Probing the topological protection of edge states in multilayer tungsten ditelluride with the superconducting proximity effect

Technical Summary: Probing the Topological Protection of Edge States in Multilayer Tungsten Ditelluride

Problem Statement
Three-dimensional second-order topological insulators (SOTIs) are predicted to host topologically protected one-dimensional (1D) helical states at the hinges of a crystal. In multilayer tungsten ditelluride (WTe$_2$), a transition-metal dichalcogenide with strong spin-orbit coupling, the topology is theorized to combine type-II Weyl semimetal and SOTI characteristics. While Josephson interferometry has previously detected 1D supercurrents along the crystal edges of WTe$_2$, these measurements could not definitively distinguish between topologically protected helical states and trivial, disorder-induced states. The critical missing evidence is the immunity to backscattering, a hallmark of helical transport. Specifically, the current-phase relation (CPR) of a junction carrying topological states is predicted to exhibit a "sawtooth" shape with a jump at $\pi$ phase difference, whereas trivial diffusive junctions display smooth, sinusoidal CPRs. However, verifying this requires isolating the edge contribution from the dominant bulk carriers and ensuring the measurement is not distorted by environmental factors.

Methodology
The authors fabricated Superconducting Quantum Interference Devices (SQUIDs) on high-quality multilayer WTe$_2$ flakes (residual resistivity ratio > 1000). The device architecture consists of two parallel Josephson junctions:

1. Reference Junction: Located in the bulk of the crystal surface, designed with an hourglass geometry to ensure a large critical current ($I_{bulk}^c$) and a diffusive transport regime.
2. Edge Junction: Located on the crystal edge (specifically the $a$-oriented edge), with a much smaller critical current ($I_{edge}^c$).

The superconducting contacts were formed by sputter-depositing a Pd/Nb bilayer, utilizing the formation of a superconducting PdTe$_x$ compound to ensure good interface quality. By operating in an asymmetric SQUID configuration where $I_{bulk}^c \gg I_{edge}^c$, the total critical current modulation in a magnetic field serves as a direct probe of the edge junction's CPR. The researchers measured the differential resistance ($dV/dI$) as a function of DC bias current and magnetic flux at temperatures ranging from 20 mK to 700 mK. They extracted the CPR by high-pass filtering the interference patterns and analyzed the harmonic content via Fast Fourier Transform (FFT).

Key Results
The study focused on six fabricated SQUIDs, revealing distinct behaviors:

• Symmetric and Sinusoidal Cases: SQUID A exhibited a symmetric interference pattern, while SQUIDs B1, B2, D1, and D2 showed small, sinusoidal modulations. These were attributed to either symmetric bulk-bulk interference or edge junctions dominated by trivial, diffusive bulk states, likely due to longer junction lengths or poor contact to the 1D states.
• The Ballistic Edge Case (SQUID C): The most significant result was observed in SQUID C, which featured the shortest edge junction (600 nm).
  • Sawtooth CPR: The interference pattern revealed a distinct sawtooth-shaped modulation, indicating a CPR with a sharp jump at $\pi$ phase difference.
  • Robustness: The edge critical current ($I_{edge}^c \approx 200$ nA) was remarkably robust, decreasing by only 25% as temperature rose from 20 mK to 700 mK and remaining largely unaffected by magnetic fields up to hundreds of Gauss. In contrast, the bulk junction current decayed rapidly with both temperature and field.
  • Harmonic Content: FFT analysis showed up to six harmonics at low temperatures, decaying as $1/n$, consistent with a ballistic junction. This contrasts with the $1/n^2$ decay expected for diffusive junctions.
  • Quantitative Parameters: Fitting the harmonic decay yielded a Thouless energy for the edge junction of $E_{edge}^{Th} \approx 560 \pm 75$ $\mu$eV (approx. 6.5 K), two orders of magnitude larger than the bulk junction ($E_{bulk}^{Th} \approx 72$ mK). The intrinsic transmission coefficient was estimated at $t \approx 0.89$.
  • Channel Count: The magnitude of the supercurrent suggests it is carried by at least three helical channels. The decay of the critical current with in-plane magnetic field indicates the edge states are confined to a lateral width of less than 2 nm in the $c$-direction and 20 nm in the $(a,b)$ plane.

Significance and Claims
The paper claims that the observation of a sawtooth-shaped CPR in SQUID C provides a "tell-tale sign" of the SOTI character of multilayer WTe$_2$. The authors argue that the robustness of the CPR shape against interface imperfections (indicated by the high transmission $t$) and the preservation of the jump at $\pi$ are signatures of topological protection, which forbids backscattering in helical states. The high Thouless energy and the specific field dependence confirm that the supercurrent is carried by quasi-ballistic channels over 600 nm, a distance significantly longer than the coherence length of trivial states in the disordered bulk.

The authors modestly conclude that while the results strongly support the hypothesis of a SOTI phase in multilayer WTe$_2$, the presence of trivial bulk carriers in other samples (leading to sinusoidal CPRs) highlights the difficulty of isolating these states. They posit that the 1D helical states in WTe$_2$ are likely located at atomic step edges and that multilayer samples offer a viable platform for exploring these states, provided the junctions are short enough to maintain phase coherence and the contacts are optimized to access the edge channels. The work does not propose new applications but establishes a critical experimental benchmark for identifying topological protection in 3D SOTIs.