Universal response of the type-II Weyl semimetals phase diagram
P. Rü?mann, A. P. Weber, F. Glott, N. Xu, M. Fanciulli, S. Muff, A. Magrez, P. Bugnon, H. Berger, M. Bode, J. H. Dil, S. Blügel, P. Mavropoulos, P. Sessi
UUniversal response of the type-II Weyl semimetals phase diagram
P. R¨ußmann, A. P. Weber,
2, 3
F. Glott, N. Xu,
2, 3
M. Fanciulli,
2, 3
S. Muff,
2, 3
A. Magrez, P. Bugnon, H. Berger, M. Bode, J. H. Dil,
2, 3
S. Bl¨ugel, P. Mavropoulos, and P. Sessi ∗ Peter Gr¨unberg Institut and Institute for Advanced Simulation,Forschungszentrum J¨ulich and JARA, 52425 J¨ulich, Germany Institute of Physics, ´Ecole Polytechnique F´ed´eralede Lausanne, 1015 Lausanne, Switzerland Swiss Light Source, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland Physikalisches Institut, Experimentelle Physik II,Universit¨at W¨urzburg, Am Hubland, D-97074 W¨urzburg, Germany (Dated: June 5, 2017) a r X i v : . [ c ond - m a t . m e s - h a ll ] J un bstract The discovery of Weyl semimetals represents a significant advance in topological band theory.They paradigmatically enlarged the classification of topological materials to gapless systems whilesimultaneously providing experimental evidence for the long-sought Weyl fermions. Beyond funda-mental relevance, their high mobility, strong magnetoresistance, and the possible existence of evenmore exotic effects, such as the chiral anomaly, make Weyl semimetals a promising platform todevelop radically new technology. Fully exploiting their potential requires going beyond the mereidentification of materials and calls for a detailed characterization of their functional response,which is severely complicated by the coexistence of surface- and bulk-derived topologically pro-tected quasiparticles, i.e., Fermi arcs and Weyl points, respectively. Here, we focus on the type-IIWeyl semimetal class where we find a stoichiometry-dependent phase transition from a trivial to anon-trivial regime. By exploring the two extreme cases of the phase diagram, we demonstrate theexistence of a universal response of both surface and bulk states to perturbations. We show thatquasi-particle interference patterns originate from scattering events among surface arcs. Analysisreveals that topologically non-trivial contributions are strongly suppressed by spin texture. Wealso show that scattering at localized impurities generate defect-induced quasiparticles sitting closeto the Weyl point energy. These give rise to strong peaks in the local density of states, which liftthe Weyl node significantly altering the pristine low-energy Weyl spectrum. Visualizing the micro-scopic response to scattering has important consequences for understanding the unusual transportproperties of this class of materials. Overall, our observations provide a unifying picture of theWeyl phase diagram. x W − x Te family as promisingcompounds [15–18]. One of the most interesting aspects of type-II Weyl materials is thepossibility to continuously tune their topological properties by acting onto their stoichiom-etry [18]. The resulting phase diagram offers an ideal platform to explore the functionalresponse of the electronic properties and the existence of unifying trends within the Weylphase.The Mo x W − x Te family is characterized by a layered structure which crystallizes in a T d orthorhombic cell lacking inversion symmetry at low temperatures (space group P mn )[19]. The transition metal (W and Mo) planes are separated by Te bilayers as shown inFig. 1(a). Adjacent Te–(W/Mo)–Te trilayers are weakly bound by van der Waals forcesthus offering a natural cleaving plane. As a result, the surface exposed after cleaving isalways Te-terminated. As illustrated in Fig. 1(a), the Te atoms occupy two inequivalent3 x E WTe MoTe WP Mo x W Te WP WP c MoTe WTe Mo doping8 WPs 8 WPs 4 WPs
Strain, SOC strength, … surface resonancetrivial topological arc k x k y WP da b top surfacebottom surface Te1Te2W/Mo1W/Mo2 Te1 Te2 FIG. 1: a Crystal structure of MoTe and WTe ; b Atomically resolved STM topography of theWTe surface. The existence of two inequivalent Te sites labelled Te1 and Te2 with Te1 beingslightly higher than Te2 gives rise to the column-like character visible in the STM image. c Schematic representation of the type-II Weyl phase diagram. Although being topologically trivial,WTe is found to lie very close to the topological phase transition. Consequently, small latticedistortions can easily drive WTe into a Weyl phase. By starting from WTe and substitutingW with Mo, the system enters into a progressively more stable Weyl phase. d Evolution of thetopological Fermi arcs as a function of the Mo concentration. By increasing the Mo concentration,topological Fermi arcs (red lines) become progressively larger. sites, labelled Te1 and Te2, with one slightly protruding over the other. This is reflected inthe column-like character visible in the atomically resolved STM image reported in Fig. 1(b).Density functional theory calculations indicated how, by starting from WTe —which liesclose to a topological phase transition—and substituting W with Mo, the system enters intoa progressively more stable Weyl phase as schematically illustrated in Fig. 1(c) [18]. In4articular, by increasing the Mo concentration, the Weyl points become well-separated inreciprocal space and thus cannot easily be annihilated by small perturbations [18]. The largerWeyl point separation has direct consequences for the surface, with topological Fermi arcsgetting progressively larger [see red lines in Fig. 1(d)]. In this respect, it is worth noticingthat the surface electronic structure of these compounds is substantially complicated by theconcomitant existence of trivial surface states. Although they do not form open arcs, trivialstates partially overlap with the projected bulk electronic structure, giving rise to surfaceresonances [dashed lines in Fig. 1(d)] which are characterized by a reduced surface spectralweight. As a result, their pure surface state part (blue line) can effectively mimic an openarc-like contour.Experimentally investigating this phase diagram proved problematic. This is mainly dueto two reasons: (i) Contrary to type-I Weyl materials, in type-II, the projection of theWeyl points onto the surface is overlapping with several bulk-derived trivial states, therebycomplicating the discrimination of topological Fermi arcs from other states of trivial origin[20–27]; (ii) The Weyl points are expected to appear at energies above the Fermi level [15–18], where they are inaccessible to conventional photoemission techniques [20–27]. In thiscontext, while a general consensus exists over the topological nature of the MoTe arcs[20–22], the situation appears highly controversial for WTe . Although surface arcs haveclearly been observed in several photoemission studies, their topological or trivial natureis highly debated with different studies reaching conflicting conclusions [23–26]. Ab initio calculations also show that—while MoTe is well inside the Weyl regime [28]—WTe is inclose proximity to a Weyl phase transition [23]. This can give rise to Weyl points of oppositechirality which are very close in reciprocal space and thus can easily be annihilated by verysmall lattice distortions induced by strain or temperature as discussed in Ref. 23. Therefore,it is particularly important to investigate the existence of universal signatures spanning theentire Weyl phase diagram, e.g. by comparing the two extreme cases, MoTe and WTe .Here, we visualize the response of MoTe and WTe and discuss the results in terms ofWeyl nodes and Fermi arcs. We identify the existence of a universal response of these systemsto perturbations, which is found to be composition-independent. In particular, we reportthe emergence of well-defined quasiparticle interference patterns originating from surfacearcs. Contrary to earlier studies [22]: (i) we can clearly resolve their open contour and (ii)demonstrate that topological Fermi arc contributions are strongly suppressed because of their5 q ace fd WTe2 MoTe2 a b -5612800.1 0.3 0.5 12-560.10.3 E - E F ( m e V ) E - E F ( m e V ) -0.2 0.0 0.2-0.2 0.0 0.2 s u r f a c e w e i g h t HighLow s p e c t r a l i n t e n s i t y HighLow s u r f a c e w e i g h t HighLow s p e c t r a l i n t e n s i t y HighLow
FIG. 2: a,b
Calculated bulk band structure at k z = 0, c,d theoretical and e,f experimental Fermisurface for WTe and MoTe , respectively. In (a), the dotted lines show the band structure along k y at the particular k x where the electron and hole pockets approach each other closest. In (b), thedotted lines show the band structure along k y at the specific k x where Weyl points appear (greenand red circles, reflection-symmetric around k x = 0). The experimental Fermi surfaces have beenobtained by angle-resolved photoemission spectroscopy at a temperature T <
60 K using photonenergies of 20 and 24 eV for MoTe and WTe , respectively. spin texture which protects them from back-scattering [22, 29]. Furthermore, in line withthe theoretical predictions of the response of Weyl semimetals, we reveal the emergence ofnew quasiparticles arising at the Weyl point energy, which lift the density of states minimumassociated to the Weyl node [30, 31]. 6 q -0.40-0.20.20.4 0.4 0.60.20-0.2-0.4-0.6-0.40-0.20.20.4 0.4 0.60.20-0.2-0.4-0.6 q -0.40-0.20.20.4 0.4 0.60.20-0.2-0.4-0.6-0.40-0.20.20.4 0.4 0.60.20-0.2-0.4-0.6 a bc d efWTe MoTe FIG. 3: a Differential conductance d I /d U map close to the Fermi ( E − E F = 10 meV); b Fourier-transformed d I /d U map and c theoretically calculated quasi-particle interference pattern obtainedon WTe . d Differential conductance d I /d U map close to the Fermi ( E − E F = 10 meV); e Fourier-transformed d I /d U map and f theoretically calculated quasi-particle interference patternobtained on MoTe . The quasi-particle interference pattern is dominated by scattering eventsbetween opposite Fermi arcs. Fig. 2(a,b) report the electronic band structure of bulk WTe and MoTe at k z = 0,respectively, where due to the crystal symmetry Weyl points are possible [15], calculatedwith the density-functional-based full-potential relativistic Korringa-Kohn-Rostoker Green-function method [32–35], respectively. Computational details can be found in supplementarymaterial. The combined effects of the inversion asymmetry of the crystal structure andthe strong spin-orbit coupling characterizing these materials give rise to a spin-polarizedband structure [36]. However, whereas Weyl points are emerging in MoTe [see green andred dots in Fig. 2(b) which identify one pair of Weyl points of opposite chirality], a gapbetween electron and hole pockets is clearly visible for WTe [Fig. 2(a)], indicating thetrivial character of this compound. In this context, it is worth noticing that the electronicstructure of WTe is very delicate. As discussed in Ref. 18, small changes of the latticeconstant can drive the system into a non-trivial state hosting 8 Weyl points, proving theclose vicinity of a topological phase transition. Despite these differences highlighted byband structure calculations, our angle-resolved photoemission data reported in Fig. 2(e,f)reveal the presence of arc-like electronic structures in both compounds (see dashed lines).An overall agreement between theory and experiments is found. Although it is tempting7 F = +10 meV Surface state topology vvxvxv
Sca � ering rate FT-dI/dUmap E F = +10 meVE F = +60 meV E F = +10 meV E F = -50 meV E F = -50 meVE F = +60 meV a b c d High s u r f a c e w e i g h t Low High I n t e n s i t y Low 0.37 p s - / a t . % Energy cut
LowHigh T r a n s i t i o n r a t e Lowtopologicaltrivial B a n d c h a r a c t e r E F = +60 meV E F = -50 meV FIG. 4: a Schematic representation of the scattering events among Fermi arcs in MoTe , b the-oretically calculated surface-projected constant energy cuts, c experimentally obtained Fouriertransformed quasi particle interference patterns and d theoretically calculated scattering rate. In(b-d) the black dots identify the Weyl points. All results are given for three representative energies:below the Fermi (top panels), close to the Fermi (middle panels), close to the position of the Weylpoints, where the extension of the topologically non-trivial arcs is maximized (bottom panels). to assign the arc-like features in Fig. 2(e,f) to topological Fermi arc states, comparisonwith calculated constant-energy contours reported in Fig. 2(c,d) reveal a more complicatedscenario. Indeed, ab initio calculations reveal that several electronic features coexist withina very small part of the first Brillouin zone which cannot fully be resolved by photoemissionexperiments [37]. It is worth stressing that, as discussed above, an analysis of our calculationsreveals that most of the arcs visible in Fig. 2(c,d) are of trivial origin and that only for MoTe also topological Fermi arcs are present. This proves that the observation of surface arcs is anecessary, but not a sufficient condition to unequivocally imply the existence of Weyl points.To disentangle trivial from topological Fermi arcs contributions, we performed quasi-particle interference experiments. This technique makes use of the standing-wave patterngenerated by elastic scattering of electronic states at surface defects and has been proven to8 WTe2MoTe2 Experiment Theory
ImpurityWTe ImpurityWTe ImpurityMoTe ImpurityMoTe d I / d U ( a r b . un i t s ) E-E F (meV) L D O S ( a r b . un i t s ) d I / d U ( a r b . un i t s ) L D O S ( a r b . un i t s ) E-E F (meV) E-E F (meV) E-E F (meV) a c eb d f FIG. 5: a,b
Topographic images acquired on WTe and MoTe . In both cases, intrinsic defectsare present on the surface. c,d Comparison of scanning tunneling spectroscopy data taken ona defect-free area (blue line) and by positioning the tip on top of antisites defects (W and Mosubstituting Te) revealing the emergence of quasiparticle resonances close the Weyl point energy(see discussion in the text). e,f
Ab-initio calculated local density of states on the pristine surface(blue line) and on top of an antisite defect (red line) confirm the experimental findings. Greenvertical lines are used as marker to identify the position of the peak maximum. be a powerful method to test the properties of topological materials [38–43]. Contrary toconventional photoemission spectroscopy, it gives access to both occupied and unoccupiedelectronic states thereby providing a complete spectroscopic characterization of all relevantelectronic features around the Fermi energy. This is particularly important for type-II Weylmaterials, where Weyl points are theoretically expected to emerge above the Fermi level [15–18]. Fig. 3(a) and (d) report differential conductance d I /d U maps acquired in close proxim-ity of the Fermi level ( E − E F = 10 meV) on WTe and MoTe , respectively. Their Fouriertransformations—reported in panels (b) and (e)—allow to conveniently analyze scatteringchannels in reciprocal space. Our results reveal the emergence of clear arc-like interferencepatterns as indicated by the black arrows in both compounds which develop along the q x direction at approximately q x = 0 . and MoTe are reported in Fig. 3(c) and (f). As for the experimental results, anarc-like feature is clearly visible in both cases along the q x direction (see dashed lines), whichoriginates from intra-arc scattering among opposite Fermi arcs. This assignment is furthersupported by direct comparison with photoemission data [cf. Fig. 2(e,f), where the scat-tering vector q corresponds to the distance connecting opposite arcs]. Indeed, momentumconservation requires k f = k i + q , where k i and k f are the wave vectors of initial and finalstates and q is the scattering vector connecting them. A quantitative comparison with theexperimental constant energy contours reported in Fig. 2(e,f) allows to directly link theseinterference phenomena to intra-arc scattering among opposite Fermi arcs. Furthermore,contrary to photoemission data, the higher surface sensitivity of STM unequivocally provesthe open contour character of the arcs.In this context, it is worth noticing that the close proximity of WTe to a Weyl transitionallows one to safely exclude any significant contribution of topological Fermi arcs to theobserved interference patters for this compound. Even when considering a slightly distortedstructure hosting Weyl points, they would be so close in reciprocal space that the exten-sion of topological Fermi arcs connecting them would be negligible. This is not the case inMoTe where—by progressively moving towards the energy position of the Weyl points—topologically non-trivial Fermi arcs span a much larger fraction of the Brillouin zone. Thisis illustrated by the schematic representation reported in Fig. 4(a) and the theoretically cal-culated constant energy cuts displayed in Fig. 4(b). As shown in panel (a), by progressivelyincreasing the energy (from top to bottom panel) the trivial part (blue line) of the arc isabsorbed into the bulk electronic band structure whereas the topologically non-trivial arc(red line) dominates the scene. This transition from trivial to topology-dominated Fermiarcs is experimentally investigated in Fig. 4(c). At energies well below the Weyl points (up-per panel) only trivial arcs exist. As a consequence, the experimental data show only weakinterference patterns. This can be traced back to the combined effect of (i) the V-shapevisible in the calculated constant energy contour which results in poor nesting conditionsand (ii) the overlap with projected bulk pockets making these states surface resonances thatare much less localized at the surface than “real” surface states [cf. top panel of Fig. 4(a)].10y going up in energy, the surface states become well separated from bulk states [cf. middlepanel of Fig. 4(a)] while simultaneously getting progressively larger, and thus occupying alarger fraction of the Brillouin zone. As a result, a well-defined arc-like interference patternappears (see middle panel) which—according to calculations—is dominated by trivial states.By moving to even higher energies we move closer to the Weyl points and the topologicalFermi arcs prevail [cf. bottom panel of Fig. 4(a)]. This process is associated to a significantflattening of the arc, where several equivalent vectors can connect opposite parallel segmentsof the Fermi arcs [see grey arrows in Fig. 4(a), lower panel]. Despite this scenario supportingoptimal nesting conditions, the scattering intensity drops (see bottom panel). Comparisonwith ab initio calculations reveals that this effect is directly linked to the spin texture ofthe topological Fermi arcs. Indeed, as schematically illustrated by the blue arrows in Fig. 4(a), at E − E F = +60 meV the spin polarization is basically pointing in opposite directionsfor opposite topological Fermi arcs segments (spin-resolved constant-energy contours are re-ported in the supplementary material). This spin texture results in an effective protectionagainst scattering. Such a behavior is reminiscent of the forbidden backscattering originallydiscussed in Rashba systems [45] and, more recently, in topological insulators [39]. In thepresent case, however, the presence of large parallel segments with opposite spin polarizationsignificantly extend the protection well beyond time-reversal symmetry partner states.These conclusions, which are based on a combined analysis of the spin-resolved bandstructure and quasi-particle interference mapping, are further quantitatively supported bythe calculated scattering rates reported in Fig. 4(d). By progressively increasing the energy,the surface arcs occupy a larger fraction of the Brillouin zone. Consequently, more scatteringvectors connecting opposite arcs become possible and the scattering intensity rises (see redline in the middle panel). However, once the topological Fermi arcs set in (lower panel) thescattering intensity drops because of the discussed spin-texture protection mechanism.Finally, we discuss the effect impurities have on Weyl nodes. Recent theoretical pre-dictions showed that a common characteristic of Dirac-like materials is the emergence ofimpurity-induced quasiparticles which lift the Dirac node [31]. This behavior has been re-cently confirmed in topological insulators, where impurity resonances induced by magneticdopants have been shown to effectively fill the gap which is expected to open at the Diracpoint in magnetically ordered samples [46]. A similar behavior has been proposed to arisein Weyl semimetals. In particular, scattering at localized impurities is expected to lift the11eyl node by inducing new quasiparticle resonances close to the Weyl point energy [30, 31].The emergence of these quasiparticles has been theoretically proposed as a signature of aWeyl phase. This has been experimentally investigated in Fig. 5, where panels (a) and (b)report topographic images acquired on the two different compounds, i.e., WTe and MoTe .Intrinsic defects highlighted by arrows are visible in both cases. They have been identifiedas anti-sites (W or Mo atoms substitute Te in the topmost layer) and are commonly foundin transition metal dichalcogenides [47–49]. Fig. 5(c,d) reports STS data acquired on bothmaterials by positioning the tip on top (red line) and far away from a defect (blue line).The minimum, visible in proximity to the Fermi level on defect-free areas, highlights thesemimetallic character of these compounds. However, on top of defects a strong peak isvisible in both materials which lifts the local density-of-states minimum.These defect-induced quasiparticle resonances appear very close to the energy whereWeyl points are expected to emerge (see vertical green line which highlights the peak max-imum). These findings are supported by the ab initio -calculated local-density-of-states atthe impurity atom reported in Fig. 5(e) and (f) for WTe and MoTe , respectively. Bothreveal the emergence of quasiparticle resonances located at energies which are in excel-lent agreement with the experiments. In particular, in MoTe the experimentally observedimpurity-induced quasiparticle resonance is positioned at the calculated Weyl point energy( E − E F = +48 meV). As discussed above, for WTe our calculations predict a trivial materialnear to a topological phase transition into a Weyl phase. In this case, the impurity-inducedresonances emerge close to the very narrow energy gap between electron and hole pockets, atan energy where Weyl points are expected to emerge according to Refs. 15, 23, 26. This ob-servation provides strong evidence that topological Weyl transitions are continuous smoothtransitions of the global bulk band structure. It follows that, although topological indexeschange when driving the system through a topological quantum phase transition, the bandstructure-dependent experimental observables—such as the impurity-induced resonances re-ported here—can be continuously tuned, and are not characterized by any discontinuity,i.e., on/off behavior. In this sense, Weyl phase transitions appear to behave similarly totopological insulators phase transitions where, by approaching the critical point at whichthe bulk band structure becomes inverted, spin-polarized helical surface states progressivelyemerge within the bulk gap [50].We would like to stress that the relevance of our observations goes well beyond topological12and-structure aspects. It is long known that disorder, and especially resonant impurities,can significantly impact onto transport properties. Even recently, the presence of defectshas been invoked to be at the origin of both positive as well as negative magnetoresistanceeffects in topological semimetals [51, 52]. In this context, our observations contribute byproviding a detailed microscopic picture of the resonant scattering off impurities in type-IIWeyl semimetals. In particular, we demonstrate that intrinsic defects significantly alter thelocal density of states close to the Weyl points, ultimately changing the low-energy spectrumof Weyl semimetals. We conclude that the presence of defects cannot be overstressed andsuggest that they play an important role in determining the fascinating transport propertiesof this class of materials [53, 54].Overall, we reveal the existence of a universal response of the type-II Weyl semimet-als phase diagram. We show that surface arcs dominate the interference pattern, with thetopological Fermi arc contribution being strongly suppressed by its spin texture. In agree-ment with theoretical predictions, we also demonstrated that impurity-induced quasiparticleresonances emerge close to the energy where Weyl points are expected. Our observationshighlight that the functional response of both surface and bulk states to perturbations inthis class of materials does not depend on whether we have passed the Weyl phase transitionor we are simply close to it. This allows to infer the existence of a stoichiometry-independentresponse to perturbations for type-II Weyl, providing a unifying picture of the type-II Weylphase diagram.This work was supported by the Deutsche Forschungsgemeinschaft within SPP 1666(Grant No. BO1468/21-1 and MA4637/3-1) and through SFB 1170 “ToCoTronics” (projectA02). P.R., P.M., and S.B. acknowledge financial support from the VITI project of theHelmholtz Association as well as computational support from the JARA-HPC Supercom-puting Centre at RWTH Aachen University. ∗ corresponding author: [email protected][1] M. Z. Hasan and C. 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