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1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot7610001, Israel. 2Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden,Germany.*These authors contributed equally to this work.†Corresponding author. Email: haim.beidenkopf@weizmann.ac.il

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Resolving the topological classification of bismuth withtopological defectsAbhay Kumar Nayak1*, Jonathan Reiner1*, Raquel Queiroz1, Huixia Fu1, Chandra Shekhar2,Binghai Yan1, Claudia Felser2, Nurit Avraham1, Haim Beidenkopf1†

The growing diversity of topological classes leads to ambiguity between classes that share similar boundaryphenomenology. This is the status of bulk bismuth. Recent studies have classified it as either a strong or ahigher-order topological insulator, both of which host helical modes on their boundaries. We resolve thetopological classification of bismuth by spectroscopically mapping the response of its boundary modes to ascrew-dislocation. We find that the one-dimensional mode, on step-edges, extends over a wide energy rangeand does not open a gap near the screw-dislocations. This signifies that this mode binds to the screw-dislocation,as expected for a material with nonzero weak indices. We argue that the small energy gap, at the time reversalinvariant momentum L, positions bismuth within the critical region of a topological phase transition between ahigher-order topological insulator and a strong topological insulator with nonzero weak indices.

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INTRODUCTIONTopological materials host unique boundary states that cannot berealized as stand-alone electronic systems but only on the bound-aries of a topologically classified bulk. These include the chiral edgemodes of the quantum Hall state, the helical modes on the surfaceof a topological insulator (TI), Fermi-arc bands on the surface of aWeyl semimetal, andMajorana modes in a topological superconductor(1–8). Bulk topological defects in the crystalline structure also exhibita similar bulk boundary correspondence. The surface termination ofa bulk topological defect necessarily hosts unique crystallographicstructures. One example is the surface step edge (gray shaded inFig. 1A) that emanates from the surface termination point of a bulkscrew dislocation (SD). The only way to realize a step edge that ter-minates at a point on the surface, away from the surface boundaries,is by its continuation into the bulk in the form of an SD. The lengthof the Burgers vector, which enumerates the winding of the bulk SD,dictates also the height of its corresponding step edge (9). Crystallinetopological defects, such as SDs and disclinations, also serve as effectiveboundaries and can potentially bind topologically protected electronicboundary modes. Soon after the discovery of the symmetry-protectedtopological electronic phases, a general theory of topological defectsand their associated gapless modes has been developed (10–16).Examples of these bound states are theMajoranamodes at vortex endsin chiral superconductors (17, 18) and one-dimensional (1D) helicalmodes at SDs in weak TIs, as proposed by Ran et al. (10). The lattercase of an SD has been reinvestigated later on by Slager et al. (14). Theyproposed a classification scheme that takes into account not onlytime-reversal symmetry but also crystalline space-group symmetries(19). Hence, it takes into account additional cases where gaplessmodes are formed but are not included in the formulation proposedbyRan et al. (10). In this classification, dislocations play a fundamentalrole as probes to distinguish different classes of topological states(14, 16). Although theoretically it is quite established that SDs

should bind helical modes in certain types of TIs, direct experimen-tal signatures for their existence are still lacking. Recently, an ex-perimental signature of increased conductivity has been reportedin Bi1−xSbx single crystals that was attributed to the introductionof SDs by plastic deformation (20). Here, we use scanning tunnel-ing microscopy (STM) to locally detect SDs on the (111) surface ofbismuth (Bi). Using careful spectroscopic mappings, we explore thelocal response of the various boundary electronic states in Bi(111)to the unique surface perturbations imposed by SDs. We providefirst direct spectroscopic evidence for the existence of helical modeson an SD. The correspondence between the structure of the topo-logical defect and the topology class of its associated electronicmode (10, 12, 14, 15, 19, 20) further allows us to determine thetopological classification of Bi as a strong TI with nonvanishingweak indices.

Bi, an extensively studied low-density semimetal, has a rich historyof intriguing quantum phenomena like electron fractionalization (21),nematic quantum Hall phases (22, 23), and unconventional super-conductivity (24). Because of its large spin-orbit coupling, it plays afundamental role in several topological materials, such as the firstrealized 3D strong TI, Bi1−xSbx (3, 25, 26), which shows almost iden-tical 3D band structure to that of pure Bi (27), and the canonical TIs,Bi2Se3 and Bi2Te3, which host a single surface Dirac band (28–31).Nevertheless, the topological classification of pure Bi has remained,thus far, rather ambiguous. Many theoretical models (3, 32–34) in-dicate that Bi has a trivial Z2 topological classification. Since bulk Biis inversion symmetric, its strong topological index, n0, and the weakindices, n = (n1, n2, n3), can be calculated from the parity eigenvaluesof the occupied bands at time reversal invariant momentum (TRIM)points (G,T, and three inequivalent L andX points, shown in Fig. 1B)(3, 32). While most theoretical calculations suggest that both thestrong and the weak indices are zero (3, 32, 34), some calculationsargue for nonvanishing strong and weak indices (35, 36). The originof the ambiguity in the topological classification of Bi lies in the ex-tremely small energy gap at the L TRIM point (34), which in calcula-tion is found to be on the order of 5 to 15meV (Fig. 1C). Experimentally,several studies demonstrate phenomenology, which implies non-trivial topology in Bi, both in bulk crystals and in thin films, as wellas in its 2D limit of a bilayer (35, 37–42). Direct visualization of the

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electronic structure of Bi around the �M surface energy gap (39, 42)suggests that the 2D surface bands may have topological origin. Abilayer of Bi was among the first materials to be classified as a 2DTI (43). Later, 1D edge modes were observed in thin films of Bi(40, 44–46) and on step edges of cleaved (111) surfaces of bulk Bicrystals (38). Various interpretations have been ascribed to thestep-edge 1D edge modes. These include trivial spin-split bands ofthe 1D Rashba type (47), helical edge modes of a 2D TI (38), and,more recently, helical hinge modes of a 3D higher-order TI (41).Helical 1D edge modes are counter-propagating time-reversed part-ners of each other, which form a Kramer’s pair. As a consequence, itis impossible to introduce any backscattering between the two states,unless time reversal symmetry is broken. This is the origin of thetopological protection of helical edge states. We show here, by theaid of SDs, the robust behavior of the 1D edge mode in Bi both atthe SD and along the step that emanates from it. This phenomenol-ogy of the 1D edge mode at the SD is consistent with a topologicalclassification of Bi as a strong TI with nonzero weak indices. Thesmall L gap positions Bi on the verge of a topological phase transitionaccompanied by a gap closure between a strong TI with nonzeroweak indices and a higher-order TI. We argue that approximatelyhelical 1D edge modes, on the step edge and the SD, appear at thecritical vicinity of the topological phase transition on either side of it(48, 49).

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RESULTSWe measured Bi samples derived from crystals of GdPtBi grown inBi flux (seeMaterials andMethods) (50).We have found that cleavingthese crystals under ultrahigh vacuum conditions repeatedly exposessurfaces of residual crystalline Bi due to its anisotropic rhombohedralstructure. The slight lattice mismatch with GdPtBi lends the opportu-nity to investigate the properties of bulk Bi in the vicinity of SDs.Nevertheless, apart from the nucleation of isolated SDs, the Bi atomicsurface structure and band structure we measured were found tobe rather unaffected by the GdPtBi substrate (see section S1). Thesamples were measured at 4.2 K in a commercial Unisoku STM.A representative topographic image of a freshly cleaved surface, consist-ing of a flat terrace terminated by step edges along high-symmetrycrystallographic directions, is shown in Fig. 1D. The atomically re-solved topographic image shows the underlying triangular latticeindicative of a pristine (111) cleave plane of Bi, thus allowing itsspectroscopic studies.

Surface electronic structure of Bi(111)We next identify the spectral features of the (111) surface of Bi.A typical differential conductance (dI/dV) spectrum measured on theterrace, away from any scatterers, is shown in Fig. 1E. The variouspeaks in the dI/dV spectrum represent the extrema of the 2D surfacebands expected on Bi(111) (38). The intermediate dips around 100

Fig. 1. Surface and edge spectrum of Bi(111). (A) Schematic illustration of an SD in a lattice. (B) Bulk Brillouin zone (BZ) of rhombohedral Bi and its (111) surface–projected 2D BZ and zigzag edge–projected 1D BZ, with all TRIM points indicated. (C) Bulk band structure of Bi obtained from ab initio calculations using the modifiedBecke-Johnson (MBJ) exchange potential and generalized gradient approximation shown by the solid and dashed lines, respectively. Inset shows the band structurearound the energy gap (5 meV) at L calculated with MBJ. (D) Large-scale STM topography of a Bi terrace with crystallographic step edges. Inset shows atomically resolvedimage overlaid with Bi bilayer crystal structure. (E) dI/dV spectroscopy given in arbitrary units (au) as measured at a point on the surface far away from impurities. (F) dI/dVmapmeasured normal to the step edge (marked with a black arrow) along the gray dotted line in (D) showing the modulated LDOS and the 1D topological edge mode (marked bycurly brackets). (G) Fourier transform of (F) showing the energy dispersion of scattering wave vectors along �G � �M and the corresponding calculated spin-selective scatteringprobability (SSP). (H) Calculated zigzag edge–projected LDOS showing 1D edge states dispersing along ~G � ~M (see fig. S4).

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and −100 meV correspond to a suppressed local density of states(LDOS) within the bulk gaps at the T and L TRIM points, respectively(calculation in Fig. 1C). To resolve some of these surface bands, wehave measured the dI/dV map, shown in Fig. 1F, taken normal tothe step edge (along the gray dashed line in Fig. 1D). It shows strongspatial modulations that disperse in energy. These modulations arequasiparticle interference (QPI) patterns the electrons embed in theLDOS due to scattering off the step edge. Its Fourier analysis (Fig. 1G,left) reveals clear dispersing modes that agree well with the calculatedspin-selective scattering probability (shown on the right panel). Thisdetailed comparison indicates that the dispersing QPI patterns origi-nate from the 2D surface bands that disperse between the �G and �Msurface TRIM points in full agreement with previous studies of the(111) surface of bulk Bi crystals (38, 51). The topological nature ofthese 2D surface bands depends on the topological classification ofBi: Trivial surface states in the case of Bi are higher-order TI, whiletopological Dirac states in the case of a strong TI. The qualitativedifference between the two is restricted to the immediate vicinity ofthe small L gap andmerely amounts to whether they all disperse thereinto the bulk valence band or, across the gap, between the valence andconduction bands. This phenomenological distinction in the 2D sur-face states is hard to resolve spectroscopically due to the small L gap.However, the response of the 1D edgemode, found onBi step edges, toSDs allows lifting of this ambiguity and revealing the topological clas-sification of Bi.

We therefore investigate the 1D edge mode on step edges and itstopological origin. Amid all the energy-dispersing QPI modes inFig. 1F, a nondispersing increased LDOS is bound along the stepedge. This 1D bound state spans the energy interval of both the T bulkgap at 30 to 180 meV and the L bulk gap at lower energies around−100 meV (both are marked with curly brackets). This extendedenergy dispersion is consistent with our ab initio calculations showing(red line in Fig. 1H) that the edgemode indeed disperses from ~G to ~M(the edge projections of the bulk T and L TRIM points, respectively;see Fig. 1B) and extends throughout the −200- to 200-meV energyrange. The topological nature of such a 1D edge mode in Bi has beendiscussed in several previous reports (37, 38, 41, 46, 52), where var-ious distinct origins have been ascribed to its formation including 2DTI (38), higher-order TI (41), and nontopological Rashba metal (47).However, all of these accounted for the appearance of the helical edgemode within the T gap alone and disregarded its extended energydispersion that we note here down to lower energies where theL energy gap resides. This is true also for the energy dispersionof the 2D surface modes in Bi(111). These energy and momentumextents suggest that the L energy gap has an important role in thetopological classification of Bi.

Spectroscopic mapping of an SDTo resolve the topological origin of the extended edge mode in Bi, weexamined its response to crystallographic defects. A topographicimage of a single Bi bilayer high step edge, terminating at an SD onthe Bi(111) surface, is shown in Fig. 2A. The height of the step edgeassociated with an SD can generally be of several unit cells, nu. Thisheight reflects the SD Burger’s vector, b = nu, which classifies thetopological invariant of the defect. Wemap the dI/dV spectrum alongthe last atomic row of the step edge, across the SD and onward,extending into the terrace (along the dotted line in Fig. 2A). An atom-ically resolved topographic profile along this line cut is shown inFig. 2B, further revealing a missing atom along the step edge. In

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Fig. 2. Topographic and spectroscopic mapping of an SD. (A) Topographicimage of a clean (111) surface consisting of an SD (solid black arrow) with a singleBi bilayer long Burger’s vector. (B) Topographic line cut along the black dottedline shown in (A). (C) Spectroscopic map measured along the dotted line in (A)showing QPI pattern for the 2D surface states and a nearly unhindered edgemode on the atomically corrugated step edge terminating at the SD. (D andE) dI/dV line cuts along the black dashed lines shown in (C). The shadowed regionsin (B) to (D) correspond to the segment where the STM tip images the atomic va-cancy states and the segment at which it descends from the top to the bottomterraces where the tunneling matrix element from tip to sample may change.

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the corresponding dI/dV spectrum (Fig. 2C), we find an increaseddensity of state all along the step edge at the same energies that show asuppressed density of states within the T and L gaps on the open ter-race (marked with curly brackets). The added density of states,contributed by the edge mode, is nicely captured in the dI/dV profilesin Fig. 2 (D and E) from the two energy ranges (marked by dotted linesin Fig. 2C), signifying the presence of the 1D edge mode. The densityof states of the edge mode remains fairly constant down to the lastatom before the SD, as well as at the vicinity of the vacancy on the stepedge, although both serve as strong scatterers for the 1D electronicstates. The small amplitude modulations in dI/dV along the step edgein Fig. 2 (D and E) are commensurate with the atomic positions re-solved in topography in Fig. 2B and accordingly do not disperse inenergy. The robustness of the 1D edge mode stands in sharp contrastto the scattering of the 2D surface states we find on the open terraceboth from the remote step edge and from the SD, giving rise to a faintquantized pattern in Fig. 2C (see also fig. S2).

Since the spatial distribution of the wave function of the edge statecan vary with energy, we also spectroscopically map the surroundingof the SD.We overlay in Fig. 3A the 2D dI/dVmappings of the energy-resolved LDOS at the SD vicinity on a topographic height map (seealso fig. S3). At high energies (Fig. 3B), we find metallic surfaces onthe SD, step edge, and adjacent terraces. Below 180 meV (Fig. 3C),we detect the edge mode localized on the edge of the top terrace, allacross the step edge, down to the SD. In a broad range of energies, weobserve an increased LDOS right on the SD (Fig. 3D), which agreeswith the continuation of the mode into the SD. We observe similarbehavior also at lower energies around −100 meV corresponding tothe L energy gap (see fig. S3), where the mode seems to be localizedslightly lower on the step edge.

The absence of backscattering and hybridization of the 1D edgemode along the step edge followed by its apparent channeling intothe SD provides two important observations: It presents first spec-troscopic signature for the long sought prediction of perfectly con-ducting channels on an SD, and it reflects the phenomenology of astrong TI with weak indices rather than of a higher-order TI. For amaterial characterized by nonvanishing weak indices, n = (n1, n2, n3),a bulk SD with Burger’s vector b is theoretically expected to bind ahelical mode if n · b = p(mod 2p) (10, 14). When the space-groupsymmetry is fully taken into account, there are cases in which thegapless SD states are formed even when this condition is not satisfied(14). Yet, for Bi with a space group of R�3M, the above condition canbe simply applied for the formation of helical modes (20). Hence, theexpected phenomenology on a single unit cell high step edge, termi-nated by an SD, is of a 1D helical mode that runs along the step edgeand continues into the SD line, with no backscattering and withoutgapping, as guaranteed by time reversal symmetry. This is consistentwith our observations. In contrast, a higher-order TI is characterizedby a null strong index, n0 = 0, and null weak indices, n = (0,0,0).Hence, an SD in a higher-order TI is not expected to bind a helicalmode, unless, as was shown recently, its Burger’s vector is of a frac-tion of a unit cell height (53). Therefore, the expected phenomenol-ogy in the case of a higher-order TI is of even number of helicalmodes on the step edge that would presumably gap out or showssome signatures of back scattering near the impurity and the SD.Even if the pair of hinge modes avoids hybridization along the stepedge, the SD would necessarily scatter them one to another and leadto their gradual gapping in its vicinity. The absence of these signa-tures, and moreover the existence of a slight increase in the LDOS on

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Fig. 3. Imaging the localization of the edge mode in the vicinity of the SD.(A) Topographic image of the region around an SD. (B to D) Spatially resolveddifferential conductance map in the vicinity of the SD. The increased LDOS onthe step edge and the SD shown in (C) and (D) correspond to a segment ofthe 1D edge mode. The black dashed line represents the position of the linecut shown in Fig. 2C.

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the SD, favors their identification as helical modes of a strong TI withnonzero weak indices rather than as higher-order TI hinge states.

While the phenomenology in the vicinity of the SD strongly sug-gests the existence of nonvanishing weak indices, this is inconsistentwith most of theoretical models of Bi (3, 32, 34, 41), including our abinitio calculations in the absence of strain. These find the same bandordering at all TRIM points in Bi, leading to trivial Z2 topologicalindices, n0 = 0, n = (0,0,0). These conclusions, however, rely on theL energy gap that in calculation is found to be as small as 5 meV(Fig. 1C). This small energy gap makes the calculation of the bandordering at the L point challenging. Recent studies suggest, for in-stance, that this energy gap is in fact inverted with respect to theenergy gap at the T TRIM point, which classifies Bi as a strong TI,n0 = 1, with nonzero weak indices, n = (1,1,1) (35, 36, 42). In thiscase, theweak TI-associated helical edgemodes are expected to bind toSDs, and the 2D surface states of Bi are identified with the Dirac sur-face band of a strong TI, providing a compelling unified resolution tothe combined boundary phenomenology observed in Bi.

Strain analysisWenext characterize in detail the SD and its effect on its surroundings.We extract the strain field induced by the SD directly from the gradi-

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ent of the measured topography, ∇z(x, y). Its phase, arg ( ∇z), shownin Fig. 4A, confirms the azimuthally oriented strain profile that windsaround the SD axis. Its magnitude,∣∇z∣, displayed in Fig. 4B, shows aradially decaying strain profile as the displacement of the lattice, b,distributes over growing circumference. Cuts of the strain magnitudealong the two directions, marked with arrows in Fig. 4C, demonstratethe anisotropy of the strain distribution. Ninety percent of the strainrelaxes along the direction of the step edge within 8 nm (blue arrow),while 60° away from this direction (orange arrow), it relaxes withinonly ~3 nm. This anisotropic strain pattern on the surface suggeststhat the SD axis is tilted away from the <111> direction normal tothe surface. A single bilayer long Burger’s vector along the <111>direction is incompatible with the ABC stacking of the Bi bilayers(Fig. 4C, inset). A surface normal SD would thus result in excessstrain and would be energetically unfavorable. The two observa-tions are consistently resolved by asserting an SD oriented withthe <001> direction, along which the periodicity is indeed of asingle bilayer. Such an SD is also expected to bind a helical modeas n · b = p.

Our ab initio calculations for the response of Bi to a uniformstrain (Fig. 4D) show that strain can also promote band inversionat the L TRIM point (36, 54, 55) that would substantiate a strong

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Fig. 4. Strain analysis in the vicinity of the SD and modeling. (A) Phase and (B) magnitude of the shear strain field around an SD calculated from the gradient of thetopography in Fig. 3A, showing its azimuthal radially decaying profile. The amount of strain within the dashed line in (B) is sufficient to invert the band ordering aroundthe L energy gap and to induce a topological phase. (C) Radial cuts along different directions from the SD of the strain magnitude. Inset shows the (001) direction of theBurger’s vector piercing the ABC stacked Bi bilayers. (D) Ab initio calculation of the band structure evolution around the L gap under uniform strain. A topological phasetransition occurs at about 3.5% strain.

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TI, n0 = 1, phase with nonzero weak indices, n = (111), in ab initiocalculation (see fig. S5 for more details). The 3 to 4% strain neededfor inducing the topological phase transition in calculation is a smallfraction of the shear strain applied at the vicinity of the SD. Thisexemplifies the ability of a bulk topological defect to embed around it acylindrical volume of an induced topological phase, as marked by thedashed line in Fig. 4B. Nevertheless, in our spectroscopic mappings,we do not identify any signature of gap closing and reopening awayfrom the SD axis, which would necessarily accompany such ashear-induced transition. We thus conclude that Bi is uniformlyclassified as a strong TI with nonzero weak indices. We also notethat recent experiments on single crystals of Bi seem to identifyvalley symmetry breaking due to spurious strain fields in thesamples (22), which alone may promote the Z2 topological classi-fication in Bi.

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DISCUSSIONWe argue that the observed phenomenology on all boundaries of Biincluding surfaces, step edges, and SDs is strongly influenced by thesmall energy gap at the L TRIM point. This small energy gap po-sitions Bi in the vicinity of a topological phase transition betweena higher-order TI and a strong TI with nonvanishing weak indices.The surface and edge states that disperse between the correspondingprojections of the bulk T and L TRIM points on theZ2 trivial side ofthe phase transition are the precursors of the helical boundariesmodes that would be firmly established beyond the critical transitionpoint (for instance, under application of strain or Sb doping). DopingBi with Sb was shown to invert the bulk energy gap at L while hardlyaffecting the surface state dispersion around the �G point (27). Hence,those edge modes are approximately helical and protected frombackscattering already on the Z2 trivial side of the topological phasetransition. Their penetration depth, the critical length associated withthe phase transition, is expected to diverge as the energy gap narrows(39, 56). Their exact topological nature is, therefore, ill-defined in afinite sample (57). A similar boundary mode evolution was indeed re-ported across the topological phase transition from the 3D TI BiSe2 toa trivial insulator with the substitution of Se with S (58–60). Similarreasoning was also applied to the edgemodes of a bilayer silicene (61).In both cases, approximate helical boundary modes were shown toemerge on approaching the topological quantum phase transitionfrom the trivial phase.

In summary, we have mapped the surface and edge spectra of Bi(111). We find surface and edge modes that disperse in energy andmomentum between the projections of the bulkT and L TRIMpoints.The edge mode withstands crystallographic irregularities withoutshowing any signs of backscattering and gapping. In particular, theedge mode seems to bind to bulk SD as expected for TI with non-vanishing weak indices. These observations provide first spectroscopicsignature for the existence of helical mode on an SD and, by this, allowus to determine the topological nature of Bi. Our findings disfavor therecent identification of the edge mode as a hinge mode of a higher-order TI class, which, in the absence of protection, would be highlysusceptible to hybridization by these structural perturbations.We ar-gue that the small scale of the bulk L energy gap, which in our abinitio calculation amounts to 5 meV, positions Bi within the criticalregion of a topological phase transition to a strong TI with non-vanishing weak indices. It also facilitates strain-induced transitioninto the topological phase in the vicinity of the SD, as well as

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throughout the crystal by spurious crystallographic defects. This de-monstrates that SD can serve not only as a probe for the topologicalclassification of materials but also as a tool to manipulate and alterthis classification.

MATERIALS AND METHODSSample growthSingle crystals of GdPtBi were grown by the solution growth methodfrom Bi flux in Ta tube. The extra Bi flux was separated from thecrystals by decanting the Ta tube well above the melting point ofBi. The complete procedure of crystal growth is described in (50). Inthis procedure, most of the crystals were found at the bottom of thecrucible where they stick. The grown crystals were removed with thehelp of a stainless steel knife after cutting the tube for further char-acterizations. After a careful study, we realized that Bi inclusions areinvariably present in the 10th of a millimeter of the bottom of theGdPtBi crystals.

STM measurementsThe STM measurements were carried out using commercial Pt-Irtips. The tips were characterized on a freshly prepared clean Cu(111) single crystal. This ensured a stable tip with reproducibleresults. The dI/dVmeasurements were performed using the standardlock-in technique. The map parameters for Fig. 1F are V = 200 mV,I = 900 pA, f = 413 Hz, and ac root mean square (rms) amplitude =1 mV. The map parameters for Fig. 2C and Fig. 3 (B to D) are V =200 mV, I = 500 pA, f = 1373 Hz, and ac rms amplitude = 3 mV.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/11/eaax6996/DC1Section S1. Effect of substrate magnetic order on BiSection S2. dI/dV mapping of the SDSection S3. Edge states of Bi nanoribbonSection S4. Model of shear strainFig. S1. Effect of the magnetic order of the GdPtBi substrate on Bi.Fig. S2. QPI of 2D surface states from SD.Fig. S3. Direct visualization of the topological edge mode localized on the SD.Fig. S4. The model used to simulate Bi step edge.Fig. S5. Effect of shear strain on the electronic properties of Bi.Reference (62)

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Acknowledgments: We acknowledge valuable discussions with P. W. Brouwer, L. Trifunovic,B. Andrei Bernevig, R. Ilan, and J. Cano. Funding: H.B. acknowledges support from the UnitedStates–Israel Binational Science Foundation (BSF; grant number 2016389), the HelmsleyCharitable Trust (grant number 2018PG-ISL006), the German-Israeli Foundation (GIF; I-1364-303.7/2016), and the European Research Council (ERC) under the European Union’s Horizon

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SC I ENCE ADVANCES | R E S EARCH ART I C L E

2020 research and innovation program (grant agreement no. 678702). Author contributions:A.K.N. and J.R. acquired and analyzed the data. H.B. and N.A. conceived the experiments.H.B., N.A., and A.K.N. wrote the manuscript with substantial contributions from all authors. B.Y.,R.Q., and H.F. modeled the system. C.S. grew the material in C.F.’s group. Competinginterests: The authors declare that they have no competing interests. Data and materialsavailability: All data needed to evaluate the conclusions in the paper are present in thepaper and/or the Supplementary Materials. Additional data related to this paper may berequested from the authors.

Nayak et al., Sci. Adv. 2019;5 : eaax6996 1 November 2019

Submitted 17 April 2019Accepted 14 September 2019Published 1 November 201910.1126/sciadv.aax6996

Citation: A. K. Nayak, J. Reiner, R. Queiroz, H. Fu, C. Shekhar, B. Yan, C. Felser, N. Avraham,H. Beidenkopf, Resolving the topological classification of bismuth with topological defects.Sci. Adv. 5, eaax6996 (2019).

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Resolving the topological classification of bismuth with topological defects

Avraham and Haim BeidenkopfAbhay Kumar Nayak, Jonathan Reiner, Raquel Queiroz, Huixia Fu, Chandra Shekhar, Binghai Yan, Claudia Felser, Nurit

DOI: 10.1126/sciadv.aax6996 (11), eaax6996.5Sci Adv 

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