Edge conduction in monolayer WTe₂

Abstract

A two-dimensional topological insulator (2DTI) is guaranteed to have a helical one-dimensional edge mode^{1,2,3,4,5,6,7,8,9,10,11} in which spin is locked to momentum, producing the quantum spin Hall effect and prohibiting elastic backscattering at zero magnetic field. No monolayer material has yet been shown to be a 2DTI, but recently the Weyl semimetal WTe₂ was predicted¹² to become a 2DTI in monolayer form if a bulk gap opens. Here, we report that, at temperatures below about 100 K, monolayer WTe₂ does become insulating in its interior, while the edges still conduct. The edge conduction is strongly suppressed by an in-plane magnetic field and is independent of gate voltage, save for mesoscopic fluctuations that grow on cooling due to a zero-bias anomaly, which reduces the linear-response conductance. Bilayer WTe₂ also becomes insulating at low temperatures but does not show edge conduction. Many of these observations are consistent with monolayer WTe₂ being a 2DTI. However, the low-temperature edge conductance, for contacts spacings down to 150 nm, never reaches values higher than ∼20 μS, about half the predicted value of e²/h, suggesting significant elastic scattering in the edge.

Main

Experimental work on 2DTIs to date has focused on quantum wells in Hg/CdTe (refs 4,5,6,7) and InAs/GaSb (refs 9,10,11) designed to achieve an inverted bandgap. These heterostructures show edge conduction as anticipated^13,14, but they also present some puzzles. One is that the conductance at low temperatures is not perfectly quantized, becoming small in long edges¹³ and showing mesoscopic fluctuations as a function of gate voltage^5,7,10. This is inconsistent with the predicted absence of elastic backscattering at zero magnetic field, although several possible explanations have been put forward for the discrepancy^{15,16,17,18,19,20}. Another is that the edges show signs of conducting even at high magnetic field^21,22, contrary to expectations that helical modes, protected by time-reversal (TR) symmetry at zero field, should Anderson-localize once this symmetry is broken. An additional complication is that non-helical edge conduction may also be present, due for instance to band bending when a gate voltage is applied²³.

Identification of a natural monolayer 2DTI would be helpful for elucidating and exploiting TI physics. Band structure calculations predict that certain monolayer materials are topologically nontrivial¹². An example is monolayer WTe₂, which has the T′ structure illustrated in Fig. 1a. Three-dimensional WTe₂, in which T′ monolayers are stacked in the orthorhombic T_d structure, has recently attracted attention as a type-II Weyl semimetal^24,25 that exhibits an extreme non-saturating magnetoresistance^26,27 related to the closely balanced electron and hole densities^28,29,30. Calculations suggest that the monolayer will likewise be a semimetal^12,30, its Fermi surface comprising two electron pockets (green) and one hole pocket (grey) as shown in Fig. 1b, with areal densities n = p ≈ 1.6 × 10¹³ cm⁻². If this is correct then the helical edge modes are always degenerate with bulk states (Fig. 1c). In contrast, in a 2DTI the edge modes span a bandgap and cannot be mixed by TR-invariant perturbations, so that they dominate transport when the Fermi energy E_F is in the gap. Here we present evidence that at low temperatures monolayer WTe₂ exhibits an insulating bulk state and edge conduction, and describe the properties of the edge conduction, including its dependence on gate voltage, magnetic field, temperature, contact separation, and bias. We then compare the behaviour with that expected for helical modes with disorder expected to be present at the monolayer edge.

Figure 1: Two-terminal characteristics of WTe₂ devices.

a, Structure and lattice constants of monolayer WTe₂. Tungsten atoms form zigzag chains along the a-axis. b, Sketch of its calculated Fermi surface, showing two electron (green) and one hole (grey) pockets. c, Sketch of calculated bands in a strip of monolayer WTe₂. The spin-resolved helical edge modes (red and blue lines) are predicted to be degenerate with bulk states (grey and green) at all energies. d, Optical image of monolayer WTe₂ device MW1. Scale bar, 5 μm. e, Schematic two-terminal measurement configuration, indicating also the voltage applied to the few-layer graphene top gate (grey). The pink region is the monolayer WTe₂. f–h, Temperature dependence of the characteristics for similar contact pairs on a trilayer (L = 0.20 μm, W = 3.4 μm), bilayer (L = 0.26 μm, W = 3.1 μm), and monolayer (L = 0.24 μm, W = 3.3 μm) device (MW1), respectively. Here the differential conductance G_diff is measured with a small (3 mV) d.c. bias to suppress effects of a zero-bias anomaly in the 1.6 K sweep for the monolayer. The inset to f compares the temperature dependence of the conductance minimum in the three cases.

Figure 1f–h shows representative two-terminal measurements of the differential conductance G_diff of encapsulated trilayer, bilayer, and monolayer devices, respectively. Each of these devices has a row of contacts along one edge of the WTe₂ sheet, as visible in the optical micrograph of monolayer device MW1 in Fig. 1d. For these particular measurements, two contacts and the gate were connected as shown in Fig. 1e and a small (3 mV) d.c. bias was superposed on the 100 μV a.c. excitation. This d.c. bias affects only the lowest temperature measurement (1.6 K), by suppressing a zero-bias anomaly (ZBA) of uncertain origin, as will be explained later.

On cooling from 300 K the trilayer (Fig. 1f) shows metallic behaviour at all V_g, the conductance rising steadily before saturating at the lowest temperatures, consistent with the behaviour²⁶ of bulk WTe₂. The bilayer (Fig. 1g) develops a strong V_g dependence with a sharp minimum near V_g = 0, while remaining metallic at large V_g. The minimum drops steadily and below ∼20 K it broadens and reproducible mesoscopic fluctuations appear. The monolayer (Fig. 1h) first develops a similar but wider minimum, but below ∼100 K the minimum stops dropping and instead broadens into a plateau of conductance, here at ∼16 μS, on which there are mesoscopic fluctuations. The inset to Fig. 1f compares the temperature dependence in the three cases. We will show below that the plateau seen only in the monolayer is due to edge conduction remaining when the bulk becomes insulating below ∼100 K.

Edge conduction is normally detected using nonlocal measurements⁶ such as those shown in Fig. 2b. Here we apply a small excitation V₀ between contacts 2 and 6 on opposite edges of monolayer device MW2 (Fig. 2a, left image) that has approximate Hall-bar geometry, and we detect the nonlocal voltage V_nl induced between contacts 4 and 5 which are far out of the normal current path. At low T and small V_g, V_nl/V₀ grows large, suggesting that in this regime most of the current follows the edge. At higher T or larger V_g, V_nl/V₀ falls off as more current takes the direct path through the bulk.

Figure 2: Distinguishing edge and bulk conduction.

a, Optical images of the electrodes for monolayer devices MW2 (left) with a Hall-bar layout and MW3 (right) designed to distinguish edge and bulk contributions. Red dashed lines show positions of the monolayer sheets transferred onto them. Scale bars are 5 μm. Inset: AFM image of the electrodes. b, Nonlocal measurements on device MW2 in the configuration shown in the inset (V₀ = 100 μV at 11.3 Hz). c–e, Measurements on MW3 (the contact separation along the edge is 0.22 μm; the pincer spacing is 0.28 μm). c, As indicated by the insets, the black trace is the two-terminal conductance between the outer contacts and the blue trace is I/V measured with the centre contact grounded (T = 10 K). In the insets, red and blue arrows indicate edge and bulk current paths, respectively. d, Same measurement as for the blue trace in c but at 1.6 K, on a logarithmic scale, showing the very weak effect of a perpendicular magnetic field of 10 T. e, Effect of in-plane magnetic field B_∥ = 14 T on I/V between adjacent contacts (T = 10 K). The red trace is the magnitude of the decrease.

Although this measurement indicates that the current follows the edge, the shape of the WTe₂ flake in device MW2 is unsuitable for quantitative separation of bulk and edge contributions. To address this, we designed monolayer device MW3 (Fig. 2a, right image) which employs a series of alternating pincer-shaped contacts overlapping one straight edge of a monolayer flake, as shown schematically in the insets to Fig. 2c. The two-terminal linear conductance between a pair of pincers (black trace in Fig. 2c), here at 10 K, behaves similarly to a pair of adjacent contacts in device MW1 (Fig. 1h). However, if the smaller rectangular contact interposed between them is grounded (blue trace in Fig. 2c), so that any current flowing near the edge is shorted out, I/V is suppressed nearly to zero around V_g = 0. This confirms that around V_g = 0 most of the current flows near the edge. However, for V_g larger than about ±2 V, conduction does occur through the two-dimensional bulk, directly across the gap between the pincers. Figure 2d shows measurements of the same quantity at 1.6 K, on a logarithmic scale, both with (green trace) and without (blue trace) a perpendicular magnetic field B_⊥ = 10 T. Near V_g = 0 at this temperature the bulk conductance is unmeasurably small. Below ∼100 K it is approximately activated, while above it rises roughly linearly with T (see Supplementary Information 5). The effect of the perpendicular magnetic field is small; the same is true for an in-plane magnetic field.

In Fig. 2e the black trace is a measurement at zero magnetic field between two adjacent contacts, using the configuration shown in the inset where the rightmost contact (not shown) is grounded to eliminate current along paths not directly between the adjacent contacts. The blue trace is the same measurement done with an in-plane field B_∥ of 14 T. Since the bulk conductivity is almost immune to magnetic field, the decrease in I/V must be associated with the edge. Near V_g = 0, where the bulk is insulating, I/V drops nearly to zero, implying that the edge conduction is strongly suppressed by the magnetic field. In addition, the magnitude of the drop, plotted in red, is similar at all V_g. This implies that the edge makes a roughly constant contribution to the conductance, independent of gate voltage and bulk conductivity.

The edge conduction can be isolated by working at T and V_g low enough that bulk conduction is negligible, corresponding for example to the plateau region in Fig. 1h. Then the section of edge between each pair of adjacent contacts behaves as an independent two-terminal conductor. This is demonstrated by the effect of grounding the central contact shown in Fig. 2c, and also by the fact that 2- and 4-terminal measurements on an edge in this regime give identical results (Supplementary Information 3). Figure 3 shows the effects of magnetic field and temperature on the linear-response conductance between two adjacent contacts in device MW2, which we call G_edge to emphasize that there is negligible bulk contribution. Figure 3a shows the T dependence at zero field. On cooling from 50 K to 10 K, G_edge increases, but below ∼10 K at typical V_g it decreases again as the mesoscopic oscillations grow. The inset shows the T dependence at a particular V_g where G_edge stays level below 10 K. Figure 3b shows the effect of B_∥, oriented as shown in the inset to Fig. 3c, at 1.6 K. Figure 3c shows B_∥ sweeps at V_g = 0 for a series of temperatures. At moderate B_∥ and T the behaviour approximates the activated function $G_{\text{edge}}=G_0{\text{e}}^{-\alpha B_{\left|\right|}/T}$, where G₀ = 17 μS and α = 5, plotted as the red dashed lines. The effect of perpendicular field (B_⊥) is similar but weaker, as illustrated in the lower inset.

Figure 3: Temperature and magnetic field dependence of the edge conductance.

a, Temperature dependence of the conductance G_edge between adjacent contacts versus gate voltage at zero magnetic field. (Device MW2, L = 1.3 μm, V = 100 μV a.c.) Inset: T dependence at V_g = +0.08 V. b, Effect of in-plane magnetic field B_∥ at 1.6 K. c, Sweeps of B_∥ at V_g = 0 for various temperatures. Dashed lines plot G₀ exp(−αB/T) with G₀ = 17 μS, α = 5. Upper inset: orientation of B_∥ relative to the edge. Lower inset: Comparing perpendicular and in-plane fields at 1.6 K (V_g = 0).

The edge conduction is often highly nonlinear at small biases. Figure 4a shows a typical I–V curve at B_∥ = 0 (black) and at 10 T (blue). Figure 4b shows the corresponding differential conductance. At B_∥ = 0 there is a sharp dip in dI/dV at V = 0, or zero-bias anomaly³¹ (ZBA), whereas at 10 T there is a sharp threshold for current flow. All edges show some dip at 1.6 K, but its size varies between different edges and as a function of V_g. It always deepens as T decreases, as illustrated in Fig. 4c. The mesoscopic fluctuations that grow on cooling are linked to the ZBA. When a small d.c. bias is applied to suppress the anomaly, the fluctuations are also suppressed and we see a flatter edge conduction plateau, which is more representative of the generic behaviour of all devices factoring out the ZBA. This is why in Fig. 1 we plotted G_diff = dI/dV at V = 3 mV.

Figure 4: Statistics, nonlinear properties, and simple picture of edge conduction.

a, I–V curves at B_∥ = 0 (black) and 10 T (blue) for device MW1, L = 1.5 μm (V_g = 0, T = 1.6 K). b, Corresponding differential conductance. A zero-bias anomaly (ZBA) is seen at B_∥ = 0. c, Temperature dependence of the ZBA at V_g = −0.68 V. d, Gate-averaged linear-response edge conductance for 19 adjacent-contact pairs (blue, device MW1; black, MW2; red, MW3; green, MW4) at 10 K (solid circles) and 1.6 K (open circles). The error bars in each case show the full range of values spanned by the mesoscopic fluctuations. e, Cartoon picture of combined effects of smooth static disorder and a magnetic field-induced gap along the edge.

In Fig. 4d we compile measurements for 19 adjacent-contact pairs in four different monolayer devices, colour coded by device, at zero magnetic field. The edge length L, which ranges from 0.16 to 5.5 μm, was estimated from atomic force microscope (AFM) images. For each edge we show the linear conductance, averaged over a window of V_g in which the bulk contribution is negligible, at 10 K (solid circles) and 1.6 K (open circles). At both temperatures the average conductance tends to decrease with L, but the trend is rather weak compared with the large, seemingly random variations. The edges with the weakest T dependence in this range also have the highest conductance, ∼20 μS.

We now discuss the compatibility of the above observations with the scenario of a helical edge mode, in comparison with a trivial edge mode or carrier accumulation due to band bending. First, the monolayer edge conductance is roughly independent of V_g, and therefore chemical potential, over the entire accessible range (Fig. 2e). This is consistent with a single gapless mode, and not with carrier accumulation due to band bending or a trivial edge mode. Second, we see no edge conduction in bilayers (Fig. 1g). This can be explained by the fact that TR symmetry does not prohibit backscattering at the bilayer edge if the electron changes layer (the pair of coupled edges is not helical), whereas band-bending effects should be similar to those in a monolayer. Third, the conductance is dramatically suppressed by B_∥ (Fig. 3c), consistent with the expectation that elastic backscattering is allowed once TR symmetry is broken.

It is also in keeping with a single mode that the linear edge conductance G_edge never exceeds the quantized value of e²/h = 38.7 μS expected in the low-temperature limit when elastic backscattering is completely prohibited (Fig. 4d). It is also encouraging that conductance of this order can occur for micron-scale edges in spite of potentially strong disorder at the torn edge of the exfoliated monolayer. On the other hand, even when the T dependence is small and for the shortest edges, G_edge reaches only about half e²/h at a peak. One possible factor is imperfect transmission between the metal contacts and the edge. (Spin relaxation in the metal contacts when current is exchanged with a helical edge might play a role.) Another is backscattering from multiple magnetic impurities^16,17,18,19 or puddles in the disorder potential²⁰, suggested to explain deviations from quantization observed in the quantum well systems. If some form of backscattering is allowed at points in a quantum wire it is natural for a ZBA to develop due to interaction effects, such as occurs in a helical Luttinger liquid^32,33,34. Nevertheless, we do not observe the quantized conductance that would be a definitive signature of a 2DTI.

The approximately activated behaviour, ${\text{e}}^{-\alpha B_{\left|\right|}/T}$, in an in-plane magnetic field suggests the opening of a gap, Δ_B = gμ_BB_∥, resulting in some form of hopping conduction. Here μ_B is the Bohr magneton and g = αk_B/μ_B ≈ 7.5 is an effective g-factor. In hopping there are many possible mechanisms for magnetoresistance^35,36, but some are ruled out by the atomic-scale thinness of the sample and by the relatively weak dependence on V_g. The following simple picture captures most of the observed behaviour.

We suppose that there is indeed a single helical edge mode that follows the physical edge of the monolayer and effectively experiences a large but smooth disorder potential, for example due to fixed charges. As a result, the energy at which the left- and right-going branches are degenerate fluctuates up and down along the edge, passing through E_F at multiple points. This situation is sketched in Fig. 4e. If some inter-branch scattering is possible in spite of the helical protection, it is likely to be strongest at these ‘weak points’ where no momentum transfer is required. At B_∥ = 0 in some edges the average linear conductance G_edge is not much less than e²/h, and so the scattering must be weak, yet we see large, rapid mesoscopic fluctuations (Fig. 3a). If the origin of these fluctuations is quantum interference, then since no Feynman paths enclose magnetic flux in a one-dimensional wire we expect no corresponding fluctuations as a function of B_⊥, as is the case (see Fig. 3c inset). As T decreases, the scattering from the weak points strengthens at energies near E_F due to interaction effects^37,38, producing the ZBA. Edges longer than a few hundred nanometres (see Fig. 4d) are not phase-coherent and so have smaller conductance due to classical addition of resistance. Also, as T rises the coherence length will decrease, consistent with the fact that G_edge tends to decrease with T when the ZBA is small (see Fig. 3a, above ∼6 K). A magnetic field opens a gap Δ_B in the helical modes, as sketched in the lower part of Fig. 4e. Electrons at E_F now encounter this gap at the weak points, which they can pass by activation, thus introducing the factor ${\text{e}}^{-{\alpha }_{\text{B}}/k_{\text{B}}T}$ in the conductance.

We note that the bulk insulating behaviour seen in monolayers below ∼100 K (see Fig. 2d and Supplementary Information 5) could involve electron–hole correlations, as for example in an excitonic insulator^39,40, but it is hard to study quantitatively by standard transport techniques because of the edge conduction. If monolayer WTe₂ is indeed a 2DTI, with helical edge conduction at temperatures as high as 100 K, it will afford new opportunities in the realms of topological and low-dimensional science. On the one hand, unlike the electrostatic confinement in quantum wells the edge of an exfoliated monolayer is abrupt, and its orientation, roughness, and chemical details are important, especially for mesoscopic effects. Control of these factors may be possible by passivation or epitaxial growth⁴¹. On the other hand, the band structure can be tuned by chemical substitution or applying strain, and the electronic properties can be probed by surface techniques such as scanning tunnelling spectroscopy. As a monolayer it can also be combined with layered magnets, semiconductors, and superconductors; for example, to manipulate spin polarization or to create Majorana modes^3,42.

Methods

Device fabrication.

hBN crystals were mechanically exfoliated under ambient conditions onto substrates consisting of 285 nm thermal SiO₂ on highly p-doped silicon. 14–30-nm-thick hBN flakes were used for the lower dielectric and 5–12-nm-thick flakes for the upper dielectric (see Supplementary Table 1 in Supplementary Information 1 for details). Pt or Pd metal contacts (no substantial difference was found between these two metals) were deposited at ∼5 nm thickness on the lower hBN by standard e-beam lithography and metallized in an e-beam evaporator followed by acetone lift-off. The upper hBN was picked up using a polymer-based dry transfer technique⁴³ and then moved into a glove box with oxygen level below 0.5 ppm along with the lower hBN/contact structure. Flux-grown WTe₂ crystals²⁷ were exfoliated inside the glove box and a monolayer flake was optically identified and quickly picked up with the upper hBN before transferring onto the lower hBN/contacts to complete the stack. Thus the WTe₂ was fully encapsulated before removing from the glove box. After dissolving the polymer, a few-layer (3–5 nm thick) graphene flake was transferred onto the BN/WTe₂/BN stack as a top gate (except for device MW2, in which the top gate was transferred after the last metallization process). Finally, another step of e-beam lithography and metallization (Au/V) was used to define wire-bonding pads connecting to the metal contacts and the top gate.

In all the measurements presented, we only biased the top gate to V_g, while the substrate gate was grounded. Assuming the electron/hole density of states is not too small, the change in electron–hole density imbalance simply depends on the capacitance, Δ(n − p) = C_tgΔV_g/e. Here C_tg is the areal capacitance corresponding to the top gate, C_tg = ε_rε₀/d_hBN, where ε_r ≈ 4 for hBN (ref. 44) and d_hBN is the thickness of the upper hBN flake.

Conductance measurements.

The conductance measurements shown in Figs 2 and 3 and in Supplementary Information 8 were made with a 100 μV a.c. excitation at 11.3 Hz and no d.c. bias, that is, in linear response. In MW1 and MW3, at low temperature and small gate voltages, conduction was found to be entirely along the shorter edge between the contacts. Conduction along the other edge, passing around the entire flake, was negligible, as determined by grounding a third contact and seeing that it had no measurable effect. In MW2, detailed studies showed that the reason V_nl/V₀ approaches unity at low T (Fig. 2b) is that in this particular device the conductance of edge 4–5 is suppressed more than that of the others by a ZBA, so that contact 4 becomes effectively connected only to contact 2, and 5 to 6, at low temperatures.

By analysing the variation with contact spacing in device MW3 above 150 K or at large gate voltages we were able to extract an approximate resistivity per square of the two-dimensional bulk, and a contact resistance to the bulk of ∼2 kΩ (see Supplementary Information 4). However, at low temperatures the contact resistance becomes too high for this procedure. Such difficulties may be overcome in the future by doping the contacts or by finding a way to make a Corbino geometry.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.