A two-dimensional topological insulator (2DTI) is guaranteed to have a helical one-dimensional edge mode1,2,3,4,5,6,7,8,9,10,11 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.
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 anticipated13,14, 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 voltage5,7,10. This is inconsistent with the predicted absence of elastic backscattering at zero magnetic field, although several possible explanations have been put forward for the discrepancy15,16,17,18,19,20. Another is that the edges show signs of conducting even at high magnetic field21,22, 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 Td structure, has recently attracted attention as a type-II Weyl semimetal24,25 that exhibits an extreme non-saturating magnetoresistance26,27 related to the closely balanced electron and hole densities28,29,30. Calculations suggest that the monolayer will likewise be a semimetal12,30, 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 EF 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 1f–h shows representative two-terminal measurements of the differential conductance Gdiff 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 Vg, the conductance rising steadily before saturating at the lowest temperatures, consistent with the behaviour26 of bulk WTe2. The bilayer (Fig. 1g) develops a strong Vg dependence with a sharp minimum near Vg = 0, while remaining metallic at large Vg. 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 Vnl induced between contacts 4 and 5 which are far out of the normal current path. At low T and small Vg, Vnl/V0 grows large, suggesting that in this regime most of the current follows the edge. At higher T or larger Vg, Vnl/V0 falls off as more current takes the direct path through the bulk.
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 Vg = 0. This confirms that around Vg = 0 most of the current flows near the edge. However, for Vg 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 Vg = 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 Vg = 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 Vg. 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 Vg 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 Gedge 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, Gedge increases, but below ∼10 K at typical Vg it decreases again as the mesoscopic oscillations grow. The inset shows the T dependence at a particular Vg where Gedge 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 Vg = 0 for a series of temperatures. At moderate B∥ and T the behaviour approximates the activated function Gedge = G0e−αB∥/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.
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 Vg. 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 Gdiff = dI/dV at V = 3 mV.
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 Vg 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 Vg, 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 Gedge 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, Gedge 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 impurities16,17,18,19 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 liquid32,33,34. Nevertheless, we do not observe the quantized conductance that would be a definitive signature of a 2DTI.
The approximately activated behaviour, e−αB∥/T, in an in-plane magnetic field suggests the opening of a gap, ΔB = gμBB∥, resulting in some form of hopping conduction. Here μB is the Bohr magneton and g = αkB/μB ≈ 7.5 is an effective g-factor. In hopping there are many possible mechanisms for magnetoresistance35,36, but some are ruled out by the atomic-scale thinness of the sample and by the relatively weak dependence on Vg. 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 EF 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 Gedge 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 EF due to interaction effects37,38, 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 Gedge tends to decrease with T when the ZBA is small (see Fig. 3a, above ∼6 K). A magnetic field opens a gap ΔB in the helical modes, as sketched in the lower part of Fig. 4e. Electrons at EF now encounter this gap at the weak points, which they can pass by activation, thus introducing the factor e−ΔB/kBT 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 insulator39,40, 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 modes3,42.
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 Vg, 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) = CtgΔVg/e. Here Ctg is the areal capacitance corresponding to the top gate, Ctg = εrε0/dhBN, where εr ≈ 4 for hBN (ref. 44) and dhBN is the thickness of the upper hBN flake.
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 Vnl/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.
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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We thank J. Yan (Oak Ridge National Laboratory) for providing the WTe2 crystals, and A. Andreev, L. Glazman, J. Maciejko, K. Matveev, J. Moore, B. Spivak, J. Väyrynen and D. Xiao for discussions. The major part of this work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, Awards DE-SC0002197 (D.H.C.) and DE-SC0012509 (X.X.). T.P. and Z.F. were supported by AFOSR FA9550-14-1-0277. P.N., J.F. and some facilities were supported by NSF EFRI 2DARE 1433496.
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Nature Communications (2018)