principal c-axis of the crystal. The CSL formation, as schematically drawn in Fig. 1(a), was detected in real space images and reciprocal scattering data by using Lorentz microscopy in CrNb₃S₆.¹² Neutron¹⁶ and resonant magnetic x-ray¹⁷ scattering experiments are also useful for identifying the CHM and CSL in a reciprocal space.

The CSL exhibits robust phase coherence at the macroscopic length scale.¹² Thus, nontrivial characteristics appear in various physical properties. Indeed, the CSL shows giant magnetoresistance (MR) due to a proliferation of chiral solitons,¹⁸ discretization MR effect,¹⁹ robust response of chiral solitons under oblique magnetic fields,²⁰ nonreciprocal electrical transport,²¹ and collective elementary excitation of the CSL up to a frequency of sub-terahertz.²² Moreover, very anisotropic soliton defects appear in the CSL system when decreasing the magnetic field H.^23–25 Such coherent, topological, and collective nature of the CSL could be useful for spintronic device applications, such as memory multi-bits and 6G communications technologies using the CSL.^26–28 

The MR is one of the feasible methods for identifying the CSL. In particular, when reducing the sample dimensions, discretized MR appears because of the countable nature of chiral solitons in the CSL.¹⁹ In addition, the surface barrier emerges upon the penetration of chiral solitons into the system in the H decrease process, as schematically drawn in Fig. 1, because of the phase coherence of the CSL.²⁹ The strength of the surface barrier is quantified by solving a one-dimensional (1D) chiral sine-Gordon model in a semi-infinite system, which describes well the CSL system realized in the monoaxial chiral helimagnets, such as CrNb₃S₆. The ratio of the magnetic field H_b, where the surface barrier disappears, to the critical magnetic field H_c takes a constant value (H_b/H_c = 4/π² ∼ 0.405).²⁹ In the experiments, the presence of a surface barrier could be detected as a sudden jump of a physical quantity at a particular strength of the magnetic field (H_jump) when decreasing H. The values of H_jump/H_c, which were experimentally obtained by using the MR in CrNb₃S₆ (e.g., 0.416, 0.405, and 0.408 in a particular micrometer-sized crystal),²⁹ showed an excellent agreement with 4/π² expected for the 1D chiral sine-Gordon model. This coincidence is regarded as evidence of the presence of CSL. Importantly, the surface barrier is an intrinsic origin of hysteresis for the CSL system that exhibits continuous phase transition of nucleation-type,³⁰ as discussed in the Bean–Livingston barrier^31,32 for Abrikosov quantized vortices in type-II superconductors. The surface barrier has been evaluated quantitatively for the first time in the CSL system in CrNb₃S₆.²⁹ The importance of the surface barrier and related surface twist structure was also discussed in cubic chiral helimagnets in the study of magnetic Skyrmions.^33,34

Recently, CrTa₃S₆ and related TMDs, which form the same crystal structure as that of CrNb₃S₆, have also attracted attention because of the possible emergence of chiral helimagnetism.^35–37 CrTa₃S₆ was initially reported as a ferromagnetic compound.^38,39 However, reexamination has revealed that CrTa₃S₆ exhibits the CHM and CSL.^35–37 Now, it turns out that CrTa₃S₆ has a helimagnetic period of 22 nm and a large H_c of 1.2–1.6 T, while 48 nm and 0.2 T, respectively, in CrNb₃S₆. The discretization MR effect was observed in the cleaved CrTa₃S₆ samples,⁴⁰ as reported in CrNb₃S₆.⁴¹ However, there has been no experimental report on the surface barrier effect in CrTa₃S₆. It is not clear whether the surface barrier works among the chiral helimagnets hosting the CSL.

In this paper, we investigate the magnetic and transport properties of CrTa₃S₆ from the viewpoint of the surface barrier effect during the CSL formation. To scrutinize this unique property, MR and magnetization measurements were performed with micrometer-sized and bulk crystals with different dimensions with regard to the c-axis direction. The obtained results demonstrate the existence of a surface barrier in CrTa₃S₆ crystals. Specifically, characterizing the surface barrier is useful for identifying the CSL system in chiral magnetic materials.

Single crystals of CrTa₃S₆ were obtained by chemical vapor transport (CVT) technique in a temperature gradient using iodine I₂ as a transporting agent.^16,42 The polycrystalline powders, synthesized by the gas phase method with a mixture of Cr, Ta, and S in the molar ratio of x_nominal: 3: 6 (x_nominal is the nominal amount of Cr), were placed at one end of an evacuated silica tube and then heated in the electric tube furnace under the fixed temperature gradient from 1100 to 1000 °C for two weeks. The bulk crystals were grown at the other end of the silica tube. The grown crystals have the shape of a hexagonal plate of around 0.5 to 1.0 mm in diameter and 100 μm in thickness.

The magnetization of the obtained bulk crystals was examined using a SQUID magnetometer (Quantum Design MPMS3). Magnetoresistance (MR) measurements were performed with the micrometer-sized specimens of CrTa₃S₆ crystals, which were prepared from the bulk CrTa₃S₆ crystal by using a focused ion beam (FIB) system.²¹ The size of the specimens was evaluated using a scanning electron microscopy system. The MR data were collected by the standard four-terminal method using a physical property measurement system (Quantum Design PPMS). Note that H was applied in the direction parallel to the sample plane of bulk and microfabricated crystals, as described below, so as to reduce demagnetizing field and extrinsic metastability effects.⁴³ 

Three configurations of the platelet specimens with regard to the c-axis direction were prepared for the MR measurements. In the first case, a c-plane sample was fabricated, as shown in Fig. 3(a). The dimension of this platelet sample No. 1 is 12 × 4 × 0.1 μm³, where the shortest length corresponds to the direction of the c axis. This length limits the maximum number of the solitons in the CSL, and thus, the discretization effect was observed in the present CrTa₃S₆ crystal, as seen in CrNb₃S₆.²⁸ Another platelet samples No. 2 and No. 3 were fabricated with the c axis being along the longitudinal direction of the sample plane, as shown in Figs. 3(b) and 3(c). The size of sample No. 2 is 9 × 1 × 10 μm³, where the longest length corresponds to the c-axis direction. This sample shape is similar to that of CrNb₃S₆ used for the demonstration of the surface barrier.²⁹ Sample No. 3 has an elongated geometry with a dimension of 2 × 1 × 19 μm³ (//c axis).

For determining the optimum condition for the crystal growth, it should be noted that the magnetic property of the grown crystals is very sensitive to x_nominal of the powder precursor. For instance, in the case of CrNb₃S₆ crystal growth,⁴⁴ the amount of Cr measured directly in the single crystals was found to be smaller than x_nominal. The x_nominal was determined to be 1.11 so as to obtain the ideal crystals of CrNb₃S₆ without Cr defects.

The optimization of CrTa₃S₆ crystal growth was performed by using powder precursors with x_nominal from 1.00 to 1.50. Imprints of the CSL formation were successfully obtained in the crystals grown with x_nominal = 1.29, while ferromagnetic response appeared in other crystals with different x_nominal values.

Figure 2(a) shows a peak anomaly of the magnetization of the obtained CrTa₃S₆ crystal at around 150 K with a magnetic field H of 0.1 T applied in a direction perpendicular to the c axis. Here, the critical temperature of the helimagnetic phase transition T_c is defined at a peak top of the magnetization. The T_c values decrease with increasing the H strength, as shown in Fig. 2(b). Note that the T_c of 150 K is 10 K higher than those reported in the previous studies.^36,37 Such a variation of the T_c values indicates that the crystals used in this study may have a small amount of Cr defects, reminiscent of the dome-shaped profile of the relationship between T_c and x_nominal discussed in CrNb₃S₆.⁴⁴ 

To see the presence of chiral solitons in the obtained CrTa₃S₆ crystals via the discretization effect, the MR was examined in the c-plane thin sample with H applied in the direction perpendicular to the c axis. First, the MR full loop was taken by cycling H between zero and above the critical magnetic field (defined as H_sat in the experiments), where all the chiral solitons escape from the sample and magnetic moments are likely to be saturated. Then, the MR minor loops were collected by sweeping H below H_sat.

All the MR data are presented in the same panel in Fig. 3(a). It is clear that, in the H increase process of the MR full loop, the MR exhibits a gradual negative change associated with a reduction of the number of chiral solitons, whereas it shows a sudden jump at H_jump in the H decrease process. In addition, six discrete MR values appear in a series of MR minor loops. Taking into consideration the helical period of 22 nm in CrTa₃S₆,³⁵ the thickness of the present MR sample was calculated to be ∼110 nm, which is consistent with the value estimated from the device fabrication.

Another feature is that the ratio of H_jump/H_sat is found to be 0.362. Although this value is slightly smaller than the theoretical value²⁹ 4/π², such a large hysteresis may indicate the influence of the surface barrier against the penetration of chiral solitons into the present CrTa₃S₆ crystal.

The presence of a surface barrier was first demonstrated in the micrometer-sized platelet CrNb₃S₆ crystals with the c-axis orienting within the plane, in which the experimental data of H_jump/H_sat are in excellent agreement with the theoretical value 4/π². The slight discrepancy found in Fig. 3(a) may be ascribed to the difference in the sample geometry. In this respect, it is worth examining the MR behavior in terms of the surface barrier in the CrTa₃S₆ sample with dimensions similar to those of the CrNb₃S₆ sample used in the previous study.²⁹ 

Figure 3(b) shows that such a CrTa₃S₆ sample (No. 2) indeed shows the MR hysteresis behavior. Note that H is applied in the direction perpendicular to the c axis and within the plane so as to eliminate the demagnetization effect. To precisely determine the H_jump and H_sat values, the MR measurements were performed five times repeatedly.

In the H increase process, the MR change is well fitted by the CSL density, which is derived from the chiral sine-Gordon model and plays a role as an order parameter of the CSL formation. On the other hand, in the H decrease process, the MR shows a sudden change at H_jump. Note that the position of H_jump and the amplitude of the MR change at H_jump were reproducible in all five MR measurements. Moreover, the ratio H_jump/H_sat was averaged to be 0.407, which is quite consistent with the theoretical value (4/π² ∼ 0.405). These features are consistent with those observed in the CrNb₃S₆ sample.^19,27,29

Figure 3(c) shows the MR data in sample No. 3 with the elongated geometry. The data were collected three times. The sudden change of the MR occurs at H_jump reproducibly. The ratio H_jump/H_sat was averaged to be 0.392, which is slightly smaller than the theoretical value.

The present MR data in the three different CrTa₃S₆ samples strongly support that the surface barrier works against the penetration of chiral solitons. Specifically, the CSL formation was successfully demonstrated in the CrTa₃S₆ crystals via the MR measurements.

Interestingly, the hysteresis behavior is observed in the magnetization curves even in the bulk CrTa₃S₆ crystals, as shown in Fig. 4. A typical geometry of the bulk crystals is a platelet shape with the c plane being of about 100 μm in thickness, as presented in the optical photographs in Fig. 4. The magnetization curves at 5 K show the downward convex behavior in the H increase process, which is regarded as evidence of the CSL formation.^5–7,18,27 On the other hand, in the H decrease process, the magnetization decreases linearly until H reaches down to the first H_jump. Sharp drops of the magnetization appear at H_jump1 and H_jump2 in the crystal No. 4, while a drastic drop occurs at H_jump1 in the crystal No. 5. The positions of H_jump1 and H_jump2 were confirmed to be reproducible in the repeated measurements. This behavior is reminiscent of the MR behavior, as discussed in the micrometer-sized CrTa₃S₆ crystals in Fig. 3, and indicates that the surface barrier hampers the penetration of chiral solitons into the bulk crystal.

To see an indication of the surface barrier, the dependence of the magnetization curves on temperature was examined, as shown in Figs. 5(a) and 5(b). The hysteresis remains small at temperatures in the vicinity of T_c, while large hysteresis accompanying a sharp drop of the magnetization becomes evident with cooling temperature.

Figures 5(c) and 5(d) show the ratio of H_jump/H_sat as a function of temperature in the bulk CrTa₃S₆ crystals No. 4 and No. 5, respectively. It is found that the H_jump/H_sat values at 5 K reduce to 0.60 and 0.66 in crystals No. 4 and No. 5, respectively. These values are still larger than the theoretical value of 4/π² expected for the analytical model of the surface barrier.²⁹ Nevertheless, the behavior observed in the present CrTa₃S₆ crystals is totally different from the magnetization data previously reported in bulk crystals of CrNb₃S₆^45,46 and CrTa₃S₆,³⁶ where H_jump/H_sat was, respectively, kept to be 0.82–0.91 and 0.93 even at low temperatures.

The ratio H_jump/H_sat is summarized in terms of the sample dimensions. For the crystals with a c-axis length of around 10 μm, which contain hundreds of chiral solitons, H_jump/H_sat shows a good agreement with the theoretical value, as seen in Fig. 6(a). When the c-plane area is normalized by the c-axis length, the experimental values tend to deviate from the theoretical value in the samples with a large normalized area, as shown in Fig. 6(b). In this respect, an elongated geometry along the helical axis is likely to be favorable for the surface barrier.

The surface barrier is an intrinsic effect in the CSL system with a clean surface.²⁹ In this respect, the discrepancy from the theoretical value may be ascribed to imperfect edges of the hexagonal-shaped bulk crystals, as shown in Fig. 4, which is quite different from the ideal surface theoretically treated in the 1D chiral sine-Gordon model. Conversely, reproducible large hysteresis appears in the present bulk CrTa₃S₆ crystals. Specifically, the chiral solitons in CrTa₃S₆ exhibit less extrinsic metastability than those in CrNb₃S₆, indicating that CrTa₃S₆ is an ideal material hosing the robust CSL.

The quality of the crystal may influence the effectiveness of the surface barrier. Note that the T_c and H_sat values in the present CrTa₃S₆ crystals are larger than those reported in the literature. Indeed, T_c is 10 K higher than the reported values, as described above, and H_sat is 1.65 T at 5 K, which is 0.35 T larger than that in the previous work.³⁶ It was already found in CrNb₃S₆ that T_c and H_sat decrease when the amount of Cr intercalation deviates from the ideal unity⁴⁴ and is closely correlated with the strength of symmetric and antisymmetric exchange interactions. Importantly, the symmetric exchange interaction perpendicular to the c-axis J_⊥ is enlarged in the present CrTa₃S₆ crystals because T_c is correlated with the strength of J_⊥.⁴⁷  J_⊥ also works to enhance the phase coherence of the CSL,²⁸ which is in favor of working the surface barrier even in the bulk crystals. Clarifying the relationship between the strength of the exchange interactions and the surface barrier would be an interesting issue in the CSL system.

The surface quality of the c-plane may also be a key element governing the strength of the surface barrier. In the experiments, samples No. 1 to No. 3 were prepared by using FIB fabrication, and samples No. 4 and No. 5 have an as-grown wide c-plane surface. A clear difference was not found in terms of the surface quality, but rather, the controllability of the surface barrier was evident in the dependence on thickness and aspect ratio, as seen in Fig. 6. The comparison of the surface barrier strength using various types of the surface, such as a freshly cleaved surface^40,41 and a sharp crystal edge, would promote the understanding of the surface barrier in the CSL system.

One of the unknown characteristics in the present CrTa₃S₆ crystals is a reduction of the magnetic moment at H_sat to 1.6 μ_B/Cr, which is almost a half smaller than the value expected for the isolated Cr³⁺ ion. The magnetic moment increases monotonically above H_sat and reaches 3.0 μ_B around 10 T with a linear extrapolation. This behavior is different from that reported in the previous work.³⁶ The reason for such a discrepancy in CrTa₃S₆ remains to be clarified. Inferred from an electronic structure of CrNb₃S₆, itinerant electrons composed of Ta and S atoms may interact with localized electrons of Cr atoms in an electronic structure of CrTa₃S₆. However, the picture of localized electrons in the Cr atoms has not been fully validated yet. In this connection, it would be interesting to examine an electronic structure of CrTa₃S₆ using x-ray magnetic circular dichroism (XMCD) spectroscopy together with density functional theory (DFT) calculations to evaluate the degree of hybridization between Ta 5d and Cr 3d orbitals, as discussed in CrNb₃S₆.⁴⁸ 

In summary, we demonstrate the CSL formation by characterizing the surface barrier in CrTa₃S₆ crystals. This observation indicates that the surface barrier effect occurs among TMDs hosting the CSL and induces nontrivial physical responses, such as discretized MR, reflecting the topological nature of the CSL.

We thank Yusuke Kato for the fruitful discussion. This work was supported by the JSPS KAKENHI under Grant Nos. 19H05822, 19H05826, 23H01870, and 23H00091.

The authors have no conflicts to disclose.

K. Mizutani: Data curation (equal); Investigation (equal). J. Jiang: Data curation (equal); Investigation (equal). K. Monden: Data curation (equal); Investigation (equal). Y. Shimamoto: Data curation (equal); Investigation (equal). Y. Kousaka: Data curation (lead); Funding acquisition (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (equal). Y. Togawa: Data curation (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.