Strain-tuned topological phase transitions in van der Waals heterostructures

Introduction

Van der Waals assembly made it possible to combine atomically thin crystals without the lattice-matching constraints of conventional epitaxy [1,2]. For topological materials, this is especially attractive because the electronic phase can depend sensitively on thickness, stacking, symmetry, electric field, interlayer coupling, and mechanical deformation. The same weak interlayer bonding that enables dry transfer also allows strain and twist to be used as design parameters rather than as uncontrolled defects.

Topological insulators are defined by band topology rather than by a conventional order parameter [3,4]. In inversion-symmetric systems, parity eigenvalues at time-reversal-invariant momenta provide a compact way to evaluate the Z2 invariant [5]. In real materials, however, topological character has to be connected to measurable properties: a bulk gap, protected boundary states, and robustness against perturbations that preserve the relevant symmetries. Three-dimensional Bi2Se3-family compounds provided an early large-gap platform with a single surface Dirac cone [6,7], while two-dimensional transition-metal dichalcogenides opened a route to quantum spin Hall physics in layered materials [8].

Monolayer 1T'-WTe2 is a particularly useful test case. Theory predicted a quantum spin Hall phase in distorted octahedral transition-metal dichalcogenide monolayers [8], and subsequent transport and spectroscopy experiments reported edge conduction and quantum spin Hall behaviour in monolayer WTe2 [9,10,11]. The same material can also be driven into superconductivity by electrostatic gating [12], which makes it a compact platform for studying the competition between topology, carrier density, disorder, and interaction effects.

Strain is a natural control knob because topological phase transitions occur when a bulk gap closes and reopens with changed band ordering. Prior work has shown strain control of WTe2 band topology and related phase transitions in layered tellurides and Bi2Se3-family films [13,14,15,16,17]. The challenge is to make that control operational in a van der Waals device: the strain must be reversible, calibrated at the active flake, and small enough to avoid cracking contacts or changing disorder irreversibly.

Here we report a fictional but experimentally realistic study of strain-tuned topology in hBN-encapsulated 1T'-WTe2 heterostructures. The aim is not to claim a universal strain threshold for WTe2. It is to show how transport, nonlocal response, strain metrology, and topological-band calculations can be combined to distinguish a semimetal-to-topological-insulator crossover from a true topological-to-trivial transition.

Device architecture

Devices were fabricated from bulk 1T'-WTe2 crystals exfoliated inside a nitrogen glovebox with oxygen and water levels below 0.5 ppm. Monolayer and bilayer candidates were identified by optical contrast and confirmed by low-frequency Raman modes after encapsulation. Each active flake was sandwiched between 18-35 nm hBN layers and placed on a thin graphite back gate. The stacks were assembled by dry pickup to minimise polymer contact with WTe2. hBN was chosen because it provides a clean, flat, weakly charged environment for two-dimensional transport devices [18].

One-dimensional edge contacts were patterned by reactive-ion etching through the hBN/WTe2/hBN stack followed by evaporated Ti/Pd/Au. The contact geometry followed the same principle used for high-quality graphene heterostructures, where edge contact reduces current crowding and protects the encapsulated channel [19]. Devices used six- and eight-terminal Hall-bar layouts with channel lengths from 1.5 to 4.8 um and widths from 0.7 to 2.2 um.

The completed heterostructures were bonded to micromachined piezoelectric bending chips using a low-modulus epoxy patterned outside the active channel. The actuator applied uniaxial strain approximately along the WTe2 tungsten-chain direction. The angular misalignment between the actuator axis and the crystallographic chain direction was below 7 deg for the monolayer devices and below 11 deg for bilayers. Two devices were intentionally mounted at 45 deg to test anisotropy; those devices showed weaker transitions and are reported only in the supplementary data.

Three classes of devices were measured: nine monolayer WTe2 stacks, five near-zero-twist WTe2 bilayers, and four WTe2/MoTe2 heterobilayers. The heterobilayers were included to test whether an adjacent distorted dichalcogenide layer shifts the WTe2 phase boundary through hybridisation and dielectric screening. Because the MoTe2 layer introduced additional disorder in several devices, the main quantitative claims are based on monolayer and WTe2 bilayer samples.

Strain calibration

The applied actuator voltage is not the same as the strain in the WTe2 layer. Strain transfer depends on epoxy thickness, hBN thickness, flake position, substrate curvature, and thermal contraction during cooldown. We therefore used three independent calibration methods. First, arrays of 40 nm gold markers were patterned on the top hBN surface near the active channel. Marker displacement was measured optically at room temperature and by scanning electron microscopy on sacrificial chips after low-temperature cycling. Second, Raman spectra were collected as a function of actuator voltage and compared with the measured marker strain. Third, finite-element models were used to estimate the strain loss between the top hBN marker plane and the encapsulated WTe2 layer.

The final strain value assigned to each transport trace is the posterior mean from a simple calibration model combining marker displacement, Raman peak shift, and finite-element transfer. The uncertainty includes actuator hysteresis and cooldown drift. The reversible strain range was -0.72% to +1.04% for monolayers and -0.65% to +0.88% for bilayers. Above about +1.1%, two devices developed irreversible contact resistance changes and were excluded from phase-boundary fitting.

Strain inhomogeneity across the channel was small but not negligible. Finite-element simulations estimated a root-mean-square variation of 0.04% in the central 60% of the channel and up to 0.12% near contacts. This inhomogeneity broadens any sharp transition. For that reason, all quoted critical strains are defined from the midpoint of a fitted conductance or gap-closing crossover rather than from a single trace.

The strain axis matters. WTe2 has low in-plane symmetry, so strain along and across the tungsten-chain direction does not produce equivalent band shifts. Devices mounted away from the chain direction showed smaller changes in bulk activation gap and nonlocal response. This anisotropy is consistent with first-principles studies that identify strain as an efficient way to tune the semimetal-topological-insulator balance in monolayer 1T'-WTe2 [13,14].

Electronic-structure calculations

Density-functional calculations were performed for monolayer WTe2, bilayer WTe2, and WTe2/MoTe2 heterobilayers under biaxial and uniaxial strain. Spin-orbit coupling was included self-consistently. The main calculations used a semilocal functional with van der Waals correction; selected structures near phase boundaries were checked with a screened hybrid functional. Because absolute gaps in WTe2 are sensitive to exchange-correlation treatment, we use the calculations primarily to track gap closing, band inversion, and topological invariant rather than to claim a precise experimental gap.

Maximally localised Wannier functions were built from W d, Mo d, and Te p states using the standard Wannier formalism [20]. Surface and ribbon spectra were computed from the Wannier tight-binding models with iterative Green-function methods implemented in WannierTools [21]. Z2 indices were evaluated by tracking hybrid Wannier charge centres using Z2Pack [22]. For inversion-symmetric monolayer structures, parity eigenvalues were used as a cross-check [5].

The unstrained monolayer calculation gives a small inverted direct gap near Gamma and a small indirect band overlap, placing the material close to the boundary between a semimetal and a quantum spin Hall insulator. Tensile strain along the chain direction reduces the indirect overlap and opens a transport gap without changing the calculated Z2 invariant. Compressive strain has a different effect: it drives the inverted bands at Gamma together, closes the direct gap, and reopens it with normal ordering. The calculated topological-to-trivial transition occurs at -0.52% strain in the semilocal calculation and -0.66% in the hybrid-functional check.

Bilayer calculations show a narrower topological window because interlayer hybridisation splits the band-edge states. In near-zero-twist bilayers, the topological-to-trivial transition shifts to larger compressive strain. In aligned bilayers, edge-state spectral weight is reduced and a small hybridisation gap appears in the ribbon calculation. This explains why bilayer transport signatures are expected to be less quantised and more contact dependent than monolayer signatures.

Transport protocol

Transport measurements were performed in a dilution refrigerator and in a variable-temperature insert from 1.6 to 120 K. Unless otherwise stated, local and nonlocal conductance were measured with 3-7 nA low-frequency AC excitation. Gate voltage was swept slowly enough to avoid charge hysteresis in hBN. Actuator voltage was stepped at fixed temperature, then returned to zero after each strain sequence to check reversibility.

The primary local observable is the four-terminal longitudinal conductance at the gate voltage where the bulk channel is least conducting. The primary nonlocal observable is the voltage response measured several micrometres from the current path. In an ideal quantum spin Hall device, nonlocal transport reflects helical edge channels; in real WTe2 devices it can also include contact leakage, puddles, residual bulk conduction, and inhomogeneous strain. We therefore use nonlocal response as one part of the phase assignment, not as a standalone proof.

For each device, we classify strain regimes using four criteria: the sign and magnitude of the bulk activation gap from temperature sweeps, the local conductance minimum, the nonlocal-to-local resistance ratio, and the calculated topological invariant for the corresponding strain. A regime is labelled topological insulating only when the bulk channel is activated and nonlocal response remains strong. A regime is labelled trivial insulating when the bulk channel is activated but nonlocal response falls to the contact background and the calculation gives normal band ordering.

Contact resistance was monitored with redundant voltage probes. Devices whose contact resistance changed by more than 20% after a strain cycle were excluded from the critical-strain fit but retained in the reproducibility table. This is conservative, but necessary because contact changes can mimic the loss of edge conduction.

Monolayer phase diagram

At zero applied strain, six of nine monolayer devices showed semimetallic temperature dependence near charge neutrality: the conductance minimum decreased from 20 to about 20 K and then saturated at lower temperature. The nonlocal signal was present but weak, consistent with residual bulk conduction shunting the edge channel. This agrees with the view that monolayer WTe2 is close to the boundary between a semimetal and a quantum spin Hall insulator [9,10,11,13].

Tensile strain along the tungsten-chain direction opened a clearer insulating gap. At +0.62% strain, the median activated gap extracted between 12 and 45 K was 7.8 meV, compared with 2.1 meV at zero strain. The nonlocal-to-local resistance ratio increased by a factor of 3.4, and the response remained detectable up to 74 K in the best device. The values are not perfectly quantised, but their geometry dependence is consistent with edge-dominated conduction rather than with homogeneous bulk leakage.

The tensile regime should be described carefully. The calculation indicates that the Z2 index remains non-trivial while the indirect band overlap is removed. This is therefore not a topological phase transition in the strict sense. It is a semimetal-to-topological-insulator crossover: strain makes the existing inverted band structure experimentally visible as a better-insulated quantum spin Hall channel.

Compressive strain produced the actual topological transition. As strain became more compressive, the activated gap first decreased, then closed within experimental resolution near -0.58 +/- 0.08%, and then reopened. Beyond -0.66%, local conductance again became insulating, but the nonlocal response collapsed to the contact background. The hybrid-Wannier calculation assigned this reopened gap to a trivial Z2 phase. The coincidence of gap closure, edge-response loss, and calculated band-order reversal is the strongest evidence for strain-driven topology switching in the device set.

The phase boundary was reversible over at least 18 strain cycles in four devices. The critical conductance traces showed modest hysteresis, approximately 0.05% strain, consistent with actuator hysteresis rather than structural phase switching in the WTe2. Raman spectra after cycling did not show new peaks associated with oxidation or structural degradation.

Bilayers and heterobilayers

Near-zero-twist WTe2 bilayers behaved differently from monolayers. At zero strain, they showed lower nonlocal response and weaker gate-tunable insulating behaviour. Tensile strain still increased the conductance minimum and produced a partial activation gap, but the nonlocal response remained smaller than in monolayers. The inferred topological-to-trivial transition under compression occurred at -0.91 +/- 0.10%, significantly more negative than the monolayer value.

This shift is consistent with interlayer hybridisation. In the calculation, bilayer coupling modifies the band-edge states at Gamma and increases the strain required to reverse their ordering. The edge spectral function shows reduced weight at the outermost layer because the two monolayer-like edge modes hybridise. That hybridisation does not eliminate all boundary conduction in weakly coupled bilayers, but it makes the transport signature more fragile.

WTe2/MoTe2 heterobilayers were less reproducible. Two devices showed a monotonic strain-dependent gap but weak nonlocal response at all strains. One device displayed a transition-like collapse of nonlocal signal near -0.8%, but the contact resistance also changed during the same sweep. We therefore treat the heterobilayer results as suggestive only. The calculations indicate that MoTe2 proximity can shift the WTe2 band edge by tens of meV, but small twist-angle and local-relaxation differences produce large device-to-device variation.

The bilayer data are still useful because they show that "van der Waals heterostructure" is not a single platform. Monolayer WTe2 in hBN is close to an ideal strain-tuned quantum spin Hall system. Adding another active layer introduces hybridisation, screening, moire relaxation, and disorder. These effects can be useful knobs, but they also blur the clean monolayer phase diagram.

Strain-defined interfaces

Two devices were patterned so that the active WTe2 channel crossed a strain gradient generated by an asymmetric actuator beam. At a fixed global actuator voltage, one half of the channel was calculated to be in the topological insulating regime while the other half was in the trivial insulating regime. The interface position could be shifted by changing actuator voltage.

Transport across this strain-defined boundary showed a conductance enhancement relative to uniformly trivial settings. In the best device, the two-terminal conductance along the calculated interface was 0.41 e2/h at 1.8 K after subtracting a contact background. The value is far from ideal and cannot be claimed as quantised edge transport. However, it is consistent with the existence of a one-dimensional conducting channel at a topological domain wall.

The interface experiment is important conceptually because it avoids etching a physical edge. A strain boundary should host a mode if the Z2 invariant changes across it, while many trivial edge defects are absent. The practical limitation is that the boundary is broad: finite-element modelling gives a transition width of 130-180 nm, comparable to or larger than the expected edge-state localisation length. Sharper strain gradients would be needed for cleaner domain-wall devices.

This strain-interface idea may be more useful in future suspended or membrane-based heterostructures. In the current bonded devices, the same epoxy layer that enables reversible strain also smooths strain gradients. There is therefore a trade-off between device robustness and spatial control.

Discussion

The main result is that small, reversible strain can move monolayer 1T'-WTe2 through two distinct regimes. Tensile strain removes semimetallic bulk overlap and strengthens edge-dominated transport while leaving the band inversion intact. Compressive strain closes and reopens the gap with changed band ordering, producing a true topological-to-trivial transition. Combining both observations in the same devices helps explain why WTe2 reports can appear inconsistent: a sample can be topologically inverted yet poor as a transport insulator if residual bulk overlap is present.

The distinction between a topological transition and a transport crossover is not semantic. A semimetal-to-topological-insulator crossover should improve nonlocal response without changing the Z2 invariant. A topological-to-trivial transition should require a bulk gap closing and should remove protected edge response after reopening. The data satisfy this pattern within experimental uncertainty, but only because strain calibration and modelling are included. Edge conductance alone would not be enough.

The results sit naturally beside earlier strain-tuning work in Bi2Se3 and ZrTe5-family materials [15,16,17]. The difference is that the van der Waals device geometry allows reversible strain to be combined with electrostatic gating and local transport. This makes it possible to move a single mesoscopic device across a phase boundary rather than comparing separate strained crystals.

The article also points to a design rule for topological van der Waals devices: active-layer simplicity matters. Encapsulation and graphite gates improve disorder, but adding another electronic layer changes hybridisation and edge-state visibility. Twisted bilayer WTe2 remains interesting because interlayer coupling can be tuned by twist and stacking [14], yet for robust strain switching the monolayer device is cleaner.

Limitations

The first limitation is strain metrology. Although three calibration methods were used, the active strain inside the WTe2 layer is still inferred rather than measured directly. Local strain near contacts may differ from the channel average, and small wrinkles or bubbles in hBN can create local strain pockets. The quoted critical strains should therefore be read as device-level effective values.

The second limitation is disorder. WTe2 devices are air sensitive, contact dependent, and prone to charge puddles near neutrality. Nonlocal response can be enhanced by edge conduction, but it can also be affected by inhomogeneous bulk conduction and current jetting. The multi-terminal geometry reduces this ambiguity but does not eliminate it.

The third limitation is theoretical. Density-functional calculations give a clear band-ordering trend, yet the absolute gap is functional dependent and sensitive to lattice constants. We therefore rely on gap closure and topological invariant changes rather than on a single calculated gap value. Many-body effects, excitonic physics, and strain-dependent dielectric screening are not fully captured.

Finally, the study does not demonstrate spin-resolved edge transport. A quantum spin Hall interpretation is supported by existing WTe2 literature and by the observed nonlocal response, but direct spin-momentum locking is outside the measurement set. Future work should combine strain tuning with microwave edge spectroscopy, local scanning probes, or superconducting contacts to test edge-mode character more directly.

Conclusion

hBN-encapsulated monolayer 1T'-WTe2 van der Waals heterostructures can be reversibly tuned by sub-percent uniaxial strain. Tensile strain converts a weakly semimetallic inverted system into a clearer topological insulator by suppressing bulk overlap. Compressive strain drives a gap-closing and reopening transition to a trivial insulating state, accompanied by loss of nonlocal edge response. Bilayers and heterobilayers shift and broaden the phase boundary because interlayer coupling changes the band-edge states.

The work supports strain engineering as a practical knob for topological van der Waals electronics. Its broader lesson is methodological: phase diagrams should combine calibrated strain, bulk transport, nonlocal geometry, and topological-band calculations. Without that combination, a strain-induced change in resistance can easily be mistaken for a change in topology.

Data and code availability

Raw transport traces, strain-calibration files, Raman spectra, finite-element meshes, DFT input decks, Wannier Hamiltonians, hybrid-Wannier evolution data, and plotting scripts are included in the supplementary archive. Device micrographs are redacted only where laboratory identification marks appear. Calculations were performed with spin-orbit coupling enabled, and the archived notebooks reproduce the reported Z2 indices from the supplied tight-binding models.

References

1. Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419-425 (2013).
2. Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).
3. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045-3067 (2010).
4. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057-1110 (2011).
5. Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
6. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438-442 (2009).
7. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398-402 (2009).
8. Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in two-dimensional transition metal dichalcogenides. Science 346, 1344-1347 (2014).
9. Fei, Z. et al. Edge conduction in monolayer WTe2. Nat. Phys. 13, 677-682 (2017).
10. Tang, S. et al. Quantum spin Hall state in monolayer 1T'-WTe2. Nat. Phys. 13, 683-687 (2017).
11. Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76-79 (2018).
12. Sajadi, E. et al. Gate-induced superconductivity in a monolayer topological insulator. Science 362, 922-925 (2018).
13. Zhao, C. et al. Strain tunable semimetal-topological-insulator transition in monolayer 1T'-WTe2. Phys. Rev. Lett. 125, 046801 (2020).
14. Lupke, F. et al. Quantum spin Hall edge states and interlayer coupling in twisted bilayer WTe2. Nano Lett. 22, 5674-5680 (2022).
15. Mutch, J. et al. Evidence for a strain-tuned topological phase transition in ZrTe5. Sci. Adv. 5, eaav9771 (2019).
16. Liu, Y. et al. Tuning Dirac states by strain in the topological insulator Bi2Se3. Nat. Phys. 10, 294-299 (2014).
17. Flototto, D. et al. In situ strain tuning of the Dirac surface states in Bi2Se3 films. Nano Lett. 18, 5628-5632 (2018).
18. Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nat. Nanotechnol. 5, 722-726 (2010).
19. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614-617 (2013).
20. Marzari, N. et al. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419-1475 (2012).
21. Wu, Q., Zhang, S., Song, H.-F., Troyer, M. & Soluyanov, A. A. WannierTools: an open-source software package for novel topological materials. Comput. Phys. Commun. 224, 405-416 (2018).
22. Gresch, D. et al. Z2Pack: numerical implementation of hybrid Wannier centers for identifying topological materials. Phys. Rev. B 95, 075146 (2017).