Magnetotransport signatures of Weyl physics anddiscrete scale invariance in the elementalsemiconductor telluriumNan 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 Scienceand Technology of China, 230026 Hefei, Anhui, China; bSynergetic Innovation Center of Quantum Information & Quantum Physics, University of Scienceand 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, Universityof 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, ChineseAcademy 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 February15, 2020)

The study of topological materials possessing nontrivial bandstructures enables exploitation of relativistic physics and develop-ment of a spectrum of intriguing physical phenomena. However,previous studies of Weyl physics have been limited exclusivelyto semimetals. Here, via systematic magnetotransport measure-ments, two representative topological transport signatures ofWeyl physics, the negative longitudinal magnetoresistance andthe planar Hall effect, are observed in the elemental semiconduc-tor tellurium. More strikingly, logarithmically periodic oscillationsin both the magnetoresistance and Hall data are revealed beyondthe quantum limit and found to share similar characteristics withthose observed in ZrTe5 and HfTe5. The log-periodic oscillationsoriginate from the formation of two-body quasi-bound statesformed between Weyl fermions and opposite charge centers, theenergies of which constitute a geometric series that matches thegeneral feature of discrete scale invariance (DSI). Our discoveryreveals the topological nature of tellurium and further confirmsthe universality of DSI in topological materials. Moreover, intro-duction of Weyl physics into semiconductors to develop “Weylsemiconductors” provides an ideal platform for manipulating fun-damental Weyl fermionic behaviors and for designing futuretopological devices.

Weyl semiconductor | tellurium | negative longitudinalmagnetoresistance | planar Hall effect | log-periodic oscillations

The discovery of nontrivial topological phases has reshaped ourunderstanding of solid-state materials (1–3). Besides band

structure measurements, transport properties can also providesome unique signatures to identify these exotic topological phases.For example, two well-known transport signatures in Dirac/Weylsemimetals are the chiral-anomaly induced negative longitudinalmagnetoresistance (NLMR) (4–14) and planar Hall effect (PHE)(12, 15–20). Recently, a new type of quantum oscillations featur-ing log-periodicity, wherein the extrema magnetic fields of oscil-lations constitute a geometric series, was discovered in ZrTe5 (21)and HfTe5 (22) beyond the quantum limit. Physically, these un-usual log-periodic oscillations can be attributed to the manifesta-tion of discrete scale invariance (DSI) (21, 23, 24), whereby self-reproduction occurs at specific scales that form a geometric series.The DSI character of topological materials is rooted in the two-body quasi-bound states formed between linearly dispersed quasi-particles and opposite charge centers, with the energy levels sat-isfying a geometric progression (21). In theory, log-periodicoscillations are considered a universal feature of materials host-ing Dirac/Weyl fermions with Coulomb attraction (21, 23, 24).

Until now, Dirac/Weyl physics has been generally regarded asunique to semimetals, with the Dirac/Weyl points located aroundthe Fermi level. Nevertheless, considering the high tunability andcompatibility with modern electronic industry, the semiconductorwill be a better candidate material for designing topologicalelectronic devices if exotic transport phenomena related to Dirac/Weyl fermions can be realized. Tellurium (Te) is a narrow band-gap semiconductor possessing strong spin–orbit coupling, and itlacks space inversion symmetry due to the characteristic chiralcrystal structure. As an elemental semiconducting material, Te hasbeen widely studied for its lattice dynamics, band structure,transport, and optical properties, with key discoveries transpiringseveral decades ago (25). Recently, the Weyl semimetal phase wastheoretically proposed in Te by closing its bulk band gap with

Significance

Recent intensive investigations have revealed unique elec-tronic transport properties in solids hosting Weyl fermions,which were originally proposed in high-energy physics. Up tonow, the discovered Weyl systems have been limited to semi-metal compounds. Here we demonstrate that the elementalsemiconductor tellurium is a Weyl semiconductor, with typicalWeyl signatures, including the negative longitudinal magne-toresistance, the planar Hall effect, as well as the intriguinglogarithmically periodic magneto-oscillations in the quantumlimit regime. SuchWeyl semiconductors offer a simple platformfor the exploration of novel Weyl physics and topological de-vice applications based on semiconductors and moreover con-firm the universality of discrete scale invariance in topologicalmaterials.

Author contributions: L.L. and C.Z. designed research; N.Z., G.Z., L.L., P.W., L.X., B.C., H.L.,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.analyzed data; and N.Z., G.Z., L.L., Z.W., and C.Z. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission. J.K. is a guest editor invited by theEditorial Board.

Published under the PNAS license.1N.Z., G.Z., and L.L. contributed equally to this work.2To whom correspondence may be addressed. Email: lilin@ustc.edu.cn, zfwang15@ustc.edu.cn, or cgzeng@ustc.edu.cn.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2002913117/-/DCSupplemental.

First published May 12, 2020.

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external perturbations (26, 27), which has renewed interest inexploring the possible topological characters of Te (26, 28–30).In this work, we report two transport signatures (NLMR and

PHE) to reveal Weyl-related topological properties in the self-hole-doped elemental semiconductor Te. The Weyl band charac-teristic further manifests as the logarithmically periodic magneto-oscillations in the quantum limit regime. Our results demonstratethat these “Weyl semiconductors” (semiconductors hosting Weylfermions, as schematically shown in Fig. 1A) can serve as a simpleand ideal platform for the exploration of Weyl physics and devicesbeyond conventional semimetal materials.

ResultsElectronic Band Structure of Te. The chiral structure of Te with noinversion symmetry is shown in Fig. 1B. The first-principles bandstructure of Te is shown in SI Appendix, Fig. S2 (see Materialsand Methods for calculation details), which demonstrates that Teis a narrow-gap semiconductor exhibiting strong spin–orbit cou-pling. Its band gap (∼0.38 eV) is near the corner of the Brillouinzone, that is, the H point (Fig. 1B), which is consistent with pre-vious results (26, 31). Along the high symmetric H–L line, twoWeyl points arise from the crossing of two spin-splitting valencebands, as shown in Fig. 1C. They are located below the Fermi levelby −0.20 eV (designated W1) and −0.36 eV (designated W2),respectively. The calculated band structure exhibits good overallagreement with our angle-resolved photoemission spectroscopymeasurements (SI Appendix, Fig. S3A and Note 2), although thetwo Weyl points cannot be seen clearly due to limited spectroscopyresolution.Te single crystals were grown via physical vapor deposition

(Materials and Methods). The as-grown crystals are normally hole-doped and possess a typical carrier density on the order of1016 cm−3, consistent with previous reports in which samples were

grown using a similar method (30, 32). The hole-doped characteris further verified via the angle-resolved photoemission spectros-copy results (SI Appendix, Fig. S3B and Note 2). Since there are nodetectable impurities in our Te crystals, such self-hole-dopingcharacteristics may result from Te vacancies which act as acceptors(33). Our first-principles calculation further demonstrates that thepresence of vacancies indeed prompts the Fermi level to shifttoward the valence bands (SI Appendix, Fig. S4). Below we willpresent the comprehensive magnetotransport measurement re-sults for a typical Te single-crystal sample, in which signatures ofWeyl-related physics are clearly observed.

NLMR Effect. Fig. 2A shows the temperature (T)-dependent re-sistance (R) curve of sample 6, which exhibits typical behaviors ofa doped semiconductor (as detailed in SI Appendix, Note 3).Fig. 2 B and C show the curves for the magnetoresistance [MR,defined as (R(B) − R(0))=R(0) × 100%] across a temperaturerange of 25 to 100 K under perpendicular and parallel fields,respectively. Further insights into the temperature dependenceof the MR properties (across a wide temperature range of 2 to100 K) are presented in SI Appendix, Fig. S5 and discussed in SIAppendix, Note 4. For the perpendicular case (B⊥I), the MR ispositive and exhibits clear linear behavior (Fig. 2B). In contrast,markedly negative MR behavior is demonstrated for B‖I (Fig.2C). At T = 25 K, the negative MR achieves a magnitude up to∼22% at 14 T, which is comparable to the NLMR effect reportedfor several Weyl semimetals such as WTe2 (11) and Co3Sn2S2(7). The negative MR degrades accordingly with increasingtemperature due to the thermal effect but remains observable upto ∼100 K.The occurrence of NLMR in the presence of parallel magnetic

and electric fields serves as an important signature of the chiralanomaly (13, 34) in material systems hosting Weyl fermions

rotcudnocimes lyeWlatemimes lyeW

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bc

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Fig. 1. Electronic band structure of tellurium. (A) Schematic conceptual illustrations of a Weyl semimetal and aWeyl semiconductor, respectively. Weyl pointsare 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 linewith 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 −1for W2). The calculated eigenvalues of the rotational operation at two arbitrary momentums (k1 and k2) are also marked (seeMaterials and Methods for moredetails). (D) Calculated z component of the Berry curvature mapped on the Fermi surface (EF ∼ −2 meV).

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(4, 6–12, 14). Theoretically, the longitudinal magnetoconductivitycan be described using the following equation (8):

σ(B) = (1 + CwB2) · σWAL + σN, [1]

where Cw is a positive parameter representing the chiral anomalycontribution to the conductivity and σWAL and σN are the conduc-tivities imparted by the weak antilocalization effect and conven-tional nonlinear band contributions around the Fermi level,respectively. Here σWAL = σ0 + a

B

√, σ−1N = ρ0 + A ·B2, and σ0 is

zero-field conductivity. As demonstrated in Fig. 2D, Inset (SI Ap-pendix, Fig. S8A), the experimental data can be well-fitted via Eq.1, and the extracted Cw increases monotonically with decreasingtemperature down to 15 K (Fig. 2D). The chiral coefficient Cw canbe depicted using the formula Cw ∝ v3Fτv=(T2 + μ2=π2) (10), thetheoretically expected temperature-dependent behavior for thechiral anomaly, where vF is the Fermi velocity, τv is the chirality-changing scattering time, and μ is the chemical potential relative tothe Weyl points.To gain further insight into the observed negative MR, we also

studied its angular-dependent character. Fig. 2E shows the MRfor different angles (θ) of B with respect to I as measured at 25 K.Rotating the magnetic field away from B⊥I prompts the positiveMR magnitude to decrease accordingly. Signatures of negativeMR occur for θ< 25° and peak at θ ∼ 0° (B‖I). In addition, theCw obtained from the fitted data reveals linear dependence ofcos2θ (Fig. 2F and SI Appendix, Fig. S8B), which is consistent withobservations in topological semimetals (12). Similar angular-dependent MR behavior is observed at temperatures down to2 K, although the negative MR is largely outweighed by the enor-mous positive MR background attributed to magnetic freeze-out(35) (SI Appendix, Fig. S9).

Thus far, the negative MR observed in Te crystals under B‖Icould be well explained via the chiral anomaly, reproducingprevious observations of Weyl semimetals (for data from addi-tional samples see SI Appendix, Figs. S10 and S11). Otherpossible origins, such as weak localization and current jetting,were examined carefully and eventually ruled out (SI Appendix,Note 6).

PHE. The PHE, which manifests as the appearance of in-planetransverse voltage when the in-plane magnetic field is not ex-actly parallel or perpendicular to the current, is another im-portant transport signature of Weyl physics (19, 20). Theconfiguration for measuring the PHE is depicted in Fig. 3A,Inset. Even though a conventional Hall component may stillexist in such a setup due to misalignment between the actualrotation plane and the sample plane, it can be eliminated bysimply averaging the Hall resistances measured under positiveand negative magnetic fields. Fig. 3A shows the symmetrizedplanar Hall resistivity ρxy vs. φmeasured at 25 K over a range ofmagnetic fields (φ is the angle between the directions of the in-plane magnetic field and the current). The in-plane anisotropicMR [AMR, defined as (R(φ) − R(φ = 90°))=R(φ = 90°) × 100%]was measured simultaneously, and the results are plotted in SIAppendix, Fig. S13A. Both the planar Hall resistivity and the in-plane AMR display a 180° periodic angular dependence, withthe AMR reaching its maximum and minimum at 90° and 0°,respectively, while ρxy does so at 135° and 45°, respectively.These observations are well consistent with typical charactersof the PHE.Theoretically, PHE arising from the chiral anomaly could be

fitted via the following equation (17) (SI Appendix, Note 7):

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0-3 T

-2)

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525 K

Cw (1

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-2)

cos2θ

CBA

FED

Fig. 2. NLMR effect. (A) Temperature-dependent resistance of sample 6. The red dashed line indicates the rough boundary between the extrinsic region (T >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-efficient Cw, which can be well fitted by using Cw ∝ v3F τv=(T2 + μ2=π2). (Inset) Field dependence of the longitudinal conductivity at 25 K (black dots) and thecorresponding 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-dependent Cw at 25 K, which is linearly dependent on cos2θ.

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ρxy = −Δρchiral sinφ cosφ + b Δρchiral cos2 φ + c, [2]

where the first term is the intrinsic PHE due to the chiralanomaly and the second and third terms are, respectively, the in-plane anisotropic MR and longitudinal offset caused by the Hallconfiguration misalignment. The quantity Δρchiral = ρ⊥ − ρ‖ is theanisotropic resistivity originating from the chiral anomaly. Fig.3A shows that Eq. 2 provides an excellent fit for the experimentalρxy across a range of magnetic fields. For a more quantitativePHE analysis, we extracted the field-dependent Δρchiral at 25 K.The plot in Fig. 3C clearly demonstrates that the obtained Δρchiralincreases monotonically with increasing magnetic field. For B <3 T, Δρchiral is proportional to B2. In the relatively high fieldregime (B > 4.5 T), Δρchiral exhibits nearly linear variation withB and displays no evidence of saturation up to B = 14 T. Similarfield dependence has been previously reported for the PHE intopological semimetals (17, 18).Fig. 3B shows the temperature-dependent PHE behavior (see

SI Appendix, Fig. S13B for the AMR data taken simultaneously).With increasing temperature, the extracted Δρchiral decreasesaccordingly until it becomes negligible at T ∼ 100 K (Fig. 3D).Such behavior is remarkably consistent with that of the Cw shownin Fig. 2D, suggesting the same underlying physical origin of both

the PHE and NLMR in the present Te crystals (see SI Appendix,Note 4 for further discussion).The above-discussed experimental observations of the NLMR

and PHE in Te crystals adhere well to the chiral-anomaly mech-anism in materials possessingWeyl fermions. On the other hand, ithas been theoretically proposed that both the NLMR and PHEcan be observed in a system with nonzero Berry curvature at theFermi level (36, 37). One notices that the nonzero Berry curvatureis also induced by either inversion or time-reversal symmetrybreaking, which is the same to the realization of Weyl fermions (3,38). Therefore, both mechanisms originate from symmetry break-ing. To explore the band topology of Te, the Berry curvature (Ω) ofthe highest occupied valence band is calculated. The Fermi level(EF) is set at ∼2 meV below the valence band maximum accordingto the typical carrier densities of samples at 25 K (SI Appendix, Fig.S14A). Nonzero Berry curvature on the Fermi surface, shown inFig. 1D, is dominated by the contribution of the Weyl point W1(see SI Appendix, Fig. S14B and Note 8). This gives a strong evi-dence of correlation between the observed NLMR and Weylphysics. Recent theoretical works have predicted that Te can beturned into a Weyl semimetal via closing the band gap by applyinghigh pressure (26, 27). In contrast, here we reveal that the pristinesemiconductor Te can be considered a “Weyl semiconductor,”

0 60 120 180 240 300 360-4

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2 K 3 K 5 K 7 K 10 K 15 K 20 K 25 K 30 K 40 K 50 K 75 K 100 K

BA

DC

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

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since Weyl fermions and related exotic transport properties can bedirectly realized without gap closing.

Log-Periodic MR/Hall Oscillations. Unusual properties of materialsystems under the quantum limit, wherein all of the carriers arecondensed to the lowest Landau level, have been a topic of in-terest since their discovery (21, 22, 39–44). As a promising can-didate for “Weyl semiconductor” with a relatively small carrierdensity and unique band topology, Te offers a suitable platformfor the study of Weyl physics beyond the quantum limit. For thestudied Te sample 6 with carrier density of ∼3.9 × 1016 cm−3 at2 K, the estimated critical field (BQ) at which the system enters thequantum limit is ∼4.4 T (SI Appendix, Note 9), which is readilyachievable. In fact, the MR curves for B⊥I obtained in the lowtemperature range of 2 to 10 K (SI Appendix, Fig. S5B) exhibit anoscillatory character even above BQ, which cannot be attributed tothe conventional Shubnikov–de Haas effect.To reveal the feature of these unexpected magneto-oscillations,

we conducted magnetotransport measurements of sample 6 in twohigh-magnetic-field facilities (Materials and Methods). Fig. 4Ashows the MR data taken under static fields at 1.8 K, which exhibitoscillations with more periods. Following previous studies (21, 22),a suitable background (SI Appendix, Fig. S15A) obtained viasmoothing was subtracted to better distinguish the oscillations(Materials and Methods). The results are presented in Fig. 4A,Bottom with dashed vertical lines marking the positions of peaks/dips. Interestingly, the interval between adjacent dashed linesgrows larger as the magnetic field increases. For better pre-sentation, the extrema fields (Bn) corresponding to these peaks/dips were extracted, and 1/Bn and log(Bn) were plotted as afunction of the extrema index (n), respectively (Fig. 4C). Herethe peaks and dips are assigned to integers and half-integers,

respectively. The clear deviation from the linear dependence of1/Bn on n further demonstrates the inability of the Shubnikov–deHaas effect to account for the observed oscillations. In contrast,the well-defined linear behavior observed for the log(Bn) vs. ncurve indicates the log-periodicity. Note that the point at ∼3.1 Tdeviates from the linear trend, possibly because this field is belowthe quantum limit, or perhaps as a result of some unavoidableanomalies during the background subtraction (SI Appendix, Note9). Fig. 4B shows the MR data taken across a wider field range at2.5 K using pulsed fields. These data exhibit more oscillations thando the data shown in Fig. 4A, while both the low field data and theextracted Bn are consistent across both datasets (see also Fig. 4C).Fig. 4D presents the Hall data obtained simultaneously with

the MR data from Fig. 4A. The log-periodic character is also ev-ident from a comparison of the index dependence of the 1/Bn andlog(Bn) curves (Fig. 4E). The fast Fourier transform for log(ΔRxx)and log(ΔRxy) shows nearly identical periods, indicating the sameorigin for both the observed MR and Hall oscillations.As aforementioned, the oscillations with peculiar log-periodicity

have been experimentally observed only recently, and only inZrTe5 (21) and its sister compound HfTe5 (22). The phenomenonis regarded as the manifestation of DSI in solid-state systems andis theoretically considered to be universal in topological materialshosting Dirac/Weyl fermions with Coulomb attraction (21–24).The underlying mechanism can be summarized as follows (21, 24):Dirac/Weyl fermions in an attractive Coulomb potential possess acontinuous scaling symmetry. In the supercritical case α> 1(α = e2=4π«0ћvF is the fine-structure constant, where e is the el-ementary charge, «0 is the vacuum permittivity, ћ is the reducedPlanck constant, and vF is the Fermi velocity), these masslessDirac/Weyl fermions tend to form two-body quasi-bound stateswith opposite charge centers, like charged impurities. Owing to

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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 of1/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 thepositions 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 MRoscillations 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-periodicoscillations 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).

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the relativistic energy–momentum dispersion relation of a Dirac/Weyl band, the energies of the quasi-bound states constitute ageometric series that matches the general feature of DSI. Theincreasing magnetic field enables the quasi-bound states with dif-ferent energies approaching the Fermi energy continuously. Theas-induced scattering between free carriers and quasi-bound statesthus influences the transport properties (21), manifesting as thelog-periodic oscillations for the measured MR and Hall data.Here we would like to attribute the observed log-periodic

oscillations in Te crystals to the DSI after excluding anotherpossible origin (SI Appendix, Note 9). The quasi-bound states arelikely formed between Weyl fermions located near the top ofvalence bands and opposite charge centers. Since the self-hole-doping characteristic may result from Te vacancies acting as ac-ceptors (33), such vacancies are likely to serve as the negativecharge centers that exert Coulomb attraction on the positive Weylfermions (see SI Appendix, Fig. S4 for the calculation results).For the log-periodic oscillations, the scale factor (λ) for the

DSI can be defined as Bn/Bn+1. As such, this quantity is esti-mated to be ∼2.33 for sample 6, as based on linear fitting of thelog(Bn) vs. n plot (Fig. 4C). This value of λ is consistent withthat obtained from the fast Fourier transform analysis, whichis ∼2.10 (∼2.28) for the MR (Hall) oscillations (Fig. 4F). In ref.21, the semiclassical quantization condition leads to the DSI

with λ = e2π=s0. Here s0 =(Zα)2 − κ2

√, which can be simplified

to s0 =α2 − 1

√when we consider the charge number Z = 1 and

the lowest angular momentum channel with κ = ±1 (21). Thus,the fine-structure constant (α) is calculated to be ∼7.5, far ex-ceeding 1/137 due to the small Fermi velocity of Te crystals.According to α = e2=4π«0ћvF, the Fermi velocity vF is calculatedto be ∼2.9 × 105 m/s, which is comparable to the value (∼1.9 ×105 m/s) estimated from the calculated band structure. The smallvF ensures the supercritical condition, promoting the formationof quasi-bound states with DSI (21).With increasing temperature, the amplitudes for both the MR

and Hall oscillations decline accordingly before becoming in-discernible at temperatures above 15 K (SI Appendix, Fig. S16and Note 10). Similar behaviors are also observed in othersamples (SI Appendix, Fig. S17). This can be attributed to thethermionic excitation that weakens the Coulomb attraction.However, compared to the cases of ZrTe5 and HfTe5, the criticaltemperature above which the oscillations disappear is muchsmaller for the present Te crystals, implying the relatively smallbinding energy of the as-formed quasi-bound states (as discussedin SI Appendix, Note 10).

ConclusionThe present work clearly demonstrates the realization of Weyl-related properties in a narrow-gap semiconductor with strongspin–orbit coupling and without inversion symmetry, thus greatlybroadening the scope of topological materials. Furthermore, thehighly tunable electronic performance of Weyl semiconductorsenables further manipulation of Weyl fermions through variousmeans commonly adopted in semiconductor electronics, such aselectrostatic gating and optical illumination. Moreover, the ob-servation of the log-periodic magneto-oscillations further confirmsthe universality of DSI in topological materials hosting Dirac/Weylfermions with Coulomb attraction, thus endowing Te with a novelplatform for the study of intriguing ground states beyond thequantum limit. The successful introduction of Weyl physics intosemiconductor systems therefore offers a new dimension for thefuture design of topological semiconductor devices.

Materials and MethodsGrowth of Te Single Crystals. High-quality Te single crystals were grown usinga physical vapor transport technique. High-purity (99.999% pure) Te powderwas loaded into a quartz tube, and a small amount of C powder was added

to remove trace oxygen. The quartz tube was heated to 1,000 °C over thecourse of 5 h and subsequently maintained at 1,000 °C for 1 h. Then, thetube was cooled at a rate of 20 °C/h to 400 °C and 300 °C for the hot andthe cold zone, respectively. After maintaining the set temperature for 2 wk,the tube was slowly cooled to room temperature, and rod-like silvery crystals(typical dimensions 5 × 0.3 × 0.1 mm3) were obtained (Fig. 2A, Inset).

Structure and Composition Characterization. The structure of Te single crystalswas measured by a Rigaku SmartLab X-ray diffractometer (XRD) at roomtemperature using monochromatic Cu-Kα radiation (λ = 1.5418 Å). A typicalXRD spectrum is shown in SI Appendix, Fig. S1A. In the wide range spectrum,

only the sharp peaks from the Te (1010) lattice plane are observed. Themicrodiffraction XRD was measured by Rigaku D/Max-RAPID II with Cu-Kαradiation (λ = 1.5418 Å) at room temperature. The large curved imagingplate detector with a 210° aperture allows a two-dimensional diffractionimaging over a broad 2θ range, such that the detection of many latticeplanes is possible without breaking the single crystal. As also shown in SIAppendix, Fig. S1A, the measured data match well with the standard pow-der XRD pattern of Te. SI Appendix, Fig. S1B shows the high-resolutionscanning transmission electron microscopy (Thermo Fisher Themis G2 60-300)imaging and selective area electron diffraction pattern of the exposed sur-face, which both further demonstrate the high quality of the single crystal atthe atomic scale. The crystalline long axis, that is, the growth direction, is the[0001] direction. The lattice parameters were determined to be a = b = 4.48Å, c = 6.00 Å. The chemical composition of the crystals was examined byan energy dispersive spectrometer (INCA detector; Oxford Instruments)attached to a field-emission scanning electron microscope (Sirion 200;FEI) operated at 20 kV. No impurity elements were detected (SI Appendix,Fig. S1C).

Magnetic and Transport Property Measurements. The magnetic propertieswere analyzed using a Quantum Design SQUID-VSM system with the mag-netic field up to 7 T and the temperature down to 2 K. For electricaltransport measurements, electron beam lithography was utilized to definethe electrode pattern on the surface of the Te crystals, then 10 nm Pd and50 nm Au were deposited as electrode materials using the e-beam evapo-ration method. Silver epoxy was subsequently used to affix the wires to thePd/Au contacts. The measurements were performed in a Quantum DesignPhysical Property Measurement System with the magnetic field up to 14 T.The planar Hall measurements were performed via rotating the sample in afixed magnetic field. The angle of 105° is the starting point for the mea-surement, and the mechanical backlash of the rotator normally leads toslight discontinuity at an angle of around 105° in the planar Hall data. High-field transport measurements were carried out in the High Magnetic FieldLaboratory 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(Fig. 1F, Inset), the current was applied along the long axis (the [0001] di-rection) during all transport measurements conducted in this study.

Angle-Resolved Photoemission Spectroscopy Measurements. The angle-resolved photoemission spectroscopy measurements were performed atbeamline 9U (Dreamline) of the Shanghai Synchrotron Radiation Facilitywith a Scienta Omicron DA30L analyzer. The angle resolution was 0.1°, andthe combined instrumental energy resolution was better than 20 meV. Thephoton energy used in our experiments ranged from 25 to 90 eV, and the

measurements were conducted at 7 to 20 K. The (1120) surface of the singlecrystals was cleaved, corresponding to the ky–kz plane in the hexagonalBrillouin zone. The Fermi level of the Te samples was compared to a goldfilm reference which was evaporated onto the sample holder. All of themeasurements were conducted under a vacuum better than 7 × 10−11 mbar.

Electronic Band Structure Calculations. First-principles calculations were car-ried out within the framework of density functional theory using the ViennaAb initio Simulation Package (45). All of the calculations were performedwith a plane-wave cutoff of 450 eV on the 9 × 9 × 9 Γ-centered k-mesh, andthe convergence criterion of energy was 10−6 eV. The generalized gradi-ent approximation with the Perdew–Burke–Ernzerhof functional (46) wasadopted to describe electron exchange and correlation. During structureoptimization, all atoms were fully relaxed until the force on each atom wassmaller than 0.01 eV/Å. In order to obtain a reliable band gap, the HSE06hybrid functional (47) was used in the band structure calculation. The ro-

tational symmetry g = {C2x

00 2

3} is a generator along the H–L line. Including

the spin–orbit coupling, the rotational symmetry g satisfies g2 = −1, so its

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eigenvalues are ±i. The first-principles calculated eigenvalues for g at twoarbitrary momentums (k1 and k2) are shown in Fig. 1C. The eigenvalueswitches between the two bands at k1 and k2. Thus, there must exist a bandcrossing point with twofold degeneracy between k1 and k2. Hence, the Weylpoint of W1 is protected by rotational symmetry. It is well known that theKramers degeneracy is a twofold degeneracy at time-reversal-invariantmomentum. Here, the Weyl point of W2 is at L point, which is a time-reversal-invariant momentum. Hence, the Weyl point of W2 is protectedby time-reversal symmetry. To accurately calculate the Berry curvature in theBrillouin zone, we have fitted a Wannier tight-binding Hamiltonian by usingthe WANNIER90 package (48). The p orbitals of Te are used as the initialprojectors for Wannier Hamiltonian construction. The chirality of a Weylpoint is calculated by integrating the Berry curvature on a surface enclosingthe Weyl point.

Data Processing. Due to the nonlinearity of Hall data, the carrier density wasextracted from the linear part of the low-field Hall resistivity. Attempts touse the two-carrier fitting model were unsuccessful. For the log-periodicoscillations, the background was obtained through smoothing the MR/Hall

data. The second derivative method is a widely adopted approach toconfirm the exact positions of the extrema fields (21). We thus comparedthe extracted oscillations obtained via these two methods and obtainedconsistent results (SI Appendix, Fig. S15), demonstrating the validity of ourdata processing.

Data Availability. All data are available in the main text or SI Appendix.

ACKNOWLEDGMENTS. This workwas supported by the National Natural ScienceFoundation of China (Grants 11974324, U1832151, 11804326, 11774325, and21603210), the Strategic Priority Research Program of Chinese Academy ofSciences (CAS) (Grant XDC07010000), the National Key Research and Develop-ment Program of China (Grants 2017YFA0403600 and 2017YFA0204904), theAnhui Initiative in Quantum Information Technologies (Grant AHY170000),Hefei Science Center CAS (Grant 2018HSC-UE014), the Anhui Provincial NaturalScience Foundation (Grant 1708085QA20), the Fundamental Research Funds forthe Central Universities (Grants WK2030040087 and WK3510000007), and theScience, Technology, and Innovation Commission of Shenzhen Municipality(Grant KQTD2016022619565991).

1. M. Z. Hasan, C. L. Kane, Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045(2010).

2. X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83,1057 (2011).

3. N. P. Armitage, E. J. Mele, A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

4. J. Xiong et al., Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science350, 413–416 (2015).

5. H. Li et al., Negative magnetoresistance in Dirac semimetal Cd3As2. Nat. Commun. 7,10301 (2016).

6. C.-L. Zhang et al., Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fer-mion semimetal. Nat. Commun. 7, 10735 (2016).

7. E. Liu et al., Giant anomalous Hall effect in a ferromagnetic Kagomé-lattice semi-metal. Nat. Phys. 14, 1125–1131 (2018).

8. X. Huang et al., Observation of the chiral-anomaly-induced negative magnetoresis-tance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).

9. M. Hirschberger et al., The chiral anomaly and thermopower of Weyl fermions in thehalf-Heusler GdPtBi. Nat. Mater. 15, 1161–1165 (2016).

10. Q. Li et al., Chiral magnetic effect in ZrTe5. Nat. Phys. 12, 550–554 (2016).11. Y. Wang et al., Gate-tunable negative longitudinal magnetoresistance in the pre-

dicted type-II Weyl semimetal WTe2. Nat. Commun. 7, 13142 (2016).12. C. Y. Guo et al., Evidence for Weyl fermions in a canonical heavy-fermion semimetal

YbPtBi. Nat. Commun. 9, 4622 (2018).13. D. T. Son, B. Z. Spivak, Chiral anomaly and classical negative magnetoresistance of

Weyl metals. Phys. Rev. B 88, 104412 (2013).14. A. A. Burkov, Chiral anomaly and diffusive magnetotransport in Weyl metals. Phys.

Rev. Lett. 113, 247203 (2014).15. N. Kumar, S. N. Guin, C. Felser, C. Shekhar, Planar Hall effect in the Weyl semimetal

GdPtBi. Phys. Rev. B 98, 041103 (2018).16. H. Li, H.-W. Wang, H. He, J. Wang, S.-Q. Shen, Giant anisotropic magnetoresistance

and planar Hall effect in the Dirac semimetal Cd3As2. Phys. Rev. B 97, 201110 (2018).17. P. Li, C. H. Zhang, J. W. Zhang, Y. Wen, X. X. Zhang, Giant planar Hall effect in the

Dirac semimetal ZrTe5. Phys. Rev. B 98, 121108 (2018).18. F. C. Chen et al., Planar Hall effect in the type-II Weyl semimetal Td-MoTe2. Phys. Rev.

B 98, 041114 (2018).19. A. A. Burkov, Giant planar Hall effect in topological metals. Phys. Rev. B 96, 041110

(2017).20. S. Nandy, G. Sharma, A. Taraphder, S. Tewari, Chiral anomaly as the origin of the

planar Hall effect in Weyl semimetals. Phys. Rev. Lett. 119, 176804 (2017).21. H. Wang et al., Discovery of log-periodic oscillations in ultraquantum topological

materials. Sci. Adv. 4, eaau5096 (2018).22. H. Wang et al., Log-periodic quantum magneto-oscillations and discrete-scale in-

variance in topological material HfTe5. Natl. Sci. Rev. 6, 914–920 (2019).23. H. Liu, H. Jiang, Z. Wang, R. Joynt, X. Xie, Discrete scale invariance in topological

semimetals. arXiv:1807.02459 (6 July 2018).24. M. Sun, Magnetic manifestation of discrete scaling symmetry in Dirac semimetals.

Phys. Rev. B 98, 205144 (2018).25. E. Gerlach, P. Grosse, Eds., The Physics of Selenium and Tellurium (Springer, 1979).

26. M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, Weyl node and spintexture in trigonal tellurium and selenium. Phys. Rev. Lett. 114, 206401 (2015).

27. S. Murakami, M. Hirayama, R. Okugawa, T. Miyake, Emergence of topological semi-metals in gap closing in semiconductors without inversion symmetry. Sci. Adv. 3,e1602680 (2017).

28. S. S. Tsirkin, I. Souza, D. Vanderbilt, Composite Weyl nodes stabilized by screw sym-metry with and without time-reversal invariance. Phys. Rev. B 96, 045102 (2017).

29. K. Nakayama et al., Band splitting and Weyl nodes in trigonal tellurium studied byangle-resolved photoemission spectroscopy and density functional theory. Phys. Rev.B 95, 125204 (2017).

30. T. Ideue et al., Pressure-induced topological phase transition in noncentrosymmetricelemental tellurium. Proc. Natl. Acad. Sci. U.S.A. 116, 25530–25534 (2019).

31. V. B. Anzin, M. I. Eremets, Y. V. Kosichkin, A. I. Nadezhdinskii, A. M. Shirokov, Mea-surement of the energy gap in tellurium under pressure. Phys. Status Solidi A 42,385–390 (1977).

32. T. Ikari, H. Berger, F. Levy, Electrical properties of vapour grown tellurium singlecrystals. Mater. Res. Bull. 21, 99–105 (1986).

33. S. Tanuma, The effect of thermally produced lattice defects on the electrical prop-erties of tellurium. Sci. Rep. Res. Inst. Tohoku Univ. Ser. A 6, 159–171 (1954).

34. H. B. Nielsen, M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in acrystal. Phys. Lett. B 130, 389–396 (1983).

35. Y. Yafet, R. Keyes, E. Adams, Hydrogen atom in a strong magnetic field. J. Phys. Chem.Solids 1, 137–142 (1956).

36. B. A. Assaf et al., Negative longitudinal magnetoresistance from the anomalous N =0 Landau level in topological materials. Phys. Rev. Lett. 119, 106602 (2017).

37. S. Nandy, A. Taraphder, S. Tewari, Berry phase theory of planar Hall effect in topo-logical insulators. Sci. Rep. 8, 14983 (2018).

38. A. Burkov, Weyl metals. Annu. Rev. Condens. Matter Phys. 9, 359–378 (2018).39. H. L. Stormer, D. C. Tsui, A. C. Gossard, The fractional quantum Hall effect. Rev. Mod.

Phys. 71, S298–S305 (1999).40. Y. Liu et al., Zeeman splitting and dynamical mass generation in Dirac semimetal

ZrTe5. Nat. Commun. 7, 12516 (2016).41. C.-L. Zhang et al., Magnetic-tunnelling-induced Weyl node annihilation in TaP. Nat.

Phys. 13, 979–986 (2017).42. B. J. Ramshaw et al., Quantum limit transport and destruction of the Weyl nodes in

TaAs. Nat. Commun. 9, 2217 (2018).43. C.-L. Zhang et al., Signature of chiral fermion instability in the Weyl semimetal TaAs

above the quantum limit. Phys. Rev. B 94, 205120 (2016).44. K. Behnia, L. Balicas, Y. Kopelevich, Signatures of electron fractionalization in ultra-

quantum bismuth. Science 317, 1729–1731 (2007).45. G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals

and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).46. J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made

simple. Phys. Rev. Lett. 77, 3865–3868 (1996).47. J. Paier et al., Screened hybrid density functionals applied to solids. J. Chem. Phys. 124,

154709 (2006).48. A. A. Mostofi et al., wannier90: A tool for obtaining maximally-localised Wannier

functions. Comput. Phys. Commun. 178, 685–699 (2008).

Zhang et al. PNAS | May 26, 2020 | vol. 117 | no. 21 | 11343

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