Discovery of Weyl semimetal has motivated the interest for investigating topological electronic structure and related transport properties of various semimetal materials.^[2,3] Of particular interest is ferromagnetic (FM) semimetal in which the breaking of time reversal symmetry can generate large Berry-curvature-bearing Weyl nodes and strong intrinsic anomalous Hall effect (AHE) thereupon.^[4] Very recently, the existence of Weyl quasi-particles and topological nontrivial surface states has been observed in FM semimetals Co₃Sn₂S₂ and Co₂MnGa in spectroscopic experiments.^[5–7] These magnetic Weyl semimetals manifest a large, intrinsic AHE which is also believed to be a signature of the existence of topological Weyl nodes.^[5,8] Understanding the mechanism of AHE and its correlation to the topological nontrivial band structure is crucial for the material design aiming for future electronic devices working based on the anomalous effect.

Heusler compounds represent a large family of materials crystallizing in closely related structures and many of them have been considered as the candidates of magnetic Weyl semimetal.^[9,10] These candidates include half-Heusler GdPtBi and full-Heusler Co₂MnGa,^[5,11] both of which exhibit large AHE. The Co-based full Heusler compounds have attracted much attention because they own some advantages such as strong magnetism and tunability which may promise their future applications in electronic devices. Their magnetic ground state is controlled by the interatomic distance and follows the well-known Slater–Pauling rule,^12-14 leading to a relatively high Curie temperature (T_C). Moreover, changing their crystalline symmetry can tune the AHE from null to very large while keeping the magnetization the same.^[15] When the number of the valence electrons equals 26, the Co-based full Heusler compounds manifest a particular half-metallic Weyl semimetal state, i.e., the spin-up electron bands form a topological nodal line and Weyl nodes while the spin-down electron bands possess a fully opened gap.^[15–17] These 26 e compounds include ZrCo₂Sn, TiCo₂Sn, and VCo₂Ga, whose electronic structures have been well calculated.^[16–18]

In this paper, we exam the magnetic and electric transport properties of single-crystalline vanadium-substituted ZrCo_1.6Sn. Previous studies on the poly-crystalline parents compounds, ZrCo₂Sn and VCo₂Sn, reported their FM ground state.^[19–22] Recent electronic structure calculations unveiled a pair of Weyl nodes 0.6 eV above the Fermi level for ZrCo₂Sn.^[16] Moreover, the calculation also suggested that 10% of V-substitution on TiCo₂Sn and 30% of Nb-substitution on ZrCo₂Sn will shift the Fermi level to the Weyl nodes energy while keeping the main band topology unchanged.^[16] However the AHE of the pristine and Nb-substituted ZrCo₂Sn has not been analyzed thoroughly, although the single crystal growth, magnetic and electric transport properties have been reported.^[23–25]

Realizing the difficulty for growing single-crystalline TiCo₂Sn,^[26,27] we successfully grew single crystals of Zr_1−xV_xCo_1.6Sn with the L2₁ Heusler phase structure from molten tin flux. We found that their Curie temperature and magnetic moment are enhanced by V-substitution while the resistivity changes from bad-metal-like to semiconductor-like at x = 0.35. Their AHE, which can be well fitted by Tian–Ye–Jin (TYJ) scaling,^[1] is also enhanced by V-substitution. Moreover, we find an apparent electron–electron-interaction (EEI) induced quantum correction in longitudinal resistivity at low temperature. On the other hand, the AHE, which is dominated by the intrinsic term, is not corrected.

Single crystals of Zr_1−xV_xCo_1.6Sn were grown in tin-flux with zirconium powder (99.8 %, Alfa Aesar), vanadium dendritic crystals (99.9 %, GRINM), cobalt powder (99.8 %, Alfa Aesar), and tin lumps (99.99 %, GRINM) as starting materials with the molar Zr : V : Co : Sn ratio of (1−x) : x : 2 : 19. The mixture was placed in an alumina crucible and then sealed in a fused silica ampoule with 0.02 MPa argon gas. The ampoule was slowly heated to 1100 °C in furnace and kept at this temperature for 12 h, followed by a slow cooling to 1000 °C at a rate of 2 °C/h. At this temperature, large single crystals were separated from molten tin by centrifugal decanting. Every single crystal for electrical and magnetic measurement was polished. The thickness is about 0.1 mm or less and no residual tin was visible. With increasing V substitution, the crystals’ shape evolves from a truncated octahedron to a well-faceted cube displaying large (100) faces [insets in Fig. 1(b)]. At the same time, the crystal size scales down with increasing x. For x > 0.35, the crystals are not large enough to be handled for electrical transport measurement and therefore we are limited for x ≤ 0.35 in this study.

Fig. 1.

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	Fig. 1. (a) Power XRD results for all studied Zr1−xVxCo1.6Sn samples (asterisks mark the impurity peaks of tin flux). (b) The refined lattice constant a (red squares) and the measured vanadium concentration x by EDS against the starting x. Inset: Photos of two representative samples of x = 0 and 0.2.

Power x-ray diffraction (Cu-Kα) measurements were conducted at room temperature with a Rigaku MiniFlex 600 diffractometer to confirm the crystal structure. The compositions of Zr, V, and Co for the series were determined by energy dispersive x-ray spectroscopy (EDS) in an X-Max 80. DC magnetization was measured by a Quantum Design superconducting quantum interference device magnetometer in a variety of applied fields (μ_0H ≤ 6 T) and temperatures (T = 2–300 K). The samples were mounted and fixed by GE varnish on a brass stick, with fields perpendicular to the face of (100). AC electrical resistivity measurements were performed in a Quantum Design physical property measurement system (T = 2–300 K, −9 T≤ μ_0H ≤ 9 T). A standard four-wire method was adopted using silver paste as the contacts. The resistivity and Hall effect measurements were conducted with fields normal to and currents within the (100) plane.

Figure 1(a) shows the powder x-ray diffraction (XRD) patterns which match a standard Heusler L2₁ phase () for all six samples x = 0, 0.1, 0.2, 0.25, 0.3, and 0.35. The peaks shift to higher angles as x increases, signifying a decreasing lattice constant. No unrelated phases were observed other than a few weak peaks of tin flux. The lattice constant, a, linearly drops with increasing x [Fig. 1(b)], which is consistent with the fact that V has smaller atomic radius than Zr. This linear lattice parameter change reflects a successful substitution of Zr atoms by V atoms which is also evidenced by EDS measurement. As shown in Fig. 1(b), the measured x from EDS and the starting x correspond well with each other within standard deviation. We notice that the lattice constant for the pristine sample is close to that of ZrCo_1.6Sn reported in previous study.^[23] This Co vacancy is the same as our EDS measurements on the Co composition for the whole series. Moreover, the saturated moment μ_sat and T_C are close to the reported values for ZrCo_1.6Sn as well.^[23] Therefore we determine that the whole series have about 20% Co vacancy and should be noted as Zr_1−xV_xCo_1.6Sn.

Figures 2(a) and 2(b) show temperature dependent magnetization under an applied magnetic field μ_0H = 1 T in a field-cooled (FC) mode and magnetic isotherms at 2 K for the whole series, respectively. The magnetization for all samples shows a similar temperature dependence, signifying the transition from paramagnetic (PM) to FM state at around 200 K. The magnetic isotherms at 2 K show a typical profile of soft ferromagnet with a quickly saturated magnetic moment in an external field about 3 kOe. No obvious hysteresis loops can be discerned in any samples (data not shown here). It is noteworthy that the magnetization for x = 0.35 slightly increases at higher field, instead of being saturated like the other samples. Figure 2(c) summarizes the magnetic parameters including Curie–Weiss temperature θ_C, T_C, effective moment μ_eff, and μ_sat with respect to x. The θ_C and μ_eff are derived from the Curie–Weiss law fitting at high temperature while the T_C is determined by the magnetic isotherms near the T_C (Arrott’s plot) (data not shown here). All the parameters are slightly enhanced by increasing x with the exception of x = 0.35, whose μ_eff, T_C, and θ_C are slightly smaller than those for x = 0.30. Previous studies on polycrystalline Co₂Ti_1−xV_xSn suggested that the magnetic moment of Co does not linearly change with respect to x.^[37,38] Instead, the Co moment reaches maximal when x is between 0.2 and 0.4. We infer the anomalous behavior for x = 0.35 is due to the fact that this sample is near the maximal point. Further study is needed for elaborating this unusual change.

Fig. 2.

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Fig. 2. (a) Temperature dependent magnetization of Zr1−xVxCo1.6Sn under an applied field of 1 T. (b) Field dependent magnetization at 2 K. (c) Curie–Weiss temperature θC, ferromagnetic Curie temperature TC, effective moment μeff, and saturated moment μsat for different x. (d) The zero-field temperature variation of longitudinal resistivity for all samples. The dotted line connecting low temperature upturn points is a guide to the eyes.

Figure 2(d) shows the normalized temperature dependent resistivity [ρ(T)/ρ(300 K)] from 2 K to 300 K for all samples. All the ρ(T) curves possess a negative temperature coefficient at high temperature and then they show a broad hump at around 200 K where the FM transition takes place. Below T_C, the samples of x ≤ 0.3 have a bad-metal-like temperature dependent profile which is ended by an upturn at low temperature. With increasing x, the ρ(T) curve gradually evolves from bad-metal-like below T_C to semiconductive at x = 0.35 for the whole temperature range. The low-temperature upturn becomes more evident and moves to higher temperature with increasing x, as denoted by the dotted line in Fig. 2(d). As shown below, the upturn is attributed to a quantum correction effect.

Figure 3 shows the Hall resistivity (ρ_yx) against external magnetic field for three representative samples at selected temperatures. They all display the same trend as the M(H) curves at 2 K, signifying a strong AHE. The rapid saturated anomalous signal is followed by a normal Hall signal with a gentle, positive slope which reflects the dominating hole carrier. As temperature increases, the anomalous Hall signal weakens and finally fades away above T_C.

Fig. 3.

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	Fig. 3. (a)–(c) The Hall resistivity versus external field at selected temperatures for three representative samples x = 0, 0.25, and 0.35, respectively. (d) Temperature dependence of anomalous Hall resistivity for all series of samples.

Empirically, ρ_yx comprises two terms^[28]

In order to shed light on the mechanisms of AHE, the experimental data are fitted with the scaling laws in which the influence of M(T) is also taken into account.^[29–32] We find that the data can be well fitted by a modified TYJ scaling^[1,33,34]

Figure 4 shows that R_S is linearly dependent on for all samples. For clarity, the data points above and below the low-temperature upturn are shown as solid and hollow symbols, respectively. These two segments lie almost on the same fitting line for one sample, signifying the validity of the TYJ scaling for a whole temperature range below T_C. The relatively large ρ_xx0 ( > 400 μΩ⋅cm) and small RRR ( < 2) in Zr_1−xV_xCo_1.6Sn imply that the impurity scattering prevails over phonon contribution. The validity of the TYJ scaling is consistent with the conclusion that the inelastic scattering from phonon plays a negligible role in skew scattering in comparison to the impurity scattering.^[1,33,34]

Fig. 4.

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	Fig. 4. The dependence of RS for each sample in which solid and hollow symbols denote the data above and below the low-temperature upturn, respectively. Solid lines are linear fit according to Eq. (2).

The fitting coefficients a′ and b, representing the extrinsic and intrinsic contributions, respectively, are plotted against x in Fig. 5(a). We notice that b remains positive for all x while a′ shows sign reversal at x = 0.25. The reason of the sign change for the extrinsic AHC is unknown, but it may be related to the dominant contribution changing from side-jump to skew-scattering or vice versa. At this point, the two extrinsic terms can not be disentangled. The intrinsic AHC and extrinsic AHC at 2 K are shown as a function of x in Fig. 5(b). The intrinsic AHC doubles from 30 Ω⁻¹⋅ cm⁻¹ to 60 Ω⁻¹⋅ cm⁻¹ with increasing x. These values are comparable to the calculated intrinsic AHC for 26e full Heusler VCo₂Ga and TiCo₂Sn.^[10,18]

Fig. 5.

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	Fig. 5. (a) The fitting parameters a′ and b as functions of the doping ratio x based on Eq. (2). (b) The variations of intrinsic AHC and extrinsic AHC at 2 K and carrier densities as x changes.

We firstly discuss the crystalline and magnetic properties of Zr_1−xV_xCo_1.6Sn. As reported by Kushwaha et al.,^[23] the tin-flux grown ZrCo₂Sn crystals inevitably have Co vacancies which significantly suppress the T_C and μ_sat. The Co vacancies also affect the density of states of the minority spins in electronic structure.^[35] Our measurements reveal that the properties of pristine ZrCo_1.6Sn are close to those reported by Kushwaha et al.^[23] and the whole series show the same Co vacancies. We also notice that the lattice parameters in polycrystalline ZrCo₂Sn (6.25 Å) and VCo₂Sn (5.98 Å) differ in 0.27 Å,^[21] which is approximately triple the change of a from x = 0 to 0.35. This linear change of a is consistent with our EDS measurements and verifies that Co vacancies are unchanged for different x.

Previous study showed that polycrystalline VCo₂Sn has T_C = 105 K and μ_sat = 1.2 μ_B/F.U.^[21] and therefore a naive expectation is that the magnetism will be suppressed by V-substitution in ZrCo_1.6Sn. In contrast, we observed that the T_C and μ_sat slightly increase until x = 0.35 in Zr_1−xV_xCo_1.6Sn, which conforms to the well-know Slater–Pauling rule^[36] but contradicts the expectation. Please note this T_C change is also distinguished from the case of Zr_1−xNb_xCo₂Sn^[25] in which the Nb substitution leads to the weakening of ferromagnetism. This unexpected change of magnetization in our case is consistent with the previous studies on Co₂Ti_1−xV_xSn^[37,38] in which the saturation magnetization initially increases with x until x = 0.4 and then drops. We speculate that it may be related to the chemical pressure difference while the 3d characteristics of V atoms may also play a role. Further elaboration on the whole series from x = 0 to 1 is needed for understanding this puzzle.

As theoretically predicted,^[16] ZrCo₂Sn has stable Weyl nodes located slightly above the Fermi level which can be tuned by alloying 30 % Nb. Although the Co vacancies strongly modify the magnetic properties, the Weyl nodes should not be annihilated by the disorder as they are protected by the crystalline symmetry.^[23] In Zr_1−xV_xCo_1.6Sn we observed an enhanced intrinsic AHE which may be correlated with Weyl nodes energy approaching to the Fermi level. According to the previous calculation,^[18] VCo₂Sn is not a half metal and it has an AHC as large as −1500 Ω⁻¹⋅cm⁻¹. However we do not observe a transition from positive to negative in the AHE for Zr_1−xV_xCo_1.6Sn till x = 0.35. As shown in Fig. 5(b), the hole density (p) is gradually reduced with increasing V substitution until it reaches 5 × 10²¹ cm⁻³ for x = 0.35. This p-type behavior indicates that the chemical potential is not beyond the Weyl nodes energy yet. We expect that the AHC and carrier density will change significantly when x > 0.35.

In the last part, we clarify the low-temperature upturn in resistivity in Fig. 2(a) and the influence on the AHE. In general, the electron–electron Coulomb interaction (EEI) and weak localization (WL) are supposed to account for the resistivity upturn.^[39–42] Their influence can be represented as quantum corrections to the normal electron transport^[43]

Fig. 6.

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	Fig. 6. The T1/2 dependence of (a) normalized relative changes in longitudinal resistivities ρxx near and below the upturns, (b) normalized changes of anomalous Hall coefficients RS and (c) anomalous Hall conductivities for all studied samples.

It is well-known that the WL contributes to the extrinsic AHC^[40,44] while the EEI will not lead to any correction to the AHC.^[29,45,46] However recent experiment observed a large correction to the AHC originating from EEI in HgCr₂Se₄.^[39] In contract, our measurements on Zr_1−xV_xCo_1.6Sn show no correction to the AHC by EEI. Further study on various systems is needed for elaborating the relation between the quantum corrections and AHE.

In summary, we measured the magnetic and transport properties for FM Weyl semimetal candidate Zr_1−xV_xCo_1.6Sn at x ≤ 0.35. Their AHE can be well-fitted by the TYJ scaling law which yields a dominated intrinsic term. The intrinsic AHC is slightly enhanced by increasing x. It will be interesting to investigate the solid solutions with more V-substitution but different crystal growth technique is needed.