# Direct observation of linear dispersion close to the Fermi level in the
topological semimetal WTe2 through Landau quantization at atomic scale

Raquel Sánchez-Barquilla  Laboratorio de Bajas Temperaturas y Altos Campos
Magnéticos, Departamento de Física de la Materia Condensada, Instituto Nicolás
Cabrera and Condensed matter Physics Center (IFIMAC), Unidad Asociada UAM-
CSIC, Universidad Autónoma de Madrid, E-18049, Madrid, Spain  Francisco Martín
Vega  Laboratorio de Bajas Temperaturas y Altos Campos Magnéticos,
Departamento de Física de la Materia Condensada, Instituto Nicolás Cabrera and
Condensed matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC,
Universidad Autónoma de Madrid, E-18049, Madrid, Spain  Alberto M. Ruiz
Instituto de Ciencia Molecular (ICMol), Universidad de Valencia, Catedrático
José Beltrán 2, 46980 Paterna, Spain  Na Hyun Jo∗ Ames Laboratory and
Department of Physics & Astronomy, Iowa State University, Ames, IA 50011
Present address: Department of Physics, University of Michigan, Ann Arbor, MI
48109, USA  Edwin Herrera  Laboratorio de Bajas Temperaturas y Altos Campos
Magnéticos, Departamento de Física de la Materia Condensada, Instituto Nicolás
Cabrera and Condensed matter Physics Center (IFIMAC), Unidad Asociada UAM-
CSIC, Universidad Autónoma de Madrid, E-18049, Madrid, Spain  José J. Baldoví
Instituto de Ciencia Molecular (ICMol), Universidad de Valencia, Catedrático
José Beltrán 2, 46980 Paterna, Spain  Masayuki Ochi  Forefront Research
Center, Osaka University, Toyonaka, Osaka 560-0043, Japan  Department of
Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan  Ryotaro Arita
RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
Sergey L. Bud’ko  Ames Laboratory and Department of Physics & Astronomy, Iowa
State University, Ames, IA 50011  Paul C. Canfield  Ames Laboratory and
Department of Physics & Astronomy, Iowa State University, Ames, IA 50011
Isabel Guillamón  Laboratorio de Bajas Temperaturas y Altos Campos Magnéticos,
Departamento de Física de la Materia Condensada, Instituto Nicolás Cabrera and
Condensed matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC,
Universidad Autónoma de Madrid, E-18049, Madrid, Spain  Hermann Suderow
Laboratorio de Bajas Temperaturas y Altos Campos Magnéticos, Departamento de
Física de la Materia Condensada, Instituto Nicolás Cabrera and Condensed
matter Physics Center (IFIMAC), Unidad Asociada UAM-CSIC, Universidad Autónoma
de Madrid, E-18049, Madrid, Spain

###### Abstract

We study the topological Weyl type-II semimetal WTe2 via Scanning Tunneling
Microscopy (STM) and Density Functional Theory calculations (DFT). We
succesfully determine the band structure at zero field close to the Fermi
level by comparing quasiparticle interference (QPI) experiments with DFT.
Under magnetic fields, we observe Landau level quantization on atomic scale
measurements and find a level sequence evidencing a linearly dispersing
portion of the band structure. Our results establish the long sought
connection between Weyl cones and Landau quantization in WTe2. Atomic scale
Landau quantization emerges as a powerful probe of topological surface states
for semimetals, superconductors and topological insulators.

Electrons under magnetic fields in a solid perform quantized motion at a plane
perpendicular to the magnetic field. The bulk boundary correspondence leads to
topologically protected surface states in materials presenting non trivial
crossing in the band structure as Dirac or Weyl points [1, 2, 3, 4, 5, 6, 7,
8, 9, 10]. Landau quantization of these surface bands can lead to chiral
conduction channels [1, 2, 3, 4, 11, 5, 6, 7, 12, 13].

WTe2 is a Van der Waals layered semimetal with conical tilted crossings
leading to Weyl fermions [14, 15, 16, 17, 12, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29]. Slight distortions that preserve symmetry lead to a higher
order topological insulator driven by double band inversion [30].

Landau quantization leads to pronounced features in WTe2 in practically all
probes under magnetic fields [15, 31, 32]. In few and single layers, anomalous
magnetic transport properties as the quantum spin Hall effect possibly related
to helical edge states have been observed [33, 34, 35, 36]. The connection
between Landau levels and the linearly dispersing band structure close to Weyl
points or at surface Dirac points remains however difficult to establish.

Figure 1: (a) Atomic structure of the surface of WTe2. Dashed black rectangle
is the unit cell. W atoms are shown in violet and Te atoms in green. The
surface is composed of Te atoms, the W atoms lie at the next plane. (b) Band
dispersion of WTe2 calculated using Density Functional Theory (DFT). W-derived
bands are shown in violet and Te-derived bands in green. Here we show only the
dispersion along the $a$ direction ($k_{x}$). The band structure along another
direction is provided in the Supplementary Material (Fig. S2). (c) Topographic
STM image of WTe2. Black horizontal scale bar is 2 nm long. In the upper right
we show a zoom where we have superposed the atomic structure (horizontal black
scale bar is 0.5 nm long). In the lower right panel we show the Fourier
transform, with the main crystal Bragg peaks surrounded by white circles
(white horizontal scale bar is 3 nm-1 long). Black arrows provide the in-plane
lattice directions.

Scanning Tunneling Microscopy (STM) seems adequate to study the band structure
at zero field through quasiparticle interference (QPI) at impurities or
defects. Landau levels, on the other hand, have been observed with STM only in
a few compounds as graphene or bismuth and in two-dimensional electron gases
[37, 38, 39, 40, 41, 42, 43, 44, 45]. Both, extremely clean samples and well
defined two-dimensional single or few band electronic properties are needed.
The energy between two Landau levels $\Delta E$ is, due to the small effective
masses in WTe2, in the meV range and thus accessible to STM. But the band
structure of WTe2 is quite intricate, with several overlapping electron and
hole bands [15, 16, 12, 18, 19, 20, 23, 24]. Here we make detailed atomic
scale STM experiments at zero and under magnetic fields on WTe2. We determine
the band structure at zero field by comparing QPI and Density Functional
Theory (DFT) calculations, and unveil Landau quantization under magnetic
fields.

We performed STM measurements using a home-made STM described in Ref. [46] and
the software described in Ref. [47]. For image and spectroscopy treatment we
used the software described in Refs. [47, 48]. The samples were grown from a
Te-rich binary melt, as described in Refs [49, 19]. The crystals are plate-
like with typical dimensions of 2 mm $\times$ 0.1 mm $\times$ 0.01 mm, and the
crystallographic $c$-axis perpendicular to the crystal surface. We exfoliated
the sample in-situ in cryogenic conditions by gluing a post and pushing it at
low temperatures using the in-situ slider described in Ref. [46]. This method
allows to obtain large, clean and flat surfaces, showing atomic rows and
interstitial sites, similar to those reported before [23, 24, 50, 51, 29]. We
succesfully cleaved eight crystals, in which we changed the field of view more
than a hundred times. Experiments were made in a helium bath cryostat at 4.2
K. Using an in-situ sample positioning system [46], we studied hundreds of
fields of view, some with defects and impurities, where we took the QPI data,
and others completely free of impurities, where we observed Landau
quantization.

We performed first-principles band structure calculations based on DFT with
the generalized gradient approximation with the Perdew-Burke-Ernzerhof
parametrization [52] and the full-potential (linearized) augmented plane-wave
plus local orbitals method including the spin-orbit coupling as implemented in
the WIEN2k code [53, 54]. We took the experimental crystal structure from Ref.
[55]. The maximum modulus for the reciprocal lattice vectors
$K_{\mathrm{max}}$ was chosen so that $rK_{\mathrm{max}}=8$, where the muffin-
tin radius $r$ was 2.5 Bohr for both W and Te atoms.

The surface atomic structure is defined by the Van der Waals gap and consists
of layers where the topmost atomic lattice is formed by arrangements of Te
atoms. In our STM images, we identify the Te atomic positions and the
underlying W atoms in between (see Fig. 1(a,c)). The surface presents a very
small number of defects, consisting mostly on W-Te substitutions and
vacancies. As shown in the Supplemental Material (Fig. S1) there is
approximately one defect per 700 atoms, which gives $10^{12}$ defects per
cm-2, a value that is compatible with the extremely high residual resistance
ratio reported in the same crystals and the neat observation of quantum
oscillations in macroscopic experiments [19, 18, 56].

These defects are scattering centers and provide the oscillatory pattern in
the density of states that we can image through QPI. The result of our QPI
measurements is shown in Fig. 2(a). We compare the Fourier transform of the
scattering pattern with the band structure obtained from DFT calculations in
Fig. 2(b). The scattering vectors along $q_{x}$ join portions of the band
structure located at $k_{x}$ and $k_{y}=0$. For example, intraband scattering
joining states at $\pm k_{x}$ gives $q=2k_{x}$ (as the light violet arrow
$\mathbf{q}_{2}$ on top of Fig. 2(a,b)). Intraband scattering joining states
at either side of a pocket provides $q=\delta k_{x}$ where $\delta k_{x}$ is
the pocket size (as the dark green arrows $\mathbf{q}_{6}$ in the lower middle
of Fig. 2(a,b)). More details are provided in the Supplemental Material, Fig.
S2. We observe scattering patterns, elongated along the $a$-direction ($x$ in
Fig. 2(a,b)), which is the direction of the Te chains (Fig. 1(b), see
Supplemental Material, Fig. S2, for results out of the $a$ direction).

Figure 2: (a) Color map showing the scattering intensity (color scale
following the bar on the top) as a function of the bias voltage and the
scattering along $q_{x}$. $q_{x}$ corresponds to the $a$ direction in real
space. More data and details are provided in the Supplemental Material.
Colored dots trace the different scattering vectors in the bias
voltage-$q_{x}$ plane. (b) The band structure calculated by DFT is shown as
colored lines as a function of the energy and the wavevector $k_{x}$. Violet
and green are for W- and Te-derived bands respectively. The scattering vectors
shown in (a) are represented in (b) as pairs of colored dots using the same
color code. Arrows represent some of the main scattering vectors, also marked
as $\mathbf{q}_{1}$ to $\mathbf{q}_{11}$.

Starting at large and positive bias voltages we find a large scattering vector
(light violet arrow, $\mathbf{q}_{2}$) in Fig. 2(a)) due to intraband
scattering between the outer sides of the small electron pockets shown in Fig.
2(b). This vector decreases in size when decreasing the energy, and disappears
close to 20 meV, where we also find the bottom of the small electron band
(Fig. 2(b)). We find a few smaller scattering vectors (dark violet and magenta
arrows, $\mathbf{q}_{1,3,4}$ in Fig. 2(a,b)) associated with the intra-band
scattering within electron pockets located at large $k_{x}$, and with a small
hole like portion of the band structure. A large vector due to intra-band
scattering within the electron pockets at large $k_{x}$ decreases in size when
decreasing the voltage (light violet arrow, $\mathbf{q}_{5}$, in Fig. 2(a,b)),
and the corresponding features mix with other scattering vectors. At small
negative bias voltages we find scattering from the same band again (light
violet arrow, $\mathbf{q}_{7}$, in Fig. 2(a,b)), with a considerably reduced
size. We also find a small interband scattering vector between electron and
hole bands (bright violet $\mathbf{q}_{10}$ in Fig. 2(a)). For the Te derived
bands we find a vector due to intra-band scattering of two hole bands (green
arrow $\mathbf{q}_{6}$ in Fig. 2(a,b)) and large scattering vectors associated
to intraband scattering in a hole like band at smaller energies (green arrow
$\mathbf{q}_{8,9}$ in Fig. 2(a,b)). The band structure below the Fermi level
is mostly compatible with angular resolved photoemission experiments probing
filled states [51, 19, 18]. Thus, in all, our QPI measurements provide a
rather complete view of the band structure of WTe2 around the Fermi level.

A detailed comparison between theory and experiment (Supplemental Material,
Fig. S3) shows evidence for surface states at the energy positions predicted
by calculations. Nevertheless, it is difficult to distinguish these surface
states from bulk features in the scattering pattern. As we shall see below,
Landau quantization provides further insight.

Figure 3: Conductance vs bias voltage at 11 T, 12 T and 14 T. A background has
been removed and curves shifted for clarity. In the inset we show the
amplitude of the oscillatory component as circles and the separation between
peaks as crosses. The dashed line provides the Landau quantization expected
for a mass of $m^{*}=0.22m_{e}$. Figure 4: (a) Tunneling conductance vs bias
voltage obtained at 14 T. In the inset we plot the index of the Landau level
$n$ as a function of the voltage. $n$ is arbitrarily set to zero at the start
of the curve for large positive bias voltages. The blue line is obtained when
fitting the Landau level index to a straight line for positive bias voltages,
and the red line is obtained when fitting data for negative bias voltages. (b)
We plot as circles the difference between the red line and the observed Landau
level index, obtained from the inset in (a). Red dashed line shows the result
using two linearly dispersing Dirac cones. (c) Schematic representation of two
perfectly compensated electron (violet) and hole (light violet) bands that
cross at two Weyl points (marked by the arrow). Weyl points arise in WTe2 on
crossing linearly dispersing bands above the Fermi level and at a certain
angle to a crystalline direction, marked in the figure by $k_{Weyl}$ [12]. The
dispersion relation is linear at Weyl points [57]. (d) Schematic
representation of a projection of linear dispersing bands, as occuring on two
Weyl points located close in energy. For clarity, we only show one half of the
cone around each Weyl point. Landau level quantization for a magnetic field
applied along the c-axis is schematically shown as horizontal colored lines.
Level separation is given by the linear dispersion relation. (e) As a black
line, we show the k-integrated density of states as a function of the energy,
considering smeared Landau levels and the overlap between bands. The sequence
of Landau levels, when counting from low to high energies, is maximally
shifted (by $1/2$) in the energy range where electron and hole bands overlap.

The presence of defects is in principle detrimental to observe Landau
quantization [58]. A relevant scale for obtaining Landau levels is the
magnetic length $\ell_{B}=\sqrt{\hbar/eB}$, which is related with the radius
of the cyclotron orbit $r_{n}=\sqrt{2n+1}\ell_{B}$. To resolve the Landau
levels in the density of states, the distance between the defects must be of
the order or larger than $\ell_{B}$. At 14 T, the Landau orbits are of the
order of $r_{0}\approx\ell_{B}\approx$ 7 nm, which is much lower than the mean
free path $\ell$, so that, in principle, one can expect the observation of
Landau levels in the tunneling conductance under a magnetic field. However,
the tunneling conductance obtained by spatially averaging results over large
fields, even when these are completely free of impurities (Supplemental
Material, Fig. S4), presents no significant changes as a function of the
magnetic field. This absence of Landau levels in the tunneling conductance
cannot be due to impurity scattering. Instead, it must be related to the
intricate band structure with multiple bands.

To address this issue, we note that tunneling with STM occurs through the
overlap between sample and tip wavefunctions [59]. The tunneling overlap on
different atomic sites is different, involving different portions of the band
structure [60, 61, 62, 63]. As we show in Fig. 1(c), we observe Te atomic rows
and distinct contrast in between Te sites. Thus, we can expect tunneling
dominated by Te derived bands on the Te sites and by W derived bands in
between. If we take tunneling conductance curves at a single atomically
defined location, we observe strong oscillations of the tunneling conductance
as a function of the bias voltage (Fig. 3). The amplitude of the oscillations
increases with magnetic field. From the Fourier Transform of the tunneling
conductance vs bias voltage curves, we obtain a single peak that corresponds
to $\Delta E\propto\hbar\omega_{c}$, the separation in energy between the
levels. Using $\omega_{c}=\frac{eB}{m^{*}}$ we obtain an effective mass
$m^{*}=0.22m_{e}$ (inset of Fig. 3). The effective mass obtained is similar to
the values reported before and compatible with the band dispersion observed
through QPI described above. Notice that all the bands crossing the Fermi
level have very similar effective masses, so the separation between the levels
is not expected to change significantly for the different bands. For example,
a difference of 0.04$m_{e}$ results in a difference in energy of only 1.2 meV.
This, and other broadening effects, discussed in the Supplemental Material,
influence the shape of the peaks, which are about 5 mV wide. Furthermore, the
phase of the oscillation changes with the atomic position, leading to
characteristic patterns that can be reproduced using a shift in the Landau
level sequence due to different band selectivity at each atomic site (see
Supplemental Material, Fig. S5). Thus, it is possible to observe Landau levels
in the tunneling conductance of a material with a three dimensional dispersion
and multiple bands, provided that there are no large differences in the
effective mass between different bands and the atomic scale tunneling
selectivity is sufficiently well developed.

To gain further insight we plot in Fig. 4(a) the atomically resolved tunneling
conductance vs bias voltage obtained at 14 T, where the signal is largest.
When numbering the levels and plotting the resulting quantum numbers $n$ as a
function of the bias voltage, we observe at first sight a straight line. But a
closer look shows that, although the slope of $n$ vs the bias voltage is the
same for all bias voltages, there is a shift at positive bias. To better see
this shift, we fit the $n$ vs bias voltage to a straight line for large
positive and negative bias voltages, and show the subtraction in Fig. 4(b),
where we observe a shift by $1/2$. To explain this we recall Landau
quantization around Weyl crossings as schematically shown in Fig. 4(c-e). Weyl
semimetals are characterized by a band crossing producing at least two points
with linear dispersion close to the Fermi level (Fig. 4(c)). The level
quantization along the linearly dispersing band structure follows a
$\sqrt{|E-E_{0}|}$ dependence, where $E_{0}$ is the crossing point. A density
of states measurement includes contributions from both bands and leads to a
mixed Landau level sequence (Fig. 4(e)). As discussed above, due to three
dimensional band dispersion, lifted band degeneracy or the influence of
impurities, the Landau levels broaden. As a consequence, the lowest Landau
levels of each band appear as a single peak in the density of states (Fig.
4(e)). However, when counting the Landau level index $n$ on a given energy
window, we observe a shift of $1/2$ within the energy window where both bands
overlap (Fig. 4(b,e)). This shift does not occur when dispersion relations are
quadratic, because it is caused by the enhanced level separation close to
$E_{0}$ from the $\sqrt{|E-E_{0}|}$ dependence. Thus, our data show that there
are linear crossings within the studied energy window.

It was shown that Landau quantization in type-II Weyl semimetals depends on
the angle between the direction of the line joining the Weyl nodes and the
magnetic field [64, 65, 66]. There is an angle for which the Landau levels
collapse due to the distortion of circular orbits induced by the Weyl tilt
[64, 65, 12, 66, 67, 68, 69]. This suggests that the amplitude of Landau
levels and the sequence we observe here should change considerably with the
direction of the magnetic field. There are 16 predicted Weyl points, most at
finite $k_{x}$ and $k_{y}$ and some at finite $k_{z}$ [12]. To the best of our
knowledge, there is yet no available STM operating on a rotating field above
10 T. If this is achieved, studying the angular dependent of Landau levels in
WTe2 could be used to trace the multiplicity of Weyl nodes.

The consequence of Weyl crossings at the surface of bulk WTe2 could become
visible in studies of heterostructures. Thanks to the layered structure of
transition metal dichalcogenides, heterostructures of WTe2 with layered
superconductors and magnets have been fabricated [70, 71, 72]. These provide
in-gap states in proximity induced superconductivity, non-reciprocal Josephson
currents, a Josephson current comparable to a single chiral conduction channel
at hinge states, or the possibility to tune the magnetization of a magnetic
layer on top of WTe2 [70, 71, 72, 73, 74]. Our observation of Landau quantized
states and Weyl cones suggests the presence of chiral edge modes at the
surface. This is strongly supported by the observation of increased amplitude
Landau oscillations in the energy range where surface states are expected (see
Supplemental Material, Fig. S6). The chiral edge modes have topologically
protected chiral currents moving in opposite directions when the magnetic
field and the current is applied perpendicular to the planes. It would be
interesting to address transport studies in heterostructrues based on WTe2
with the new perspective opened by our data.

It is noteworthy that WTe2 presents ferroelectric properties, being one of the
few polar metals known so far [75, 76, 77]. In few and single layers of WTe2 a
gap is observed under magnetic fields [35]. The origin of this gap opening is
under debate. A charge density wave and a Coulomb gap have been considered as
possibilities [78, 34]. Both should be characteristic distortions of the band
structure in the single or few layer limit. Our results confirm that this
system, being very sensitive to distortions, has a linearly dispersing band
structure in the semimetallic phase.

In summary, we have determined the band structure close to the Fermi level in
WTe2 using QPI, obtaining an excellent agreement with DFT calculations. We
demonstrate the existence of linear dispersion in WTe2 due to Weyl nodes at
crossing points of electron and hole bands. Our data show that Landau level
quantization can be observed at the surface of materials with a three-
dimensional dispersion relation, provided that there is well differentiated
tunneling on different atomic positions. This opens interesting prospects,
which might help understanding magnetic field properties of materials having
unconventional surface states as PtSn4 or NdBi, or potentially strong
dependencies of the band structure with the magnetic field, as UTe2 [8, 9, 79,
80, 81, 82].

### Acknowledgments.

This work was supported by the Spanish Research State Agency
(PID2020-114071RB-I00, CEX2023001316-M, TED2021-130546B-I00,
PDC2021-121086-I00), by the Comunidad de Madrid through program NANOMAGCOST-CM
(Program No.S2018/NMT-4321), by the European Research Council PNICTEYES grant
agreement 679080, and by VectorFieldImaging grant agreement 101069239. We
acknowledge collaborations through EU program Cost CA21144 (Superqumap). We
acknowledge segainvex for design and construction of STM and cryogenic
equipment. Work at Valencia received support from the European Union
(ERC-2021-StG-101042680 2D-SMARTiES) and the Gen-T programme of the
Generalitat Valenciana (CIDEXG/2023/1). A.M.R. thanks the Spanish MIU
(FPU21/04195). Work at the Ames National Laboratory was supported by the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division. Ames National Laboratory is operated for
the U.S. Department of Energy by Iowa State University under Contract No. DE-
AC02-07CH11358.

## References

  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Wieder  _et al._ [2022] B. J. Wieder, B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, L. Elcoro, A. A. Soluyanov, C. Felser, T. Neupert, N. Regnault, and B. A. Bernevig, Topological materials discovery from crystal symmetry, Nature Reviews Materials 7, 196 (2022).
  * Nagaosa  _et al._ [2020] N. Nagaosa, T. Morimoto, and Y. Tokura, Transport, magnetic and optical properties of weyl materials, Nature Reviews Materials 5, 621 (2020).
  * Yan and Felser [2017] B. Yan and C. Felser, Topological materials: Weyl semimetals, Annual Review of Condensed Matter Physics 8, 337 (2017).
  * Armitage  _et al._ [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * Lv  _et al._ [2021] B. Q. Lv, T. Qian, and H. Ding, Experimental perspective on three-dimensional topological semimetals, Rev. Mod. Phys. 93, 025002 (2021).
  * Mun  _et al._ [2012] E. Mun, H. Ko, G. J. Miller, G. D. Samolyuk, S. L. Bud’ko, and P. C. Canfield, Magnetic field effects on transport properties of PtSn4, Phys. Rev. B 85, 035135 (2012).
  * Wu  _et al._ [2016a] Y. Wu, L.-L. Wang, E. Mun, D. D. Johnson, D. Mou, L. Huang, Y. Lee, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, Dirac node arcs in PtSn4, Nature Physics 12, 667 (2016a).
  * Inoue  _et al._ [2016] H. Inoue, A. Gyenis, Z. Wang, J. Li, S. W. Oh, S. Jiang, N. Ni, B. A. Bernevig, and A. Yazdani, Quasiparticle interference of the Fermi arcs and surface-bulk connectivity of a Weyl semimetal, Science 351, 1184 (2016).
  * Yu  _et al._ [2023] T. Yu, Z. Luo, and G. E. Bauer, Chirality as generalized spin–orbit interaction in spintronics, Physics Reports 1009, 1 (2023), chirality as Generalized Spin-Orbit Interaction in Spintronics.
  * Soluyanov  _et al._ [2015] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Type-II Weyl semimetals, Nature 527, 495 (2015).
  * Moll  _et al._ [2016] P. J. W. Moll, N. L. Nair, T. Helm, A. C. Potter, I. Kimchi, A. Vishwanath, and J. G. Analytis, Transport evidence for fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2, Nature 535, 266 (2016).
  * Weyl [1929] H. Weyl, Elektron und gravitation. I, Zeitschrift für Physik 56, 330 (1929).
  * Ali  _et al._ [2014] M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong, and R. J. Cava, Large, non-saturating magnetoresistance in WTe2, Nature 514, 205 (2014).
  * Das  _et al._ [2019] P. K. Das, D. D. Sante, F. Cilento, C. Bigi, D. Kopic, D. Soranzio, A. Sterzi, J. A. Krieger, I. Vobornik, J. Fujii, T. Okuda, V. N. Strocov, M. B. H. Breese, F. Parmigiani, G. Rossi, S. Picozzi, R. Thomale, G. Sangiovanni, R. J. Cava, and G. Panaccione, Electronic properties of candidate type-II Weyl semimetal WTe2. a review perspective, Electronic Structure 1, 014003 (2019).
  * Xu  _et al._ [2015] S. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. Lee, S. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science 349, 613 (2015).
  * Wu  _et al._ [2016b] Y. Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, Observation of Fermi arcs in the type-II Weyl semimetal candidate WTe2, Phys. Rev. B 94, 121113 (2016b).
  * Wu  _et al._ [2015] Y. Wu, N. H. Jo, M. Ochi, L. Huang, D. Mou, S. L. Bud’ko, P. C. Canfield, N. Trivedi, R. Arita, and A. Kaminski, Temperature-induced lifshitz transition in WTe2, Phys. Rev. Lett. 115, 166602 (2015).
  * Wang  _et al._ [2016] C. Wang, Y. Zhang, J. Huang, S. Nie, G. Liu, A. Liang, Y. Zhang, B. Shen, J. Liu, C. Hu, Y. Ding, D. Liu, Y. Hu, S. He, L. Zhao, L. Yu, J. Hu, J. Wei, Z. Mao, Y. Shi, X. Jia, F. Zhang, S. Zhang, F. Yang, Z. Wang, Q. Peng, H. Weng, X. Dai, Z. Fang, Z. Xu, C. Chen, and X. J. Zhou, Observation of Fermi arc and its connection with bulk states in the candidate type-II Weyl semimetal WTe2, Phys. Rev. B 94, 241119 (2016).
  * Sánchez-Barriga  _et al._ [2016] J. Sánchez-Barriga, M. G. Vergniory, D. Evtushinsky, I. Aguilera, A. Varykhalov, S. Blügel, and O. Rader, Surface Fermi arc connectivity in the type-II Weyl semimetal candidate WTe2, Phys. Rev. B 94, 161401 (2016).
  * Bruno  _et al._ [2016] F. Y. Bruno, A. Tamai, Q. S. Wu, I. Cucchi, C. Barreteau, A. de la Torre, S. McKeown Walker, S. Riccò, Z. Wang, T. K. Kim, M. Hoesch, M. Shi, N. C. Plumb, E. Giannini, A. A. Soluyanov, and F. Baumberger, Observation of large topologically trivial Fermi arcs in the candidate type-II Weyl semimetal WTe2, Phys. Rev. B 94, 121112 (2016).
  * Zhang  _et al._ [2017] W. Zhang, Q. Wu, L. Zhang, S.-W. Cheong, A. A. Soluyanov, and W. Wu, Quasiparticle interference of surface states in the type-II Weyl semimetal WTe2, Phys. Rev. B 96, 165125 (2017).
  * Yuan  _et al._ [2018] Y. Yuan, X. Yang, L. Peng, Z.-J. Wang, J. Li, C.-J. Yi, J.-J. Xian, Y.-G. Shi, and Y.-S. Fu, Quasiparticle interference of Fermi arc states in the type-II Weyl semimetal candidate WTe2, Phys. Rev. B 97, 165435 (2018).
  * Pan  _et al._ [2015] X.-C. Pan, X. Chen, H. Liu, Y. Feng, Z. Wei, Y. Zhou, Z. Chi, L. Pi, F. Yen, F. Song, X. Wan, Z. Yang, B. Wang, G. Wang, and Y. Zhang, Pressure-driven dome-shaped superconductivity and electronic structural evolution in tungsten ditelluride, Nature Communications 6, 7805 (2015).
  * Kang  _et al._ [2015] D. Kang, Y. Zhou, W. Yi, C. Yang, J. Guo, Y. Shi, S. Zhang, Z. Wang, C. Zhang, S. Jiang, A. Li, K. Yang, Q. Wu, G. Zhang, L. Sun, and Z. Zhao, Superconductivity emerging from a suppressed large magnetoresistant state in tungsten ditelluride, Nature Communications 6, 7804 (2015).
  * Marini and Calandra [2021] G. Marini and M. Calandra, Light-tunable charge density wave orders in MoTe2 and WTe2 single layers, Phys. Rev. Lett. 127, 257401 (2021).
  * Sajadi  _et al._ [2018] E. Sajadi, T. Palomaki, Z. Fei, W. Zhao, P. Bement, C. Olsen, S. Luescher, X. Xu, J. A. Folk, and D. H. Cobden, Gate-induced superconductivity in a monolayer topological insulator, Science 362, 922 (2018).
  * Maximenko  _et al._ [2022] Y. Maximenko, Y. Chang, G. Chen, M. R. Hirsbrunner, W. Swiech, T. L. Hughes, L. K. Wagner, and V. Madhavan, Nanoscale studies of electric field effects on monolayer 1T-WTe2, npj Quantum Materials 7, 29 (2022).
  * Wang  _et al._ [2019] Z. Wang, B. J. Wieder, J. Li, B. Yan, and B. A. Bernevig, Higher-order topology, monopole nodal lines, and the origin of large fermi arcs in transition metal dichalcogenides $x{\mathrm{te}}_{2}$ ($x=\mathrm{Mo},\mathrm{W}$), Phys. Rev. Lett. 123, 186401 (2019).
  * Alekseev  _et al._ [2015] P. S. Alekseev, A. P. Dmitriev, I. V. Gornyi, V. Y. Kachorovskii, B. N. Narozhny, M. Schütt, and M. Titov, Magnetoresistance in two-component systems, Phys. Rev. Lett. 114, 156601 (2015).
  * Zhu  _et al._ [2015] Z. Zhu, X. Lin, J. Liu, B. Fauqué, Q. Tao, C. Yang, Y. Shi, and K. Behnia, Quantum Oscillations, Thermoelectric Coefficients, and the Fermi Surface of Semimetallic WTe2, Phys. Rev. Lett. 114, 176601 (2015).
  * Fei  _et al._ [2017] Z. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, P. Nguyen, J. Finney, X. Xu, and D. H. Cobden, Edge conduction in monolayer WTe2, Nature Physics 13, 677 (2017).
  * Jia  _et al._ [2017] Z.-Y. Jia, Y.-H. Song, X.-B. Li, K. Ran, P. Lu, H.-J. Zheng, X.-Y. Zhu, Z.-Q. Shi, J. Sun, J. Wen, D. Xing, and S.-C. Li, Direct visualization of a two-dimensional topological insulator in the single-layer $1{T}^{{}^{\prime}}-\mathrm{WT}{\mathrm{e}}_{2}$, Phys. Rev. B 96, 041108 (2017).
  * Wu  _et al._ [2018] S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Observation of the quantum spin Hall effect up to 100 Kelvin in a monolayer crystal, Science 359, 76 (2018).
  * Tang  _et al._ [2017] S. Tang, C. Zhang, D. Wong, Z. Pedramrazi, H.-Z. Tsai, C. Jia, B. Moritz, M. Claassen, H. Ryu, S. Kahn, J. Jiang, H. Yan, M. Hashimoto, D. Lu, R. G. Moore, C.-C. Hwang, C. Hwang, Z. Hussain, Y. Chen, M. M. Ugeda, Z. Liu, X. Xie, T. P. Devereaux, M. F. Crommie, S.-K. Mo, and Z.-X. Shen, Quantum spin Hall state in monolayer 1T’-WTe2, Nature Physics 13, 683 (2017).
  * Miller  _et al._ [2009] D. L. Miller, K. D. Kubista, G. M. Rutter, M. Ruan, W. A. de Heer, P. N. First, and J. A. Stroscio, Observing the quantization of zero mass carriers in graphene, Science 324, 924 (2009).
  * Yin  _et al._ [2020] J.-X. Yin, W. Ma, T. A. Cochran, X. Xu, S. S. Zhang, H.-J. Tien, N. Shumiya, G. Cheng, K. Jiang, B. Lian, Z. Song, G. Chang, I. Belopolski, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, B. Swidler, H. Zhou, H. Lin, T. Neupert, Z. Wang, N. Yao, T.-R. Chang, S. Jia, and M. Zahid Hasan, Quantum-limit Chern topological magnetism in TbMn6Sn6, Nature 583, 533 (2020).
  * Hossain  _et al._ [2024] M. S. Hossain, F. Schindler, R. Islam, Z. Muhammad, Y.-X. Jiang, Z.-J. Cheng, Q. Zhang, T. Hou, H. Chen, M. Litskevich, B. Casas, J.-X. Yin, T. A. Cochran, M. Yahyavi, X. P. Yang, L. Balicas, G. Chang, W. Zhao, T. Neupert, and M. Z. Hasan, A hybrid topological quantum state in an elemental solid, Nature (2024).
  * Feldman  _et al._ [2016] B. E. Feldman, M. T. Randeria, A. Gyenis, F. Wu, H. Ji, R. J. Cava, A. H. MacDonald, and A. Yazdani, Observation of a nematic quantum hall liquid on the surface of bismuth, Science 354, 316 (2016).
  * Hanaguri  _et al._ [2010] T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3, Phys. Rev. B 82, 081305 (2010).
  * Okada  _et al._ [2013] Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C. Dhital, M. Neupane, S. Xu, Y. J. Wang, R. Sankar, F. Chou, A. Bansil, M. Z. Hasan, S. D. Wilson, L. Fu, and V. Madhavan, Observation of Dirac node formation and mass acquisition in a topological crystalline insulator, Science 341, 1496 (2013).
  * Johnsen  _et al._ [2023] T. Johnsen, C. Schattauer, S. Samaddar, A. Weston, M. J. Hamer, K. Watanabe, T. Taniguchi, R. Gorbachev, F. Libisch, and M. Morgenstern, Mapping quantum Hall edge states in graphene by scanning tunneling microscopy, Phys. Rev. B 107, 115426 (2023).
  * Bindel  _et al._ [2017] J. R. Bindel, J. Ulrich, M. Liebmann, and M. Morgenstern, Probing the nodal structure of Landau level wave functions in real space, Phys. Rev. Lett. 118, 016803 (2017).
  * Tsui  _et al._ [2024] Y.-C. Tsui, M. He, Y. Hu, E. Lake, T. Wang, K. Watanabe, T. Taniguchi, M. P. Zaletel, and A. Yazdani, Direct observation of a magnetic-field-induced Wigner crystal, Nature 628, 287 (2024).
  * Suderow  _et al._ [2011] H. Suderow, I. Guillamon, and S. Vieira, Compact very low temperature scanning tunneling microscope with mechanically driven horizontal linear positioning stage, Review of Scientific Instruments 82, 033711 (2011).
  * Martín-Vega  _et al._ [2021] F. Martín-Vega, V. Barrena, R. Sánchez-Barquilla, M. Fernández-Lomana, J. Benito Llorens, B. Wu, A. Fente, D. Perconte Duplain, I. Horcas, R. López, J. Blanco, J. A. Higuera, S. Mañas-Valero, N. H. Jo, J. Schmidt, P. C. Canfield, G. Rubio-Bollinger, J. G. Rodrigo, E. Herrera, I. Guillamón, and H. Suderow, Simplified feedback control system for scanning tunneling microscopy, Review of Scientific Instruments 92, 103705 (2021).
  * Horcas  _et al._ [2007] I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro, WSXM: A software for scanning probe microscopy and a tool for nanotechnology, Review of Scientific Instruments 78, 013705 (2007).
  * Canfield and Fisk [1992] P. C. Canfield and Z. Fisk, Growth of single crystals from metallic fluxes, Philosophical Magazine B 65, 1117 (1992).
  * Das  _et al._ [2016] P. K. Das, D. Di Sante, I. Vobornik, J. Fujii, T. Okuda, E. Bruyer, A. Gyenis, B. E. Feldman, J. Tao, R. Ciancio, G. Rossi, M. N. Ali, S. Picozzi, A. Yadzani, G. Panaccione, and R. J. Cava, Layer-dependent quantum cooperation of electron and hole states in the anomalous semimetal WTe2, Nature Communications 7, 10847 (2016).
  * Pletikosić  _et al._ [2014] I. Pletikosić, M. N. Ali, A. V. Fedorov, R. J. Cava, and T. Valla, Electronic structure basis for the extraordinary magnetoresistance in WTe2, Phys. Rev. Lett. 113, 216601 (2014).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Blaha  _et al._ [2018] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz, R. Laskowski, F. Tran, and M. L. D., WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Vienna University of Technology, Austria) (2018).
  * Blaha  _et al._ [2020] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, WIEN2k: An APW+lo program for calculating the properties of solids, The Journal of Chemical Physics 152, 074101 (2020).
  * Mar  _et al._ [1992] A. Mar, S. Jobic, and J. A. Ibers, J. Am. Chem. Soc. 114, 8963 (1992).
  * Jo  _et al._ [2019] N. H. Jo, L.-L. Wang, P. P. Orth, S. L. Bud’ko, and P. C. Canfield, Magnetoelastoresistance in WTe2: Exploring electronic structure and extremely large magnetoresistance under strain, Proceedings of the National Academy of Sciences 116, 25524 (2019).
  * Lu  _et al._ [2015] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, Experimental observation of Weyl points, Science 349, 622 (2015).
  * Cheng  _et al._ [2010] P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, J. Jia, J. Wang, Y. Wang, B. Zhu, X. Chen, X. Ma, K. He, L. Wang, X. Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S. Zhang, and Q. Xue, Landau quantization of topological surface states in Bi2Se3, Phys. Rev. Lett. 105, 076801 (2010).
  * Tersoff and Hamann [1985] J. Tersoff and D. R. Hamann, Theory of the scanning tunneling microscope, Phys. Rev. B 31, 805 (1985).
  * Cho  _et al._ [2017] K. Cho, A. Fente, S. Teknowijoyo, M. A. Tanatar, K. R. Joshi, N. M. Nusran, T. Kong, W. R. Meier, U. Kaluarachchi, I. Guillamón, H. Suderow, S. L. Bud’ko, P. C. Canfield, and R. Prozorov, Nodeless multiband superconductivity in stoichiometric single-crystalline ${\mathrm{cakfe}}_{4}{\mathrm{as}}_{4}$, Phys. Rev. B 95, 100502 (2017).
  * Guillamon  _et al._ [2008] I. Guillamon, H. Suderow, F. Guinea, and S. Vieira, Intrinsic atomic-scale modulations of the superconducting gap of $2h\text{$-$}{\text{nbse}}_{2}$, Phys. Rev. B 77, 134505 (2008).
  * Calleja  _et al._ [2004] F. Calleja, A. Arnau, J. J. Hinarejos, A. L. Vázquez de Parga, W. A. Hofer, P. M. Echenique, and R. Miranda, Contrast reversal and shape changes of atomic adsorbates measured with scanning tunneling microscopy, Phys. Rev. Lett. 92, 206101 (2004).
  * Kim  _et al._ [2017] H. Kim, Y. Yoshida, C.-C. Lee, T.-R. Chang, H.-T. Jeng, H. Lin, Y. Haga, Z. Fisk, and Y. Hasegawa, Atomic-scale visualization of surface-assisted orbital order, Science Advances 3, eaao0362 (2017).
  * Yu  _et al._ [2016] Z.-M. Yu, Y. Yao, and S. A. Yang, Predicted unusual magnetoresponse in type-II Weyl semimetals, Phys. Rev. Lett. 117, 077202 (2016).
  * Tchoumakov  _et al._ [2016] S. Tchoumakov, M. Civelli, and M. O. Goerbig, Magnetic-field-induced relativistic properties in type-i and type-ii weyl semimetals, Phys. Rev. Lett. 117, 086402 (2016).
  * Udagawa and Bergholtz [2016] M. Udagawa and E. J. Bergholtz, Field-selective anomaly and chiral mode reversal in type-ii weyl materials, Phys. Rev. Lett. 117, 086401 (2016).
  * Li  _et al._ [2017] P. Li, Y. Wen, X. He, Q. Zhang, C. Xia, Z.-M. Yu, S. A. Yang, Z. Zhu, H. N. Alshareef, and X.-X. Zhang, Evidence for topological type-II Weyl semimetal WTe2, Nature Communications 8, 2150 (2017).
  * Wang  _et al._ [2015] L. Wang, I. Gutiérrez-Lezama, C. Barreteau, N. Ubrig, E. Giannini, and A. F. Morpurgo, Tuning magnetotransport in a compensated semimetal at the atomic scale, Nature Communications 6, 8892 (2015).
  * Potter  _et al._ [2014] A. C. Potter, I. Kimchi, and A. Vishwanath, Quantum oscillations from surface fermi arcs in weyl and dirac semimetals, Nature Communications 5, 5161 (2014).
  * Tao  _et al._ [2022] W. Tao, Z. J. Tong, A. Das, D.-Q. Ho, Y. Sato, M. Haze, J. Jia, Y. Que, F. Bussolotti, K. E. J. Goh, B. Wang, H. Lin, A. Bansil, S. Mukherjee, Y. Hasegawa, and B. Weber, Multiband superconductivity in strongly hybridized 1T-WTe2/NbSe2 heterostructures, Phys. Rev. B 105, 094512 (2022).
  * Li  _et al._ [2018] Q. Li, C. He, Y. Wang, E. Liu, M. Wang, Y. Wang, J. Zeng, Z. Ma, T. Cao, C. Yi, N. Wang, K. Watanabe, T. Taniguchi, L. Shao, Y. Shi, X. Chen, S.-J. Liang, Q.-H. Wang, and F. Miao, Proximity-induced superconductivity with subgap anomaly in type II Weyl semi-metal WTe2, Nano Letters 18, 7962 (2018).
  * Shi  _et al._ [2019] S. Shi, S. Liang, Z. Zhu, K. Cai, S. D. Pollard, Y. Wang, J. Wang, Q. Wang, P. He, J. Yu, G. Eda, G. Liang, and H. Yang, All-electric magnetization switching and Dzyaloshinskii-Moriya interaction in WTe2/ferromagnet heterostructures, Nature Nanotechnology 14, 945 (2019).
  * Choi  _et al._ [2020] Y.-B. Choi, Y. Xie, C.-Z. Chen, J. Park, S.-B. Song, J. Yoon, B. J. Kim, T. Taniguchi, K. Watanabe, J. Kim, K. C. Fong, M. N. Ali, K. T. Law, and G.-H. Lee, Evidence of higher-order topology in multilayer WTe2 from Josephson coupling through anisotropic hinge states, Nature Materials 19, 974 (2020).
  * Chen  _et al._ [2018] C.-Z. Chen, J. J. He, M. N. Ali, G.-H. Lee, K. C. Fong, and K. T. Law, Asymmetric Josephson effect in inversion symmetry breaking topological materials, Phys. Rev. B 98, 075430 (2018).
  * Fei  _et al._ [2018] Z. Fei, W. Zhao, T. A. Palomaki, B. Sun, M. K. Miller, Z. Zhao, J. Yan, X. Xu, and D. H. Cobden, Ferroelectric switching of a two-dimensional metal, Nature 560, 336 (2018).
  * Sharma  _et al._ [2019] P. Sharma, F.-X. Xiang, D.-F. Shao, D. Zhang, E. Y. Tsymbal, A. R. Hamilton, and J. Seidel, A room-temperature ferroelectric semimetal, Science Advances 5, eaax5080 (2019).
  * Bhowal and Spaldin [2023] S. Bhowal and N. A. Spaldin, Polar metals: Principles and prospects, Annual Review of Materials Research 53, 53 (2023).
  * Song  _et al._ [2018] Y.-H. Song, Z.-Y. Jia, D. Zhang, X.-Y. Zhu, Z.-Q. Shi, H. Wang, L. Zhu, Q.-Q. Yuan, H. Zhang, D.-Y. Xing, and S.-C. Li, Observation of coulomb gap in the quantum spin Hall candidate single-layer 1T’-WTe2, Nature Communications 9, 4071 (2018).
  * Gu  _et al._ [2023] Q. Gu, J. P. Carroll, S. Wang, S. Ran, C. Broyles, H. Siddiquee, N. P. Butch, S. R. Saha, J. Paglione, J. C. S. Davis, and X. Liu, Detection of a pair density wave state in UTe2, Nature 618, 921 (2023).
  * Aishwarya  _et al._ [2023] A. Aishwarya, J. May-Mann, A. Raghavan, L. Nie, M. Romanelli, S. Ran, S. R. Saha, J. Paglione, N. P. Butch, E. Fradkin, and V. Madhavan, Magnetic-field-sensitive charge density waves in the superconductor UTe2, Nature 618, 928 (2023).
  * Lewin  _et al._ [2023] S. K. Lewin, C. E. Frank, S. Ran, J. Paglione, and N. P. Butch, A review of UTe2 at high magnetic fields, Reports on Progress in Physics 86, 114501 (2023).
  * Aoki  _et al._ [2022] D. Aoki, J.-P. Brison, J. Flouquet, K. Ishida, G. Knebel, Y. Tokunaga, and Y. Yanase, Unconventional superconductivity in UTe2, Journal of Physics: Condensed Matter 34, 243002 (2022).

## Supplementary information

## 1\. Defects and surface of WTe2

Figure S1: (a) Large STM topographic image showing a region with different
types of defects. White and black arrows mark vacancy-like defects and
interstitials, respectively. Wavelike patterns can be seen coming due to
scattering at the defects. Horizontal scale bar is 2 nm long. (b,c) Zoom into
a vacancy (b) defect showing the lack of a surface Te atom and an interstitial
(c) showing a substitution of Te by W. The surface pattern caused by this
disruption is slightly asymmetric along the b-direction, as also shown in the
calculations. (d,e) Calculated local density of states of a Te vacancy (d) and
a substitution of W by Te (e). We again observe an asymmetry along the
b-direction, also found in the calculations. Note that the maps (b-e) are
rotated with respect to (a) so as to place the Te chains parallel to the
x-axis. The atomic structure is superposed to the experiments in (b,c). Violet
is for W and green for Te.

WTe2 belongs to the transition metal dichalcogenides. The crystalline
structure of these materials is usually layered. In each layer, the transition
metal is sandwiched between two chalcogen layers (either S, Se or Te). Each
Te-W-Te layer is separated by weak van der Waals interactions along the
$c$-axis. Within each layer, there are strong covalent bonds. In this
material, the W atoms form zigzag chains along the $a$ axis, as can be seen in
the atomic structure shown in Fig. 1(c) of the main text. In topography
images, we can see the Te atoms also forming chains along this direction. In
the inset of Fig. 1(c) of the main text, we provide the atomic structure
superposed to a high resolution topographic image. These features are also
shown in Fig. S1, but in a field of view showing a few defects. We can clearly
identify the Te rows and interstitials as well as vacancies. These are the two
types of defects that we can identify on the surface of this material. We find
either small point vacancies, indicated with a white arrow in Fig. S1(a) or
bump-like features that we refer to as interstitials, indicated by a black
arrow in Fig. S1(a).

To better understand the features found in the topography at defects, we
performed first-principles calculations based on density functional theory
(DFT) as implemented in the Quantum Espresso code. WTe2 presents a primitive
unit cell with 12 sites, showing an orthorhombic structure with the Space
group Pnm21 and with lattice parameters of a = 3.477 Å, b = 6.249 Å and c =
14.0179 Å. The 1T’ phase is a distorted 1T phase where the Te atoms have
distanced themselves from their hexagonal position around the W atom. The 1T’
phase can be easily recognized and differentiated from the 1T phase by the
presence of isolated zigzag chains of chalcogenide atoms that run vertically.
We investigated: (i) the formation of a Te vacancy in the surface layer and
(ii) the replacement of one Te atom by a W one providing an antisite defect.
To model the experimental system, we constructed slabs of 4$\times$4$\times$2
size containing four layers, ensuring a 13 Å separation between neighbouring
defects, which is enough to discard interactions between them.

We calculated the density of states at an energy of 100 meV and compared the
result with topography experiments made at 100 mV (Fig. S1(a)). The high
resolution topographies of a vacancy and an interstitial are shown in Fig.
S1(b,c), respectively. Calculations are shown in Fig. S1(d,e). We note that
the calculations nicely reproduce the elevated density of states along the Te
chains, giving the characteristic row-like structure of STM images in WTe2.
The row-like structure is broken at the defects. On Te vacancies (Fig. S1(b))
we find a clear black “hole” in a Te row, due to the missing Te atom. This
“hole” is also found in the calculations (Fig. S1(d)). The chain like
structure is compressed at the “hole” in experiments. This can be related to
the features perpendicular to the chains, which are observed in the
calculations (Fig. S1(d)). On the other hand, calculations present (Fig.
S1(e)) an increased density of states on interstitials (substitution of Te
with W), with a highly asymmetric structure around the additional W atom,
similar to a horseshoe. This structure is also observed in the experiment
(Fig. S1(c)). Overall, the agreement between experiment and calculation is
very good and, importantly, highlights that both W and Te derived orbitals
contribute to the tunneling at atomic scale.

In the large scale STM topography (Fig. S1(a)), we see that these features are
repeatedly observed over the image. We can estimate a density of defects of
approximately $1.3\times 10^{12}$ cm-2. Using the lattice parameters $a$ = 3.5
Å and $b$ = 6.25 Å, we can estimate that the electrons can move freely along
$\sim$ 336 unit cells, which gives a mean free path $\ell\sim$ 167.5 nm. Using
the Drude model, we find a residual resistivity of $\rho=1.95$
$\mu\Omega\cdot$ cm and mobility of $\mu=$4.26 m2/$($Vs$)$. This agrees with
previous reports and macroscopic characterization of our samples [19, 56].

In the STM topography (Fig. S1(a)) we also observe additional features at
larger length scales around defects. These are related to the scattering
pattern observed when making conductance maps at different bias voltages to
obtain the QPI results discussed in Fig. 2 of the main text and further
discussed in the following section.

## 2\. Quasiparticle interference scattering

The relation between the measured local density of states and the momentum
eigenstates $\Psi(\vec{r}_{k})$ is given by

$\displaystyle
LDOS(E,\vec{r})\propto\sum_{\vec{k}}|\Psi(\vec{r}_{k})|^{2}\delta\left(E-\epsilon(\vec{k})\right),$
(1)

where $\epsilon(\vec{k})$ is the dispersion relation of the material. When the
periodicity of the crystal is broken by the presence of impurities or defects
on the surface, electrons are scattered, producing oscillations in the $LDOS$.
These scattering processes are typically elastic. In this picture, scattering
between states with $\vec{k}_{1}$ and $\vec{k}_{2}$ gives rise to a modulation
with $\vec{q}=\vec{k}_{1}-\vec{k}_{2}$ in the LDOS that can be observed in the
tunneling conductance with the STM. The scattering between an initial state,
$i$, and a final state, $f$, is described by the Fermi golden rule,

$\displaystyle\omega(i\longrightarrow
f)\propto\frac{2\pi}{\hbar}|V(\vec{q})|^{2}N_{i}(E_{i},\vec{k}_{i})N_{f}(E_{f},\vec{k}_{f}),$
(2)

where $E_{i}=E_{f}$ for elastic scattering, $\vec{q}=\vec{k}_{f}-\vec{k}_{i}$
is the scattering vector, $N_{i}$ and $N_{f}$ the initial and final density of
states, and $V(\vec{q})$ the scattering potential.

To get some insights from calculation, we first performed DFT calculation in
the way described in the main text. After the first-principles calculation, we
constructed the Wannier functions [2, 3] using the Wannier90 software [4]. We
did not perform the maximal localization procedure to prevent the mixture of
the different spin components. We took the W-$d$ and Te-$p$ orbitals as the
Wannier basis. Fermi surfaces and autocorrelation functions were calculated
using the tight-binding model consisting of the Wannier functions. The
autocorrelation function calculated in this study was defined as

$I(E,\vec{q})=\frac{1}{2}\sum_{\vec{k},i,j}\sum_{l=0}^{3}f_{l}(\vec{k},\epsilon_{i}-E)f_{l}(\vec{k}+\vec{q},\epsilon_{j}-E)$
(3)

where $f_{l}(\vec{k},\epsilon_{i}-E)$ is the Lorentzian function,

$f_{l}(\vec{k},\epsilon_{i}-E)=\frac{1}{\pi}\frac{\gamma}{(\epsilon_{i}-E)^{2}+\gamma^{2}}\langle\sigma_{l}\rangle_{i},$
(4)

with the smearing width $\gamma$ for the $i$-th Kohn-Sham energy
$\epsilon_{i}$ and the expectation value of the Pauli matrix $\sigma_{l}$
($l=0,1,2,3$) for the corresponding eigenstate (see, e.g., Ref. [10], for a
use of the spin-decomposed spectral density). To reduce computational cost, we
constrained the summation on the $k_{z}=0$ plane. We took $\gamma=1$ meV and a
$240\times 240$ ${\bm{k}}$-mesh in the region of $|k_{x,y}|\leq 0.6\pi
a^{-1}$. The Fermi energy in all the calculated data was shifted by $-45$ meV
so as to reproduce the experimental data.

We show in Fig. S2(a) the Fourier transform of the tunneling conductance maps,
compared to calculations of the band structure (Fig. S2(b,c)). In Fig. S2(b)
we show the constant energy contours of the band structure and in Fig. S2(c)
their autocorrelation function. The latter is expected to reproduce some of
the scattering features. Nevertheless, a full agreement is not to be expected,
as the autocorrelation function just leads to a doubling in k space of the
features of the constant energy contour, or a passage from $\vec{k}$ to
$\vec{q}$ by geometrical construction, and does not account for
$k_{z}$-summation nor the scattering potential $V(\vec{q})$.

We can now discuss the comparison between bulk band structure and experiment.

The dark and light violet and the magenta points refer to scattering between
the two tiny W-derived pockets for large positive energies. Close to the light
violet point at large wavevectors, there is a convex feature which is also
seen in the autocorrelation function Fig. S2(c). This convex feature is also
there at 50 meV, but is much weaker or absent below 50 meV. As for the magenta
points, these lead to convex features in both experiment and calculations at
75 mV. The dark violet feature leads to a concave feature in the calculations.
In the experiments, there is also a concave feature, but at the same time the
curves clearly show a convex feature which is not seen in the calculations,
for example at 75 mV. As we show below, the convex feature might be due to the
surface state, although calculations locate the surface state at a slightly
larger $k_{x}$.

On the other hand, scattering from the hole Te-derived pockets provides the
green points shown for bias voltages and energies below approximately 0 meV.
We see that the dark green scattering is generally associated to a concave
feature in the autocorrelation function which is also seen in the experiment,
particularly at -25 mV. At the lowest bias voltages, the experiments show a
tendency to form convex features, which is not clearly seen in the
calculations. This might be to the surface state discussed below. The light
green points show generally convex features in experiment, except at -75 mV,
which are however more concave like in the calculations. This, again, might be
due to the surface states discussed below.

Figure S2: (a) Fourier transforms of the conductance maps on a field of view
containing defects (interstitials as well as vacancies, similar to Fig.
S1(a)). The corresponding bias voltage is shown at the left of each panel. We
show a few bias voltages, for clarity. The scattering intensity is shown by
the color scale on the top. We indicate by colored points the scattering
features that lead to the identify the scattering vectors $q_{1}$ to $q_{11}$
shown in Fig. 2(b) of the main text for $k_{y}=0$. (b) Calculated constant
energy contours of the band structure. We show by colored arrows the
scattering signal associated to the points in (a), with the same color code.
(c) Autocorrelation of the calculated constant energy contours. Again,
relevant features are shown as colored points.

## 3\. Surface states and QPI

Figure S3: (a) In the top panel we show the constant energy contour at -50 meV
of the surface band structure of WTe2. We highlight by two yellow arrows the
position of the surface state joining W-derived electron with Te-derived hole
bands. The surface state is shown separately in the bottom panel. (b) In the
top panel we show the autocorrelation function of the surface state. In the
middle panel we show the experiment at -50 mV. In the bottom panel we show the
autocorrelation function of the bulk band structure. In (c,d) we show the same
plots, for the surface state at +75 meV.

For surface-state calculation, the semi-infinite-slab tight-binding model was
used for calculating the surface spectra in a way described in the literature
[5]. The energy smearing of 0.1 meV was applied as the imaginary part of the
denominator of the Green’s function. For calculating the autocorrelation
functions for the surface, $f_{l}({\bm{k}},\epsilon_{i}-E)$ in Eq. (3) was
replaced with the calculated surface spectrum.

To address with more detail the behavior at surface states, we start by
analyzing the scattering pattern at the voltage range where surface states are
expected to show up. From calculations, this might occur around -50 meV and
around +75 meV. In Fig. S3 we show results and calculations at these energies
and bias voltages (a,b at -50 meV and c,d at +75 meV). We mark the surface
states in the constant energy contours by arrows in Fig. S3(a,c). These
surface states join the electron (W-derived) and the hole (Te-derived)
pockets. The Weyl points are expected to occur above the Fermi level at a
finite $k_{x}$ and $k_{y}$, as described in Ref. [12].

We then isolate the surface states (bottom panels of Fig. S3(a,c)) and make
the autocorrelation function of the resulting images (top panels of Fig.
S3(b,d)). We compare this with the experiment (middle panels of Fig. S3(b,d))
and with the autocorrelation from the bulk band structure (bottom panels of
Fig. S3(b,d)).

We note that Fermi arcs imply a concave X-like feature crossing exactly
$q_{x}=q_{y}=0$ in the scattering pattern, as shown in top panels of Fig.
S3(b,d) and predicted in Ref. [6]. This feature is indeed found in the
experiment (middle panels of Fig. S3(b,d)) but its extension along $q_{y}$ is
somewhat larger than in the autocorrelation function. The other features of
surface states are also observed in the experiment. The convex feature
associated to the light green point in Fig. S3(b) is clealy seen in the
experiment (middle panel of Fig. S3(b)) and the surface calculations (top
panel of Fig. S3(b)), but absent in the bulk calculations (bottom panel of
Fig. S3(b)). Furthermore, the large convex feature at the light violet point
in Fig. S3(d) is seen in experiment (middle panel of Fig. S3(d)) and surface
calculations (top panel of Fig. S3(d)). There is a similar feature in the bulk
calculations (bottom panel of Fig. S3(d)). However, there is a much stronger
convex feature on the dark violet point (middle panel), which is absent in the
bulk calculations (bottom panel of Fig. S3(d)). This feature might be
associated to the well established convex feature observed in the surface
calculations (top panel of Fig. S3(d)) at a larger wavevector.

Thus, although the precise separation between bulk and surface states remains
difficult due to the multiplicity of scattering features, in all there is
consistent evidence for the presence of surface states, further supported by
Landau quantization as discussed below.

It is difficult to compare with angular resolved photoemission (ARPES) results
which focus at higher energies and filled states [51]. But there is a general
agreement. ARPES is mostly sensitive to the hole-like Te derived pockets and
their structure well below the Fermi level. However, high resolution ARPES
provides the same convex features in the hole pockets at small $k_{x}$ that we
observe here and shows in addition that these hole pockets move below the
Fermi level when increasing temperature above 160 K [19, 18]. Furthermore, in
ARPES, there is a band at larger $k_{x}$ which crosses the Fermi level above
approximately -50 meV which was first identified as stemming from an electron
pocket [19], but could be related to the surface state discussed above,
judging from the position in reciprocal space. These surface states are indeed
measured in detail with ARPES in Ref. [18], where it is clearly shown that
these are Fermi arcs. Although the features appear at a smaller wavevector,
possibly because of slightly shifted band structure, the observed bulk and
surface band structure are mostly compatible with what we observe, including
the Fermi arcs identified in Ref. [18].

## 4\. Spatially averaged tunneling conductance vs magnetic field

Figure S4: (a) Calculated bulk band structure along the $\Gamma$ ($k_{x}$ = 0)
to $X$ ($k_{x}$ = 1) direction inside the energy range explored in the
experiment. Electron and hole bands are depicted in purple and green,
respectively. Electron bands are derived from W 5d-orbitals and hole bands
from Se 5p-orbitals. The calculations are shifted 45 meV downwards in energy
to match our experimental results. (b) Spatially averaged normalized tunneling
conductance versus bias voltage curves measured at magnetic fields ranging
from 0 T to 14 T (colored lines). The normalized density of states obtained
from the band structure in (a) is also plotted as a black line. Curves are
shifted vertically for better visualization.

Fig. S4(a) shows DFT band structure calculations in the same range than the
STM measurements, along the $\Gamma-X$ direction. Electron and hole bands are
plotted in purple and green respectively. Electron bands have a predominant
$W-5d$ character, while the hole bands have $Te-5p$ character. The resulting
momentum integrated density of states of these bands as a function of energy
is represented in black in Fig. S4(b). We compare the calculated density of
states to the spatially averaged tunneling conductance at different magnetic
fields (colored lines in Fig. S4(b)). We see that the shape of spatially
averated tunneling conductance curves vs bias voltage is similar to the shape
of the calculated density of states vs energy.

It is of interest to ask how can one observe a huge increase, by more than six
orders of magnitude, in the resistance when applying a magnetic field in WTe2,
without any relevant magnetic field induced changes in the band structure.
This shows that magnetoresistance is not caused by any change in the Fermi
surface with magnetic field, but is rather associated to perfect electron-hole
compensation, as proposed in Ref. [15, 31].

## 5\. Atomic scale maps of Landau quantized tunneling conductance

As we have discussed, our data show that Landau quantization is observed in
WTe2 with STM when taking tunneling conductance curves with atomic precision.
We show in Fig. S5 the tunneling conductance as a function of the position
along a line on top of Te atoms. We observe that the oscillations due to
Landau levels shown in Fig. 3 of the main text change as a function of the
position (Fig. S5(a)). We can obtain a similar map taking a spatially varying
density of states $N_{local}(E,x)=\sum_{n}\left[E-E_{0}-\sqrt{2e\hbar
v_{F}^{2}|n|B}+A\cos(qx)\right]^{-\frac{1}{2}}$. We use $A$ = 4.2 mV,
$v_{F}=2.8\times 10^{5}$ $m/s$ and $q=2\pi a$, where $a$ is the unit cell
parameter in the direction of the Te chains. Note that $A$ is larger than
$\hbar\omega_{c}/2$. Hence, the result is not a simple cosine wave, but an
intricate set of maxima and minima, as shown in Fig. S5(b).

Figure S5: (a) Tunneling conductance along a row following the a axis, taken
at 14 T. In the left panel we show a topography of the surface, scaled to
match the distance and atomic locations in the y-scale of the main panel. The
map was obtained on top of a Te row. (b) Same as in (a), with a map obtained
calculating the density of states as explained in the text, with parameters
$A=4.2$ mV and $q=2\pi/a$. We use the same pixel size as in the experiment and
the level separation measured for 14 T.

More information can be found if we take a closer look on the behavior under
magnetic fields. Surface features are purely two-dimensional, and the Landau
quantization should occur without any $k_{z}$ dispersion and should thus be
stronger. We can indeed find evidence of two surface states when looking at
the amplitude of the Landau levels at different atomic positions. We can see
in Figure S6 that the amplitude of the Landau oscillations for the curves
taken between Te atoms (violet arrow), which corresponds to the position of
the W atom, the conductance increases below -50 mV and above +75 mV. In Fig.
S6(b) we show the conductance curves for four consecutive W atoms, and the
grey shadow indicates the bias voltage range where the amplitude in the
conductance oscillations increases. When looking at the conductance curves
taken on top of the Te atoms (green arrow), the conductance increases below
-50 mV. Figure S6(c) shows the conductance curves taken on top of four
consecutive Te positions.

Figure S6: (a) Zoom into the topography around a chain of Te atoms. Arrows
provide schematically the atomic positions inside a unit cell where the curves
in (b,c) were taken. (b) Tunneling conductance vs bias voltage obtained over
four W atoms in four different unit cells. (c) Tunneling conductance vs bias
voltage obtained over four Te atoms in four different unit cells. The grey
background indicates the range above which two-dimensional surface states are
expected within calculations (see text).

## 6\. Properties of defects

To further understand the role of defects, we carried out first principles
calculations based on spin-polarized DFT in the plane wave formalism as
implemented in the QuantumESPRESSO package [7]. We described the exchange-
correlation energy by the generalized gradient approximation (GGA) using the
Perdew–Burke–Ernzerhof (PBE) functional [8] and standard Ultra-soft (USPP)
solid-state pseudopotentials extracted from the Materials Cloud Database. The
electronic wave functions were expanded with well-converged kinetic energy
cut-offs for the wave functions (charge density) of 50 (400) Ry. Van der Waals
interactions between layers were considered by applying semiempirical
Grimme-D3 corrections. We optimized atomic positions by using the Broyden-
Fletcher-Goldfarb-Shanno (BFGS) algorithm until the forces on each atom were
smaller than $10^{-5}$ Ry/au and the energy difference between two consecutive
relaxation steps was less than $10^{-7}$ Ry. The Brillouin zone was sampled by
a fine $\Gamma-$centered 3 $\times$ 2 $\times$ 1 k-point Monkhorst-Pack [9].
To avoid unphysical interactions between images along the non-periodic
direction, we added a vacuum of 18 Å along the $z$ direction.

For the Te vacancy defect (Fig. S7(b)) we considered two different W sites,
according to the W atom coordination. We found antiferromagnetic correlation
between the two different W sites in the entire range of U, resulting in a
global positive magnetic moment (Fig. S7(b)). From the spin-isosurface
representation, we observed that there is a tendency for a two-fold
arrangement (Fig. S7(c)), which is corroborated in the STM topography (Fig.
S1(b)). For the W interstitial defect, there are three different W
inequivalent sites, as a function of the W coordination (Fig. S7(d)). There
one can observe that W atoms around the interstitial defect carry magnetic
moments with ferromagnetic interactions between them and antiferromagnetically
coupled with the interstitial W defect for the whole range of U (Fig. S7(e)).
This results in a small magnetization. In all the defects produce pronounced
changes in the band structure at the local level which could have some
influence on the scattering potential $V(\vec{q})$, favoring certain
directions in scattering and helping to observe the shape of the band
structure out of the $a$ direction discussed above. Other than that, we expect
that the specificities of the band structure at Te vacancies and W-Te
substitutions have a small effect in the scattering results.

Figure S7: (a) Surface atomic arrangement around a defect. We highlight the
two types of W atoms around the intersitials, accroding to their bonding, with
dashed circles. (b) Magnetic moment as a function of the Hubbard U parameter.
We see that magnetic moments are essentially compensated, giving
antiferromagnetic coupling, except for large U. (c) Spin isosurface,
corresponding to a spin imbalance of 0.003 in red (spin down) and blue (spin
up) and U=5 eV. Different atomic species are shown in colors, following the
legend in (a,d).

## References

  * [1]
  * Marzari and Vanderbilt [1997] N. Marzari and D. Vanderbilt, Maximally localized generalized wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
  * Souza  _et al._ [2001] I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001).
  * Piz [2020] G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J. Lihm, D. Marchand,A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, L. Paulatto, S. Poncé, T. Ponweiser, T. Ponweiser, J. Qiao, F. Thöle, S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi and J. R. Yates, Wannier90 as a community code: new features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
  * Sancho  _et al._ [1985] M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Maximally localized wannier functions for entangled energy bands, J. Phys. F: Met. Phys. 15, 851 (1985).
  * Kourtis  _et al._ [2016] S. Kourtis, J. Li, Z. Wang, A. Yazdani, and B. A. Bernevig, Universal signatures of Fermi arcs in quasiparticle interference on the surface of Weyl semimetals, Phys. Rev. B 93, 041109 (2016).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, Quantum espresso: a modular and open-source software project for quantum simulations of materials, Journal of Physics: Condensed Matter 21, 395502 (2009).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Monkhorst and Pack [1976] H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).