TOPOLOGICAL MATTER

Quasiparticle interference of theFermi arcs and surface-bulkconnectivity of a Weyl semimetalHiroyuki Inoue,1* András Gyenis,1* Zhijun Wang,1 Jian Li,1 Seong Woo Oh,1

Shan Jiang,2 Ni Ni,2 B. Andrei Bernevig,1 Ali Yazdani1†

Weyl semimetals host topologically protected surface states, with arced Fermisurface contours that are predicted to propagate through the bulk when theirmomentum matches that of the surface projections of the bulk’s Weyl nodes.We used spectroscopic mapping with a scanning tunneling microscope to visualizequasiparticle scattering and interference at the surface of the Weyl semimetal TaAs.Our measurements reveal 10 different scattering wave vectors, which can beunderstood and precisely reproduced with a theory that takes into account the shape,spin texture, and momentum-dependent propagation of the Fermi arc surface statesinto the bulk. Our findings provide evidence that Weyl nodes act as sinks for electrontransport on the surface of these materials.

Understanding the exotic properties of quasi-particles at the boundaries of topologicalphases of electronicmatter is at the forefrontof condensed-matter physics research. Ex-amples include helical Dirac fermions on the

surface of time-reversal invariant topological in-sulators (1), which have been demonstrated to beimmune to backscattering (2–4), or the emergentMajorana fermions at the edge of a topologicalsuperconductor (5–7). Topological properties ofphases ofmatter are, however, not limited togapped

systems, and recent theoretical efforts have un-covered the possibility of topologically protectedmetallic phases. Topological Dirac semimetals(8, 9)—three-dimensional analogs of graphenewith band crossings protected by crystallinesymmetry—havebeen recently realized experimen-tally (10–12). Breaking of the inversion or time-reversal symmetry splits the Dirac crossings of thebandstructure intoWeylpoints,whichare singular-ities in theBerry curvature characterizedby aChernnumber or more colloquially monopole charge(13, 14). These semimetals host bulk quasiparticlesthat are chiral Weyl fermions. The topologicallyprotected boundarymodes ofWeyl semimetals aresurface states with a disjointed two-dimensionalFermi surface (15, 16). These so-called Fermi arcsconnect surface projections of bulk Weyl pointsof opposing Chern numbers. A notable property

of these Fermi arc states is that they can becomedelocalized into the bulk through the projectedWeyl points. Ideally, in the absence of disorder,an electron on the Fermi arcs of one surface cansink through the bulk and appear on the arcs ofthe opposing surface.Recent work has shown strong evidence that

inversion symmetry–breaking transition-metalcompounds (TaAs, TaP,NbAs,NbP) areWeyl semi-metals (17–21). Angle-resolvedphotoemission spec-troscopy (ARPES) measurements have confirmedthat the surface electronic structure of these com-pounds has a disconnected arclike topology con-necting the surface projection of 24Weyl crossingsin the bulk band structure. Here we used the scan-ning tunneling microscope (STM) to directly vi-sualize the surface states of the Weyl semimetalTaAs and examine their scattering and quantum-interference properties. Our experimental ap-proach follows similar STM studies that showedthat the spin texture (and time-reversal symmetry)protects surface states on topological insula-tors from backscattering (2–4, 22). In con-trast to these previous studies, we find that themomentum-dependent delocalization of theFermi arcs into the bulk, which is a uniqueproperty of these surface states, determines thecoherent interference properties of these sur-faces states. The surface-bulk connectivity exam-ined here underlies several other novel electronicphenomena, such as nonlocal transport thatcan occur in Weyl semimetals (23–27). A recentmagnetotransport study has observed quantumoscillations that may be associated with the non-local nature of transport, with electron orbits tra-versing through bulk Weyl nodes and surfacestates (28).To probe the properties of theWeyl semimetal’s

Fermi arc states, we have carried out STM studies(at 40 K) of in situ cleaved surfaces of TaAs singlecrystals that showatomically ordered terraces andtunneling density of states spectra consistent with

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1Joseph Henry Laboratories of Physics, Department ofPhysics, Princeton University, Princeton, NJ 08540, USA.2Department of Physics and Astronomy and CaliforniaNanoSystems Institute, University of California at LosAngeles, Los Angeles, CA 90095, USA.*These authors contributed equally to this work. †Correspondingauthor. E-mail: yazdani@princeton.edu

Fig. 1. STM topography and dI/dV spectroscopyofWeyl semimetal TaAs. (A) Illustration of the crys-tal structure of TaAs with (001) As termination. Oneunit cell contains four layers of As and Ta. The lat-tice parameters are a = b = 3.43 Å and c = 11.64 Å.(B) STM topographic image (Vbias = 500mV, Isetpoint =100 pA) of the cleaved (001) surface of TaAs. Mag-nified views on some of the pronounced impurities(right panels) show apparent C2v symmetric defor-mation of the electronic structure. (C) Spatial var-iation of the differential conductance measurementsalong a line of 30 Å, shown as a yellow line in (B),was measured with lock-in techniques employing3-mV excitations at 707 Hz. The orange curveshows the spatially averaged dI/dVspectra. An atom-ically periodic modulation of the spectra is visible.(D) Topographic image and (E) conductance mapat 120 meV (Vbias = –340 mV, Isetpoint = 80 pA) onthe same area, where the C2v symmetric nature ofthe surface electronic states is clearly visible.

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a semimetal (Fig. 1). The analysis of the atomicstep edge heights in STM topographs shows onlyone type of surface for the cleaved samples, whichis likely terminated by As atoms [see below and(29)]. Although the surface atomic structure visu-alized in the STM images has fourfold symmetry,the underlying electronic structure of this com-pound only has C2v symmetry, which is evidentfrom its overall crystal structure shown in Fig. 1A.A C4v nonsymmorphic symmetry is broken by thesurface. The anisotropic local shape of electronicsignatures caused by native surface defects shownin STM topography (Fig. 1, B andD), as well as thescattering of surface quasiparticles around suchdefects over long length scales displayed in STM

conductance maps (Fig. 1E), both show a clearC2v symmetry.Detailed information about the electronic prop-

erties of the surface states can be obtained frommeasurements of the quasiparticle interference(QPI) patterns in large-area STM conductancemaps. The Fourier transform of such STM con-ductance maps identifies scattering wave vectorsq, which connect ki and kf states on the contoursof constant-energy surface in momentum space(30). QPI measurements can be used not only tofollow the evolution of the Fermi surface probednear the surface but also to determine whethersome scattering wave vectors are forbidden as aresult of discrete symmetries (2, 23, 24). Informa-

tion on quasiparticle lifetime can also be extractedfrom QPI; however, such effects do not typicallyresult inmomentum-specific changes in the QPI,as we discuss here.Experimentally, thedetectionof scatteringwave

vectors in QPI measurements is limited by instru-mental resolution, which is determined by thestability of the instrument and the maximum av-eraging time possible during each map. In Fig. 2,we show QPI measurements on the TaAs surfaceobtained with a high-resolution, home-built STMinstrument capable of averaging for up to 6 dayswhile maintaining atomic register. The Fouriertransform of these QPI measurements (Fig. 2) al-lows us to resolve the rich array of scatteringwavevectors on this compound. Such high-resolutionmeasurements are required to resolve the entireset of allowed scattering wave vectors in this com-pound. In Fig. 2, Q to S, we also show measure-ments of the QPI features as a function of energyalong specific directions in q space, displaying con-tinuous dispersion of these features with energy,as is typical ofQPI features. Recentmeasurementson a relatedWeyl compound (NbP) at lower reso-lution have yielded a subset of QPI signals (3 outof 10 wave vectors) reported here (31).QPImeasurements from surface states ofmost

materials can be understood by starting from amodel of contours of constant energy in momen-tum space, consistent with ARPES-measured bandstructure, that can be used to calculate a jointdensity of states (JDOS) probability for scatteringas a function of momentum difference q (32). Ifthe Fermi surface is spin textured, as in the case ofhelical Dirac surface states or strongly spin-orbit–coupled systems, the experimental results can becompared with the spin-dependent scatteringprobability (SSP)maps that also take into accountthe influence of relative spin orientations ofthe initial and final states on the QPI measure-ments (2, 29). Following such an approach, weuse density functional theory (DFT) methods tocalculate the surface states for TaAs. We repro-duce both the shape (17, 20, 21) and spin texture(33–35) of the surface Fermi contours (Fig. 3A)recently measured with ARPES on this compoundand calculate the SSP map near the chemical po-tential, which can then be compared to our ex-perimental results at the same energy. In additionto the Fermi arcs, both ARPES measurementsand the DFT calculations capturing them mayinclude some features that are in fact caused bytrivial surface states (29). This approach resultsin an SSP (Fig. 3B) that resembles the overallsymmetry of the experimentally measured QPIpattern (Fig. 3F); however, it produces manymore scattering wave vectors than seen experi-mentally (such as those in the middle of the QPIzone highlighted in Fig. 3B that are missing inFig. 3F). This approach or a related one pro-posed recently (36) for understanding the data onNbP (31) also fails to capture the QPI data onTaAs at other energies (29).An accurate model of our experimental results

over a wide energy range can be achieved if weconsider not only the shape, the density of states,and the spin texture of the Fermi arcs but also

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Fig. 2. Quasiparticle interference of TaAs surface states. (A to D and I to L) Spatially resolved dI/dVconductance maps at different energies obtained on the area shown in Fig. 1B (Vbias = 240 meV,Isetpoint = 120 pA). (E to H and M to P) Symmetrized and drift-corrected Fourier transforms of thedI/dVmaps (QPI maps). Red dot indicates the (2p, 2p) point in the reciprocal space in the unit of 1/a;PSD denotes power spectral density. In the color bar, m corresponds to the mean value of the mapand s to the standard deviation. (Q to S) Energy-momentum structure of the dI/dV maps (Vbias =600 meV, Isetpoint = 80 pA) along the high-symmetry directions shown in (M).

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their momentum-dependent delocalization intothe bulk of the sample. The key conceptual idea isthat the QPI of the Fermi arcs is dominated byinitial and final momentum states that do notstrongly leak into the bulk. The degree of connec-tivity between Fermi arc surface states and thebulk is a property that depends on their momen-tum approaching the projection of Weyl pointson the surface. It is also related to the atomiccharacter of the Weyl nodes. Our DFT simulationof the TaAs electronic structure shows that themajority (~90%) of electronic states associatedwith the bulkWeyl nodes are based on Ta atomicorbitals (29). Consistent with this information, wealso find from the DFT calculation that the elec-tronic states close to the projectedWeyl points onthe surface arcs have a large component of suchorbitals. This results in the slow decay of theFermi arcs’ spectral weight associated with Taorbitals into the bulk, as compared to that asso-ciated with As orbitals (Fig. 3C) (29). This picturesuggests that to understand theQPI data on TaAs,we should project out the Fermi arc states that areassociated with Ta and focus only on states withthe As-orbital characteristics, which have theweakest connectivity to the bulk states and hencecan interfere coherently to produce theQPI patterns.The Ta orbitals in the first layer only weakly con-tribute to theQPI signal: An electron that scattersfrom the As site to a Ta site in the surface layer ismore likely to sink into the bulk Weyl states (Fig.3C). The presence of topologically trivial surfacestates [likely the inner bowtie feature in Fig. 3, Aand D; see also (29)] does not alter the projec-tion of bulk Weyl points at the surface, and allsurface states at the Fermi level with thesemomen-tawould still be strongly delocalized into the bulk.To confirm this idea, we consider a weighted

Fermi arc contour for TaAs, in which we projectthe Fermi arc states onto the As orbitals at thetopmost surface layer (Fig. 3D). This projectionresults in changes that are strongly dependenton the distance (in k space) from the Weyl point.Taking into account the delocalization into thebulk, we find strong fading of the innermostspoon-shaped Fermi arcs in the weighted Fermisurface, as compared to that shown in Fig. 3A.These short spoon-shaped sections of the Fermiarcs are close to their correspondingWeyl points,which are expected to act as sinks for electron prop-agation on the surface (because of their dominantTa orbital character). In contrast, the prominentfeatures of thisweighted Fermi surface come fromlonger bowtie-shaped arcs at the Brillouin zoneboundary, which lie far away from their corre-sponding Weyl point, with the least probabilityof sinking into the bulk (owing to their dominantAs orbital character). An electron in theweightedFermi arcs shown in Fig. 3D (or a similar onethat also includes the As projection in the secondlayer; see the supplementary materials) is in asubset of states that do not efficiently propagateaway into the bulk and remain near the surface.Without any further computation, we can un-

derstand the various wave vectors seen in the QPImeasurements by simply considering scatteringaround theweighted Fermi arc contours (Fig. 3D).

Based on the length and orientation of the pos-sible scattering wave vectors on the weightedFermi surface, we identify 10 different groups ofscattering wave vectors, Q1 to Q10, that are seenexperimentally. More detailed calculations of SSP(Fig. 3E) near the chemical potential based onthese weighted Fermi arcs also compare favorablyto the experimentally measured QPI at the sameenergy (Fig. 3F). In making this comparison, wenote that although the spin texture of the Fermiarcs and the absence of backscattering betweentime-reversed pairs of states play a role in thescattering data (a point towhichwe return below),the differences between JDOS and SSP for theFermi arcs are relativelyminor and not critical tounderstanding theQPI data [(29) and see below].Contrasting results of the SSPs using weighted(Fig. 3E) and unweighted (Fig. 3B) Fermi arcswith the experimental data (Fig. 3F) demonstratesthat at some momenta, the Fermi arc surfaceelectrons have a strong probability of sinking intothe bulk state and hence are not part of the QPIprocess. We further test this physical picture by amore exhaustive comparison of theoretical modelcalculations based on the weighted Fermi arcsand the experimental data over a wide range ofenergies (Fig. 4, A to L). As shown in this figure,ourmodel SSP calculations for theweightedFermi

arc can reproduce the finer features of the largebody of QPI data obtained in our studies. In ad-dition, isolating the contribution from the As orTa atomic sites to the QPI signal also can be usedto confirm our theoretical identification of themajor contribution of each atomic orbital to dif-ferent sections of Fermi arcs [bowtie and spoonfeatures dominated byAs andTa, respectively (29)].The agreement between theory and QPI measure-ments demonstrates the importance of account-ing for delocalization of the TaAs surface statescaused by the Weyl nature of its band structure.We now return to the role of spin texture in de-

termining the scattering properties of the Fermiarcs. Focusing on some of the finer features of theQPI data, in particular, scattering wave vectorsQ3

and Q9, and their comparison to the theory, wecan also resolve the subtle influence of spin tex-ture on the STMdata. This comparison (Fig. 4, Mto O) demonstrates that some of the duplicatefeatures in the JDOS maps caused by scatteringbetween arc states that have opposing spins aresuppressed in the SSP maps. Although almostat the limit of our experimental resolution, thecorrespondence between the finer predicted fea-tures in the SSP based on our model and thosein the experimental QPI data qualitatively con-firms that spin does play a role, albeit minor, in

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Fig. 3. Fermi arcs and qua-siparticle interference.(A) DFT-calculated Fermi arccontour of constant energyin first Brillouin zone (BZ) at+40 meV calculatedprojecting the DFT-calculated spectral density tothe top unit cell. Green dotsindicate the projectedpositions of the Weyl nodes,and arrows show thedirection of the spin on theFermi arcs. The combinationof time-reversal symmetryand C2v symmetry impliesvanishing out-of-planecomponent of the spin. FS,Fermi surface. (B) SSPderived from (A) marked byregions of strong scatteringin the middle of the BZ,which is missing in the QPIdata. (C) The integratedspectral density over the fullBZ for As and Ta separatelyas a function of layer indexshows the fast decay of Asorbital states. (D) WeightedFermi surface (WFS)calculated by projecting theelectronic states only to thetopmost As layer. TheQ vectors indicate thescattering wave vectors expected. (E) SSP based on (D) with red boxes showing the scattering wavevectors mapped in (F). (F) QPI map at 40 meV (same as in Fig. 2G) and experimentally observedgroups of Q vectors. FT-STS, Fourier transform–scanning tunneling spectroscopy.

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our QPI measurements. There are a few specialwave vectors that are strictly prohibited, owing tothe presence of time-reversal or mirror symmetry(23, 24). However, these signatures of protectedscatterings are unfortunately obscured by manyallowed overlapping wave vectors with similarlengths and are hard to resolve experimentally.Future experiments at higher resolution or onWeyl semimetal with simpler structures maybetter resolve this protection and other universalfeatures of Fermi arcs that are predicted theo-retically (37, 38).

Our work reveals that the absences of, or re-striction on, coherent scattering at certain wavevectors in both theory and experiments are not aconsequence of a symmetry-related protection forthe Fermi arc surface states, but rather followfrom the topological connection between the sur-face and bulk states. This connection can also beseen in geometries in which the sample thick-ness is less than the bulk’s scattering mean freepath, where the top and bottom surface of a Weylsemimetal would be strongly linked through theWeyl points (23, 26–28).

REFERENCES AND NOTES

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30. L. Petersen et al., Phys. Rev. B 57, R6858–R6861 (1998).31. H. Zheng et al., ACS Nano 10, 1378–1385 (2016).32. J. E. Hoffman et al., Science 297, 1148–1151 (2002).33. B. Q. Lv et al., Phys. Rev. Lett. 115, 217601 (2015).34. S. Y. Xu et al., Spin polarization and texture of the Fermi arcs

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ACKNOWLEDGMENTS

Work at Princeton was supported by Army ResearchOffice–Multidisciplinary University Research Initiative(ARO-MURI) program W911NF-12-1-0461, Gordon and BettyMoore Foundation as part of Emergent Phenomena in QuantumSystem initiative (GBMF4530), by NSF–Materials ResearchScience and Engineering Centers programs through the PrincetonCenter for Complex Materials DMR-1420541, NSF-DMR-1104612,NSF CAREER DMR-0952428, Packard Foundation, and KeckFoundation. This project was also made possible through use ofthe facilities at Princeton Nanoscale Microscopy Laboratorysupported by grants through ARO-W911NF-1-0262, ONR-N00014-14-1-0330, ONR-N00014-13-10661, U.S. Department of Energy–Basic Energy Sciences (DOE-BES)Defense Advanced ResearchProjects Agency–U.S.Space and Naval Warfare Systems CommandMeso program N6601-11-1-4110, LPS and ARO-W911NF-1-0606,and Eric and Wendy Schmidt Transformative TechnologyFund at Princeton. Work at University of California–Los Angeleswas supported by the DOE-BES (DE-SC0011978).

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/351/6278/1184/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S8References (39–41)

17 November 2015; accepted 3 February 201610.1126/science.aad8766

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Fig. 4.Comparison of the FT-STS maps with the DFTcalculation at various energies. (A to C andG to I) QPI maps (same as in Fig. 2) and (D to F and J to L) the corresponding SSP derived from theprojection of the spectral density to the topmost As. The obtained data and the calculations are inagreement over a wide energy range. (M to O) Enlargement of the Q9 and Q3 vectors in the calculatedJDOS, SSP, and QPI data at different energies. Whereas strong triple-arc structures are visible in JDOS,SSP shows a reduced number of arcs, which is consistent with the single arcs in the QPI data.

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Quasiparticle interference of the Fermi arcs and surface-bulk connectivity of a Weyl semimetalHiroyuki Inoue, András Gyenis, Zhijun Wang, Jian Li, Seong Woo Oh, Shan Jiang, Ni Ni, B. Andrei Bernevig and Ali Yazdani

DOI: 10.1126/science.aad8766 (6278), 1184-1187.351Science 

, this issue p. 1184Scienceassociated with Ta orbitals on the Fermi arcs sank into the bulk of the material.data to the predictions of density functional theory. The data could be reconciled with the theory only if electrons

thethe material TaAs. They mapped the scattering of electrons off impurities on the surface of the material and compared used a high-resolution scanning tunneling microscope to explore the properties of these states inet al.Fermi arcs. Inoue

A recently discovered class of topological materials, Weyl semimetals, have surface states in the form of so-calledSinking into the bulk of a Weyl semimetal

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