Universal response of the type-II Weyl semimetals phase diagram

TL;DR

This paper investigates the universal response of surface and bulk states in type-II Weyl semimetals, revealing a phase transition and how scattering processes influence their topological and electronic properties.

Contribution

It demonstrates a universal response in type-II Weyl semimetals' surface and bulk states across different phases, highlighting the effects of scattering and impurities on their electronic structure.

Findings

Existence of a stoichiometry-dependent phase transition from trivial to non-trivial regimes.

Universal response of surface and bulk states to perturbations.

Impurity-induced quasiparticles significantly alter the Weyl spectrum.

Abstract

The discovery of Weyl semimetals represents a significant advance in topological band theory. They paradigmatically enlarged the classification of topological materials to gapless systems while simultaneously providing experimental evidence for the long-sought Weyl fermions. Beyond fundamental relevance, their high mobility, strong magnetoresistance, and the possible existence of even more exotic effects, such as the chiral anomaly, make Weyl semimetals a promising platform to develop radically new technology. Fully exploiting their potential requires going beyond the mere identification 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 protected quasiparticles, i.e., Fermi arcs and Weyl points, respectively. Here, we focus on the type-II Weyl semimetal class where we find a stoichiometry-dependent phase transition from a trivial to a non-trivial regime. By exploring the two extreme cases of the phase diagram, we demonstrate the existence of a universal response of both surface and bulk states to perturbations. We show that quasi-particle interference patterns originate from scattering events among surface arcs. Analysis reveals that topologically non-trivial contributions are strongly suppressed by spin texture. We also show that scattering at localized impurities generate defect-induced quasiparticles sitting close to the Weyl point energy. These give rise to strong peaks in the local density of states, which lift the Weyl node significantly altering the pristine low-energy Weyl spectrum. Visualizing the microscopic response to scattering has important consequences for understanding the unusual transport properties of this class of materials. Overall, our observations provide a unifying picture of the Weyl phase diagram.

Full text

Beyond their fundamental relevance, Weyl semimetals are characterized by intriguing transport properties. Very high mobility, extremely strong magnetoresistence SNS2015 , and even more exotic phenomena such as the chiral anomaly PhysRevX.5.031023 or the possibility for Fermi arcs to tunnel through the bulk via the Weyl points have been discussed MNH2016 . Because of the robustness inherited from topological protection, these effects are raising great expectations for direct applications of these materials in spintronics and magneto-electrics.

Weyl semimetals have been first experimentally discovered in the TaAs monopnictide family, known as type-I Xu613 ; PhysRevX.5.031013 ; YLS2015 ; XAB2015 ; LYS2016 ; LXW2015 ; PhysRevLett.115.217601 . More recently, it has been suggested that once Lorentz invariance is broken, a new flavor of Weyl material can be realized, the so-called type-II, where strongly tilted Weyl cones emerge at the boundaries between bulk electron and hole pockets SGW2015 . Theoretical predictions identified the MoxW1-xTe2 family as promising compounds SGW2015 ; PhysRevB.92.161107 ; PhysRevLett.117.056805 ; CXC2016 . One of the most interesting aspects of type-II Weyl materials is the possibility to continuously tune their topological properties by acting onto their stoichiometry CXC2016 . The resulting phase diagram offers an ideal platform to explore the functional response of the electronic properties and the existence of unifying trends within the Weyl phase.

The MoxW1-xTe2 family is characterized by a layered structure which crystallizes in a $T_{d}$ orthorhombic cell lacking inversion symmetry at low temperatures (space group $Pmn$21) Clarke1978 . The transition metal (W and Mo) planes are separated by Te bilayers as shown in Fig. 1(a). Adjacent Te–(W/Mo)–Te trilayers are weakly bound by van der Waals forces thus offering a natural cleaving plane. As a result, the surface exposed after cleaving is always Te-terminated. As illustrated in Fig. 1(a), the Te atoms occupy two inequivalent sites, labelled Te1 and Te2, with one slightly protruding over the other. This is reflected in the column-like character visible in the atomically resolved STM image reported in Fig. 1(b).

Density functional theory calculations indicated how, by starting from WTe2—which lies close to a topological phase transition—and substituting W with Mo, the system enters into a progressively more stable Weyl phase as schematically illustrated in Fig. 1(c) CXC2016 . In particular, by increasing the Mo concentration, the Weyl points become well-separated in reciprocal space and thus cannot easily be annihilated by small perturbations CXC2016 . The larger Weyl point separation has direct consequences for the surface, with topological Fermi arcs getting progressively larger [see red lines in Fig. 1(d)]. In this respect, it is worth noticing that the surface electronic structure of these compounds is substantially complicated by the concomitant existence of trivial surface states. Although they do not form open arcs, trivial states partially overlap with the projected bulk electronic structure, giving rise to surface resonances [dashed lines in Fig. 1(d)] which are characterized by a reduced surface spectral weight. As a result, their pure surface state part (blue line) can effectively mimic an open arc-like contour.

Experimentally investigating this phase diagram proved problematic. This is mainly due to two reasons: (i) Contrary to type-I Weyl materials, in type-II, the projection of the Weyl points onto the surface is overlapping with several bulk-derived trivial states, thereby complicating the discrimination of topological Fermi arcs from other states of trivial origin HMO2016 ; PhysRevX.6.031021 ; DWD2016 ; PhysRevB.94.121112 ; PhysRevB.94.121113 ; PhysRevB.94.161401 ; PhysRevB.94.241119 ; BSI2016 ; (ii) The Weyl points are expected to appear at energies above the Fermi level SGW2015 ; PhysRevB.92.161107 ; PhysRevLett.117.056805 ; CXC2016 , where they are inaccessible to conventional photoemission techniques HMO2016 ; PhysRevX.6.031021 ; DWD2016 ; PhysRevB.94.121112 ; PhysRevB.94.121113 ; PhysRevB.94.161401 ; PhysRevB.94.241119 ; BSI2016 . In this context, while a general consensus exists over the topological nature of the MoTe2 arcs HMO2016 ; PhysRevX.6.031021 ; DWD2016 , the situation appears highly controversial for WTe2. Although surface arcs have clearly been observed in several photoemission studies, their topological or trivial nature is highly debated with different studies reaching conflicting conclusions PhysRevB.94.121112 ; PhysRevB.94.121113 ; PhysRevB.94.161401 ; PhysRevB.94.241119 . $Ab$ $initio$ calculations also show that—while MoTe2 is well inside the Weyl regime FN1 —WTe2 is in close proximity to a Weyl phase transition PhysRevB.94.121112 . This can give rise to Weyl points of opposite chirality which are very close in reciprocal space and thus can easily be annihilated by very small lattice distortions induced by strain or temperature as discussed in Ref. PhysRevB.94.121112, . Therefore, it is particularly important to investigate the existence of universal signatures spanning the entire Weyl phase diagram, e.g. by comparing the two extreme cases, MoTe2 and WTe2.

Here, we visualize the response of MoTe2 and WTe2 and discuss the results in terms of Weyl nodes and Fermi arcs. We identify the existence of a universal response of these systems to perturbations, which is found to be composition-independent. In particular, we report the emergence of well-defined quasiparticle interference patterns originating from surface arcs. Contrary to earlier studies DWD2016 : (i) we can clearly resolve their open contour and (ii) demonstrate that topological Fermi arc contributions are strongly suppressed because of their spin texture which protects them from back-scattering DWD2016 ; PhysRevLett.117.266804 . Furthermore, in line with the theoretical predictions of the response of Weyl semimetals, we reveal the emergence of new quasiparticles arising at the Weyl point energy, which lift the density of states minimum associated to the Weyl node PhysRevB.87.155123 ; doi:10.1080/00018732.2014.927109 .

Fig. 2(a,b) report the electronic band structure of bulk WTe2 and MoTe2 at $k_{z}=0$, respectively, where due to the crystal symmetry Weyl points are possible SGW2015 , calculated with the density-functional-based full-potential relativistic Korringa-Kohn-Rostoker Green-function method Heers2011 ; Zimmermann2016 ; Long2014 ; Zeller2004 , respectively. Computational details can be found in supplementary material. The combined effects of the inversion asymmetry of the crystal structure and the strong spin-orbit coupling characterizing these materials give rise to a spin-polarized band structure PhysRevB.94.195134 . However, whereas Weyl points are emerging in MoTe2 [see green and red dots in Fig. 2(b) which identify one pair of Weyl points of opposite chirality], a gap between electron and hole pockets is clearly visible for WTe2 [Fig. 2(a)], indicating the trivial character of this compound. In this context, it is worth noticing that the electronic structure of WTe2 is very delicate. As discussed in Ref. CXC2016, , small changes of the lattice constant can drive the system into a non-trivial state hosting 8 Weyl points, proving the close vicinity of a topological phase transition. Despite these differences highlighted by band 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 tempting to assign the arc-like features in Fig. 2(e,f) to topological Fermi arc states, comparison with calculated constant-energy contours reported in Fig. 2(c,d) reveal a more complicated scenario. Indeed, $ab$ $initio$ calculations reveal that several electronic features coexist within a very small part of the first Brillouin zone which cannot fully be resolved by photoemission experiments FN2 . It is worth stressing that, as discussed above, an analysis of our calculations reveals that most of the arcs visible in Fig. 2(c,d) are of trivial origin and that only for MoTe2 also topological Fermi arcs are present. This proves that the observation of surface arcs is a necessary, 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 pattern generated by elastic scattering of electronic states at surface defects and has been proven to be a powerful method to test the properties of topological materials RSP2009 ; PhysRevLett.103.266803 ; SRB2014 ; ZOH2014 ; JZG2014 ; PhysRevB.95.035114 . Contrary to conventional photoemission spectroscopy, it gives access to both occupied and unoccupied electronic states thereby providing a complete spectroscopic characterization of all relevant electronic features around the Fermi energy. This is particularly important for type-II Weyl materials, where Weyl points are theoretically expected to emerge above the Fermi level SGW2015 ; PhysRevB.92.161107 ; PhysRevLett.117.056805 ; CXC2016 . Fig. 3(a) and (d) report differential conductance d$I$/d$U$ maps acquired in close proximity of the Fermi level ($E-E_{\rm F}=10$ meV) on WTe2 and MoTe2, respectively. Their Fourier transformations—reported in panels (b) and (e)—allow to conveniently analyze scattering channels in reciprocal space. Our results reveal the emergence of clear arc-like interference patterns as indicated by the black arrows in both compounds which develop along the $q_{x}$ direction at approximately $q_{x}=0.4$ Å (see dashed blue lines). To shed light on the origin of these vectors, experimental data have been compared with calculated interference patterns. Our theoretical approach employs the extended-JDOS model PhysRevB.94.075137 , with the impurity’s electronic structure and scattering amplitude calculated by means KKR method. Results for WTe2 and MoTe2 are reported in Fig. 3(c) and (f). As for the experimental results, an arc-like feature is clearly visible in both cases along the $q_{x}$ direction (see dashed lines), which originates from intra-arc scattering among opposite Fermi arcs. This assignment is further supported by direct comparison with photoemission data [cf. Fig. 2(e,f), where the scattering vector $q$ corresponds to the distance connecting opposite arcs]. Indeed, momentum conservation requires $k_{f}=k_{i}+q$, where $k_{i}$ and $k_{f}$ are the wave vectors of initial and final states and $q$ is the scattering vector connecting them. A quantitative comparison with the experimental constant energy contours reported in Fig. 2(e,f) allows to directly link these interference phenomena to intra-arc scattering among opposite Fermi arcs. Furthermore, contrary to photoemission data, the higher surface sensitivity of STM unequivocally proves the open contour character of the arcs.

In this context, it is worth noticing that the close proximity of WTe2 to a Weyl transition allows one to safely exclude any significant contribution of topological Fermi arcs to the observed interference patters for this compound. Even when considering a slightly distorted structure hosting Weyl points, they would be so close in reciprocal space that the extension of topological Fermi arcs connecting them would be negligible. This is not the case in MoTe2 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. This is illustrated by the schematic representation reported in Fig. 4(a) and the theoretically calculated constant energy cuts displayed in Fig. 4(b). As shown in panel (a), by progressively increasing the energy (from top to bottom panel) the trivial part (blue line) of the arc is absorbed into the bulk electronic band structure whereas the topologically non-trivial arc (red line) dominates the scene. This transition from trivial to topology-dominated Fermi arcs is experimentally investigated in Fig. 4(c). At energies well below the Weyl points (upper panel) only trivial arcs exist. As a consequence, the experimental data show only weak interference patterns. This can be traced back to the combined effect of (i) the V-shape visible in the calculated constant energy contour which results in poor nesting conditions and (ii) the overlap with projected bulk pockets making these states surface resonances that are much less localized at the surface than “real” surface states [cf. top panel of Fig. 4(a)]. By going up in energy, the surface states become well separated from bulk states [cf. middle panel of Fig. 4(a)] while simultaneously getting progressively larger, and thus occupying a larger fraction of the Brillouin zone. As a result, a well-defined arc-like interference pattern appears (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 topological Fermi arcs prevail [cf. bottom panel of Fig. 4(a)]. This process is associated to a significant flattening of the arc, where several equivalent vectors can connect opposite parallel segments of the Fermi arcs [see grey arrows in Fig. 4(a), lower panel]. Despite this scenario supporting optimal nesting conditions, the scattering intensity drops (see bottom panel). Comparison with $ab$ $initio$ calculations reveals that this effect is directly linked to the spin texture of the topological Fermi arcs. Indeed, as schematically illustrated by the blue arrows in Fig. 4 (a), at $E-E_{\rm F}=+60$ meV the spin polarization is basically pointing in opposite directions for opposite topological Fermi arcs segments (spin-resolved constant-energy contours are reported in the supplementary material). This spin texture results in an effective protection against scattering. Such a behavior is reminiscent of the forbidden backscattering originally discussed in Rashba systems PhysRevLett.93.196802 and, more recently, in topological insulators PhysRevLett.103.266803 . In the present case, however, the presence of large parallel segments with opposite spin polarization significantly extend the protection well beyond time-reversal symmetry partner states.

These conclusions, which are based on a combined analysis of the spin-resolved band structure and quasi-particle interference mapping, are further quantitatively supported by the 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 scattering vectors connecting opposite arcs become possible and the scattering intensity rises (see red line in the middle panel). However, once the topological Fermi arcs set in (lower panel) the scattering intensity drops because of the discussed spin-texture protection mechanism.

Finally, we discuss the effect impurities have on Weyl nodes. Recent theoretical predictions showed that a common characteristic of Dirac-like materials is the emergence of impurity-induced quasiparticles which lift the Dirac node doi:10.1080/00018732.2014.927109 . This behavior has been recently confirmed in topological insulators, where impurity resonances induced by magnetic dopants have been shown to effectively fill the gap which is expected to open at the Dirac point in magnetically ordered samples SBB2016 . A similar behavior has been proposed to arise in Weyl semimetals. In particular, scattering at localized impurities is expected to lift the Weyl node by inducing new quasiparticle resonances close to the Weyl point energy PhysRevB.87.155123 ; doi:10.1080/00018732.2014.927109 . The emergence of these quasiparticles has been theoretically proposed as a signature of a Weyl 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., WTe2 and MoTe2. Intrinsic defects highlighted by arrows are visible in both cases. They have been identified as anti-sites (W or Mo atoms substitute Te in the topmost layer) and are commonly found in transition metal dichalcogenides 584907111 ; HZP2015 ; doi:10.1021/acsnano.5b05854 . Fig. 5(c,d) reports STS data acquired on both materials 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 the semimetallic character of these compounds. However, on top of defects a strong peak is visible in both materials which lifts the local density-of-states minimum.

These defect-induced quasiparticle resonances appear very close to the energy where Weyl points are expected to emerge (see vertical green line which highlights the peak maximum). These findings are supported by the ab initio-calculated local-density-of-states at the impurity atom reported in Fig. 5(e) and (f) for WTe2 and MoTe2, respectively. Both reveal the emergence of quasiparticle resonances located at energies which are in excellent agreement with the experiments. In particular, in MoTe2 the experimentally observed impurity-induced quasiparticle resonance is positioned at the calculated Weyl point energy ($E-E_{\rm F}=+48$ meV). As discussed above, for WTe2 our calculations predict a trivial material near to a topological phase transition into a Weyl phase. In this case, the impurity-induced resonances emerge close to the very narrow energy gap between electron and hole pockets, at an energy where Weyl points are expected to emerge according to Refs. SGW2015, ; PhysRevB.94.121112, ; PhysRevB.94.241119, . This observation provides strong evidence that topological Weyl transitions are continuous smooth transitions of the global bulk band structure. It follows that, although topological indexes change when driving the system through a topological quantum phase transition, the band structure-dependent experimental observables—such as the impurity-induced resonances reported 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 to topological insulators phase transitions where, by approaching the critical point at which the bulk band structure becomes inverted, spin-polarized helical surface states progressively emerge within the bulk gap XNB2015 .

We would like to stress that the relevance of our observations goes well beyond topological band-structure aspects. It is long known that disorder, and especially resonant impurities, can significantly impact onto transport properties. Even recently, the presence of defects has been invoked to be at the origin of both positive as well as negative magnetoresistance effects in topological semimetals PhysRevLett.114.117201 ; Li2017 . In this context, our observations contribute by providing a detailed microscopic picture of the resonant scattering off impurities in type-II Weyl semimetals. In particular, we demonstrate that intrinsic defects significantly alter the local density of states close to the Weyl points, ultimately changing the low-energy spectrum of Weyl semimetals. We conclude that the presence of defects cannot be overstressed and suggest that they play an important role in determining the fascinating transport properties of this class of materials MGS2014 ; YEH2016 .

Overall, we reveal the existence of a universal response of the type-II Weyl semimetals phase diagram. We show that surface arcs dominate the interference pattern, with the topological Fermi arc contribution being strongly suppressed by its spin texture. In agreement with theoretical predictions, we also demonstrated that impurity-induced quasiparticle resonances emerge close to the energy where Weyl points are expected. Our observations highlight that the functional response of both surface and bulk states to perturbations in this class of materials does not depend on whether we have passed the Weyl phase transition or we are simply close to it. This allows to infer the existence of a stoichiometry-independent response to perturbations for type-II Weyl, providing a unifying picture of the type-II Weyl phase 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” (project A02). P.R., P.M., and S.B. acknowledge financial support from the VITI project of the Helmholtz Association as well as computational support from the JARA-HPC Supercomputing Centre at RWTH Aachen University.