Magnetotransport Signatures of Weyl Physics and Discrete Scale Invariance in the Elemental Semiconductor Tellurium

Magnetotransport signatures of Weyl physics and discrete scale invariance in the elemental semiconductor tellurium

Nan Zhanga,b,c,1, Gan Zhaoa,b,1, Lin Lia,b,c,1,2, Pengdong Wangd, Lin Xiee, Bin Chenga,b,c, Hui Lif, Zhiyong Lina,b,c, Chuanying Xig, Jiezun Keh, Ming Yangh, Jiaqing Hee, Zhe Sund, Zhengfei Wanga,b,2, Zhenyu Zhanga,b, and Changgan Zenga,b,c,2

aInternational Center for Quantum Design of Functional Materials, Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, 230026 Hefei, Anhui, China; bSynergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, 230026 Hefei, Anhui, China; cChinese Academy of Sciences Key Laboratory of Strongly Coupled Quantum Matter Physics, Department of Physics, University of Science and Technology of China, 230026 Hefei, Anhui, China; dNational Synchrotron Radiation Laboratory, University of Science and Technology of China, 230029 Hefei, Anhui, China; eDepartment of Physics, Southern University of Science and Technology, 518055 Shenzhen, China; fInstitutes of Physical Science and Information Technology, Anhui University, 230601 Hefei, Anhui, China; gHigh Magnetic Field Laboratory, Chinese Academy of Sciences, 230031 Hefei, Anhui, China; and hWuhan National High Magnetic Field Center, Huazhong University of Science and Technology, 430074 Wuhan, China

Edited by Junichiro Kono, Rice University, Houston, TX, and accepted by Editorial Board Member Angel Rubio April 4, 2020 (received for review February 15, 2020) The study of topological materials possessing nontrivial band Until now, Dirac/Weyl physics has been generally regarded as structures enables exploitation of relativistic physics and develop- unique to semimetals, with the Dirac/Weyl points located around ment of a spectrum of intriguing physical phenomena. However, the Fermi level. Nevertheless, considering the high tunability and previous studies of Weyl physics have been limited exclusively compatibility with modern electronic industry, the semiconductor to semimetals. Here, via systematic magnetotransport measure- will be a better candidate material for designing topological ments, two representative topological transport signatures of electronic devices if exotic transport phenomena related to Dirac/ Weyl physics, the negative longitudinal magnetoresistance and PHYSICS the planar Hall effect, are observed in the elemental semiconduc- Weyl fermions can be realized. Tellurium (Te) is a narrow band- – tor tellurium. More strikingly, logarithmically periodic oscillations gap semiconductor possessing strong spin orbit coupling, and it in both the magnetoresistance and Hall data are revealed beyond lacks space inversion symmetry due to the characteristic chiral the quantum limit and found to share similar characteristics with crystal structure. As an elemental semiconducting material, Te has

those observed in ZrTe5 and HfTe5. The log-periodic oscillations been widely studied for its lattice dynamics, band structure, originate from the formation of two-body quasi-bound states transport, and optical properties, with key discoveries transpiring formed between Weyl fermions and opposite charge centers, the several decades ago (25). Recently, the Weyl semimetal phase was energies of which constitute a geometric series that matches the theoretically proposed in Te by closing its bulk band gap with general feature of discrete scale invariance (DSI). Our discovery reveals the topological nature of tellurium and further confirms the universality of DSI in topological materials. Moreover, intro- Significance duction of Weyl physics into semiconductors to develop “Weyl semiconductors” provides an ideal platform for manipulating fun- Recent intensive investigations have revealed unique elec- damental Weyl fermionic behaviors and for designing future tronic transport properties in solids hosting Weyl fermions, topological devices. which were originally proposed in high-energy physics. Up to now, the discovered Weyl systems have been limited to semi- Weyl semiconductor | tellurium | negative longitudinal metal compounds. Here we demonstrate that the elemental magnetoresistance | planar Hall effect | log-periodic oscillations semiconductor tellurium is a Weyl semiconductor, with typical Weyl signatures, including the negative longitudinal magne- he discovery of nontrivial topological phases has reshaped our toresistance, the planar Hall effect, as well as the intriguing Tunderstanding of solid-state materials (1–3). Besides band logarithmically periodic magneto-oscillations in the quantum structure measurements, transport properties can also provide limit regime. Such Weyl semiconductors offer a simple platform some unique signatures to identify these exotic topological phases. for the exploration of novel Weyl physics and topological de- For example, two well-known transport signatures in Dirac/Weyl vice applications based on semiconductors and moreover con- semimetals are the chiral-anomaly induced negative longitudinal firm the universality of discrete scale invariance in topological magnetoresistance (NLMR) (4–14) and planar Hall effect (PHE) materials. (12, 15–20). Recently, a new type of quantum oscillations featur- Author contributions: L.L. and C.Z. designed research; N.Z., G.Z., L.L., P.W., L.X., B.C., H.L., ing log-periodicity, wherein the extrema magnetic fields of oscil- Z.L., C.X., J.K., M.Y., J.H., and Z.S. performed research; N.Z., G.Z., L.L., Z.W., Z.Z., and C.Z. lations constitute a geometric series, was discovered in ZrTe5 (21) analyzed data; and N.Z., G.Z., L.L., Z.W., and C.Z. wrote the paper. and HfTe5 (22) beyond the quantum limit. Physically, these un- The authors declare no competing interest. usual log-periodic oscillations can be attributed to the manifesta- This article is a PNAS Direct Submission. J.K. is a guest editor invited by the tion of discrete scale invariance (DSI) (21, 23, 24), whereby self- Editorial Board. reproduction occurs at specific scales that form a geometric series. Published under the PNAS license. The DSI character of topological materials is rooted in the two- 1N.Z., G.Z., and L.L. contributed equally to this work. body quasi-bound states formed between linearly dispersed quasi- 2To whom correspondence may be addressed. Email: [email protected], zfwang15@ustc. particles and opposite charge centers, with the energy levels sat- edu.cn, or [email protected]. isfying a geometric progression (21). In theory, log-periodic This article contains supporting information online at https://www.pnas.org/lookup/suppl/ oscillations are considered a universal feature of materials host- doi:10.1073/pnas.2002913117/-/DCSupplemental. ing Dirac/Weyl fermions with Coulomb attraction (21, 23, 24).

www.pnas.org/cgi/doi/10.1073/pnas.2002913117 PNAS Latest Articles | 1of7 Downloaded by guest on September 25, 2021 external perturbations (26, 27), which has renewed interest in grown using a similar method (30, 32). The hole-doped character exploring the possible topological characters of Te (26, 28–30). is further verified via the angle-resolved photoemission spectros- In this work, we report two transport signatures (NLMR and copy results (SI Appendix,Fig.S3B and Note 2). Since there are no PHE) to reveal Weyl-related topological properties in the self- detectable impurities in our Te crystals, such self-hole-doping hole-doped elemental semiconductor Te. The Weyl band charac- characteristics may result from Te vacancies which act as acceptors teristic further manifests as the logarithmically periodic magneto- (33). Our first-principles calculation further demonstrates that the oscillations in the quantum limit regime. Our results demonstrate presence of vacancies indeed prompts the Fermi level to shift that these “Weyl semiconductors” (semiconductors hosting Weyl toward the valence bands (SI Appendix,Fig.S4). Below we will fermions, as schematically shown in Fig. 1A) can serve as a simple present the comprehensive magnetotransport measurement re- and ideal platform for the exploration of Weyl physics and devices sults for a typical Te single-crystal sample, in which signatures of beyond conventional semimetal materials. Weyl-related physics are clearly observed.

Results NLMR Effect. Fig. 2A shows the temperature (T)-dependent re- Electronic Band Structure of Te. The chiral structure of Te with no sistance (R) curve of sample 6, which exhibits typical behaviors of inversion symmetry is shown in Fig. 1B. The first-principles band a doped semiconductor (as detailed in SI Appendix, Note 3). structure of Te is shown in SI Appendix, Fig. S2 (see Materials Fig. 2 B and C show the curves for the magnetoresistance [MR, and Methods for calculation details), which demonstrates that Te defined as (R(B)−R(0))=R(0)×100%] across a temperature is a narrow-gap semiconductor exhibiting strong spin–orbit cou- range of 25 to 100 K under perpendicular and parallel fields, pling. Its band gap (∼0.38 eV) is near the corner of the Brillouin respectively. Further insights into the temperature dependence zone, that is, the H point (Fig. 1B), which is consistent with pre- of the MR properties (across a wide temperature range of 2 to vious results (26, 31). Along the high symmetric H–Lline,two 100 K) are presented in SI Appendix, Fig. S5 and discussed in SI Weyl points arise from the crossing of two spin-splitting valence Appendix, Note 4. For the perpendicular case (B⊥I), the MR is bands, as shown in Fig. 1C. They are located below the Fermi level positive and exhibits clear linear behavior (Fig. 2B). In contrast, B‖I by −0.20 eV (designated W1)and−0.36 eV (designated W2), markedly negative MR behavior is demonstrated for (Fig. respectively. The calculated band structure exhibits good overall 2C). At T = 25 K, the negative MR achieves a magnitude up to agreement with our angle-resolved photoemission spectroscopy ∼22% at 14 T, which is comparable to the NLMR effect reported measurements (SI Appendix,Fig.S3A and Note 2), although the for several Weyl semimetals such as WTe2 (11) and Co3Sn2S2 two Weyl points cannot be seen clearly due to limited spectroscopy (7). The negative MR degrades accordingly with increasing resolution. temperature due to the thermal effect but remains observable up Te single crystals were grown via physical vapor deposition to ∼100 K. (Materials and Methods). The as-grown crystals are normally hole- The occurrence of NLMR in the presence of parallel magnetic doped and possess a typical carrier density on the order of and electric fields serves as an important signature of the chiral − 1016 cm 3, consistent with previous reports in which samples were anomaly (13, 34) in material systems hosting Weyl fermions

A B c c b

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(eV) Weyl points

+i L -0.20 W1 H’ H Energy -0.25 L L +i -0.30

W2 -0.35 -i K K’ L k k H 1 2 - -1 ky (Å )

Fig. 1. Electronic band structure of tellurium. (A) Schematic conceptual illustrations of a Weyl semimetal and a Weyl semiconductor, respectively. Weyl points are indicated by the black circles. (B) Crystal structure and Brillouin zone of trigonal Te. (C) Calculated energy dispersion of electronic bands along the H–L line

with spin–orbit coupling. The two Weyl points close to the top valence bands are marked by the black circles, which have opposite chirality (+1 for W1 and −1 for W2). The calculated eigenvalues of the rotational operation at two arbitrary momentums (k1 and k2) are also marked (see Materials and Methods for more details). (D) Calculated z component of the Berry curvature mapped on the Fermi surface (EF ∼−2 meV).

2of7 | www.pnas.org/cgi/doi/10.1073/pnas.2002913117 Zhang et al. Downloaded by guest on September 25, 2021 A 100 B 200 C 10 B⊥I B//I 2 mm 5 80 150 25 K [0001] 30 K 0 40 K 50 K -5 (%) (%) Ω ) 60 75 K 25 K ( 100

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C B (T) 50 90° C 2 2 w Fitting 0 1 0 20 40 60 80 100 -10 -5 0 5 10 0.88 0.90 0.92 0.94 0.96 0.98 1.00 T (K) B (T) cos2θ

> Fig. 2. NLMR effect. (A) Temperature-dependent resistance of sample 6. The red dashed line indicates the rough boundary between the extrinsic region (T PHYSICS 20 K) and the freeze-out region (T < 20 K). (Inset) The optical image of the as-grown rod-like crystal, where the crystalline long axis lies along [0001]. (B and C) MR measured at temperatures from 25 K to 100 K for B perpendicular and parallel to I, respectively. (D) Temperature dependence of extracted chiral co- ∝ 3τ =( 2 + μ2=π2) efficient Cw, which can be well fitted by using Cw vF v T .(Inset) Field dependence of the longitudinal conductivity at 25 K (black dots) and the corresponding fitting curve (orange line) using Eq. 1.(E) MR measured at different angles (θ) between B and I. θ is better illustrated in the inset. (F) Angular- 2 dependent Cw at 25 K, which is linearly dependent on cos θ.

(4, 6–12, 14). Theoretically, the longitudinal magnetoconductivity Thus far, the negative MR observed in Te crystals under B‖I can be described using the following equation (8): could be well explained via the chiral anomaly, reproducing previous observations of Weyl semimetals (for data from addi- 2 σ(B) = (1 + CwB ) · σWAL + σN, [1] tional samples see SI Appendix, Figs. S10 and S11). Other possible origins, such as weak localization and current jetting, C where w is a positive parameter representing the chiral anomaly were examined carefully and eventually ruled out (SI Appendix, σ σ contribution to the conductivity and WAL and N are the conduc- Note 6). tivities imparted by the weak antilocalization effect and conventional nonlinear band contributions√̅̅̅̅ around the Fermi level, PHE. The PHE, which manifests as the appearance of in-plane σ = σ + a B σ−1 = ρ + A · B2 σ respectively. Here WAL 0 , N 0 ,and 0 is transverse voltage when the in-plane magnetic field is not ex- D Inset SI Ap- zero-field conductivity. As demonstrated in Fig. 2 , ( actly parallel or perpendicular to the current, is another im- pendix,Fig.S8A), the experimental data can be well-fitted via Eq. 1 C portant transport signature of Weyl physics (19, 20). The , and the extracted w increases monotonically with decreasing A D C configuration for measuring the PHE is depicted in Fig. 3 , temperature down to 15 K (Fig. 2 ). The chiral coefficient w can Inset C ∝ v3 τ =(T2 + μ2=π2) . Even though a conventional Hall component may still be depicted using the formula w F v (10), the theoretically expected temperature-dependent behavior for the existinsuchasetupduetomisalignmentbetweentheactual rotation plane and the sample plane, it can be eliminated by chiral anomaly, where vF is the Fermi velocity, τv is the chirality- changing scattering time, and μ is the chemical potential relative to simply averaging the Hall resistances measured under positive the Weyl points. and negative magnetic fields. Fig. 3A shows the symmetrized To gain further insight into the observed negative MR, we also planar Hall resistivity ρxy vs. φ measured at 25 K over a range of studied its angular-dependent character. Fig. 2E shows the MR magnetic fields (φ is the angle between the directions of the in- for different angles (θ) of B with respect to I as measured at 25 K. plane magnetic field and the current). The in-plane anisotropic Rotating the magnetic field away from B⊥I prompts the positive MR [AMR, defined as (R(φ)−R(φ = 90°))=R(φ = 90°)×100%] MR magnitude to decrease accordingly. Signatures of negative was measured simultaneously, and the results are plotted in SI MR occur for θ < 25° and peak at θ ∼ 0° (B‖I). In addition, the Appendix,Fig.S13A. Both the planar Hall resistivity and the in- Cw obtained from the fitted data reveals linear dependence of plane AMR display a 180° periodic angular dependence, with cos2θ (Fig. 2F and SI Appendix,Fig.S8B), which is consistent with the AMR reaching its maximum and minimum at 90° and 0°, observations in topological semimetals (12). Similar angular- respectively, while ρxy does so at 135° and 45°, respectively. dependent MR behavior is observed at temperatures down to These observations are well consistent with typical characters 2 K, although the negative MR is largely outweighed by the enor- of the PHE. mous positive MR background attributed to magnetic freeze-out Theoretically, PHE arising from the chiral anomaly could be (35) (SI Appendix,Fig.S9). fitted via the following equation (17) (SI Appendix, Note 7):

Zhang et al. PNAS Latest Articles | 3of7 Downloaded by guest on September 25, 2021 4 1.0 A Vxy B 2 K 25 K 3 T 3 K 5 K 7 K 10 K 2 φ Vxx 0.5 15 K B I 20 K cm) 25 K cm) 30 K μΩ 4 μΩ 0 0.0 40 K 4 50 K (10

(10 1 T 75 K xy xy 3 T ρ 100 K ρ -2 5 T -0.5 9 T 14 T Fitting -4 -1.0 0 60 120 180 240 300 360 0 60 120 180 240 300 360 φ (°) φ (°) C D 6 1.5 25 K 3 T ∝ B

cm) 4

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(10

chiral

chiral 0.5 ∝ 2 Δρ ρ B Experiment Fitting 1 Δ Fitting 2 0 0.0 036912 0 20 40 60 80 100 B (T) T (K)

ρ Fig. 3. PHE. (A) Angular-dependent xy measured under different B at 25 K. The fitting curves using Eq. 2 are also plotted (red lines). (Inset) Schematic of the φ ρ PHE measurement geometry. is the angle between the directions of the in-plane magnetic field and the current. (B) Angular-dependent xy measured at 3 T for different temperatures. (C) Magnetic field dependence of the chiral-anomaly-induced anisotropic resistivity Δρchiral obtained at 25 K. In the low field regime, Δρchiral is proportional to B2.(D) Evolution of the extracted Δρchiral at 3 T as a function of temperature.

chiral chiral 2 ρxy =−Δρ sin φ cos φ + b Δρ cos φ + c, [2] the PHE and NLMR in the present Te crystals (see SI Appendix, Note 4 for further discussion). where the first term is the intrinsic PHE due to the chiral The above-discussed experimental observations of the NLMR anomaly and the second and third terms are, respectively, the in- and PHE in Te crystals adhere well to the chiral-anomaly mech- plane anisotropic MR and longitudinal offset caused by the Hall anism in materials possessing Weyl fermions. On the other hand, it chiral configuration misalignment. The quantity Δρ = ρ⊥ − ρ‖ is the has been theoretically proposed that both the NLMR and PHE anisotropic resistivity originating from the chiral anomaly. Fig. can be observed in a system with nonzero Berry curvature at the 3A shows that Eq. 2 provides an excellent fit for the experimental Fermi level (36, 37). One notices that the nonzero Berry curvature ρxy across a range of magnetic fields. For a more quantitative is also induced by either inversion or time-reversal symmetry chiral PHE analysis, we extracted the field-dependent Δρ at 25 K. breaking, which is the same to the realization of Weyl fermions (3, chiral The plot in Fig. 3C clearly demonstrates that the obtained Δρ 38). Therefore, both mechanisms originate from symmetry break- increases monotonically with increasing magnetic field. For B < ing. To explore the band topology of Te, the Berry curvature (Ω)of chiral 2 3T,Δρ is proportional to B . In the relatively high field the highest occupied valence band is calculated. The Fermi level B > Δρchiral regime ( 4.5 T), exhibits nearly linear variation with (E ) is set at ∼2 meV below the valence band maximum according B B = F and displays no evidence of saturation up to 14 T. Similar to the typical carrier densities of samples at 25 K (SI Appendix,Fig. field dependence has been previously reported for the PHE in S14A). Nonzero Berry curvature on the Fermi surface, shown in topological semimetals (17, 18). D B Fig. 1 , is dominated by the contribution of the Weyl point W1 Fig. 3 shows the temperature-dependent PHE behavior (see (see SI Appendix,Fig.S14B and Note 8). This gives a strong evi- SI Appendix B , Fig. S13 for the AMR data taken simultaneously). dence of correlation between the observed NLMR and Weyl chiral With increasing temperature, the extracted Δρ decreases physics. Recent theoretical works have predicted that Te can be accordingly until it becomes negligible at T ∼ 100 K (Fig. 3D). turned into a Weyl semimetal via closing the band gap by applying Such behavior is remarkably consistent with that of the Cw shown high pressure (26, 27). In contrast, here we reveal that the pristine in Fig. 2D, suggesting the same underlying physical origin of both semiconductor Te can be considered a “Weyl semiconductor,”

4of7 | www.pnas.org/cgi/doi/10.1073/pnas.2002913117 Zhang et al. Downloaded by guest on September 25, 2021 since Weyl fermions and related exotic transport properties can be respectively. The clear deviation from the linear dependence of directly realized without gap closing. 1/Bn on n further demonstrates the inability of the Shubnikov–de Haas effect to account for the observed oscillations. In contrast, Log-Periodic MR/Hall Oscillations. Unusual properties of material the well-defined linear behavior observed for the log(Bn)vs.n systems under the quantum limit, wherein all of the carriers are curve indicates the log-periodicity. Note that the point at ∼3.1 T condensed to the lowest Landau level, have been a topic of in- deviates from the linear trend, possibly because this field is below terest since their discovery (21, 22, 39–44). As a promising can- the quantum limit, or perhaps as a result of some unavoidable didate for “Weyl semiconductor” with a relatively small carrier anomalies during the background subtraction (SI Appendix, Note density and unique band topology, Te offers a suitable platform 9). Fig. 4B shows the MR data taken across a wider field range at for the study of Weyl physics beyond the quantum limit. For the 2.5 K using pulsed fields. These data exhibit more oscillations than − studied Te sample 6 with carrier density of ∼3.9 × 1016 cm 3 at do the data shown in Fig. 4A, while both the low field data and the (B ) 2 K, the estimated critical field Q at which the system enters the extracted Bn are consistent across both datasets (see also Fig. 4C). quantum limit is ∼4.4 T (SI Appendix, Note 9), which is readily Fig. 4D presents the Hall data obtained simultaneously with achievable. In fact, the MR curves for B⊥I obtained in the low theMRdatafromFig.4A. The log-periodic character is also ev- SI Appendix B temperature range of 2 to 10 K ( ,Fig.S5 ) exhibit an ident from a comparison of the index dependence of the 1/Bn and B oscillatory character even above Q, which cannot be attributed to log(Bn)curves(Fig.4E). The fast Fourier transform for log(ΔRxx) the conventional Shubnikov–de Haas effect. and log(ΔRxy) shows nearly identical periods, indicating the same To reveal the feature of these unexpected magneto-oscillations, origin for both the observed MR and Hall oscillations. we conducted magnetotransport measurements of sample 6 in two As aforementioned, the oscillations with peculiar log-periodicity high-magnetic-field facilities (Materials and Methods). Fig. 4A have been experimentally observed only recently, and only in shows the MR data taken under static fields at 1.8 K, which exhibit ZrTe5 (21) and its sister compound HfTe5 (22). The phenomenon oscillations with more periods. Following previous studies (21, 22), is regarded as the manifestation of DSI in solid-state systems and a suitable background (SI Appendix,Fig.S15A) obtained via is theoretically considered to be universal in topological materials smoothing was subtracted to better distinguish the oscillations hosting Dirac/Weyl fermions with Coulomb attraction (21–24). (Materials and Methods). The results are presented in Fig. 4A, The underlying mechanism can be summarized as follows (21, 24): Bottom with dashed vertical lines marking the positions of peaks/ Dirac/Weyl fermions in an attractive Coulomb potential possess a dips. Interestingly, the interval between adjacent dashed lines continuous scaling symmetry. In the supercritical case α > 1 2 grows larger as the magnetic field increases. For better pre- (α = e =4π«0ћvF is the fine-structure constant, where e is the el- B PHYSICS sentation, the extrema fields ( n) corresponding to these peaks/ ementary charge, «0 is the vacuum permittivity, ћ is the reduced dips were extracted, and 1/Bn and log(Bn) were plotted as a Planck constant, and vF is the Fermi velocity), these massless function of the extrema index (n), respectively (Fig. 4C). Here Dirac/Weyl fermions tend to form two-body quasi-bound states the peaks and dips are assigned to integers and half-integers, with opposite charge centers, like charged impurities. Owing to

600 A 1.8 K B1000 2.5 K C 0.3 b : peak

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0.4 D E F Δ 1.8 K peak Rxx d : ΔR Ω ) 10 dip xy 0.02 ( 0.3 0.4 xy R

1 Quantum limit 10 (T) (1/T)

0.2 n n B 0.2 0.01 1/B 0 Amplitude Ω ) (

xy 0.1 -1 R

Δ -2 0.0 0.00 0.0 0 5 10 15 20 25 30 234234 0 5 10 15 B (T) n Frequency

Fig. 4. Log-periodic MR and Hall oscillations. (A and B) MR taken under static magnetic fields at 1.8 K and under pulsed fields at 2.5 K, respectively. (Bottom) The extracted oscillations after background subtraction. The positions of peaks and dips of oscillations are marked by dashed lines. (C) Index dependence of

1/Bn and log(Bn) from the MR oscillation data shown in A and B.(D) Hall resistance measured at 1.8 K and the extracted oscillations. Dashed lines mark the positions of peaks and dips. (E) Index dependence of 1/Bn and log(Bn) for the Hall oscillations. (F) Comparison of fast Fourier transform results for the MR oscillations from A and the Hall oscillations from D. The starting field of fast Fourier transform analysis was set to be ∼7 T, from which the log-periodic oscillations become distinct. The frequency corresponding to the peaks is 3.1 (2.8) for the MR (Hall) oscillations, corresponding to a period of log(B) ∼0.32 (0.36).

Zhang et al. PNAS Latest Articles | 5of7 Downloaded by guest on September 25, 2021 the relativistic energy–momentum dispersion relation of a Dirac/ to remove trace oxygen. The quartz tube was heated to 1,000 °C over the Weyl band, the energies of the quasi-bound states constitute a course of 5 h and subsequently maintained at 1,000 °C for 1 h. Then, the geometric series that matches the general feature of DSI. The tube was cooled at a rate of 20 °C/h to 400 °C and 300 °C for the hot and increasing magnetic field enables the quasi-bound states with dif- the cold zone, respectively. After maintaining the set temperature for 2 wk, the tube was slowly cooled to room temperature, and rod-like silvery crystals ferent energies approaching the Fermi energy continuously. The (typical dimensions 5 × 0.3 × 0.1 mm3) were obtained (Fig. 2A, Inset). as-induced scattering between free carriers and quasi-bound states thus influences the transport properties (21), manifesting as the Structure and Composition Characterization. The structure of Te single crystals log-periodic oscillations for the measured MR and Hall data. was measured by a Rigaku SmartLab X-ray diffractometer (XRD) at room Here we would like to attribute the observed log-periodic temperature using monochromatic Cu-Kα radiation (λ = 1.5418 Å). A typical oscillations in Te crystals to the DSI after excluding another XRD spectrum is shown in SI Appendix, Fig. S1A. In the wide range spectrum, possible origin (SI Appendix, Note 9). The quasi-bound states are only the sharp peaks from the Te (1010) lattice plane are observed. The likely formed between Weyl fermions located near the top of microdiffraction XRD was measured by Rigaku D/Max-RAPID II with Cu-Kα valence bands and opposite charge centers. Since the self-hole- radiation (λ = 1.5418 Å) at room temperature. The large curved imaging doping characteristic may result from Te vacancies acting as ac- plate detector with a 210° aperture allows a two-dimensional diffraction ceptors (33), such vacancies are likely to serve as the negative imaging over a broad 2θ range, such that the detection of many lattice charge centers that exert Coulomb attraction on the positive Weyl planes is possible without breaking the single crystal. As also shown in SI SI Appendix Appendix, Fig. S1A, the measured data match well with the standard pow- fermions (see ,Fig.S4for the calculation results). der XRD pattern of Te. SI Appendix, Fig. S1B shows the high-resolution For the log-periodic oscillations, the scale factor (λ) for the B B scanning transmission electron microscopy (Thermo Fisher Themis G2 60-300) DSI can be defined as n/ n+1. As such, this quantity is esti- imaging and selective area electron diffraction pattern of the exposed sur- mated to be ∼2.33 for sample 6, as based on linear fitting of the face, which both further demonstrate the high quality of the single crystal at log(Bn) vs. n plot (Fig. 4C). This value of λ is consistent with the atomic scale. The crystalline long axis, that is, the growth direction, is the that obtained from the fast Fourier transform analysis, which [0001] direction. The lattice parameters were determined to be a = b = 4.48 is ∼2.10 (∼2.28) for the MR (Hall) oscillations (Fig. 4F). In ref. Å, c = 6.00 Å. The chemical composition of the crystals was examined by 21, the semiclassical quantization√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ condition leads to the DSI an energy dispersive spectrometer (INCA detector; Oxford Instruments) attached to a field-emission scanning electron microscope (Sirion 200; π=s λ = e2 0 s = (Zα)2 − κ2 with √̅̅̅̅̅̅̅̅̅̅̅̅̅. Here 0 , which can be simplified FEI) operated at 20 kV. No impurity elements were detected (SI Appendix, Fig. S1C). 2 to s0 = α − 1 when we consider the charge number Z = 1 and κ =± the lowest angular momentum channel with 1 (21). Thus, Magnetic and Transport Property Measurements. The magnetic properties the fine-structure constant (α) is calculated to be ∼7.5, far ex- were analyzed using a Quantum Design SQUID-VSM system with the mag- ceeding 1/137 due to the small Fermi velocity of Te crystals. netic field up to 7 T and the temperature down to 2 K. For electrical 2 According to α = e =4π«0ћvF, the Fermi velocity vF is calculated transport measurements, electron beam lithography was utilized to define to be ∼2.9 × 105 m/s, which is comparable to the value (∼1.9 × the electrode pattern on the surface of the Te crystals, then 10 nm Pd and 105 m/s) estimated from the calculated band structure. The small 50 nm Au were deposited as electrode materials using the e-beam evapo- ration method. Silver epoxy was subsequently used to affix the wires to the vF ensures the supercritical condition, promoting the formation Pd/Au contacts. The measurements were performed in a Quantum Design of quasi-bound states with DSI (21). Physical Property Measurement System with the magnetic field up to 14 T. With increasing temperature, the amplitudes for both the MR The planar Hall measurements were performed via rotating the sample in a and Hall oscillations decline accordingly before becoming in- fixed magnetic field. The angle of 105° is the starting point for the mea- discernible at temperatures above 15 K (SI Appendix, Fig. S16 surement, and the mechanical backlash of the rotator normally leads to and Note 10). Similar behaviors are also observed in other slight discontinuity at an angle of around 105° in the planar Hall data. High- samples (SI Appendix, Fig. S17). This can be attributed to the field transport measurements were carried out in the High Magnetic Field thermionic excitation that weakens the Coulomb attraction. Laboratory in Hefei (static field, 33 T) and the Wuhan National High Mag- netic Field Center (pulsed field, 53 T). Due to the rod-like shape of Te crystals However, compared to the cases of ZrTe5 and HfTe5, the critical temperature above which the oscillations disappear is much (Fig. 1F, Inset), the current was applied along the long axis (the [0001] di- smaller for the present Te crystals, implying the relatively small rection) during all transport measurements conducted in this study. binding energy of the as-formed quasi-bound states (as discussed SI Appendix Note 10 Angle-Resolved Photoemission Spectroscopy Measurements. The angle- in , ). resolved photoemission spectroscopy measurements were performed at Conclusion beamline 9U (Dreamline) of the Shanghai Synchrotron Radiation Facility with a Scienta Omicron DA30L analyzer. The angle resolution was 0.1°, and The present work clearly demonstrates the realization of Weyl- the combined instrumental energy resolution was better than 20 meV. The related properties in a narrow-gap semiconductor with strong photon energy used in our experiments ranged from 25 to 90 eV, and the spin–orbit coupling and without inversion symmetry, thus greatly measurements were conducted at 7 to 20 K. The (1120) surface of the single broadening the scope of topological materials. Furthermore, the crystals was cleaved, corresponding to the ky–kz plane in the hexagonal highly tunable electronic performance of Weyl semiconductors Brillouin zone. The Fermi level of the Te samples was compared to a gold enables further manipulation of Weyl fermions through various film reference which was evaporated onto the sample holder. All of the −11 means commonly adopted in semiconductor electronics, such as measurements were conducted under a vacuum better than 7 × 10 mbar. electrostatic gating and optical illumination. Moreover, the observation of the log-periodic magneto-oscillations further confirms Electronic Band Structure Calculations. First-principles calculations were car- the universality of DSI in topological materials hosting Dirac/Weyl ried out within the framework of density functional theory using the Vienna Ab initio Simulation Package (45). All of the calculations were performed fermions with Coulomb attraction, thus endowing Te with a novel with a plane-wave cutoff of 450 eV on the 9 × 9 × 9 Γ-centered k-mesh, and platform for the study of intriguing ground states beyond the the convergence criterion of energy was 10−6 eV. The generalized gradi- quantum limit. The successful introduction of Weyl physics into ent approximation with the Perdew–Burke–Ernzerhof functional (46) was semiconductor systems therefore offers a new dimension for the adopted to describe electron exchange and correlation. During structure future design of topological semiconductor devices. optimization, all atoms were fully relaxed until the force on each atom was smaller than 0.01 eV/Å. In order to obtain a reliable band gap, the HSE06 Materials and Methods hybrid functional (47) was⃒ used in the band structure calculation. The ro- ⃒ Growth of Te Single Crystals. High-quality Te single crystals were grown using 2 tational symmetry g = {C2x^⃒00 } is a generator along the H–L line. Including a physical vapor transport technique. High-purity (99.999% pure) Te powder 3 was loaded into a quartz tube, and a small amount of C powder was added the spin–orbit coupling, the rotational symmetry g satisfies g2 =−1, so its

6of7 | www.pnas.org/cgi/doi/10.1073/pnas.2002913117 Zhang et al. Downloaded by guest on September 25, 2021 eigenvalues are ±i. The first-principles calculated eigenvalues for g at two data. The second derivative method is a widely adopted approach to arbitrary momentums (k1 and k2) are shown in Fig. 1C. The eigenvalue confirm the exact positions of the extrema fields (21). We thus compared switches between the two bands at k1 and k2. Thus, there must exist a band the extracted oscillations obtained via these two methods and obtained crossing point with twofold degeneracy between k1 and k2. Hence, the Weyl consistent results (SI Appendix,Fig.S15), demonstrating the validity of our point of W1 is protected by rotational symmetry. It is well known that the data processing. Kramers degeneracy is a twofold degeneracy at time-reversal-invariant momentum. Here, the Weyl point of W is at L point, which is a time- 2 Data Availability. All data are available in the main text or SI Appendix. reversal-invariant momentum. Hence, the Weyl point of W2 is protected by time-reversal symmetry. To accurately calculate the Berry curvature in the ACKNOWLEDGMENTS. This work was supported by the National Natural Science Brillouin zone, we have fitted a Wannier tight-binding Hamiltonian by using Foundation of China (Grants 11974324, U1832151, 11804326, 11774325, and the WANNIER90 package (48). The p orbitals of Te are used as the initial 21603210), the Strategic Priority Research Program of Chinese Academy of projectors for Wannier Hamiltonian construction. The chirality of a Weyl Sciences (CAS) (Grant XDC07010000), the National Key Research and Develop- point is calculated by integrating the Berry curvature on a surface enclosing ment Program of China (Grants 2017YFA0403600 and 2017YFA0204904), the the Weyl point. Anhui Initiative in Quantum Information Technologies (Grant AHY170000), Hefei Science Center CAS (Grant 2018HSC-UE014), the Anhui Provincial Natural Data Processing. Due to the nonlinearity of Hall data, the carrier density was Science Foundation (Grant 1708085QA20), the Fundamental Research Funds for extracted from the linear part of the low-field Hall resistivity. Attempts to the Central Universities (Grants WK2030040087 and WK3510000007), and the use the two-carrier fitting model were unsuccessful. For the log-periodic Science, Technology, and Innovation Commission of Shenzhen Municipality oscillations, the background was obtained through smoothing the MR/Hall (Grant KQTD2016022619565991).

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