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Vanishing quantum oscillations in Diracsemimetal ZrTe5Jingyue Wang a, Jingjing Niua, Baoming Yana, Xinqi Lia, Ran Bia, Yuan Yaob, Dapeng Yua,c,d,e,and Xiaosong Wua,c,d,e,1

aState Key Laboratory for Artificial Microstructure and Mesoscopic Physics, Peking University, Beijing 100871, China; bInstitute of Physics, Chinese Academyof Sciences, Beijing 100190, China; cBeijing Key Laboratory of Quantum Devices, Peking University, Beijing 100871, China; dCollaborative Innovation Centerof Quantum Matter, Beijing 100871, China; and eDepartment of Physics, Southern University of Science and Technology of China, Shenzhen 518055, China

Edited by John Singleton, National High Magnetic Field Laboratory, Los Alamos, NM, and accepted by Editorial Board Member Zachary Fisk July 30, 2018(received for review March 22, 2018)

One of the characteristics of topological materials is their nontriv-ial Berry phase. Experimental determination of this phase largelyrelies on a phase analysis of quantum oscillations. We study theangular dependence of the oscillations in a Dirac material ZrTe5and observe a striking spin-zero effect (i.e., vanishing oscillationsaccompanied with a phase inversion). This indicates that the Berryphase in ZrTe5 remains nontrivial for arbitrary field direction, incontrast with previous reports. The Zeeman splitting is found tobe proportional to the magnetic field based on the condition forthe spin-zero effect in a Dirac band. Moreover, it is suggestedthat the Dirac band in ZrTe5 is likely transformed into a line nodeother than Weyl points for the field directions at which the spinzero occurs. The results underline a largely overlooked spin factorwhen determining the Berry phase from quantum oscillations.

quantum oscillations | Zeeman splitting | Berry phase | Dirac semimetal |topological material

Graphene is the first Dirac material. Its most celebrated prop-erties, the linear dispersion and chirality, represent some

of the essential characteristics of massless Dirac fermions. Thelatter gives rise to an additional nontrivial Berry phase of π tothe electron wave function when it completes a closed orbit.This phase has a direct consequence on the phase of quantumoscillations. In presence of a magnetic field, electrons performcyclotron motion and form quantized Landau levels. The quan-tization condition requires an accumulated phase change of 2πnfor the wave function on completing a revolution, where n is aninteger (1). The additional Berry phase of π naturally leads to aπ phase shift to the quantum oscillations (2). A phase analysisof oscillations thus becomes a straightforward method for deter-mination of the Berry phase. It has been beautifully shown ingraphene and extensively used ever since (3, 4).

Discovery of topological materials (e.g., topological insulators,3D Dirac semimetals, and Weyl semimetals) has further proventhe usefulness of such a phase analysis method, as many of thesetopological phases feature Dirac/Weyl cones in the bulk or onthe surface and the Berry phase is closely linked to their topolog-ical properties (5, 6). It has been the most widely used methodfor determination of the nontrivial Berry phase (7–19). Contrari-wise, a shift of the oscillation phase from nontrivial π to zero hasbeen considered as an indication of a possible topological phasetransition (18–22). Note that theories have predicted a rich set oftopological phases that a 3D Dirac semimetal can be turned intowhen subjected to breaking of certain crystal symmetry or timereversal symmetry (23, 24). It would be very interesting to carryout experimental investigation of these transitions, in which theoscillation phase analysis will continue to be valuable.

In this work, we study the angular dependence of the quan-tum oscillations in a 3D Dirac material, ZrTe5. A striking effect,namely spin zero, has been observed in topological materials,manifested as vanishing oscillations at certain field directionsand a concomitant π phase shift. The result shows that a spin

zero can happen for a Dirac dispersion and highlights a largelyoverlooked spin factor in determination of the Berry phase intopological materials. Being a consequence of a destructive inter-ference between two Zeeman split bands, the very existence ofthe spin zero also strongly favors a line node phase over a Weylpoint one for ZrTe5 subject to a magnetic field along directionsclose to the a or c axes.

ResultsMagnetoresistance. An optical image of a device for both elec-trical and thermoelectric measurements is shown in Fig. 1B,Inset. The same technique has been used to study the thermo-electric response of nanostructured Cd3As2 (25). The armchairgold strip is a microheater for generating a temperature gra-dient along the sample. With the structure, both the resistivitytensor and the thermoelectric tensor can be measured. ZrTe5is an orthorhombic layered material stacked along the b axis,while the trigonal prismatic chains of ZrTe3 are along the a axis,which is usually the longest dimension in a single crystal. In thisstudy, the current/temperature gradient is always along the aaxis. The temperature T dependence of the resistivity ρ, shownin Fig. 1B, displays the characteristic broad maximum of ZrTe5at T = 124 K (26–29).

In the following, we focus on the angular dependence of themagnetoresistance (MR), defined as MR = ρ(B)−ρ(0)

ρ(0). Starting

from the b axis, the magnetic field is tilted toward the c (a) axisby an angle denoted as θbc (θba). When the field is parallel to theb axis (θbc = θba = 0), MR is over 1,200% at 14 T. When the fieldis tilted away from the b axis toward the a or c axis, MR decreases

Significance

Topological materials exhibit a nontrivial Berry phase, exper-imental determination of which heavily relies on a straight-forward phase analysis of quantum oscillations. We reportthe observation of a striking spin-zero effect in quantumoscillations of topological materials. The concomitant phaseinversion underlines a largely overlooked phase factor in pre-vious oscillation analysis of topological materials. Moreover,our results indicate that the Berry phase in ZrTe5 remainsnontrivial in the presence of a magnetic field and support afield-driven line node phase.

Author contributions: X.W. designed research; J.W., J.N., B.Y., X.L., and R.B. performedresearch; J.W., Y.Y., D.Y., and X.W. analyzed data; Y.Y. carried out transmission electronmicroscopy; and J.W. and X.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. J.S. is a guest editor invited by the EditorialBoard.

Published under the PNAS license.1 To whom correspondence should be addressed. Email: xswu@pku.edu.cn.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1804958115/-/DCSupplemental.

Published online August 27, 2018.

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A B

C

Fig. 1. Resistivity of ZrTe5. (A) MR for field in the bc plane at T = 1.5 K. θbc = 0◦ denotes the b axis, and θbc = 90◦ denotes the c axis. Inset is a schematicillustration of the geometry for the angle-dependent measurement. (B) Temperature dependence of resistivity. The anomaly, a peak, appears at T = 124 K.Inset is an optical image of a device that we used for both electric and thermoelectric measurements. (C) MR for field in the ba plane at T = 1.7 K. θba = 0◦

denotes the b axis, and θba = 90◦ denotes the a axis.

substantially, as reported by others (21, 30, 31). Particularly whenB‖a , MR is nearly zero, as shown in Fig. 1C.

On top of MR, Shubnikov–de Haas oscillations (SdHOs) arewell-resolved. To obtain the oscillatory component ∆ρ(B), asmooth background ρ0(B) has been subtracted. In Fig. 2A,∆ρ(B)/ρ0(B) is plotted as a function of 1/B when B is in the

bc plane. When B‖b, the onset of SdHOs is as low as 0.4 T,indicating a relatively high mobility. From the angular depen-dence of the oscillation frequency, the Fermi surface topographycan be mapped out (SI Appendix, Fig. S3). The Fermi surfaceis a cigar-like ellipsoid, with the longest dimension along the baxis. The Fermi wave vectors in three crystal axes are ka = 0.096,

A

B C

Fig. 2. Angular dependence of SdHOs for field in the bc plane at T = 1.5 K. (A) MR oscillations after subtracting a smooth background ρ0(B). Oscillationsalmost disappear at the angle θbc = 83.8◦. (Inset) FFT amplitude of the fundamental vs. θbc. The dashed line is a guide to eyes. (B) Landau plots for differ-ent field angles. An integer Landau index is assigned to the resistance peak, while a half-integer is assigned to the valley. To obtain the error for themagnetic field positions of the peak and valley, we have adopted twice the noise level as the error for the resistance and then find the corresponding errorfor the field position from the MR curve. Solid lines are linear fits weighted by errors in 1/B (32). (C) Intercept γ of the Landau plot as a function of θbc. Asudden change from near 1

8 to near 58 occurs at the angle between 80◦ and 85.5◦. This angle range overlaps the angle at which the spin zero happens.

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kb = 0.761, and kc = 0.124 nm−1. The strong anisotropic shapeagrees well with other studies (21, 30, 33). The frequency is smallwhen the direction of the magnetic field is not far away from theb axis. Therefore, we reach the quantum limit at a relatively lowfield: ∼ 4 T. Zeeman splitting starts to appear for the Landaulevel n = 1. There are additional weak oscillations in the quan-tum limit, which have been reported before, and the origin is notclear (21, 30).

Vanishing Quantum Oscillations. The most interesting featurecomes from the angular dependence of the oscillation amplitude.The dependence is not monotonic. At θbc ≈ 83.8◦, it becomesvery weak. This evolution can be seen more clearly in Fig.2A, Inset, where the fundamental amplitude obtained by a fastFourier transformation (FFT) is plotted against θbc . A sharpminimum at θbc ≈ 83.8◦ is evident. In Fig. 2B, a Landau plot isperformed for SdHOs at different angles. The oscillation max-ima are assigned an integer (14, 34). By a linear fit, the interceptγ of the Landau-level index n at 1/B = 0 can be obtained and isplotted as a function of θbc in Fig. 2C. When θbc < 83.8◦, γ≈ 1/8,which suggests a nontrivial Berry phase of π, in agreement withother studies (21, 30). However, it jumps to around 5/8 whenθbc > 83.8◦. This π phase change has previously been observedand attributed to a change of the Berry phase (hence, the topol-ogy of the energy band) (21, 35). Intriguingly, this critical anglecoincides with that at which the oscillation amplitude dropsto zero.

Similar angular dependence has also been observed when Bis tilted in the ba plane, as illustrated in Fig. 3. The amplitudeof the quantum oscillations displays a minimum of near zero atθba ≈ 86.5◦. The angle is slightly larger than that when B is inthe bc plane. Concurrently, the intercept γ undergoes an abruptchange from 1/8 to about 5/8. In contrast, when B is tilted withinthe ac plane, similar disappearance of SdHOs and abrupt changeof γ have not been observed, shown in SI Appendix, Fig. S4. Inaddition, we have measured the field-dependent thermopowerfor different field directions (SI Appendix, Figs. S5 and S6). As

the thermopower is proportional to the derivative of the conduc-tivity with respect to energy according to the Mott relation, itsoscillations in field are often stronger than that of the resistivity(36). Indeed, the suppression of the quantum oscillations to zerois more evident there. The critical angles are the same.

SdHOs are a consequence of formation of Landau levels,which result from the cyclotron motion of electron along closedorbits in the momentum space. In layered materials with suffi-ciently low interlayer coupling, the Fermi surface is a warpedcylinder along the interlayer direction (37–39). Thus, when thefield is perpendicular to the axis of the cylinder, the orbit willbe open, leading to collapse of Landau levels (hence, SdHOs)(1). In ZrTe5, we and others have already experimentally deter-mined the topology of the Fermi surface being an ellipsoid,which excludes any open orbit (21, 30, 33). In fact, SdHOs arepresent even when B‖a or c. An alternative way to understandthis is to look at the dependence of the Landau-level separationon the effective mass of carriers. The heavier the carriers, thesmaller the Landau-level separation is (hence, a smaller oscilla-tion amplitude). An open orbit indicates an infinite mass, whichcompletely suppresses the Landau level. The effective mass inZrTe5 increases when the field is tilted away from the b axistoward the ac plane, suggesting a monotonic decrease of theamplitude, which is inconsistent with the observed nonmono-tonic dependence. More importantly, the effect cannot explainthe phase inversion. Therefore, the Fermi surface topology is notthe origin of the vanishing amplitude.

Spin-Zero Effect. To understand the disappearance of SdHOs, weturn to the Lifshitz–Kosevich formula, according to which thefundamental oscillations are expressed as

∆ρ(B)

ρ0∝RTRDRs cos

(2π

(Bf

B+ γ

)), [1]

where RT , RD, and Rs stand for the reduction factors due totemperature, scattering, and spin splitting, respectively. Bf is

A

B C

Fig. 3. Angular dependence of SdHOs for field in the ba plane at T = 1.7 K. (A) MR oscillations after subtracting a smooth background ρ0(B). Oscillationsalmost disappear at the angle θba = 86.5◦. (Inset) FFT amplitude of the fundamental vs. θba. The red dashed line is a guide to the eye. (B) Landau plots fordifferent field angles. Solid lines are linear fits. An integer Landau index is assigned to the resistance peak, while a half-integer is assigned to the valley. (C)Intercept γ of the Landau plot as a function of θba. A sudden change from near 1

8 to near 58 occurs at the angle between 85◦ to 88◦.

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the oscillation frequency and linked to the extreme cross-sectionarea Se of the Fermi surface by the Onsager relation Bf =Se~/2πe (1). The shape of the Fermi surface can be inferredfrom the angular dependence of Bf (SI Appendix, Fig. S3).The phase factor γ= 1/2−φB/2π− δ is obtained by readingthe intercept in a Landau plot. Here, φB is the Berry phase,and δ=±1/8 in 3D systems (14). This is the relation used todetermine the Berry phase of a system (3, 4). The spin fac-tor Rs originates from lifting of the twofold spin degeneracy.When Landau levels are spin split, one will have two sets ofoscillations (i.e., two frequencies). In general, a beating patternoccurs. Rs describes the envelope of the beating pattern, Rs =cos(πEs/∆E), where Es and ∆E are the spin-splitting energyand the Landau-level separation, respectively. For normal non-realistic electrons, the Landau-level separation is independentof energy and proportional to the field, ∆E = ~ωc = ~eB/m∗,where ~ is the reduced Plank constant, ωc is the cyclotron angularfrequency, e is the electron charge, and m∗ is the effective massof electrons. When the spin splitting is due to the Zeeman effect,Es = gµBB = gBe~/2me, where g is the Lande factor, µB is theBohr magnetron, me is the electron mass, and Rs = cos

(π2

gm∗

me

)is independent of B . In other words, beating vanishes, and onlya single frequency remains. Particularly, when gm∗/me is anodd number, Rs = 0 for arbitrary field. Such a disappearance ofSdHOs is the so-called spin zero (1).

The stringent requirement for the spin zero implies thatthis effect is not commonly seen. If g and/or m∗ are stronglyanisotropic, gm∗/me may vary in a range that includes at leastone odd number. Therefore, it is possible to observe a spinzero by rotating the field. ZrTe5 has an anisotropic electronicstructure manifested by the cigar-like Fermi surface, as we havejust shown. m∗ also differs significantly for different directions(21, 30). Furthermore, the g factor has been found to be large(40–42), which helps gm∗/me to span a larger range. In thisregard, it should not be unexpected to see a spin-zero effectat a certain field angle, as observed in our study. Moreover, animportant feature of the spin zero is a simultaneous phase inver-sion. As the field rotates, gm∗/me passes an odd number. Rs

passes zero and then changes its sign, which leads to an inver-sion of the oscillation phase. The observed change of γ from1/8 to a value close to 5/8 around the zero-oscillation ampli-tude is in excellent agreement with the spin-zero effect. At thespin zero, we have gm∗/me = 1, 3, 5, 7 . . .. This is can be furthernarrowed down to gm∗/me = 1, 5, 9 . . . by recognizing a non-trivial Berry phase (Rs> 0) when B‖b. Because the n = 2 levelexhibits spin splitting (for instance, around θbc = 80.1◦ in Fig.2A), we have gm∗/me< 4; hence, gm∗/me = 1 (1). This choiceis further supported by noting that larger g values combinedwith a strong anisotropic m∗ most likely lead to multiple spinzeroes, which are at odds with experiments. The cyclotron massm∗ac = 0.026me is calculated from the temperature-dependentdamping of the oscillations. Based on the Dirac dispersionand the ellipsoidal Fermi surface obtained from the angulardependence of the oscillation frequency, the band velocity iscalculated. Then, the cyclotron mass at the spin-zero angleis determined: m∗(θbc = 83.8◦) = 0.132me (43). Therefore, theg factor is g(θbc = 83.8◦) = 7.6. Similarly, g(θba = 86.5◦) = 5.3.The g factor has been experimentally determined as 15.8–24.3 inthe b direction (21, 41, 42), but these values cannot be directlycompared with our results obtained in directions close to the acplane. A first principle calculation of the g factor has been car-ried out recently for a few topological materials, by which weobtain g(θbc = 83.8◦) = 9.6 and g(θba = 86.5◦) = 0.3 (44). Theagreement for θbc = 83.8◦ is reasonably good, while it is notfor θba = 86.5◦.

Eq. 1 represents only the dominant fundamental component,which cannot produce a spin-splitting feature in SdHOs, unless

sufficiently strong higher harmonics are included. In our experi-ment, when the field is not in the vicinity of the spin-zero angles,splitting of oscillations can be barely seen, except for some weakindication at high fields. However, as the field is tilted towardthe spin-zero angles, the splitting becomes increasingly apparent,seen in Fig. 4A. This is because higher harmonics emerge fromthe vanishing fundamental. At the spin-zero angle, the split peaksare equally spaced, proving that the phases of the fundamentaloscillations for the spin-up and spin-down Landau levels differ byπ, as expected by the spin-zero effect. Note that the spin factorfor harmonics is Rs = cos

(pπ2

gm∗

me

), where p is the harmonic

order (1). At the spin zero, where the fundamental vanishes,the second harmonic is in its maximum, which is experimentallyobserved, as seen in Fig. 4B.

Spin-Zero Effect in a Relativistic Band. Although the data agreewell with the spin-zero effect, there seems to be one caveat left.The above spin factor Rs = cos

(π2

gm∗

me

)is obtained for a nonrel-

ativistic parabolic band. For a Dirac cone, Landau levels are notequally spaced in energy and the energy separations are depen-dent on

√B , En =

√2~ev2

0nB , where v0 is the band velocity (3,4, 45). In the case of a B -linear Zeeman splitting Es = gµBB (21,23, 24, 42), a sizable splitting will result in an irregular arrange-ment between spin-up and spin-down Landau levels, as sketchedin SI Appendix, Fig. S8. Will the irregular level arrangement pre-vent a regular interference between oscillations from two spin

A

B

Fig. 4. Zeeman splitting in the vicinity of a spin zero. (A) Oscillations as afunction of the Landau index n for tilt angles around the spin-zero angleθbc = 83.8◦. The phase inversion is evident. The black dashed lines are guideto the eyes, showing the evolution of the Zeeman splitting with the tiltangle. At the spin-zero angle 83.8◦, split peaks are equally spaced. (B)FFT amplitude of the second harmonic as a function of the tilt angle. Theamplitude peaks at the spin-zero angle.

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bands (hence, a spin zero)? To answer the question, we havetaken into account a Dirac dispersion E = ~kv0 and derived theexpression for the spin factor (SI Appendix)

Rs = cos

(πEFgµB

~ev20

). [2]

Note that the cyclotron mass of a massless Dirac particle canbe written as mc =EF/v

20 . Thus, we arrive at Rs = cos

(π2

gmcme

),

which looks the same as the one for nonrelativistic electrons.Hence, oscillation interference will take place in the same way.The estimation of the g factor that we have just done in the pre-vious section remains valid. Even so, two differences are worthmentioning. First, to have a spin zero in Dirac materials, a largeg factor is required, as mc is usually very small. While g is toosmall in graphene, it is often quite large in topological materi-als due to strong spin orbit coupling (7, 8, 18, 21, 40, 41, 46, 47).Second, mc in Dirac materials is a function of the Fermi level.Therefore, Rs strongly depends on the Fermi level, which is instark contrast to that in a parabolic band. Consequently, tuningEF can be used as a knob to generate a spin zero. Because of thephase inversion accompanied with a spin zero, it should be keptin mind that an observation of such a phase inversion does notindicate a change of the Berry phase.

Implications. Some interesting conclusions can be drawn basedon the observation of the spin-zero effect in ZrTe5. First, it sug-gests that the Berry phase of ZrTe5 stays unchanged as the fieldis tilted away from the b axis toward the ac plane. In other words,the Fermi surface is topologically nontrivial for the field appliedin any direction. Second, it can be inferred that the Zeemansplitting in ZrTe5 is linear in B (i.e., the effective g factor isindependent of B). Although this seems a commonplace, it isnot necessarily the case in the presence of spin orbit coupling(48). For the 3D Dirac electrons in the ultrarelativistic limit ofthe quantum electrodynamics (QED), the energy of the Zeemansplit Landau levels is

En,σ(kz ) =±~c√

(2n + 1 +σ)eB/mc + k2z , [3]

where the Landau-level index n = 0, 1, 2 . . ., σ=±1 denotes thespin direction, cs the speed of light, and kz is the wave vectoralong the field (49, 50). When kz = 0, g = (En,↑−En,↓)/µBB

diverges as 1/√B . This effect has been observed in a Kane

fermion system, HgCdTe (50). Since En,↑=En+1,↓, the spinfactor Rs = 1 (hence, no spin zero for Dirac electrons in theultrarelativistic QED). On the contrary, the spin zero always

occurs for Kane fermions, because σ in Eq. 3 is now replacedby σ/2 (50). It would be intriguing to check the oscillations inHgCdTe. Third, the appearance of the spin zero sheds light onsome aspects of the phase into which ZrTe5, as a 3D Dirac mate-rial, is driven by a magnetic field. The 3D topological Dirac phaseis at the boundary with various topological phases (23, 24). TheDirac cone consists of two overlapping Weyl cones with oppo-site chirality and is protected by certain crystal symmetries andtime reversal symmetry. Breaking of these symmetries gives riseto different topological phases, depending on which of them isbroken and how it is broken (23, 24). For ZrTe5, it is proposedthat application of a magnetic field can drive the system intoeither a Weyl semimetal when B‖a or a line node semimetalwhen B‖b or c (42). Because a spin zero requires that the areasof the extreme orbit perpendicular to the field for the two spin-split Fermi surfaces are not equal, this immediately rules out theWeyl node scenario: two Weyl nodes are separated in momen-tum along the field, and the extreme orbits are identical in shape.However, the spin zero is compatible with a line node phase, inwhich two Fermi surfaces are separated in energy by gµBB (42).As a result, the spin-zero effect can provide clear-cut evidencefor distinguishing these topological phases.

ConclusionsIn conclusion, we observe a spin-zero effect in the Diracsemimetal ZrTe5 when the direction of the field is close to eitherthe a or c axis. The phase inversion accompanied with the effectindicates that the Berry phase remains nontrivial, in contrast toprevious reports. Analysis of the spin-zero effect in a Dirac bandsuggests that the Zeeman splitting is proportional to B when thespin zero happens. The experiment calls for caution with regardto determination of the Berry phase by quantum oscillations.

Materials and MethodsMicroplatelets of ZrTe5 were grown by a silicon-assisted chemical vaportransport method using iodine as a transport agent (33, 51, 52). The growthdetail can be found elsewhere (53). Platelets of around 100 nm in thicknesswere selected in the study. High-angle annular dark-field images taken withthe aberration-corrected transmission electron microscope (JEOL ARM200F)show great crystalline quality of our platelets (SI Appendix, Fig. S1). A stan-dard e-beam lithography process was used to fabricate Hall bar structuresand thermoelectric measurement devices. Transport measurements werecarried out using a lock-in method in a helium cryostat and a PhysicalProperty Measurement System by Quantum Design.

ACKNOWLEDGMENTS. We acknowledge insightful discussions with F. Wang,X. C. Xie, and K. Chang. This work was supported by National Key BasicResearch Program of China Grants 2016YFA0300600, 2013CBA01603, and2016YFA0300903 and National Natural Science Foundation of China (NSFC)Projects 11574005, 11774009, 11222436, and 11234001.

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