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Visualizing Type-II Weyl Points in Tungsten Ditelluride by Quasiparticle Interference Chun-Liang Lin,† Ryuichi Arafune,‡ Ro-Ya Liu,§,∇ Masato Yoshimura,† Baojie Feng,§,○ Kazuaki Kawahara,†,¶ Zeyuan Ni,∥ Emi Minamitani,∥ Satoshi Watanabe,∥ Youguo Shi,⊥ Maki Kawai,†,◇ Tai-Chang Chiang,# Iwao Matsuda,§ and Noriaki Takagi*,† †
Department of Advanced Materials Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan § Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan ∥ Department of Materials Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ⊥ Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China # Department of Physics, University of Illinois, Urbana, Illinois 61801, United States ‡
ABSTRACT: Weyl semimetals (WSMs) are classified into two types, type I and II, according to the topology of the Weyl point, where the electron and hole pockets touch each other. Tungsten ditelluride (WTe2) has garnered a great deal of attention as a strong candidate to be a type-II WSM. However, the Weyl points for WTe2 are located above the Fermi level, which has prevented us from identifying the locations and the connection to the Fermi arc surface states by using angle-resolved photoemission spectroscopy. Here, we present experimental proof that WTe2 is a type-II WSM. We measured energy-dependent quasiparticle interference patterns with a cryogenic scanning tunneling microscope, revealing the position of the Weyl point and its connection with the Fermi arc surface states, in agreement with prior theoretical predictions. Our results provide an answer to this crucial question and stimulate further exploration of the characteristics of WSMs. KEYWORDS: Weyl semimetals, topological matter, transition metal dichalcogenides, Fermi arc, quasiparticle interference, scanning tunneling microscopy, WTe2
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external perturbations in contrast to conventional surface states generated only by the discontinuity of the wave function at surfaces. WSMs are classified into type-I and type-II. In type-I WSMs, the WPs are located at the Fermi level (EF) to form a point-like Fermi surface (see Figure 1a). Such intriguing characteristics have been confirmed very recently for transition metal monopnictides such as TaAs, NbAs, and NbP, by measuring the Fermi arc SSs through angle-resolved photoemission spectroscopy (ARPES)2−4 as well as scanning tunneling microscopy quasiparticle interference (STM-QPI).11,12 Soon after the discovery of type-I WSMs, type-II WSMs were proposed theoretically.10 In type-II WSM, electron and hole pockets (EPs and HPs) form a tilted Weyl cone with a finite density of states at the WP, as shown in Figure 1b. In contrast
opology in abstract mathematics has revolutionized our conventional understanding of condensed matter physics, resulting in the emergence of exotic quantum phases such as topological insulators (TIs),1,2 topological superconductors,3,4 Dirac semimetals,5 Weyl semimetals (WSMs),6−10 etc. WSMs have been the subject of a deal of great attention because the quasiparticles in WSMs behave as Weyl Fermions, massless chiral Fermions, long sought after in high-energy particle physics. Two Weyl Fermions with opposite chirality can be combined to generate a massless Dirac Fermion. Such particles to realize the prediction in quantum field theory have not yet been discovered in nature but first observed as the quasiparticle excitation in WSMs. In WSMs, the crossing of two linear bands constitutes a Weyl point (WP). At their surfaces, unclosed surface states (SSs) emerge, called Fermi arcs. These Fermi arcs connect a pair of WPs associated with the opposite chiral charges of ±1. Similarly to the nontrivial topological SSs of the TIs, the Fermi arc SSs are protected topologically by bulk symmetry and robust from © 2017 American Chemical Society
Received: August 30, 2017 Accepted: October 23, 2017 Published: October 23, 2017 11459
DOI: 10.1021/acsnano.7b06179 ACS Nano 2017, 11, 11459−11465
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surface of an in situ cleaved WTe2 crystal. Figure 2a shows a topographic STM image acquired soon after cleaving. As shown
Figure 1. Band topologies of (a) type-I and (b) type-II WSMs near the WPs. The upper electron bands touch the lower hole bands at the WP. In type-II WSM, planes A and B indicate band dispersion along selected k paths on the surface, which do not cross in the case of plane A and cross at the WP in the case of plane B. The band dispersions in planes A and B are shown in the right panels. In plane A, the electron pocket (EP) and the hole pocket (HP) are still far away from each other.
to the type-I WSMs where the Weyl Fermions exhibit the character of Lorentz invariance13 (i.e., the characteristic same as the invariants during Lorentz transformation, such as the speed of light and the Planck’s constant), Lorentz invariance is violated in type-II WSMs due to the symmetry breaking by its tilted-cone band structure (see Figure 1b). It generates a different type of Weyl Fermion, which has just recently been defined in the condense matter, type-II WSM.10 Therefore, the type-II WSM provides a playground for the investigation of Lorentz-violating theories beyond the Standard Model,14,15 giving rise to several peculiar properties such as nonsaturating magnetoresistance.16 WTe2 and MoTe2 are strong candidates for type-II WSMs. Unclosed Fermi arc SSs for MoTe217,18 and MoxW1−xTe219,20 have been captured by ARPES and STM-QPI, confirming that these dichalcogenides are indeed type-II WSMs. In contrast, it remains an open question whether pure WTe2 is a type-II WSM, despite several ARPES measurements performed so far.21−26 The ARPES studies have shown an occupied SS (denoted as SS1) consistent with the detailed theoretical analysis by the combination of density functional theory (DFT) calculations and ab initio tight binding model obtained by maximally localized Wannier function. However, the WPs calculated for WTe2 are located above the EF, which prevents both experimental identification of WP locations and determining their connectivity with the Fermi arc SSs. These crucial issues still remain to be resolved so that it is elusive whether WTe2 is a type-II WSM. Experimental approaches other than ARPES are therefore required. The STM-QPI technique allows us to probe both occupied and unoccupied electronic states simultaneously. Its application to WSMs has successfully elucidated detailed electronic structures.27−30 Thus, STM-QPI is a suitable tool to clarify the topological properties of WTe2. Here, we identify the location of the WP and reveal its connectivity with both occupied and unoccupied SSs through detailed analysis of energy-dependent QPI patterns.
Figure 2. STM-QPI observations. (a) Topographic image of an in situ cleaved WTe2 surface (Vs = 40 mV, It = 2 nA, 50 × 50 nm2). The inset is a high-resolution topographic image (Vs = 50 mV, It = 0.5 nA, 5.6 × 5.6 nm2). (b) Top view and side view of the structure model of Td phase WTe2. (c) dI/dV mapping image taken at the sample voltage of Vs = 40 mV (It = 2 nA, 50 × 50 nm2) for the same area in panel a. (d) FFT image (QPI pattern) of panel c. The FFT image of topographic image is shown in the inset. Red circles mark the same spots observed in both FFT images of the topographic image and QPI pattern.
in the inset of Figure 2a, the diagonal lines of bright spots appear together with less-bright zigzag stripes between the lines. Figure 2b shows the structure model of WTe2. Three atomic layers constitute one unit layer, which stacks to crystallize into Td configuration; these atomic layers are the top-Te, center-W, and bottom-Te layers in WTe2, respectively. The top-Te layer is buckled, in which the Te atoms take higher and lower vertical positions. In the high-resolution STM image, the brighter lines are derived from the higher Te atoms, and the zigzag stripes are the lower Te atoms together with W atoms. Figure 2c shows the spectroscopic mapping image acquired at 40 mV. Wavy structures appear around the bright round protrusions. These protrusions are associated with defects formed mainly due to missing surface Te atoms. The wavy structures arise from the interference of quasiparticles scattered by the defects. We acquired the QPI patterns through the fast Fourier transformation (FFT) of the mapping image. Figure 2d shows a QPI pattern obtained from the mapping image shown in Figure 2c. We determined the ΓX and ΓY directions in the QPI pattern from the spots marked by the red circles in Figure 2d. These spots originate from the surface periodic lattice shown in the inset of Figure 2d. Figure 3a shows the energy-dependent evolution of QPI patterns. Each QPI pattern contains several fine structures caused by the interferences between specified pairs of incident and scattered waves with the wave vectors of k1 and k2, respectively. To reveal the relation of these fine structures with
RESULTS We measured spectroscopic mapping images as a function of sample voltage with low-temperature STM for the [001] 11460
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Figure 3. Energy-dependent evolution (EDE) of QPI pattern. (a) QPI patterns from −160 to 160 meV. (b) The EDE of cross-sectional intensity curves (CIC) cut along the X−Γ−X direction (qx = 0 corresponds to the Γ point). The labels (unit meV) on the right show the energy of curves. Some of curves are multiplied by factors (×0.3 for 50 meV, ×0.5 for 30, 40, 60, 70, 80, and 90 meV, ×0.6 for 100 and 110 meV, ×0.7 for 130 meV, ×0.8 for 140 and 150 meV, ×0.9 for 160 and 170 meV, ×3.5 for −10 meV) to avoid overlapping with other curves. Six main features observed symmetrically on the opposite side with respect to Γ point are marked from A to F.
the electronic band structure, we cut each QPI pattern along the direction specified by q = k1 − k2, and plot the crosssectional intensity curves as functions of energy (E) and q. An example of the energy-dependent evolution of the crosssectional intensity curves (denoted as EDE-CIC hereafter) is shown in Figure 3b, which is acquired by cutting the QPI patterns along the ΓX direction, q = (qx, 0). In the analysis of EDE-CIC, we at first consider the scattering of k1 = −k2 as a dominant channel. When the scattering does not explain the EDE-CIC, we consider the other scattering channels satisfying q = k1 − k2. Several recent works also suggest that the scattering on the WSM can originate from the channels in which k1 ≠ −k2.27,28 Recent spin-resolved ARPES studies of WTe2 demonstrate that even though the spin functions of +k and −k mostly take antiparallel directions, they are not orthogonal to each other.22,26 This indicates that the restriction of tying the QPI to the spin texture is not strictly enforced. Thus, we neglect the contribution from the spin texture to the QPI. In prior theoretical calculations of the QPI patterns observed for several WSMs,27−30 similar analysis reasonably explains the QPI patterns. Taking a look at the EDE-CIC shown in Figure 3b, one can see six main features (denoted as A−F). These features are observed clearly and symmetrically in both sides with respect to qx = 0. Each peak arises from the scattering specified by a pair of waves with (k1x, k1y) and (k2x, k2y) where k1y = k2y and qx = k1x − k2x.
Figure 4. Manifestation of WPs in QPI observation. (a) Contour map of EDE-CIC near feature C. Feature C is marked by the blue circle. (b, c) EDE-CIC along ΓY direction from 30 to 90 meV (bottom to top) and its derivatives, respectively. The evolution of major peaks is indicated by the dash line. (d) Calculated band structures parallel to the ΓY direction at kx = 0.20 Å−1 (left) and 0.25 Å−1 (right). The positions of the apex for HP and EP are marked by red and green arrows, respectively.
Å−1, 58 meV) in the QPI patterns. The spot observed at qCx = 0.46 ± 0.02 Å−1 reasonably coincides with 0.438 Å−1, leading to the assignment that feature C originates from the scattering between the pair of WPs. Note that we could not reveal a pair of separated spots for the WPs because two WPs are too close to each other (only 6 meV difference, which is under the energy resolution in our experiment). Figure 4b,c shows the CICs cut along the ΓY direction around 50 meV and their derivatives, respectively. The CICs reveal some shoulders within the huge peak at q = 0, and the derivatives further help us to uncover the evolution, which is worthy of notice. Figure 4c shows the several peaks around qy = 0.10 ± 0.02 Å−1. The evolution of the
DISCUSSION Visualization of the WPs. In these six features, first of all we notice feature C. As shown in the contour map in Figure 4a, feature C appears as a high intensity spot marked by the circle at qCx = 0.46 ± 0.02 Å−1 and E = 50 meV. The prior theoretical study indicates that the WP1 and WP2 are located at (kx, ky, E) = (±0.219 Å−1, ±0.038 Å−1, 52 meV) and (±0.219 Å−1, ±0.045 Å−1, 58 meV), respectively.10 The interferences relevant to the WPs with different chiralities in the opposite sides of the Brillouin zone simply should provide high intensity spots around (0.438 Å−1, 0.076 Å−1, 52 meV) and (0.438 Å−1, 0.090 11461
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with the bulk band structures cut at several values of ky from 0 to 0.08 Å−1 demonstrates that the feature D extends in the band gap region and that the feature D cannot be rationalized by the bulk electronic states. In addition, the red dashed curve in Figure 5b indicates the Fermi arc SS obtained by previous APRES.23 It also well matches the current QPI observation. These results give direct evidence to conclude that the SS1 is the Fermi arc SS connecting the WPs. Furthermore, we analyzed the EDE-CIC by cutting the QPI patterns along the directions making angle θ with respect to the ΓX direction as indicated by the scheme in Figure 6a. This
major peaks as marked by the dashed line indicates that a change happens at 50 meV. From the band structures parallel to ΓY direction as shown by Figure 4d, we notice the intervalley scattering along ΓY direction for the HP in lower energy is shorter than that for the EP in higher energy. This change from ky = 0.04 to 0.06 Å−1 well matches the shift from qy = 0.08 to 0.11 Å−1 in the QPI observation. Therefore, the shift found in Figure 4c can be rationalized as the variation in intervalley scattering from HP to EP with the increase of energy, which should happen near the WPs because the WPs are created by the touch of EP and HP. This result assures the assignment of feature C and gives kWP = 0.05 ± 0.01 Å−1. The present QPI y experiments identify the location of WPs to be (0.23 ± 0.01 Å−1, 0.05 ± 0.01 Å−1, 50 meV), providing a nice support to the calculated WPs. Connection of Fermi Arc Surface States to the WPs. The evolution of QPI patterns also captures the SS1 predicted theoretically and the connectivity with the WPs identified by the present QPI measurements. Figure 5a is an enlarged EDE-
Figure 6. Distribution of SS1 in isoenergetic kx−ky planes. (a) Illustration of angle (θ) at which the EDE-CIC is cut. (b) Schematic illustration of scatterings between the Fermi arc SS1 that can be observed at different θ. (c) Distributions of the feature D at three different energy near EF.
analysis enables us to discuss the topology of the Fermi arc SS1 in the kx−ky plane. The variation of feature D as a function of θ reflects the scattering processes as shown by the arrows in Figure 6b. Figure 6c shows the distribution of feature D at three different energies. Feature D distributes in a limited region of qy = ±0.2 Å−1 near EF. It well reproduces the previous ARPES studies showing that each SS1 disperses in the range of ky = ± 0.09 Å−1 in the isoenergetic surfaces near EF. Theoretically, another surface state (denoted as SS2) has been demonstrated near the electron bands above the EF.10,23 This state has not been identified yet by using ARPES because it runs above the EF. The present QPI measurements capture SS2. In contrast to SS1, SS2 is not very clearly seen in Figure 3b as it overlaps with feature B. As shown by an enlarged EDECIC in Figure 7a, each peak of feature B is broad, indicating that feature B arises from the superposition of several components (marked by the green bars). Overlapping a component at smaller qx (green crosses in Figure 7a) by the use of qx = 2kx with the bulk band structures provides a band branch running very close to the bulk electron bands, and as the ky increases as shown by Figures 7b−f, SS2 clearly distributes in the bulk band gap. This is reasonably matched with the
Figure 5. Manifestation of SS1 in QPI observation. (a) The enlarged EDE-CIC near feature D. The green crosses mark the peaks corresponding to feature D. They start from −60 meV and increase up to 40 meV, and the blue circle is the feature C. (b−f) Superposition of the feature D with the calculated band structure parallel to the ΓX direction at ky = 0, 0.02, 0.04, 0.06, and 0.08 Å−1. The red dash line in panel b is from the ARPES observation.19
CIC along the ΓX direction, focusing on the feature D. As marked by green crosses in Figure 5a, the feature D starts from −70 meV and crosses the EF, connecting to the WPs identified in this study. The dispersion deduced by qDx = 2kDx from the feature D nicely matches with that of SS1 determined by the previous collaborative studies of ARPES and theoretical calculations.23−25 As shown in Figure 5b−f, the superposition 11462
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Figure 8. Comparison of the Fermi arc SSs between our observation and the previous theoretical calculation.23 The crosses are the results from the QPI measurements (green and pink represent SS1 and SS2, respectively). The blue circle indicates the WPs, and the gray dash curve is the prediction from ref 23.
Figure 7. Manifestation of SS2 in QPI observation. (a) Enlarged EDE-CIC near observed SS2. The green crosses mark the peak corresponding to SS2, and the green bars mark the distribution of feature B. (b−f) Superposition of SS2 with the calculated band structures parallel to the ΓX direction at ky = 0, 0.02, 0.04, 0.06, and 0.08 Å−1.
dispersion of SS2 predicted theoretically.10,23 We are supposed to give some comments on both theoretical predictions from the current experiment results since Soluyanov et al. suggest that two surface states should be topologically protected,10 while Bruno et al. claim that they could be trivial except the short section connecting with the WPs.23 It is difficult to judge which one is correct by the present QPI experiments since two calculations show almost the same distributions of SS1 and SS2. From the two calculations, we can only conclude that the current observation is important because the connection to WPs of two Fermi arc surface states (SS1 and SS2) are clearly revealed. The comparison of both SS1 and SS2 between the QPI measurements and the theoretical prediction23 is shown in Figure 8. As a result, because both SS1 and SS2 fit the calculation reasonably, the Fermi arc SSs observed by QPI are reliable and almost reproduce the theoretical prediction. The Origins of Other Features. Finally, we discuss the origins of remaining four features (A, B, E, and F). These features arise from the bulk electronic states. Figure 9a shows the calculated bulk band structure. We cut the bulk band structure at characteristic energies and discuss the origins specified by qAx , qBx , qEx , and qFx based on the topologies of the isoenergetic cuts. We assign the origins to be the scatterings specified by the vectors shown in Figure 9b−d. Feature A originates from the scattering between the electron pockets as shown by the arrows in Figure 9b. Feature B is the intervalley scattering between two electron pockets on the opposite sides
Figure 9. Origins of scattering for other features. (a) Calculated bulk band structure along the ΓX direction. (b−d) Isoenergetic cuts at the energies marked by the dash lines in panel a. The green arrows indicate the scattering channels associated with the features A to F besides the features C and D.
of the Brillouin zone as shown in Figure 9c. Features E and F are caused by the scatterings between hole bands as shown in Figure 9d. Several electron and hole bands distribute around the EF, which provides various fine structures in the QPI patterns.
CONCLUSIONS The electronic structure of pure WTe2 is investigated through a detailed analysis of the energy-dependent evolution of QPI patterns. We identified the positions of WPs to be (kx, ky, E) = (0.23 ± 0.01 Å−1, 0.05 ± 0.01 Å−1, 50 meV), which are in a good agreement with the theoretical prediction.10 Furthermore, we also demonstrated the connection of topological Fermi arc SSs to the WPs; SS1 disperses in the occupied state and crosses the EF to connect to the WPs, and SS2 disperses all above the EF. These features nicely agree with the theoretical prediction. The appearance of WPs and the connection with topological SSs are the two decisive factors of Weyl semimetals. The 11463
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present results provide positive proof that WTe2 is a type-II WSM.
METHODS Sample Preparation and STM Measurement. We used high quality WTe2 crystals (HQ Graphene Company, The Netherlands). A piece of WTe2 sample of ∼5 mm2 area and ∼0.1 mm thick was stuck on a Cu plate and then mounted on the sample holder. A carbon tape was used for in situ exfoliating WTe2 sample to get a clean surface in an ultrahigh vacuum (UHV) chamber. The base pressure of the UHV chamber was kept under 2 × 10−10 Torr. The STM images, spectroscopic images, and tunneling spectra were acquired at 5 K. The modulation voltages used for the lock-in detection were set 10 mV and 4 mV of 616 Hz for measuring spectroscopic images and tunneling spectra, respectively. Calculation of Band Structure. All the DFT calculations were done by using Vienna ab initio Simulation Package (VASP)31,32 with the projected augmented wave (PAW) method.33 Exchange and correlation functional are described at the level of generalized gradient approximation (GGA) parametrized by Perdew−Burke−Ernzerhof.34 The cut off energy of the plane wave expansion used was 400 eV. The lattice constants and atom positions in the unit cell are taken from the literature.35 In the self-consistent calculations, the Brillouin zone was sampled with (24 × 12 × 8) k-points.
AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. ORCID
Noriaki Takagi: 0000-0002-0799-9772 Present Addresses ∇
R.-Y.L.: Institute of Physics, Academia Sinica, 128 Academia Road, Section 2, Nankang, Taipei 11529, Taiwan. ○ B.F.: Hiroshima Synchrotron Radiation Center, 2-313 Kagamiyama, Higashi-Hiroshima 739-0046, Japan. ¶ K.K.: Institute of Engineering Innovation, The University of Tokyo, Tokyo 113-8656, Japan. ◇ M.K.: Institute for Molecular Science, 38 Nishigo-naka, Myodaiji, Okazaki 444-8585, Japan. Author Contributions
C.L.L., R.A., I.M., and N.T. designed the experiments. C.L.L. carried out the experiments and analyzed the data. R.A. carried out the DFT calculations with the help of Z.N. and E.M. and coded the program for QPI analysis. C.L.L., R.A., and N.T. wrote the manuscript with feedback from all authors. Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS R.A. and N.T. acknowledge support by MEXT KAKENHI Grant Number 25110008. R.A. also acknowledges support by World 25 Premier International Research Center Initiative (WPI), MEXT, Japan. K.K. acknowledges support by JSPS KAKENHI Grant Number 15J01086. E.M. acknowledges support by MEXT KAKENHI Grant Numbers 26102017 and 15K17465. T.C.C. acknowledges support by the U.S. Department of Energy (DOE), Office of Science (OS), Office of Basic Energy Sciences, Division of Materials Science and Engineering, under Grant No. DE-FG02-07ER46383. REFERENCES (1) Hsieh, D. D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y. S.; Cava, R. J.; Hasan, M. Z. A Topological Dirac Insulator in a Quantum Spin Hall Phase. Nature 2008, 452, 970−974. 11464
DOI: 10.1021/acsnano.7b06179 ACS Nano 2017, 11, 11459−11465
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DOI: 10.1021/acsnano.7b06179 ACS Nano 2017, 11, 11459−11465
