Intrinsically enhanced anomalous Hall conductivity and Hall angle in Sb-doped magnetic Weyl semimetal Co₃Sn₂S₂

Owing to the fundamental physics and great potential in technical applications, the anomalous Hall effect (AHE) has always attracted extensive research attention.^1–3 As for the physical origin of AHE, after long-term research, it is now generally believed that AHE can be caused by two different mechanisms: extrinsic and intrinsic mechanisms.^4–8 For the intrinsic mechanism, it is strongly related to the Berry curvature of the electronic band structure in momentum space.^6,7 In addition, the intrinsic contribution of AHE depends on the position of the Fermi level (E_F), which requires that the strong Berry curvature should be near the E_F.⁹ Therefore, to obtain a large AHE, it is also necessary for the E_F of the material to be located at a suitable position. In emerging magnetic topological materials, the topological band structures, such as Weyl points and gapped nodal rings near E_F, can produce topologically enhanced Berry curvature, which brings giant intrinsic AHE.^10–18 In turn, the intrinsic AHE is also regarded as an important experimental means to explore the topological characteristics of electronic bands of materials.

From the perspective of practical applications, large AHE is very essential.^19,20 Owing to the giant intrinsic AHE induced by topologically enhanced Berry curvature, the magnetic topological materials are expected to promote the potential application of AHE in emerging spintronics. Recently, owing to large Berry curvature induced by the gapped nodal rings connecting two Weyl points near E_F, a giant intrinsic anomalous Hall conductivity AHC (∼1130 S/cm) and anomalous Hall angle AHA (∼20%) were reported in magnetic Weyl semimetal Co₃Sn₂S₂, which is the first magnetic Weyl semimetal confirmed experimentally and theoretically.^{10,11,21–24} In addition, further enhanced AHC and AHA were obtained by electron and hole doping in Co₃Sn₂S₂, respectively.^25–27 Experimental and theoretical studies show that the disorder doping of heterogeneous atoms can not only regulate the position of E_F, but also lead to the splitting and broadening of bands by breaking of local lattice translational symmetry and changing the local environment of conducting electrons, which affects the intrinsic AHC of the system.^28,29 Meanwhile, heterogeneous atoms also enhance the scattering of conducting electrons, which may introduce extrinsic AHC to the system and further enhance the total AHC.¹ Therefore, in magnetic topological materials, appropriate disorder doping can realize the enhancement of AHE.

In this Research Update, by electron doping with Sb substituting for Sn in magnetic Weyl semimetal Co₃Sn₂S₂, the enhanced AHC and AHA were obtained. Combined with transport measurements, model separations and theoretical calculations, the enhanced AHE comes from the intrinsic contribution related to Berry curvature induced by the topological band structures.

A series of single crystal samples of Co₃Sn_2−xSb_xS₂ (x = 0, 0.1, 0.3, 0.5, 0.7) were prepared by slowly cooling method.³⁰ The initial primary materials with a molar ratio of Co:S:Sn:Sb = 3:2:2−x:x were mixed and placed in alumina crucibles, which were sealed in quartz tubes. The quartz tubes were slowly heated to 673 K over 6 h and held for an extra 6 h. Then, the quartz tubes were heated to 1323 K for 6 h and kept for 6 h to get the homogeneous melts. The melts were then cooled slowly to room temperature over 70 h to obtain a series of single crystal samples. The chemical compositions of the grown crystals were determined by using energy dispersive x-ray spectroscopy (EDS) (see supplementary material, Sec. I). The magnetic properties were measured by using superconducting quantum interference device (SQUID) magnetometer. The electrical transport characteristics were measured by using the physical property measurement system (PPMS). The energy band structures and AHC of Co₃Sn_2−xSb_xS₂ were calculated theoretically based on the virtual crystal approximation (VCA) with spin–orbit coupling (SOC) (see supplementary material, Sec. II).

The crystal structure of Co₃Sn₂S₂ with a space group of R-3m (No. 166) possesses quasi-2D Co–Sn kagome layers sandwiched between S and Sn atoms, in which Sn atoms have two different positions: in-plane Sn1 and interplane Sn2, as shown in Fig. 1(a). Different from Co position substitution,^25,26 there are two possibilities when Sb replaces Sn. One is to replace in-plane Sn1 and the other is to replace interplane Sn2. To explore the more favorable one in the energy occupation case of Sb substituting for Sn, the energies of the two cases were calculated theoretically based on the VCA with SOC. Figure 1(b) shows the energy difference (ΔE) of Sb substituting for in-plane Sn1 and interplane Sn2. With the increase in Sb content, the energy of in-plane substitution is lower than that of interplane substitution, which means that Sb preferentially replaces in-plane Sn1.

Figure 1(c) shows the temperature dependence of magnetization with field cooling for H//c at H = 0.1 kOe. A sharp magnetic transition can be observed for all the thermomagnetic curves near the Curie temperature (T_C). The T_C of the system were obtained from dM/dT curves, as shown in the inset of Fig. 1(c). Figure 1(d) shows the Sb content dependence of T_C. As with other elements doping in Co₃Sn₂S₂,^25–27 the T_C of the system decreases monotonically with the increase in Sb content, from 175 K of x = 0 to 130 K of x = 0.7.

To probe the evolution of AHE of the system with the Sb substituting for Sn, the AHE behavior of Co₃Sn_2−xSb_xS₂ was studied in detail. The temperature dependence of the longitudinal resistivity (ρ_xx) shows a sharp kink at T_C, which is related to the magnetic transition, as shown in Fig. 2(a). With the increase in Sb content, the residual electric resistivity at 2 K increases monotonically from 28.8 µΩ cm of x = 0 to 200 µΩ cm of x = 0.7, which is attributed to the scattering enhancement of conduction electrons caused by heterogeneous atoms. Figure 2(b) shows the Hall resistivity ρ_yx as a function of the magnetic field at 2 K for H//c and I//ab-plane. The anomalous Hall resistivity $ρyxA$ can be obtained with a vertical intercept at zero field. With the increase in Sb content, the $ρyxA$ at 2 K increases monotonically from 0.93 µΩ cm of x = 0 to 42.6 µΩ cm of x = 0.7, as shown in Figs. 2(b) and 2(d). According to formula $σxyA=ρyxA/[(ρyxA)2+ρxx2]$⁠, the anomalous Hall conductivity $σxyA$ can be obtained. Figure 2(c) shows Sb content dependence of $σxyA$ at 2 K. The $σxyA$ of x = 0 at 2 K reaches up to 1114 Ω⁻¹ cm⁻¹, which is attributed to the strong Berry curvature induced by the topological band structures near E_F.¹⁰ With the increase in Sb content, the $σxyA$ at 2 K shows an initial increase and a subsequent decrease, forming a maximal value of 1565 Ω⁻¹ cm⁻¹ at x = 0.3. Moreover, the anomalous Hall angle AHA can be characterized by the ratio of $ρyxA/ρxx$⁠. The AHA of the system at 2 K shows a monotonous increase from an initial 3.2% to 22% of x = 0.7, as shown in Fig. 2(c).

In addition, the temperature dependence of $ρyxA$⁠, $σxyA$⁠, and AHA of Co₃Sn_2−xSb_xS₂ are shown in Figs. 2(d)–2(f), respectively. The $ρyxA$ of all samples show an increase and then a decrease with increasing temperature, in which the maximum $ρyxA$ is 69.8 µΩ cm for x = 0.7 at 95 K, as shown in Fig. 2(d) and Table I. The $σxyA$ of Co₃Sn_2−xSb_xS₂ shows a maximum of 1565 Ω⁻¹ cm⁻¹ for x = 0.3 at 2 K and then decreases with increasing temperature, and disappears above T_C, as shown in Fig. 2(e) and Table I. The AHA of system strongly depends on the synergetic change of $ρyxA$ and ρ_xx of the system. Similar to the evolution behavior of $ρyxA$ with the temperature, the AHA of the system also shows an increase and then a decrease with increasing temperature, in which the maximum AHA is 25.6% for x = 0.7 at 76 K, as shown in Fig. 2(f) and Table I. It has been proved that the giant AHC was entirely derived from the intrinsic mechanism associated with the topological band structures in Co₃Sn₂S₂.¹⁰ Here, the anomalous Hall behaviors of Co₃Sn₂S₂ and Co₃Sn_2−xSb_xS₂ with the temperature change are very similar, which may indicate that the AHC in Co₃Sn_2−xSb_xS₂ is still dominated by the intrinsic mechanism.

As per current knowledge, it is believed that the total $σxyA$ can be expressed as the sum of three terms: $σxyA=σint⋅+σsk.+σsj.$⁠, where σ_int⋅, σ_sk⋅, and σ_sj⋅ denote the contributions of the intrinsic, skew, and side-jump mechanisms, respectively.¹ In addition, the intrinsic mechanism yields the resistivity relation $ρH∝ρxx2$⁠. With regard to the extrinsic mechanism, skew scattering and side-jump effects yield the relations ρ_H ∝ ρ_xx and $ρH∝ρxx2$⁠, respectively. To qualitatively analyze the mechanism of AHC enhancement in Co₃Sn_2−xSb_xS₂, the model $log⁡ρyxA=α⁡logρxx$ was adopted, as shown in Fig. 3(a). With an increase in Sb content, the value of α decreases gradually from 2 to 1.79, as shown in the inset of Fig. 3(a). Both the intrinsic mechanism and side-jump mechanism have similar resistivity relation, which is difficult to distinguish between the two mechanisms on experimental measurements. However, the intrinsic contribution can be calculated theoretically, whereas the side-jump contribution can be estimated by the formula (e²/ha)(ɛ_SO/E_F).³¹ Wang et al. have reported that the side-jump contribution in Co₃Sn₂S₂ is very small and negligible.¹¹ Therefore, we conclude that the intrinsic contribution of AHC is still dominant in Co₃Sn_2−xSb_xS₂, while the extrinsic contribution is slightly enhanced.

To further quantitatively analyze the respective proportions of intrinsic and extrinsic contributions of AHC in Co₃Sn_2−xSb_xS₂, the TYJ models,^32–34  $ρyxA=aρxx0+bρxx2$ and $σxyA=−aσxx0−1σxx2−b$⁠, were adopted, as shown in Figs. 3(b) and 3(c), where ρ_xx0 and σ_xx0 = 1/ρ_xx0 represent residual resistivity and residual conductivity, respectively. In the Tian-Ye-Jin (TYJ) scaling models, the first term on the right-hand side of the equation corresponds to the extrinsic contribution, and the second term corresponds to the intrinsic contribution. Based on the TYJ model, a linear relationship is expected between $ρyxA$ and $ρxx2$⁠. Figure 3(b) shows a very good linear fitting in the temperature range of 10–40 K. Meanwhile, Fig. 3(c) shows the $σxyA$ vs $σxx2$ curves in the temperature range of 10–40 K. It can be clearly seen that each curve can be fitted by a linear relationship. The intercept b of each line on the longitudinal axis represents the intrinsic AHC contribution. Based on the TYJ models, the extrinsic and intrinsic contributions of AHC at 2 K were quantitatively separated, as shown in Fig. 3(d). It can be observed that with the increase in Sb content, the intrinsic $σxyA(int.)$ shows a same trend as the total measured AHC. The $σxyA(int.)$ first increases and then decreases with the increase in Sb content, reaching a maximum value of 1498 Ω⁻¹ cm⁻¹ at x = 0.3. The intrinsic and total AHC values are almost consistent, with only a slight deviation at x = 0.7. These results confirm that the intrinsic contribution to AHC is dominant in Co₃Sn_2−xSb_xS₂, as analyzed in Fig. 3(a). In addition, the fitting of x = 0.1 shows a weak deviation of only ∼5 Ω⁻¹ cm⁻¹ at low temperatures, as shown in Fig. 3(c), which indicates that a slight extrinsic side-jump contribution may be introduced.³⁴ Therefore, the qualitative and quantitative separation results of AHC in Co₃Sn_2−xSb_xS₂ indicate that the enhanced AHC is dominated by the intrinsic mechanism related to the electronic band structure.

According to the existing research on doping of Co₃Sn₂S₂, it is found that both electron doping and hole doping can enhance the intrinsic AHC of Co₃Sn₂S₂, as shown in Fig. 4. Since the AHC of pristine samples prepared by different methods are different, the AHC of pristine samples shown in Fig. 4 are selected from the respective studies of Refs. 10, and 25–27. For the pristine composition, the giant AHC was entirely derived from the intrinsic mechanism associated with the strong Berry curvature, which is mainly dominated by the gapped nodal rings, which has been confirmed in Refs. 10, 25, and 27. In addition, the theoretically calculated energy-dependent AHC shows the position of E_F is already optimal for the integrated Berry curvature, implying that electron doping is not expected to further enhance the intrinsic AHC.¹⁰ The intrinsic AHC and energy band evolution as Sb content increases based on VCA calculations with SOC shows the E_F rises significantly with the increase in Sb content; on the contrary, the intrinsic AHC decreases monotonically (see supplementary material, Sec. II). However, the enhanced AHC dominated by the intrinsic mechanism is observed in the experiment.

According to the Kubo formula,⁹ the intrinsic AHC is sensitive to the electron occupation determined by the position of E_F, and the topological band character around the E_F. Energy band calculations in this work based on VCA (see supplementary material, Sec. II), which considers a symmetry-preserved primitive cell composed of “virtual” atoms, do not contain the modulation effect of the energy bands caused by disorder doping. It is inevitable that disorder doping will lead to the breaking of local lattice translational symmetry and change the local environment of conducting electrons in real materials, which affects the energy bands and intrinsic AHC of the system.^28,29 In Co₃Sn₂S₂, the E_F is located around the gapped nodal rings, implying that the gapped nodal rings exist above and below the E_F. A small amount of electron doping will cause the E_F to still remain in the gapped nodal rings. Meanwhile, disorder doping leads to the splitting and broadening of bands, which enhance the Berry curvature and intrinsic AHC. Therefore, in this work, the enhancement of intrinsic AHC is mainly attributed to the enhanced Berry curvature induced by bandgap narrowing caused by the disorder doping effect, similar to the case in CoNiSnS.²⁵ The effect of disorder doping is common in chemical doping systems. For Co₃Sn₂S₂, whether electron doping or hole doping, the intrinsic AHC of the system is significantly enhanced as long as the E_F is still located in the gapped nodal rings after doping, as shown in Fig. 4.

In summary, a giant enhanced AHE was observed in magnetic Weyl semimetal Co₃Sn_2−xSb_xS₂, with a maximum AHC of ∼1600 Ω⁻¹ cm⁻¹ at x = 0.3 and AHA of ∼26% at x = 0.7. Based on the qualitative and quantitative analysis of the intrinsic and extrinsic mechanism separation models, i.e., logarithm and TYJ models, the separation results indicate that the enhanced AHC comes from the intrinsic mechanism related to the Berry curvature of electronic band structures, while the contribution of the extrinsic mechanism is very weak. According to the existing research on doping of Co₃Sn₂S₂, a small amount of electron doping still makes the E_F around the gapped nodal rings. At the same time, the disorder doping leads to the splitting and broadening of bands, which enhance the Berry curvature and intrinsic AHC. Our work provides a new scenario for the acquisition of large AHC and AHA, and helps to promote the potential applications of emerging magnetic topological materials in topological spintronics.

See the supplementary material for the energy dispersive x-ray spectroscopy of samples, and theoretical calculations of bands and intrinsic AHC based on VCA considering SOC.

This work was supported by the Fundamental Science Center of the National Natural Science Foundation of China (Grant No. 52088101), the Beijing Natural Science Foundation (Grant No. Z190009), the National Natural Science Foundation of China (Grant Nos. 12104280, 11974394, and 12174426), the National Key R&D Program of China (Grant No. 2019YFA0704900), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (CAS) (Grant No. XDB33000000), the Key Research Program of CAS (Grant No. ZDRW-CN-2021-3), and the Basic Research Plan of Shanxi Province (Grant No. 20210302124160).

The authors have no conflicts to disclose.

J. Shen and S. Zhang contributed equally to this work.

Jianlei Shen: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). Shen Zhang: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). Tingting Liang: Investigation (equal). Jing Wang: Investigation (equal). Qingqi Zeng: Formal analysis (equal); Investigation (equal); Writing – review & editing (supporting). Yibo Wang: Data curation (equal); Investigation (equal); Software (supporting); Validation (equal). Hongxiang Wei: Funding acquisition (equal); Resources (equal); Writing – review & editing (equal). Enke Liu: Conceptualization (lead); Funding acquisition (lead); Investigation (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead). Xiaohong Xu: Conceptualization (equal); Investigation (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

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