Quantum oscillations from networked topological interfaces in a Weyl semimetal

Orthorhombic Td–MoTe2

The first-order structural transition separating the T’ and Td phases in MoTe2 has a distinct pressure dependence (Fig. 1a). At ambient pressure, the inversion-symmetric T’ phase is stable at room temperature, only transforming into the noncentrosymmetric Td phase when cooled below roughly 250 K3,12. Neutron diffraction allows the determination of the relative volume fraction of these phases under different conditions12. As pressure increases, the transition temperature decreases. At pressures higher than 0.8 GPa, a completely different phenomenon emerges, where a roughly balanced mixture of the T’ and Td phases stabilizes over an appreciable temperature range, and crucially, extends to the lowest measured temperatures. The existence of this frozen mixed-phase region is stabilized by the lack of sufficient entropy at these suppressed temperatures for atoms to move to their lowest-energy configuration, implying that there is a dominant extrinsic transformation energy barrier between two energetically nearly degenerate structures12.

Fig. 1: Pressure–temperature phase diagram of MoTe2.

a Pressure–temperature phase diagram of MoTe2. The green and blue symbols delineate the extent of 100% volume fraction of T’ and Td phases, respectively, determined through neutron scattering. The onset of full volume superconductivity coincides with the end of the Td phase at 0.8 GPa 12, where a topological interface network (TIN) is observed. b Pressure dependence of the quantum oscillation frequency, with different branches labeled by Greek letters. The numbers correspond to the effective mass, which changes slightly with pressure. The quantum oscillations with the strongest pressure dependence, α and β in the Td phase, and η in the T’ phase, correspond to extremal orbits on the large electron pockets. In the TIN region, these disappear and are replaced by a completely distinct set of oscillations arising from topological interface states. Representative quantum oscillations (c) at ambient pressure (Weyl semimetal in Td), d 0.9 GPa (TIN), and e 1.8 GPa (higher-order topology in T’). Clear changes in the quantum oscillations reflect significant changes in the electronic structure.

The basic components underlying the Weyl semimetallic state of the low-pressure, low-temperature Td phase are a large hole pocket centered on the Brillouin zone and two neighboring electron pockets along the Γ –X direction8,9,10,13. The hole pocket is observed in angle-resolved photoemission spectroscopy (ARPES)8,9,10, but is not apparent in SdH measurements14. Prominent quantum oscillations observed in the Td phase arise from orbits associated with the electron pocket14. Figure 2a, b shows magnetoresistance and SdH oscillations at ambient pressure, in which these are clearly seen. As the fast Fourier transform (FFT) explicitly shows (Fig. 2c), the beating seen in Fig. 2b is due to two similar frequencies, Fα = 240.5 T and Fβ = 258 T, the result of symmetry-allowed spin–dorbit splitting. First-principles calculations identify these frequencies with the larger extremal kz = 0 cross sections of the electron pocket.

Fig. 2: Quantum oscillations in the magnetoresistance in MoTe2 at ambient pressure.

a The longitudinal MR of the bulk Td-MoTe2 measured at ambient pressure with a magnetic field parallel to the c axis. b The corresponding SdH oscillations after subtraction of a second-order polynomial background. c The fast Fourier transform (FFT) spectra show three Fermi surfaces with oscillation frequencies at Fγ = 32.5 T, Fα = 240.5 T, and Fβ = 258 T. The inset shows the α and β orbits on the calculated electron pocket. d Fit to the SdH oscillation at 1.8 K yield nontrivial Berry’s phases ϕγ = π, ϕα = 0.88π, and ϕβ = 0.88π. Details of the fit are discussed in the Supplementary Information. e The effective masses of the carriers are obtained through temperature-dependent Lifshitz–Kosevich fits.

Modeling of the SdH oscillations yields a remarkably good fit (Fig. 2d) to the experimental SdH by the Bumps program15 (Supplementary Figs. 1523, Supplementary Table 5, Supplementary Notes 9 and 10). Notably, all of the oscillations feature a π Berry’s phase16, consistent with a Weyl topology (Supplementary Figs. 37, 14, and Supplementary Note 2). The effective band masses are light and slightly less than previously reported14,17, as shown in Fig. 2e. As a function of pressure, the electron pockets increase modestly in size due to lattice compression, but the nontrivial phase shift is maintained throughout the Td phase (Supplementary Tables 14). This trend is consistent with first-principles calculations as shown in Fig. 3a and b, which indicates the persistence of Weyl nodes up to 1.4 GPa (Supplementary Fig. 6 and Supplementary Note 2).

Fig. 3: Comparison between the calculated and experimental quantum oscillation frequencies for Td and T’ phases.

a, b Calculated in-plane quantum oscillation frequencies for the Td and T’ phases. The SdH oscillation frequencies arising from the electron pockets increase with the same pressure dependence in both phases. c A comparison between calculated and measured frequencies shows excellent quantitative agreement. The discontinuity in measured values is due to the jump in c axis dilation between phases. d, e Calculated bulk Fermi surfaces of the Td and T’ phases.

Monoclinic T’– MoTe2

In the T’ phase at pressures of 1.2 GPa and greater, one main frequency Fη replaces the two frequencies Fα and Fβ. The Fermi surface in the T’ phase closely resembles that of the Td phase, with the exception that the centrosymmetry of the T’ phase nullifies the spin–orbit splitting of the bands that contribute to this electron pocket (Fig. 3d, e). This similarity leads to a common pressure dependence of the measured electron pocket frequency in both phases, consistent with the pressure evolution of the calculated Fermi surface (Fig. 3c, Supplementary Fig. 8, and Supplementary Tables 14). The main feature not captured by the calculations is a pronounced discontinuity in band structure between Td and T’ phases due to the discontinuous c axis dilation at the structural transition, which is not modeled (Fig. 3c). Unexpectedly, our SdH oscillations analysis shows that a π Berry’s phase also exists in the centrosymmetric T’ phase (Supplementary Table S6). After an ARPES study identified surface states in this phase13, it was identified theoretically with an unusual type of nontrivial topological state11. Our results are consistent with this prediction, and further have the exciting implication that the high-pressure superconductivity in the T’ phase may be inherently topologically nontrivial, as suggested by μSR measurements at these pressures18.

Natural topological interface network (TIN)

A structural mixed region exists over a range of pressures and temperatures in between the bulk Td and T’ phases (Fig. 1a). It consists of an approximately balanced partial volume fraction of Td and T’. We emphasize that no other structural phases or ordered superstructures are apparent from neutron diffraction measurements (Supplementary Figs. 1 and 2). It would, therefore, be expected that any measured SdH oscillations in the mixed region would consist of a superposition of Td and T’ signals, but we do not observe oscillations from either phase. We conclude that the mixed region is sufficiently disordered that SdH oscillations from both the Td and T’ phases are suppressed due to increased electron scattering.

In light of this, it is completely unexpected that a distinct set of SdH oscillations appears (Fig. 1b, d). This is a robust effect; the oscillations in the mixed phase always appear on both increasing and decreasing applied pressures through the critical range, over multiple cycles, confirming their intrinsic nature. A different band structure in the mixed region is inferred from the presence of frequencies corresponding to changed Fermi surfaces, and a change in effective mass and much weaker oscillation amplitude relative to bulk Td and T’, as shown in Fig. 4a–f. These quantum oscillations reflect features typical of topologically protected states, namely, persistence in the presence of strong disorder, as well as π Berry’s phases (Supplementary Tables 5 and 6). Because there are no additional structural phases in the mixed region, only Td and T’, these topological states must be surface states of the bulk phases, which in this case, exist at their interfaces. Thus, the multiple natural interfaces stabilized in MoTe2 by the first-order nature of the structural transition serve as the foundation for a completely different type of electronic system in MoTe2: a natural topological interface network (TIN).

Fig. 4: Quantum oscillations in the magnetoresistance in MoTe2 under pressure.

Temperature dependence of the SdH oscillations of MoTe2 measured at (a) 0.6 GPa in the Td phase, (c) 0.9 GPa in the topological interface network (TIN), and (e) 1.8 GPa in the T’ phase. Note the large changes in oscillation amplitude at different pressures. Corresponding fast Fourier transform (FFT) spectra of (b) Td, (d) TIN, and (f) T’ emphasize the dramatic changes in quantum oscillation frequency, due to changes in the dominant electronic orbits, as pressure tunes through the different phases. g Schematic of the topological interface network (TIN) showing the 3D mixed Td–T’ microstructure. The relatively weak but coherent signals are robust against pressure-induced disorder and only come from the connected interfaces between grains of Td and T’. This TIN heterostructure can parametrically increase the number of surface channels and is a promising approach to increasing the surface-to-volume ratio of mixed-phase topological materials. The SdH signal is only related to cyclotron motion in the ab plane since electric current (I) and magnetic field (B) are along a and c crystallographic axis.

In the TIN, due to the layered structure of both Td and T’ phases, the ab plane is preserved, and the largest grain boundaries fall along the ab plane, which is the orientation probed by the SdH measurements (Fig. 4g). In MoTe2, a naturally generated heterostructure provides an interesting demonstration of topological transport protection. The lateral dimensions of the interfaces are the same as those of the bulk grains, based upon which one naively expects similar damping of the SdH oscillations from the interfaces. The absence of quantum oscillations from bulk Td and T’ phases illustrates that the scattering from the TIN microstructure is significant for the bulk bands. Yet the clear SdH oscillations from the interfaces prove that the interfacial states have lower scattering than the bulk, and are a sign of their topologically nontrivial nature. In other words, the interfacial signal has been amplified by suppressing the bulk SdH oscillations through grain boundary scattering, and increasing the interface volume.

Density-functional theory (DFT) calculation of Td–T’ slab model

First-principles calculations offer additional insight into the electronic structure of the TIN and its stability. We investigated a variety of possible mixed phases that might describe the mixed region, including Td-like phases with different MoTe2 stacking orders, periodic superstructures, and finite slabs of Td and T’ phases and their interfaces as shown in Fig. 5a and b (detailed calculations are discussed in the Supplementary Figs. 913 and Supplementary Notes 46). Calculations demonstrate that the only model consistent with the experimentally determined high-frequency oscillations of ~1 kT (Fig. 4d) is a system consisting of only Td–T’ interfaces, the TIN (Fig. 5d).

Fig. 5: The topological interface network Td–T’ superstructure model.

a Six layers of Td–T’ periodic superstructure, consisting of three layers of Td and T’ phases with L–L interface. b Three layers of Td and T’ slabs, separated (top) and joined (bottom). c Fermi surface obtained from separated (top) and joined slabs (bottom). d—top: The difference in the Fermi surfaces of the separated (c—top) and joined slabs (c—bottom), directly indicating the states due to the Td–T’ interface. Similarly, (d—bottom) shows the interface Fermi pockets from the periodic superstructure shown in (a). The middle panel in (d) shows the quantum oscillations from the Td–T’ joint slab calculations (b—bottom) compared with the experimental frequencies, which are represented as Gaussian curves with equal but arbitrary intensities.

The Fermi-surface pockets associated with the interface are identified in two different but complementary calculations. Consider a periodic superstructure of Td and T’ phases, as shown in Fig. 5a. Surprisingly, the Td–T’ interface has lower energy if the MoTe2 layers at the interface have the same type of Te distortion, labeled as L or R, compared to the L–R–L–R-type stacking order found in bulk Td and T’ phases. Since L- or R-type planes cannot be converted to each other by simply sliding the planes, this sort of interface presents a large energy barrier to the removal of stacking faults. The fact that this LL-type interface has lower energy than the interface with L–R stacking explains the stability of the mixed intermediate phase. Comparing the Fermi surfaces of the pure Td and T’ phases with the Td–T’ superstructure yields distinct Fermi-surface pockets with quantum oscillation orbits of 0.16 kT and 1.0 kT (bottom panel of Fig. 5d) in excellent agreement with our measurements.

Even though the periodic superstructure calculations of the Td–T’ interface are consistent with the TIN model, the interface pockets have some dispersion due to interactions between the periodic images of the interface. To eliminate this effect, consider two slabs of Td and T’ initially separated by about 9.5 Å (Fig.5b). These two slabs are brought together to form the Td–T’ interface in a supercell where the combined Td–T’ slab is separated by ~16.3 Å, large enough to avoid any image interactions (Fig. 5b, bottom). Comparing the Fermi surfaces of the isolated slabs (Fig. 5b) and combined slabs with the interface (shown in Fig. 5c), yields Fermi-surface pockets associated with the interface (Fig. 5d, top).

The agreement is excellent between the surface states from the periodic and finite slab calculations. These 2D quantum oscillation orbits are in impressive agreement with the experimental results (middle panel of Fig. 5d). Thus, the slab calculations directly link the observed high-frequency oscillation in the mixed region to interface states, as a TIN would produce.

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