Sample dependence studies of the Kondo Weyl semimetal YbPtBi

Single crystals of YbPtBi were prepared using a Bi self flux. Here in this paper, we have presented seven samples, labeled S_a to S_g, which have different carrier densities. Note that S_g is S13 in Ref. 8, while S_e corresponds to S11. For the samples S_a to S_d, the Bi-flux used for the single crystal growth was systematically reduced. To further decrease the carrier density, an increasing ratio of gold doping is used for S_e to S_g with a maximum starting amount of 5%, keeping the Bi content unchanged. The magnetotransport was measured in a Quantum Design Physical Property Measurement System (PPMS) using a sample rotator and a 9 T superconducting magnet. To apply the standard four probe method for the measurements, four Pt wires were attached to the samples using silver paint.

Figure 1(a) presents the temperature dependence of the resistivity of REPtBi (RE= Ce, Gd, Sm, Yb). Besides YbPtBi, all the other materials show typical semiconducting behavior at high temperature, indicating a small carrier concentration. The resistivity of YbPtBi, however, displays no upturns upon cooling across the whole temperature range, which suggests the larger carrier concentration. This is further indicated by previously reported Hall resistivity measurements, where the carrier densities of CePtBi, GdPtBi and SmPtBi are about two orders of magnitude smaller than YbPtBi.^8,10–12 This significant difference is a result of the peculiar band structures of these compounds. For CePtBi, GdPtBi and SmPtBi, the band structure calculation shows that they are all zero-gap semiconductors,¹³ where the Fermi level is close to the gap.^10–13 The resulting Fermi surface would be rather small, and hence the carrier density of those materials is also very small. It has been shown in GdPtBi¹⁰ that with applying field, due to the strong Zeeman effect, the spin degeneracy will be lifted and pairs of bulk crossing points will be formed. Further theoretical and experimental studies confirmed that these crossing points are Weyl nodes, as displayed in the upper panel of Fig. 1(b).¹⁰ Meanwhile, for YbPtBi, theoretical calculations and ARPES experiments demonstrate a unique band structure⁸ as illustrated in the lower panel of Figure 1(b). Unlike the other members of the REPtBi family, the electron band forms a band crossing with the hole band, and therefore no band gap is present. The Fermi level of the flux grown samples is usually close to the bulk band crossing points, which are found to be triply degenerate, and can be split into Weyl nodes upon applying an external magnetic field.⁸ This specific band structure allows for a much larger hole band Fermi surface, which then results in the larger carrier density of YbPtBi. Moreover, we found that similar to other REPtBi compounds,¹⁰ with the modification of crystal growth conditions, we can systematically tune the chemical potential of YbPtBi, and such tuning allows for the exploration of the influence of Weyl fermions on the transport properties.⁸ We show below how this systematic tuning can be used to reveal the presence of the chiral anomaly in YbPtBi.

To examine the effects of Au-doping in YbPtBi, we have performed the energy-dispersive spectroscopy to obtain the atomic ratios. As displayed in Fig. 2(a), for S_b, no Au element is detected, and the elements ratio of Yb:Pt:Bi is close to 1:1:1, as expected for YbPtBi single crystals. However for S_g, as shown in Fig. 2(b) the presence of Au is clearly detected, and the derived Au:Pt ratio is about 5.2%, which is very close to the starting ratio of 5%. Figure 2(c) displays Hall resistivity measurements on seven different samples. A systematic change of the Hall coefficient between different samples can clearly be seen. The greater the amount of Bi-flux used during the synthesis procedure, the larger the hole carrier density, which suggests a change of defects or impurities level via changing the flux ratio. On the other hand, Au doping shifts the Fermi level up, resulting in a reduced hole carrier density. A clear peak is observed in the low field range for S_d to S_g, suggesting the presence of a small pocket of electron-type carriers, which is consistent with the previous reports.^8,14 The Hall resistivity of S_e and S_f is therefore analyzed with a two band model, while the other samples are analyzed using the single band model, this analysis reveals a tunable carrier density from n_H = 1.7 × 10²⁰ cm⁻³ to 3.5 × 10²⁰ cm⁻³ and n_e = 0.2 × 10¹⁹ cm⁻³ to 1.3 × 10¹⁹ cm⁻³. Meanwhile, the mobility of hole band μ_h is larger than 20cm²/Vs while the mobility of the electron band μ_e is much larger at around 10000cm²/Vs, which much larger than μ_h, and is again consistent with previous reports.^8,14 We also measured the longitudinal MR of various samples with both the field and current applied along the [100] direction. Negative MR appears for S_b to S_e, which was shown in the previous study to provide clear evidence for the chiral anomaly.⁸ The evolution of the negative MR ratio with changing carrier density is summarized in Fig. 2(e). A clear peak in the Hall resistivity near zero field appears at the point where the electron pocket start to emerge, which corresponds to the samples with the strongest signature of the chiral anomaly. These results can be consistently explained by considering the band structure,⁸ as illustrated in Fig. 2(f). Upon increasing the chemical potential, the size of the hole pocket continuously shrinks. At a certain point, the Fermi level crosses the electron pocket, and further shifting up of the Fermi level results in a continuous increase of the electron-type carrier density, which is consistent with the results of Hall resistivity measurements. On the other hand, the strength of the chiral anomaly strongly depends on the energy difference between the Fermi level and Weyl nodes, since when they are closer together, the influence of the Weyl nodes on the magnetotransport properties will be stronger.^10,15 The fact that the strongest negative longitudinal MR occurs where the electron band appears suggests that the Weyl nodes are located close to the bottom of the electron band. Note that there are other possible origins for the appearance of a negative longitudinal MR. This is particularly the case for a magnetic system such as YbPtBi, since the suppression of inelastic scattering upon the application of a magnetic field can also contribute to a negative longitudinal MR. However in such cases, the strength of the negative longitudinal MR is not expected to strongly depend on the carrier density. Since the negative MR in YbPtBi is found to have a significant dependence on the carrier concentrations, and hence the position of the Fermi level, this suggests that this affect is likely to originate from the chiral anomaly, as was also found for isostructural GdPtBi.¹⁰ 

The Hall resistivity of different samples at various temperatures are displayed in Figure 3. An abrupt increase of Hall resistivity near zero field is observed in both S_e and S_f which is absent for S_b and S_c. This is due to the aforementioned differences in the electronic structure of different samples, as S_b and S_c only possess one hole band, while S_e and S_f have both electron and hole bands. Meanwhile, it can clearly be seen that this low field feature gets more pronounced at higher temperature. One possible reason for this is a shift of the chemical potential with increasing temperature, which can result in a change of carrier density and thus the enhancement of the low field signature. Another explanation is that at higher temperatures, more electrons can be thermally excited to the Fermi level and contribute to the transport properties. In addition to this small feature, there is also a clear non-linear signature at low temperatures, which was found to correspond to the signature of the anomalous Hall effect.⁸ 

Figure 4(a) displays the total Hall resistivity (solid line) and the contribution from the normal Hall resistivity (dashed line) of S_b, S_c, S_e and S_f. A clear non-linear component appears in every sample, which corresponds to the contribution from the anomalous Hall effect. Note that this is different from the previously discussed signature originating from the electron band, since the electron band contribution shows a much stronger increase with field and only appears in a narrower field range close to zero field. Moreover, the two features show very different temperature dependences, as with increasing temperature, the anomalous Hall effect, which arises due to the Berry curvature in momentum space, is reduced,¹⁶ while the electron band contribution gets more pronounced. Figure 4(b) presents the anomalous Hall effect after subtracting the normal part. A clear peak can be observed for all samples centered around 3 T. This is very similar to the signature observed in the magnetic Weyl semimetal GdPtBi.^16,17 In the case of some non-collinear antiferromagnets, this peak is attributed to the topological Hall effect caused by the spin structure,^18,19 which cannot be applied for YbPtBi since the Hall resistivity is measured above T_N. As a result, we attribute this to the complex Berry curvature arising from the Weyl nodes. We also note that the peak value increases from S_b to S_f, which suggests that there is also a close relationship between the topological Hall effect and carrier density, the origin of which still needs to be clarified in future studies.

In summary, we studied the sample dependence of the Kondo Weyl semimetal YbPtBi. A strong sample dependence of the chiral anomaly is observed, which can be well explained based on the calculated band structure. We also report the temperature dependence of the Hall resistivity of different samples. Sharp changes close to zero field are observed for samples with two carrier bands, which is consistent with previous studies.^8,14 The anomalous Hall effect is also examined for several different samples. The presence of a topological Hall effect above T_N is ascribed to being caused by the complex Berry curvature in momentum space due to the presence of Weyl points. We find that although this anomaly is observed in all measured samples, the magnitude in the anomalous signal also shows a dependence on the carrier density. The observation of a topological Hall effect at low temperatures within the heavy fermion state indicates that the Weyl nodes are still present, but further studies are required to reveal how exactly the associated electronic bands are modified by the strong hybridization.

This work was supported by the National Key R&D Program of China (No. 2017YFA0303100, No. 2016YFA0300202), the National Natural Science Foundation of China (No. U1632275, No. 11474251) and the Science Challenge Project of China (No. TZ2016004).