## Introduction

2D transition metal dichalcogenides present an exciting platform for realizing emergent phases having non-trivial topology and strong correlations, and are prime candidates for hosting topological edge states, topological superconductivity, and fractional excitations1. After more than a decade of intense search, monolayer WTe2 (ML-WTe2) (Fig. 1a–e) has recently emerged as a 2D time-reversal invariant topological insulator, i.e., a quantum spin Hall (QSH) insulator2,3,4,5. While bulk WTe2 is a semimetal, band structure calculations of a monolayer indicate that it is a narrow-gap semiconductor exhibiting inverted bands that give rise to the QSH phase (Fig. 1f)6,7. Transport studies have confirmed quantized two-terminal edge conductance up to 100 K, and microwave impedance microscopy measurements have been used to observe edge states8,9.

The simplest mechanism to tune the properties of 2D materials is by gating, i.e., the application of an electric field through a dielectric layer. Naively, gating may be expected to add or remove carriers from the film by shifting the bands with respect to the Fermi energy (rigid band shift) as is observed in graphene10. However, gating could have non-trivial effects on the band structure due to the presence of the out-of-plane electric field, or changes in effective screening. For example, out-of-plane electric fields break inversion symmetry and can lead to a spin splitting of the bands, like the Rashba effect. In WTe2, electrostatic gating has been used to tune the circular photogalvanic effect11 and induce an exotic superconducting phase12,13. Despite the far-ranging interest in this system and extensive transport studies, the exact gap size of ML-WTe2, the dispersion of the edge states within the gap, as well as the effect of gating on the band structure have not yet been established. Indeed, the experimental literature reports a puzzlingly large variation in the observed gap sizes, which has yet to be explained2,3,5,7,8,11,12,13,14. Furthermore, recent studies have suggested that correlation effects may play a critical role in controlling band structure as well as gating dependent effects. Hence, comprehensive gating-dependent spectroscopic data is urgently needed to understand the key elements that control the low-energy physics of the system, and scanning tunneling microscopy (STM) and spectroscopy (STS) are ideal tools to probe gate-induced effects on the band structure and edge states.

In this work, we present a combined experimental-theoretical study on ML-WTe2. High-quality monolayer WTe2 films grown by molecular beam epitaxy (MBE) were studied by STM at 4 K. STM STS reveals clear bulk gaps with edge states that agree with density functional theory (DFT)—Heyd–Scuseria–Ernzerhof (HSE) calculations. We further find that the gap magnitude is highly tunable with gating. This gap tunability is quantitatively explained using a parameter-free first-principles DFT-derived tight-binding model. The calculations show that the conduction band states are predominantly located on the top and bottom layers with opposite spin-momentum locking, while the valence band states are localized to the interior tungsten layer. Application of an electric field changes the energy spacing between states in the three layers, resulting in a field-controllable spin-momentum selected gap.

## Results and discussion

### Film structure and topography

Monolayer WTe2 has a distorted orthorhombic 1 T′ structure which preserves inversion symmetry (Fig. 1a). Figure 1b shows a large scale STM image of ML-WTe2 grown on graphene on SiC. These films were transferred directly from the MBE to the STM using a vacuum suitcase (see Methods for details). Multiple monolayer islands of ~20–30 nm size, which correspond to the typical island size of MBE-grown WTe2, can be seen in this field of view. Zooming into a representative island, we can resolve the characteristic stripes corresponding to protruding Te atoms of the top surface (for example, Fig. 1c, d). The observed lattice corresponds well to 1 T′-WTe2. A representative bulk spectrum averaged over an area of 5 by 5 nm2 on one island (Fig. 1e) exhibits a band gap of approximately 40 meV.

### DFT calculations

To compare the experimentally measured gap to standard band theory calculations, we carry out DFT calculations using the Perdew–Burke–Ernzerhof (PBE)15 and Heyd–Scuseria–Ernzerhof exchange-correlation functional (HSE06)16 with and without spin-orbit interactions to compute the electronic structure of monolayer WTe2. The results are summarized in Fig. 1f. As is common for small gap semiconductors such as Ge, PBE obtains a negative gap, while HSE06 obtains a gap of around 60 meV, within the range of the experimental observations. The bands are inverted due to SOC according to HSE06, and thus we expect the existence of topological helical edge states.

As a benchmark for the topological electronic structure, we identify the bulk and edge states of an island (shown in Fig. 2a) with no applied gate voltage. Upon tracking the spectra from the center of the island to the edge, we observe a bulk insulating gap and an emergence of a finite density of states in the nominal insulating gap as the tip approaches the edge. The in-gap density of states arises from the helical edge modes expected for a QSH insulator. The edge modes can be seen more clearly in dI/dV maps shown in Fig. 2d, thus confirming the existence of states which are highly localized to the edge near the Fermi energy, which we assign to be helical QSH edge states. We find that the spatial extent of the edge modes increases as the energy of the modes approaches the bulk band edges. This trend can be seen on both sides of the Fermi energy as plotted in Fig. 2c. This is an expected feature since topological edge modes must eventually merge into the bulk bands at a point beyond which their localization length diverges17,18. In our data, we find that the edge states are maximally localized at the Fermi level and merge into the bulk at the conduction band minima (CBM) and valence band maxima (VBM) values.

Further characterization of the ML-WTe2 films shows that the bulk gap varies considerably from island to island (Fig. 3b–d). As measured by the difference between the CBM and VBM (Eg = Ecbm-Evbm), the bulk gap can vary between 0 meV and 60 meV for different islands within the same film. The gap is however quite homogeneous within any given island as shown in Supplementary Fig. 3c. This large variation in gap sizes on different islands is not easy to explain, especially since the topographies of areas exhibiting different gaps show no differences in structure or defect concentration. A survey of the literature shows similarly large variations in gap magnitudes between different samples with no consensus on the low temperature gap. A recent study suggests that large strains can lead to gap variations in this system19. However, the samples in that study were deliberately grown to induce strain which is not the case for our samples or other reports. Another possible explanation is that the gap variations arise from trapped ions in the SiC substrate creating local electric fields. Such trapped ions have been previously observed in other films grown on SiC20 and could result from the high energy RHEED electrons used to monitor film growth. So, while our measurements substantiate earlier conflicting experimental determinations of the band gap magnitude, the causes of the variations in gap magnitude remain unknown. Using the gated-STM capability we can systematically investigate the origin of these gap variations by measuring the response of ML-WTe2 to electric fields.

### Effect of gating on band structure

Our gating setup is shown in Fig. 4a where ML- WTe2 was grown via MBE on chemical-vapor-deposited (CVD) graphene transferred onto 300 nm SiO2/Si, with the Si substrate serving as the back gate electrode (see Methods and Supplementary Fig. 1 for details). Figure 4c, d shows a topographical STM image of a representative WTe2 island on graphene on SiO2. With this setup, we can apply positive and negative back gate voltages up to a maximum of 80 V. Naively, we expect that the applied electric field would simply modify the carrier concentration resulting in a rigid shift of the Fermi level relative to the bulk bands. The signature of this effect would be a rigid shift of the measured dI/dV spectrum relative to the Fermi energy. As shown in Fig. 4b, we find that gating does indeed move the Fermi energy toward the conduction/valence band consistent with electron/hole doping for positive/negative voltages respectively. However, gating has another, more dominant effect. As shown in Fig. 4b, e, f, with increasing positive gate voltage the band gap increases, while for negative gate voltages the band gap decreases. Hence, we find an inherent sensitivity of the WTe2 insulating gap to (even modest) electric fields. At face value, it appears that positive and negative voltages have opposite effects on the gap. However, we will show below that this observed behavior is because the STM signal is dominated by the density of states from the top layer of the sample. In reality, both signs of the gate voltage decrease the net band gap.

### Tight-binding model

To gain an understanding of these gating-induced band structure changes, (see methods section), we fit a tight-binding model to the DFT band structure using Wannier interpolation for our analysis. At zero gate electric field (see second panel of Fig. 5a), the CBM are doubly degenerate at $$k_ \pm \simeq \pm \left( {0.15\frac{{2\pi }}{a},0} \right).$$ At $$k_ +$$, the spin $$\uparrow$$ electron (i.e., the spin pointing in the z-direction) is localized on the top Te atom due to strong spin orbit coupling, while the spin $$\downarrow$$ electron is localized on the bottom Te atom. Consistent with time-reversal symmetry, this is reversed in the $$k_ -$$ direction, as noted on the diagram.

Next, we model the applied electric field by adding a term $$\left( {H\prime } \right)$$ to the tight-binding model, $$H = H_0 + H^\prime = H_0 + \mathop {\sum }\nolimits_i eE \cdot z_i$$, where zi represents the z-coordinate of the ith Wannier center, computed as the expectation value of the position operator on the Wannier function. Figure 5a shows the 2D bulk band structures for different applied gate electric fields, and we see that the electric field breaks the spatial inversion symmetry and spin-splits the doubly degenerate CBM at $$k_ \pm \simeq \pm \left( {0.15\frac{{2{\uppi}}}{a},0} \right)$$. The spin-splitting is consistent with preserved time-reversal, but broken inversion symmetry. For negative gate voltages the conduction band minimum in the $$k_ +$$ direction has spin $$\downarrow$$, while the minimum in the $$k_ -$$ direction has spin $$\uparrow$$. For the opposite electric field, this situation is reversed. From our model, an applied electric field always reduces the (indirect) band gap, irrespective of the sign of the gate voltage. To explain the STM gate dependence, we consider the density of states on the top layer. In Fig. 5a, the bands are colored according to the projection of the electronic states onto the top Te atoms, which we expect to be the most accessible to the STM probe. For a negative gate voltage, the states in the conduction band near $$k_ \pm$$ that are pushed downward by the spin splitting are primarily located on the top Te atoms while the states pushed upward are primarily on the bottom Te atoms. For positive gate voltage, the situation is reversed. Since we expect the STM to have a preferential coupling to the top Te atoms this scenario predicts that the STM-observed gap should decrease for negative gate and increase for positive gate, in clear alignment with the experimental results.

In Fig. 5b, c we plot the gap and the calculated $${{{\mathrm{d}}}}I/{{{\mathrm{d}}}}V$$ spectra projected on the top Te atoms, as a function of electric field. The spectra are obtained by calculating the projected density of states (PDOS) of the Wannier functions of only the top Te atoms, i.e.,$${{{\mathrm{d}}}}I/{{{\mathrm{d}}}}V|_{\it{\epsilon }} \propto \mathop {\sum }\nolimits_i \left| \langle{\phi _i{{{\mathrm{|}}}}\psi } \rangle\right|^2$$, where i runs through all the Wannier orbitals of the top Te atoms. The results of the calculations shown in Fig. 5b, c agree well with the behavior of the experimentally measured gap with electric field as shown in Fig. 4f. For the 300 nm SiO2 gate oxide, the 80 V gate voltage results in an electric field on the order of 100 mV Å1. A more accurate determination of the electric field strength is difficult because the thickness of the SiO2 can show variations and graphene can induce some degree of screening. Nevertheless, using the value of 100 mV Å1 as an estimate, we find that the change in the bulk gap with gate voltage (as measured from the top surface) to be 0.377(4) meV mV−1, which is in excellent agreement with the experimental value of 0.39(2) meV mV−1.

As an additional check, we confirm this mechanism for the decrease in gap magnitude with gate electric field by considering a minimal four-band tight-binding model constructed from a crystal symmetry analysis and fitted to first-principles calculations of the band structure21. Consistent with the DFT-derived tight-binding model, we find that the spin-splitting of the conduction band by the electric field is responsible for the observed behavior of the gap. We discuss this minimal model and present calculations of the surface gap electric field dependence in the Supplementary Information.

Our work highlights the sensitivity of the band structure of monolayer 1 T′-WTe2 to electric fields, which cause large changes in the magnitude of the bulk band gap. Electronic structure calculations show that the effects due to electric fields can be primarily attributed to broken inversion symmetry and strong spin-orbit coupling akin to the Rashba effect, rather than electron-electron interaction effects. These systematic studies resolve the puzzlingly large variations in the gap magnitudes that have been previously observed in monolayer 1 T′-WTe2 (see Supplementary Fig. 3). Indeed, we expect that the gap variations observed in the literature in the monolayer thin films may be attributed to electric fields arising from variations in the local potential induced by the substrate. This is consistent with our STM data on monolayer films grown without RHEED, which show far smaller variations in gap magnitude as shown in Supplementary Fig. 4c. Finally, it is worth highlighting our finding that the gate electric field can generate a spin-splitting of tens of meV in the 2D bulk bands. This substantial spin-splitting provides a new avenue to realizing proximity induced non-trivial superconductivity and Majorana bound states in a tunable system.

## Methods

### Film growth

Graphene-on-SiC substrates were fabricated directly on crystalline SiC in vacuum, providing a high-quality graphene surface for subsequent epitaxial growth. WTe2 films grown on graphene on SiC were transferred directly to the STM using a vacuum suitcase without exposure to air.

For gating studies, commercial graphene on SiO2 substrates was used. These substrates were prepared by transferring monolayer CVD graphene onto a p-doped Si wafer coated with a 300 nm thermal SiO2. Monolayer WTe2 was grown using a custom MBE setup. Prior to growth, Gr/SiO2/Si substrates were annealed in ultra-high vacuum (UHV) for 10 h at 450 °C. The film was grown by co-evaporation of elemental W (3 N purity) and Te (6 N purity) while the substrate was held at T = 280 °C. Te was evaporated from a Knudsen cell with a rate of 0.01 Å s−1. W was evaporated using an electron beam evaporator at a rate of 0.18 Å h−1. Monolayer film with 60% coverage was grown in 5 h, thus having the rate of approximately one monolayer per 8.5 h. Such slow growth and 200:1 Te to W flux ratio were necessary to grow high-quality monolayer films. After growth, the film was annealed in UHV at 300 °C for 1 h, cooled down to room temperature and capped with 15 nm of Te. The capped film was then transferred to the gating sample holder, which is a three-lead holder, one of which serves as the STM bias lead and the other two are connected to the heating W filament (see Supplementary Fig. 1). Once transferred to the STM system, the sample was ion-milled for 10 s using an Ar+ beam at 400 V voltage and 4 µA beam current and then annealed at 200 °C for 1 h to remove the Te cap. The samples were immediately transferred into the STM stage kept at 4 K.

### DFT details

The DFT calculations are performed using Quantum ESPRESSO 6.522,23. In order to compute the detailed band structure, we use Wannier interpolation. We derived a tight-binding model on a 56-spinor Wannier basis (W: s and d, Te: s and p) using Wannier9024,25. The tight-binding Hamiltonian is constructed and solved using the open-source code TBmodels26.