Weyl fermions in a canonical heavy-fermion semimetal YbPtBi
Topological insulators which are insulating in the bulk yet conducting on the surface RMPTopo (), and gapless Dirac and Weyl semimetals Dirac1 (); Weyl1 (); Weyl2 (), where the chiral excitations of the latter are manifestations of Weyl fermions Weyl1 (); Weyl2 (), have been extensively studied in weakly correlated electron systems. Consequently, it is of interest to examine Weyl fermions when the electronic correlations are strong. Here, we report electronic structure calculations, ARPES, magnetotransport and calorimetric measurements of the canonical heavy fermion semimetal YbPtBi YbBiPt1991 (); Hundley1997 (), where we find that the band structure hosts triply degenerate points, yielding Weyl nodes in an applied magnetic field. For 20K - 170K, the chiral anomaly is detected in the magnetotransport, but this contribution becomes negligible when the electronic correlations become stronger at lower temperatures. However, the topological Hall effect still provides evidence for a Berry curvature caused by the Weyl points, and the specific heat shows the expected signature of Weyl nodes SiHFEeyl (). These results suggest that YbPtBi is a Weyl heavy fermion semimetal, where the bands hosting Weyl points are renormalized at low temperatures due to the Kondo interaction. Our findings open up the opportunity to explore the interplay between topology and strong electronic correlations, particularly when tuning through a quantum critical point Mun2013YbPtBi ().
A rich variety of phenomena have been discovered in gapless topological materials, such as those exhibiting Dirac-fermion excitations near the points of linear crossings of bands close to the Fermi energy Dirac1 (). The breaking of either spatial inversion symmetry or time reversal symmetry splits the degeneracy of the Dirac points, leading to a pair of topologically protected Weyl points Weyl1 (); Weyl2 (). Weyl fermions have been found to cause distinct experimental signatures such as the chiral anomaly in transport measurements Nielsen1983 (); ChiralAnomTheor1 (); WeylTaAsChirAnom1 (), a topological Hall effect GdPtBiAHE (); nakatsuji2015large (); li2018momentum () and Fermi arcs ArcRev ().
Weyl fermions have mainly been studied in weakly correlated electron systems, while strong electronic correlations are frequently found to lead to novel electronic properties beyond those of simple metals or insulators, and heavy fermion systems are the prototype examples showing phenomena characteristic for strongly correlated electron systems. Here, due to strong hybridization between the -electron and conduction-band states, below the Kondo temperature (), the electronic bands in the vicinity of may become strongly renormalized, showing a strong “-character” and a huge enhancement of the quasiparticle mass. When the chemical potential lies within the hybridization gap, insulating behavior is found at low temperatures and in the topological Kondo insulators, such as has been proposed for SmB, the resulting electronic structure is topologically non-trivial, again leading to conducting states on the surface SmB6a (). It is therefore of particular interest to look for topological heavy fermion semimetals with gapless excitations, i.e. Weyl fermions in the presence of strongly renormalized bands. Such a Weyl-Kondo semimetal phase has been predicted from the periodic Anderson model with broken inversion symmetry SiHFEeyl (). While it was proposed that CeBiPd displays the low temperature thermodynamic signatures of a Weyl-Kondo semimetal SiHFEeyl (); dzsaber2016tuning (), other signatures of Weyl fermions such as the chiral anomaly have not been reported. A Weyl heavy fermion state was also proposed for CeRuSn from ab initio calculations CeRu4Sn6Theor (), but no experimental evidence for Weyl fermions has been demonstrated. Consequently, whether Weyl fermions exist in the presence of strong electronic correlations needs to be determined experimentally. Furthermore, the influence of electronic correlations on Weyl fermions is to be explored, specifically how such a system evolves from high temperatures, where the -electrons are well localized, to low temperatures where there is a strong Kondo interaction and a reconstruction of the electronic bands.
The cubic half Heusler compounds can be tuned by elemental substitution from trivial to topological insulators Canfield1991RBiPt (); chadov2010tunable (). It was recently found that the half Heusler GdPtBi, which has a strongly localized -electron shell, shows evidence for Weyl fermions in an applied magnetic field due to the presence of the chiral anomaly GdPtBiChiral () and topological Hall effect GdPtBiAHE (). Here we examine the isostructural compound YbPtBi. Although at high temperatures the Yb -electrons are localized similar to GdPtBi, upon cooling YbPtBi becomes a prototypical heavy-fermion semimetal YbBiPt1991 (); Hundley1997 (); TwoChanYbPtBi (), where the enormous Sommerfeld coefficient of J/mol K well below K demonstrates the enhanced effective mass of the charge carriers YbBiPt1991 (). This compound is therefore highly suited to look for Weyl fermions which are strongly affected by electronic correlations. At higher temperatures, the band structure of YbPtBi can be calculated treating -electrons as core states, as displayed in Fig. 1. The bulk Fermi surface consists of hole pockets centered at the -point and electron pockets slightly away from . Along -L, the four-fold degenerate state splits into two non-degenerate hole bands, and doubly degenerate electron bands, mainly consisting of Yb- and Bi- orbitals. The bands cross the two hole bands near , forming two triply degenerate fermion points lv2017observation (). Under a magnetic field, each triply degenerate point will further split into a Weyl point and a trivial crossing, with energies close to the bottom of the electron bands. The projected bulk band structure calculation is in excellent agreement with the ARPES measurements [Fig. 1(b)], where both the hole and electron bands near can be clearly resolved. We emphasize here that the direct observation of both electron and hole pockets guarantees the existence of the triply degenerate fermion points.
Magnetotransport measurements were performed to look for the chiral anomaly associated with Weyl fermions (Fig. 2). Figure 2a shows the field dependence of the resistivity of YbPtBi at selected temperatures with a current along  and a magnetic field applied parallel and perpendicular to . For temperatures between 25 K and 170 K, the longitudinal magnetoresistance () is positive at low fields but becomes negative in the higher field region, while the transverse magnetoresistance () is positive, which is evidence for the chiral anomaly. The negative longitudinal magnetoresistance cannot be explained by either current jetting (Supplementary Fig. 2) Currentjet (), nor the sample anisotropy since similar behavior is found for other current directions (Supplementary Fig. 3). The negative longitudinal magnetoresistance above 20 K could be well fitted using a conductivity (Fig. 2a and Supplementary Fig. 4), where is the chiral constant and is due to the weak antilocalization ChiralAnomTheor1 (); WeylTaAsChirAnom1 (). As shown in Fig. 2b, the temperature dependence of is well fitted with the expected behavior of , where is the chirality-changing scattering time and is the chemical potential ZrTe5Chiral (), yielding m/s and meV. for various angles between and are displayed in Fig. 2c as a function of , where the high field linear behavior indicates a contribution, while the very small values lead to a negligible component (Fig. 2b). As displayed in Fig. 2d (and Supplementary Fig. 5), the extracted shows the expected angular dependence of . Therefore both the angle and temperature dependence of the magnetoresistance are highly consistent with the presence of a chiral anomaly in YbPtBi.
Meanwhile either by changing the Bi flux concentration or by Au doping, the carrier concentration can be tuned. The Hall resistivity for various samples shows that more strongly hole doped samples exhibit one band behavior with larger hole densities (), but upon electron doping, is shifted and eventually crosses the electron bands, leading to two band behavior (Figs. 2e and 2f, Supplementary Figs. 6 and 7). As shown in Fig. 2f, in the vicinity of the crossover between one and two band behavior, the negative longitudinal magnetoresistance is greatest. For more strongly electron or hole doped samples, no negative magnetoresistance is seen at elevated temperatures, indicating that this negative longitudinal magnetoresistance arises when is close to the Weyl points (Figs. 1a and 2g). Measurements of the transverse resistivity (with the voltage measured perpendicular to ) for fields rotated in the plane of the voltage drop and () provide an alternative method for probing the chiral anomaly which is much less sensitive to spin scattering than the magnetoresistance (Fig. 2h) PHEGdPtBi (). For two samples with evidence for the chiral anomaly in the magnetoresistance (S7 and S9), the oscillation amplitude of is greatly enhanced above 20 K, while this remains small for the more electron-doped sample, which is another signature of the chiral anomaly in samples where is near the band crossing. Interestingly, at 2 K the oscillations have very small amplitudes and are not sample dependent. This suggests that evidence for the chiral anomaly disappears from these measurements at low temperatures, leaving only a small contribution likely from the orbital magnetoresistance. Similar conclusions are drawn from the magnetoresistance in Fig. 2(a), which at low temperatures is negative at all , and the behavior is well accounted for by single impurity Kondo scaling schlottmann1989some () (Supplementary Fig. 9). This disappearance can be related to the drop of the effective Fermi velocity to as the quasiparticles gain mass in the heavy fermion state, since and therefore decreasing will greatly reduce the chiral anomaly contribution.
However, even when is small, the Berry curvature induced by the Weyl points can still contribute to the anomalous Hall effect (AHE). Figure 3a shows the anomalous contribution to the Hall resistivity after subtracting the ordinary band part; the data are taken from measurements of sample S6 which exhibits single band behavior and evidence for the chiral anomaly. This AHE contains a contribution proportional to the magnetization (dashed lines) and a term related to the topology . The latter gives rise to the peak in Fig. 3a at low temperatures, while the former dominates at higher temperatures due to an increased resistivity (Supplementary Fig. 8). After subtracting , the topological Hall angle is obtained and is displayed in Fig. 3b. Here a peak can be resolved up to temperatures of at least 30 K, with a large maximum in of 0.18 at 0.3 K, similar to the peak values found in the magnetic Weyl semimetals GdPtBi (0.17) GdPtBiAHE () and MnSn (0.4) li2018momentum (). These results strongly indicate that even at low temperatures the Berry curvature from the Weyl points is still manifested in the anomalous Hall effect.
Evidence for the presence of Weyl points in the heavy fermion state is also found in specific heat measurements. While in zero field there is an upturn of prior to the onset of antiferromagnetic order in zero-field at 0.4 K (Supplementary Fig. 10) YbBiPt1991 (); Mun2013YbPtBi (), for larger applied fields reaches a maximum before decreasing at lower temperatures. However, as also shown by the solid lines in Fig. 3c, the low temperature at higher fields deviates from a spin-1/2 resonance-level model for Kondo impurity systems (Supplementary discussion) SCHOTTE197538 (), where two levels of width are split by a Zeeman field. This model can be widely applied in heavy fermion systems, both in the coherent heavy Fermi liquid state and the dilute limit Pikul2012 (). In higher fields, of the Kondo impurity model becomes nearly temperature independent at low temperatures, but the data are instead well described by a dependence of the specific heat, [Fig. 3d], which was proposed for a Weyl-Kondo semimetal SiHFEeyl (), as a result of the linear dispersion in the vicinity of the Weyl nodes. We note that this term is too large to arise from acoustic phonons. With increasing field there is a decrease of the Sommerfeld coefficient and an increase of , consistent with the applied field reducing the effective mass of the quasiparticles (Supplementary Table 1). However, even at T a value of mJ/mol K is obtained, indicating that a significant mass enhancement persists. Correspondingly, low effective Fermi velocities are obtained from fitting the data of ms at 7 T and ms at 13 T, which are significantly reduced compared to the Fermi velocity of ms estimated from at 50 K (Fig. 2f).
Based on the above experimental findings, we propose the diagram shown in Fig. 4 to describe the Weyl fermions in YbPtBi. At high temperatures there are Weyl nodes formed from the conduction bands, while the electrons are well localized. At lower temperatures, the strong band renormalization due to hybridization enhances the effective quasiparticle mass, which modifies the dispersion of the bands in the vicinity of the topologically protected Weyl points, as shown schematically in the diagram. The renormalization also leads to a greatly reduced effective Fermi velocity compared to the bare band value, which eventually causes the disappearance of the chiral anomaly in transport measurements, but the emergence of a specific heat contribution SiHFEeyl () . Importantly, there is evidence for the Berry curvature associated with the Weyl nodes from the anomalous Hall effect, which can be detected in both the intermediate and low temperature regimes.
Our results highlight the existence of Weyl fermions in YbPtBi, where we find evidence for their modification as the Kondo coupling is strengthened upon lowering the temperature. To elucidate the detailed response of the Weyl fermions to the increasing electronic correlations, further theoretical and experimental studies of YbPtBi are highly desirable. However, although the bulk Yb -bands are found to lie close to from soft x-ray ARPES (Supplementary Fig. 11), resolving the hybridized bands deep inside the heavy fermion state is still challenging. We note that the case of YbPtBi is different from that of both CeSb guo2016cesb () and GdPtBi GdPtBiChiral (), where the bands hosting Weyl fermions do not have a significant -electron contribution. On the other hand, it is of great interest to look for this kind of dichotomy in other potential Weyl heavy-fermion semimetals, such as CeBiPd where a similarly small was inferred from the specific heat dzsaber2016tuning (), yet evidence for the chiral anomaly at elevated temperatures has not yet been reported. Furthermore, the strength of the Kondo interaction in heavy fermion systems can be readily tuned by non-thermal control parameters such as pressure and magnetic field, and in particular, a quantum critical point can be reached in YbPtBi at a critical field of 0.4 T Mun2013YbPtBi (). Therefore, our findings open up the opportunity to explore the exciting relationship between Weyl fermions, electron-electron correlations and quantum criticality.
Methods Single crystals of YbPtBi were prepared using a Bi self flux canfield1992growth (). Elemental Yb, Pt and Bi were combined in a range of molar ratios from 1:1:7-1:1:20 and heated to 1150C, before being slowly cooled to C at a rate of C/hour. For some samples, Au was also added up to a maximum ratio of Au:Pt of 1:19. The single crystal quality and orientation were checked using Laue diffraction, (Supplementary Fig. 1 and methods).
The magnetotransport was measured using the four-probe method in a Quantum Design Physical Property Measurement System (9T-PPMS) with the sample rotation option, where Pt wires were attached to the sample. As shown in Supplementary Fig. 2, for some samples multiple voltage contacts were made, so as to rule out current inhomogeneities and the current jetting effect. Specific heat measurements were performed using a 14T-PPMS using a He option, while magnetization measurements were carried out using the vibrating sample magnetometer (VSM) option. Hall effect measurements for determining the anomalous Hall effect were performed in a He cryostat with a 15 T magnet.
ARPES measurements (Fig. 1) were performed at the Advanced Light Source, BL7 micro-ARPES beamline. The (111)-oriented YbPtBi samples were cleaved in-situ and measured at around 20 K with 75 eV photons. The surface termination (either Yb or Bi terminated) is determined by core level analysis. The typical domain size after cleavage is only a few tens of for the Yb termination. The soft X-ray ARPES measurements (SI) were performed at the ID29, Advanced Photon Source.
The DFT calculations were performed with plane-wave basis and projected augmented wave method as implemented in VASP. The -electrons are treated as core-states in these calculations. To ensure convergence, plane-waves up to 480 eV and 12x12x12 -centered K-mesh was employed. The generalized gradient approximation is known to overestimate the band inversions in crystal, therefore we have employed modified Becke-Johnson potentials to calculate the band structure.
Acknowledgments We would like to thank Qimiao Si, Joe Thompson, Fuchun Zhang, Jianhui Dai and Yi Zhou for valuable discussions. We also thank Dr. J. McChesney and Dr. F. Rodolakis Simoes for beamline support during the soft ARPES measurements. 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). The ALS and APS are supported by the Office of Basic Energy Sciences of the U.S. DOE under Contracts No. DE-AC02-05CH11231 and No. DE-AC02-06CH11357.
Additional information Correspondence and requests for materials should be addressed to H. Q. Yuan (email@example.com)
Author contributions The project was conceived by C.Y.G. and H.Q.Y.. The crystals were grown by F.W.. Magnetotransport and specific heat measurements were performed by C.Y.G. and F.W., and analyzed by C.Y.G., F.W., M.S., F.S., and H.Q.Y.. Electronic structure calculations were carried out by C.C.. ARPES measurements were performed and analyzed by Z.Z.W, A.B., C.J., E.R., and Y.L.. C.Y.G., M.S., F.S., C.C., Y. L. and H.Q.Y. wrote the manuscript.
Competing financial interests The authors declare no competing financial interests.
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Figure 1. Existence of triply degenerate fermion points in the high temperature phase of YbPtBi: DFT and ARPES. a, Bulk band structure of YbPtBi from density functional theory (DFT) calculations along the  direction. Blue curves are doubly degenerate states, while the red and cyan curves represent the non-degenerate hole states. b, Comparison between the projected bulk band structure calculations and ARPES measurements, which are highly consistent. The comparison was made for the Yb-terminated (111) surface along the in-plane direction (). The flat bands at energies of -0.9 and -2.1 eV are the surface bands from the topmost Yb layer (Supplementary Fig. 11).
Figure 2. Evidence for the chiral anomaly in YbPtBi at elevated temperatures from magnetotransport. a, Field dependence of the resistivity of YbPtBi at four temperatures for fields parallel and perpendicular to the current . The solid lines show fits for at elevated temperatures, taking into account the chiral anomaly and the weak antilocalization, as described in the text. b, Temperature dependence of the chiral constant and the weak antilocalization -coefficient. The solid line shows the fitted temperature dependence of ZrTe5Chiral (). c, Conductivity of sample S7 versus for an angle between the field and current, which exhibits the expected dependence at high fields. d, Normalized chiral constant obtained from fitting the high field conductivity, where shows a cos dependence. e, Hall resistivity and longitudinal magnetoresistance () for different samples, where S11-S13 were Au doped. f, The strength of the negative longitudinal magnetoresistance as [, as a function of the hole density from fitting the Hall resistivity (Supplementary Fig. 7). For samples with two-band behavior, the electron density is also shown. g, Illustration of for different samples where upon electron doping, eventually intersects the electron bands, very close to the triply-degenerate fermion points. Samples S5-S11 (thick lines) show evidence for the chiral anomaly. h, Transverse resistivity of three samples, where a field of 9 T is rotated in the plane containing the voltage drop and current, where at the field and voltage are parallel.
Figure 3. Evidence for Weyl fermions in the heavy fermion state of YbPtBi from the anomalous Hall effect and specific heat. a, Anomalous contribution to the Hall effect of sample S6 obtained from subtracting the ordinary one band Hall resistivity. The Anomalous Hall effect contains two terms, which is proportional to the magnetization, and the topological term . The dashed lines show just obtained from analyzing the data together with the measured magnetization (Supplementary Fig. 8). b Topological Hall angle as a function of field, after subtracting . A clear peak is observed at temperatures up to at least 30 K, giving evidence for the Berry curvature induced by the Weyl points. c, Specific heat as at 5 T, 7 T, 9 T, and 13 T where the solid lines show the results of fitting a Kondo resonance model SCHOTTE197538 (). The deviation from the model at low temperatures shows clear evidence for an additional low energy contribution. For comparison the zero-field specific heat of non-magnetic LuPtBi from Ref. Mun2013YbPtBi () is also displayed. d, Specific heat in-field, as vs , showing that the low temperature behavior is well accounted for by a dependence (solid lines) which is the expected behavior for Weyl heavy fermion semimetals SiHFEeyl ().
Figure 4. Schematic phase diagram for Weyl fermions in heavy fermion systems. Illustration of the evolution of the contributions from different experimental probes of Weyl fermions in heavy fermion systems; the chiral anomaly , electronic specific heat , and topological Hall effect . When the electronic bands in the vicinity of the Weyl points become heavy at low temperatures, the massive reduction of the effective Fermi velocity leads to the chiral anomaly contribution becoming undetectable, yet gives a significant contribution from Weyl nodes to the electronic specific heat , which is otherwise not detectable in weakly correlated materials. Meanwhile the topological Hall effect which arises from the Berry curvature can be detected both when the -electrons are well localized, as well as deep inside the heavy fermion state.