An emergence of chiral helimagnetism or ferromagnetism governed by Cr intercalation in a dichalcogenide CrNb₃S₆

A chiral helimagnetic (CHM) material exhibits a helical magnetic structure with a single handedness. Its formation stems from a competition between symmetric Heisenberg and antisymmetric Dzyaloshinskii–Moriya (DM) exchange interactions. The antisymmetric nature of the DM interaction is strongly coupled to a chiral crystalline structure of chiral helimagnets.^1,2 Chiral helimagnets have attracted much attention because of the emergence of nontrivial chiral magnetic textures such as chiral magnetic soliton lattice (CSL)^3–7 and chiral magnetic vortices called magnetic Skyrmions.^8,9 The presence of such chiral magnetic order unique to chiral helimagnets has been observed via neutron scattering or electron microscopy in the recent decade.^10–12 

As envisioned by Dzyaloshinskii, the CSL is a superlattice of chiral spin twists with uniform periodicity, as shown in Fig. 1(a). Under an external magnetic field applied in the direction perpendicular to the helical axis (principal c-axis of the crystal), the harmonic chiral helimagnetic (CHM) order at zero magnetic field transforms into the nonlinear CSL. Importantly, the CSL period can be controlled continuously by the strength of the magnetic field. Another feature is that the CSL exhibits robust phase coherence at macroscopic length scale.¹² Because of these intriguing characteristics, the CSL shows a variety of interesting physical responses, as exemplified by giant magnetoresistance (MR), due to the proliferation of magnetic solitons,¹³ discretization effect of MR,¹⁴ presence of a surface barrier for soliton penetration,¹⁵ and collective resonant dynamics up to a frequency of sub-terahertz.¹⁶ Such coherent, topological, collective nature of the CSL could be utilized for spintronic device applications.^17,18

The number of chiral helimagnets hosting the CSL is still limited. One reason comes from the difficulty in synthesizing helimagnetic materials with a suitable chiral crystalline structure. Another reason could be found in how to detect the CSL. The CHM period at zero magnetic field L(0) is determined by the ratio of the Heisenberg exchange interaction J_‖ and DM interaction D along the helical axis. L(0) is typically about tens of nanometers or longer, which corresponds to some tenth nanometers inverse or less in a wave-vector k space. Unfortunately, the Q resolution of neutron scattering experiments using a thermal neutron source is not high enough to separate fundamental Bragg and magnetic satellite peaks. Apparently, the CSL has a longer period than the CHM and thus the CSL detection using neutron scattering technique becomes much harder. In this connection, there has been a possibility that some compounds of CHM order might be misinterpreted as those of ferromagnetic order in previous studies.

Transition-metal dichalcogenides are one of the appropriate materials that could host the CSL. More precisely, an intercalation of 3d transition metal ion T into the mother compound MX₂ (M: Nb or Ta, X: S or Se), provides a rich variety of magnetic properties. In the case of disulfide, it is known that, depending on the quantity of intercalated T ions, two different compounds, TM₄S₈ and TM₃S₆, are synthesized. For clarifying the connection of mother materials, these compounds are sometimes described as T_1/4MS₂ and T_1/3MS₂, respectively. The former has an achiral crystalline structure, while the latter has a chiral crystalline structure with a space group P6₃22. In the literature before the 1980s, it was reported that TM₃S₆ showed a variety of magnetic behaviors; paramagnetism for TiNb₃S₆, VNb₃S₆, TiTa₃S₆,and VTa₃S₆, antiferromagnetism for FeNb₃S₆, CoNb₃S₆, NiNb₃S₆, CoTa₃S₆,and NiTa₃S₆, and ferromagnetism for CrNb₃S₆, MnNb₃S₆, CrTa₃S₆, MnTa₃S₆,and FeTa₃S₆.^19–24 In the present study, we mainly focus on CrNb₃S₆, which is now known as one of the representative chiral helimagnets for hosting the CSL. Interestingly, other chiral physical properties unique to chiral materials such as electrical magneto-chiral effect²⁵ and chiral-induced spin selectivity (CISS)²⁶ are discussed in CrNb₃S₆.

In earlier studies around the 1970s, CrNb₃S₆ was reported to show ferromagnetic order.^21–24 In the 1980s, Miyadai and Moriya investigated a magnetization process in terms of the formation of the magnetic helix²⁷ They pointed out that it could be interpreted as a formation of the CHM order in CrNb₃S₆ and directly observed a diffractive satellite peak that corresponds to the CHM structure with 48 nm by means of small angle neutron scattering.²⁸ However, at that time, the field dependence of magnetization was regarded as a discontinuous phase transition between the CHM phase and the forced ferromagnetic one, without taking any intermediate state. Precisely speaking, the magnetization curve changes continuously toward a critical magnetic field H_c, as shown in Fig. 2(a). Such a continuous change of the magnetization should be interpreted as a scenario of CSL formation, which has been properly discussed in the 2000s.^7,29,30 Eventually, the CSL was directly observed using Lorentz microscopy.¹² 

Direct observation of the CSL revealed unexpected characteristics: the CSL is robust and stable with macroscopic phase coherence. To realize this property, one-dimensional chains of the chiral twists are strongly correlated with each other, as shown in Fig. 1(b). The pitch of the chiral twists is governed by the ratio of J_‖ and D along the c-axis, while the coherency arises from an in-plane (ab-plane) ferromagnetic exchange interaction J_⊥. In CrNb₃S₆, the strength of J_⊥ is as large as the magnetic transition temperature T_c and much larger than J_‖.³¹ Therefore, the value of T_c could be a good indicator of the phase coherency of the CSL.

Although TM₃S₆ has been investigated for decades, it is still difficult to control the quantity of intercalated T. In the case of CrNb₃S₆, the physical properties, including T_c, take different values depending on the samples, as shown in Table I. For example, Parkin reported the ferromagnetic order within the basal plane based on the magnetization of CrNb₃S₆ with T_c = 120 K.²³ Miyadai found the chiral helimagnetism of CrNb₃S₆ with T_c = 127 K.²⁸ 

When the CrNb₃S₆ sample shows chiral helimagnetism, the magnetization curve shows a sharp peak as a function of temperature, as shown in Fig. 2(b). However, a monotonous change of the magnetization appears alternatively in the samples of ferromagnetism as reported in the literature.^23,32–35 It would be important to examine the sample quality in terms of the quantity of the Cr ions. Some studies evaluated the quantity of defects in the Cr sites by means of quantitative analysis using single crystal x-ray diffraction^32,34 and electron probe microanalysis (EPMA).^33,35 However, there has been no systematic investigation about a correlation between the defects and the magnetic behavior.

The importance of controlling the quantity of intercalated T can be seen in another example of FeTa₃S₆ and MnNb₃S₆. It is known that they show ferromagnetic order at around 40 K.²³ However, in some papers,^34,38 a coexistence of additional magnetic order was reported to occur at 100 and 160 K in Fe and Mn disulfide compounds, respectively. These values are likely to correspond to the transition temperatures of MnNb₄S₈³⁹ and FeTa₄S₈.⁴⁰ Studies using crystals with a smaller T_c and/or those with an impurity phase of TM₄S₈ may bring confusing results.

In this Research Update, we examine how the quantity of the Cr intercalation influences the magnetic property of single crystals of CrNb₃S₆ in order to establish a method for growing the intercalation-controlled single crystals. Indeed, we found that the amount of Cr intercalation was an essential factor in controlling the quality of the crystals. In the experiments, we grew single crystals of Cr_xNb₃S₆ with x = 0.89 ∼ 1.03 and investigated a correlation between the amount of Cr intercalation and the magnetic property. The magnetization anomaly, which appears in the temperature dependence as evidence of the CSL formation, was observed only in the narrow region of x = 0.98 ∼ 1.03. These results mean that even a small number of defects in T sites in TM₃S₆ will affect the quality of single crystals significantly.

The crystal structures of the intercalated system of 2H-MS₂ depend on the intercalation amount of a 3d transition metal,^19,20,22 as shown in Fig. 3. CrNb₄S₈ forms a centrosymmetric crystal structure with a space group of P6₃/mmc. It has one Cr atom per four units of NbS₂ and the unit cell along the a-axis is twice that of 2H-NbS₂, as seen in Fig. 3(c). The location of the Cr atoms differs in CrNb₃S₆, which forms a chiral crystal structure with a space group of P6₃22. It has one Cr atom per three units of NbS₂, while the unit cell along the a-axis is $3$ times longer than that of 2H–NbS₂, as shown in Fig. 3(d).

In CrNb₃S₆, the Cr ion is normally intercalated at 2c (1/3, 2/3, 1/4) Wyckoff position. However, there are reports^19,22 that the Cr ions take the positions of not only 2c, but also 2b (0, 0, 0) and 2d (2/3, 1/3, 1/4). In the case of CrNb₃Se₆, which has the same structure as that of CrNb₃S₆, only 5% of Cr occupy Wyckoff 2c position, but 90% occupy Wyckoff 2b (0, 0, 0) position.⁴¹ Resultantly, the magnetic structure is not a CHM, but a ferromagnetic one. Similar discussion may hold for CrNb₃S₆ and MnNb₃S₆ with smaller T_c. In particular, when CrNb₃S₆ and MnNb₃S₆ samples exhibit ferromagnetic behavior in the magnetization without any imprint of the CSL formation, 3d transition metal ions do not fully occupy the 2c position.^32,34

The polycrystalline samples of Cr_xNb₃S₆, used for single crystal growth afterward, were synthesized by the gas phase method. Powders of Cr, Nb, and S, mixed in a molar ratio of x_nominal:3:6, were sealed in an evacuated silica tube. The powder in the silica tube was placed in an electric tube furnace at 1000 °C for a week. Several polycrystalline specimens with x_nominal from 1.00 to 1.32 were prepared. The polycrystals were evaluated by powder x-ray diffraction experiments with Cu Kα radiation (Rigaku MultiFlex) and magnetization measurements using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design MPMS-5).

The single crystals were obtained by chemical vapor transport (CVT) technique in a temperature gradient using iodine I₂ as the transporting agent. The polycrystalline sample, synthesized with a mixture of Cr, Nb, and S in the molar ratio of x_nominal:3:6, was placed at one end of an evacuated silica tube and then heated in the electric tube furnace with a temperature gradient for two weeks. The single crystals were grown at another end of the silica tube located at the lower temperature. As described in the following, the growth condition affects the amount of Cr intercalation and thus the magnetic behavior. The molar ratio of Cr, Nb, and S in a single crystalline sample was determined by quantitative analysis using electron probe microanalyzer (JEOL JXA-8200S). The ZAF correction method (Z: atomic number factor, A: absorption factor, F: characteristic fluorescence correction) using standard specimens of each element was carefully applied to the raw data. To examine the locational variation of the molar ratio in the specimen, EPMA data were obtained in ten different positions in each specimen. Then, the averaged values of the molar ratio were derived for all the specimens together with the error bar including information of the standard deviation between the positions. The amount of Cr intercalation x_EPMA in a crystal was determined by the molar ratio between Cr and Nb. As for the amount of S, it took almost the ideal molar ratio of Nb and S. Thus, we discuss the defects on the Cr sites in this study. The magnetic property was evaluated by the magnetization in the presence of small magnetic field orienting in a direction perpendicular to the c-axis using SQUID magnetometers (Quantum Design MPMS-5 and MPMS3).

The powder x-ray diffraction experiments revealed that the polycrystals with x_nominal from 1.00 to 1.32 had no impurity phases since all the diffraction peaks in each sample were indexed to the CrNb₃S₆ structure. The magnetic property was evaluated by the temperature dependence of magnetization. As shown in Fig. 4, all the specimens exhibit ferromagnetic response. The value of T_c of the polycrystal with x_nominal = 1.00 is 130 K. T_c decreases with increasing x_nominal, and eventually drops down to 30 K when x_nominal becomes 1.32. However, super-high-resolution powder neutron diffraction studies with the x_nominal = 1.00 specimen exhibit magnetic satellite peaks as evidence of the CHM order.⁴² Thus, the magnetization measurements with polycrystalline specimens are not useful for identifying the CSL formation in polycrystals and finding the optimum x_nominal for single crystal growth.

As for single crystal growth, single crystals with a hexagonal plate shape were obtained in the growth conditions used in the experiments. The actual amount of Cr intercalation in each specimen was evaluated by EPMA. It was found that the Cr intercalation x_EPMA of the single crystal strongly depends on x_nominal of the polycrystals as well as on the growth conditions, such as the temperature gradient in the furnaces. Here, to find an appropriate growth condition for obtaining the single crystals of CrNb₃S₆, we show a growth strategy by optimizing x_nominal under a fixed experimental condition of crystal growth. Figure 5 shows the amount of Cr intercalation x_EPMA of single crystalline Cr_xNb₃S₆ grown from polycrystals with x_nominal under the fixed temperature gradient 1100 to 1000 °C. Note that the distance between the highest and lowest temperature locations and the density of I₂ inside the silica tube were fixed to be 16 cm long and 5 mg/cm³, respectively.

When the single crystals were grown from polycrystalline samples with x_nominal = 1, the x_EPMA of the single crystals was 0.933(2), indicating the presence of defects in the Cr sites. x_EPMA of the crystals increases in proportion to the x_nominal of polycrystals up to x_nominal = 1.14 and is nearly saturated at 1.019(3) above it. Judging from the actual composition determined by EPMA analysis, single crystal growth should be performed using polycrystals with x_nominal ∼ 1.1 so as to reduce the generation of defects on the Cr sites. Let us note that the linear correlation between x_nominal and x_EPMA makes it easy to optimize the crystal growth condition. On the other hand, in the case of optimizing the temperature setting in the furnace by using the polycrystals with a fixed value of x_nominal, it is difficult to find a good combination of many experimental parameters such as temperature values and gradients at high and low temperature locations and the distance between them. In the present synthesis condition, the obtained crystals exhibit a uniform distribution of Cr over the specimen because the error bars of the Cr amount in the EPMA analyses were quite low as shown below. Typical values were 100 times smaller than those reported by Mao et al.³⁵ shown in Table I. In this respect, the values of error bars are likely to reflect the quality of crystals.

Figure 6(a) shows the temperature dependence of magnetization in single crystals with different x_EPMA. Here, a small magnetic field is applied in a direction perpendicular to the c-axis, and the amplitude of the magnetization in each specimen was normalized by the magnetization at 80 K. The single crystals show CSL or ferromagnetic behavior depending on the values of x_EPMA. The presence of chiral helimagnetism could be demonstrated by identifying a sharp peak anomaly of the magnetization as a function of temperature,^7,30 as shown in Fig. 2. T_c of the chiral helimagnet crystal is defined at a peak top of the magnetization, while T_c of the ferromagnetic order is determined by an extrapolation of the steepest portion of the magnetization toward zero magnetization. The CSL behavior is observed in the crystals with x_EPMA from 0.98 to 1.03. The single crystals with x_EPMA = 0.982(2) show the CSL response with the highest T_c of 133 K. On the other hand, when the value of x_EPMA is a few percent larger or smaller than 1, the ferromagnetic behavior appears with low values of T_c. For example, the single crystal with x_EPMA = 0.933(2), obtained from the polycrystal with x_nominal = 1, exhibits the ferromagnetic behavior with T_C = 102 K. This value is much lower than the values of T_c for chiral helimagnets. Note that, as shown in Fig. 6(b), the amplitude of the magnetization at 80 K in x_EPMA = 0.933(2) is 24 times larger than that in x_EPMA = 0.982(2). Such a large difference in the magnetization intensity can be regarded as another criterion in distinguishing the ferromagnetic response from the CSL one. For the absolute distinguishing of the CSL and ferromagnetic phases, further diffractive studies using x ray, electron, and neutrons are required, as described below.

To summarize the experimental results, Fig. 7 shows a correlation between the Cr intercalation x_EPMA and magnetic transition temperature T_c in single crystalline $CrxEPMA$Nb₃S₆. Note that most of the present data, denoted by circles and squares, were taken from crystals grown in the same experimental condition with the fixed temperature gradient. The data at x = 0.887(2), 0.915(2), 0.993(2), and 0.995(6) were obtained from crystals in different experimental condition. CSL behavior is observed in a narrow region of x_EPMA from 0.98 to 1.03. T_c takes the highest value at 0.98 and 0.99. When x_EPMA is larger or smaller than 0.98 ∼ 0.99, T_c is likely to decrease monotonously.

In growing single crystals of CrNb₃S₆, the nominal amount of Cr, x_nominal, is one of the most important and controllable parameters. As demonstrated in the present study, the Cr intercalation x of single crystals is smaller than the x_nominal in polycrystals used for crystal growth. Resultantly, Cr-defected crystals with x < 1 are synthesized without an optimization of the growth condition. In the present study, we found that the Cr intercalation x of single crystals is proportional to x_nominal up to some amount of x_nominal (up to x ∼ 1.02 in our experimental condition). Thus, the optimization of x_nominal enables us to obtain CrNb₃S₆ single crystals without defects on the Cr sites.

Note that the magnetic properties are influenced by the Cr intercalation x of single crystals as well. As shown in Fig. 7, the highest T_c of 133 K for chiral helimagnetism is obtained for x = 0.98 and 0.99. A deviation of the x values from the optimum gradually decreases the value of T_c. When x is three or four percent smaller than the optimized x, the single crystals do not show chiral helimagnetism but exhibit ferromagnetism. When increasing the x value, T_c for chiral helimagnetism decreases and ferromagnetism appears in some crystals. This behavior reminds us of a dome-shaped distribution of the T_c of high-T_c oxide superconductors against a carrier doping. The value of T_c is proportional to the strength of J_⊥, which stabilizes the coherency of the chiral magnetic order in CrNb₃S₆.³¹ In this sense, the amplitude of J_⊥ and the CSL robustness can be controlled by the Cr intercalation x.

CSL behavior was observed in crystals with excessive intercalation of Cr, by 3%. The magnetic properties in an over-intercalated regime would be worth investigating in terms of the robustness of the CSL formation. In the present experimental condition, the Cr intercalation x was nearly saturated at around 1.02, as shown in Fig. 5. However, the polycrystalline powders in Fig. 4 show a gradual decrease of T_c for the x_nominal values from 1.00 to 1.32. Thus, there is a possibility of obtaining single crystals with such over-intercalated Cr ions. We stress that small angle neutron scattering experiments will directly probe the ferromagnetic or chiral helimagnetic structure of such single crystals with a variety of intercalation amounts.

Now we know that the single crystals of CrNb₃S₆ with smaller values of T_c have a deficient or excessive intercalation of Cr ions. Namely, we can estimate the amount of Cr intercalation x simply by measuring the magnetization. In this respect, magnetization gives us important information for evaluating the quality of the obtained single crystals. To find the growth condition for obtaining high-quality single crystals, the nominal amount of Cr should be optimally parameterized with the fixed experimental condition related to the furnace setting. Similar ideas can be applied for growing single crystals of other kinds of TM₃S₆ compounds.

As listed in Table I, some studies have discussed the deficiency of ions in CrNb₃S₆. Mao et al. pointed out the influence of defects of sulfur (S) elements against a drastic reduction of T_c.³⁵ However, as shown in Table I, in their analysis using one particular single crystal that showed ferromagnetic response at 56 K, they obtained x = 0.99(12) within one sigma error (thus x ranges from 0.87 to 1.11). Based on the results in Fig. 7, T_c can be less than 60 K for the crystal with x = 0.87. Thus, such a low T_c can be ascribed to the defects of the Cr ions even within the experimental error bars. To avoid misleading conclusions, systematic examination is required using several crystals with different number of defects of the target elements.

Han et al. reported that their single crystal of CrNb₃S₆ had the defects of Cr of 12.5%.³³ The temperature dependence of magnetization showed ferromagnetic response at 125 K, which is inconsistent with the present results in Fig. 7. Since there was no description on the experimental error of the defect evaluation, the data might be taken at only one location in the crystal. Interestingly, their crystal exhibited an indication of CSL formation with much lower H_c in the magnetic field dependence of the magnetization.

The correlation between the Cr intercalation x and T_c, shown in Fig. 7, is in good agreement with the results obtained by single crystal x-ray diffraction.^32,34 Dyadkin et al. reported that a ferromagnetic single crystal of Cr_xNb₃S₆ with T_c = 92 K had defects in the Cr sites. For this crystal, the crystal structure analysis gave 0.942 for the x value.³² As shown in Fig. 7, our analysis reveals that crystals with T_c = 92 K have about 0.92 for x, which roughly reproduces the results by Dyadkin et al.

Hall et al. reported x = 1.00(3) for the Cr_xNb₃S₆ single crystal with T_c of ∼118 K.³⁴ The crystal structure analysis told that 95.2(2)% of Cr occupied the Wyckoff 2c position, while the remaining Cr occupied the Wyckoff 2b position. Although the total amount of the Cr intercalation took an ideal value, the temperature dependence of magnetization showed ferromagnetism rather than CSL formation. Interestingly, the field dependence of magnetization was not simply regarded to be of ferromagnetic origin, although H_c was eight times smaller than the typical value shown in Fig. 2(a).

Similar examples of single crystals can be found in Fig. 7. The crystals with x = 1.019(3) and 1.033(1) exhibit CSL behavior, while the one with x = 1.020(3) shows a ferromagnetic response. The latter is quite similar to the crystal examined by Hall et al. Even though the crystals mentioned above have almost the same amount of the Cr intercalation x, the ferromagnetic behavior appears in a particular piece of the single crystal, which may be caused by a delicate balance of the locations of the Cr ions in the single crystals. Namely, the deficiency of the Cr ions on the 2c site leads to the additional intercalation on the 2b site. Such disorders on the Cr sites may decrease the amplitude of the DM interaction significantly, which could be monitored by the value of H_c since it is proportional to D²/J_‖. In this connection, it would be interesting to investigate the relationship between the intercalation ratio between 2b and 2c sites and H_c in addition to the evaluation of T_c.

In order to discuss the dome-shaped behavior of T_c in detail, additional experiments will be necessary. Similar behavior is exhibited by other transition metal sulfides, such as ferromagnetic Fe-intercalated tantalum sulfide, and is discussed from the viewpoint of oscillatory characteristics of the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction.⁴⁵ The Cr–Cr distance determined by single crystal x-ray diffraction measurements and the carrier concentration by Hall measurements will be helpful in understanding the mechanism behind the dome-shaped behavior. In this Research Update, the ferromagnetic or chiral helimagnetic order is estimated by the existence of the sharp anomaly around T_c. Such magnetic structures can be distinguished by the existence of magnetic satellite peaks in neutron diffraction experiments. Regarding T_c, a precise value can be determined by Arrott plot analysis of the field dependence of magnetization³⁶ as well as by neutron diffraction and heat capacity measurements. In the present study, most crystals grown with x > 1 reveal CSL behavior, while a particular crystal with x = 1.020(3) shows a ferromagnetic response. This exception may be caused by disorders on the Cr sites, the presence of which can be probed by single crystal x-ray diffraction measurements, as observed in ferromagnetic CrNb₃Se₆.⁴¹ 

We show how to find the appropriate growth condition of a chiral helimagnet Cr_xNb₃S₆ by optimizing the nominal amount of Cr in crystal growth. As shown in Fig. 5, the actual Cr intercalation x in single crystals is smaller than the nominal amount of Cr x_nominal of polycrystals used in crystal growth. x dramatically affects the magnetic properties and has a correlation with T_c. Therefore, the amount of Cr intercalation is able to be estimated only by the magnetization measurements. As shown in Fig. 7, the single crystals with x being 0.98 and 0.99 show CSL behavior with the highest T_c of 133 K. Thus, the intercalation-controlled single crystals were successfully obtained using the present approach. Only several percent of defects of the Cr ions prevent the CSL formation and alternatively stabilize the ferromagnetic order probably because of the reduction of the strength of the DM interaction. In this sense, it is very important to check the amount of Cr intercalation x in crystals.

TM₃S₆ compounds will attract more attention because of studies on the CSL formation made recently in CrTa₃S₆.^42–44 The strategy for the crystal growth demonstrated for CrNb₃S₆ in the present study can be applied to synthesizing other TM₃S₆ compounds. The synthesis strategy can be categorized into two cases. The first case is that the compound has a correlation between the amount of T intercalation and T_c, which is the same situation as that for CrNb₃S₆. Even when the intercalation amount is smaller or larger than the ideal value, the crystal keeps the TM₃S₆ phase. However, deficient or excessive intercalation affects the physical properties. For example, such a tendency can be found in the T_c distribution as a function of the intercalation amount for CrNb₃S₆, as shown in Fig. 7. Fortunately, in this case, the growth condition can be optimized by choosing the nominal composition so as to give the highest T_c.

The second case is that the mixture of TM₃S₆ and TM₄S₈ is synthesized in the crystal. When the intercalation amount is smaller than the ideal, the deficient amount of T does not decrease T_c, but stabilizes some amount of TM₄S₈ phase. For example, MnNb₃S₆ and MnNb₄S₈ have different T_c at 40 and 100 K,^23,30,39 respectively. The magnetization of the MnNb₃S₆ crystal frequently shows two-step transitions at both temperatures.³⁴ In such cases, the volume fraction between MnNb₃S₆ and MnNb₄S₈ can be determined by the temperature dependence of magnetization. Then, the actual amount of Mn ions can be estimated by the volume fraction as well. The remaining task is the optimization of the growth condition so as to eliminate the TM₄S₈ phase in the crystal.

The next challenge in synthesizing TM₃S₆ compounds will be the development of the method of enantiopure crystal growth. Non-reciprocal magneto-chiral response and CISS effect are recently observed in CrNb₃S₆.^25,26 These findings shed light on the importance of synthesizing mono-chiral crystals because these phenomena are coupled with the crystallographic chirality of the compounds. However, it is still difficult to stabilize only left- or right-handed chiral crystal structure in synthesizing inorganic chiral materials because it is inevitable to prevent the formation of racemic-twinned crystals, having left- and right-handed crystalline domains within a specimen. Indeed, the CrNb₃S₆ crystals obtained by the CVT method form the racemic-twinned grains.²⁶ In some inorganic compounds such as NaClO₃, TSi, CsCuCl₃,and YbNi₃Al₉, recent progress in growth techniques makes it possible to obtain enantiopure crystals with the desired handedness.^46–51 However, such methods are not applicable in the synthesis of TM₃S₆,and thus a new method will be required to obtain enantiopure crystals. These challenges will be a key factor for observing new physical properties coupled with the handedness of the crystals.

This work was supported by JSPS KAKENHI Grant Nos. 15H05885, 19KK0070, JP19H05822, JP19H05826, and 20H02642.

The authors have no conflicts to disclose.

Y. Kousaka: Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). T. Ogura: Data curation (equal); Investigation (equal); Methodology (supporting). J. Jiang: Data curation (supporting); Investigation (supporting). K. Mizutani: Data curation (supporting); Investigation (supporting). S. Iwasaki: Data curation (supporting); Investigation (supporting). J. Akimitsu: Data curation (supporting). Y. Togawa: Data curation (supporting); Funding acquisition (supporting); Writing – original draft (supporting); Writing – review & editing (equal).

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