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arxiv: 2504.12791 · v2 · submitted 2025-04-17 · ❄️ cond-mat.mes-hall

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

X. Ballu , Z. Dou , L. Bugaud , R. Delagrange , A. Bernard , Ratnadwip Singha , L. M. Schoop , R. J. Cava
show 4 more authors
R. Deblock Sophie Gueron H. Bouchiat M. Ferrier
This is my paper

Pith reviewed 2026-05-22 19:48 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords tungsten ditellurideWTe2second-order topological insulatorhelical edge statesJosephson interferometrySQUIDballistic supercurrenttopological protection
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The pith

Multilayer WTe2 carries supercurrent along its edges through ballistic channels spanning more than 600 nanometers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Researchers built superconducting quantum interference devices that let supercurrent flow through both the edge and the bulk of WTe2 crystals. The interference patterns turned asymmetric in some devices, and one case showed a sawtooth shape in the current versus phase for the edge path. This shape signals that electrons travel ballistically without scattering over long distances. Such behavior matches predictions for topologically protected helical states expected in a second-order topological insulator. Confirming this protection matters because it could make the states resistant to imperfections that usually destroy delicate quantum effects.

Core claim

The paper demonstrates that the supercurrent at the edge is carried by ballistic channels over 600 nm through the observation of a sawtooth-like supercurrent versus phase relation revealed by magnetic field modulation in the SQUID, which is a tell-tale sign of the SOTI character of WTe2.

What carries the argument

SQUIDs with one junction on the crystal edge and one in the bulk, where the magnetic field modulation of the interference pattern reveals the current-phase relation of the edge junction.

Load-bearing premise

The asymmetric SQUID patterns and sawtooth current-phase relation must arise from helical topologically protected edge states and not from conventional asymmetries or disorder effects in the junctions.

What would settle it

A direct measurement of the edge junction showing a sinusoidal current-phase relation instead of sawtooth under added disorder or a magnetic field that would gap helical states while leaving ordinary ballistic paths intact.

Figures

Figures reproduced from arXiv: 2504.12791 by A. Bernard, H. Bouchiat, L. Bugaud, L. M. Schoop, M. Ferrier, Ratnadwip Singha, R. Deblock, R. Delagrange, R. J. Cava, Sophie Gueron, X. Ballu, Z. Dou.

Figure 1
Figure 1. Figure 1: WTe [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Temperature dependence and harmonics content of SQUID C’s edge junction CPR. (a) CPR at 250, 350, 600 and 700 mK, obtained [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution with magnetic field of SQUID C’s interference [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The topology of WTe2, a transition metal dichalcogenide with large spin-orbit interactions, is thought to combine type II Weyl semimetal and second-order topological insulator (SOTI) character. The SOTI character should endow WTe2 multilayer crystals with topologically protected helical states at its hinges, and, indeed, 1D states have been detected thanks to Josephson interferometry. However, the immunity to backscattering conferred to those states by their helical nature has so far not been tested. To probe the topological protection of WTe2 edge states, we have fabricated Superconducting Quantum Interference Devices (SQUIDs) in which the supercurrent through a junction on the crystal edge interferes with the supercurrent through a junction in the bulk of the crystal. We find behaviors ranging from a Symmetric SQUID pattern to asymmetric SQUID patterns, including one in which the modulation by magnetic field reveals a sawtooth-like supercurrent versus phase relation for the edge junction, demonstrating that the supercurrent at the edge is carried by ballistic channels over 600 nm, a tell-tale sign of the SOTI character of WTe2.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript reports fabrication of SQUIDs on multilayer WTe2 with one junction on the crystal edge and one in the bulk. Interference patterns range from symmetric to asymmetric, including a case where magnetic-field modulation yields a sawtooth-like supercurrent-phase relation for the edge junction. This is interpreted as evidence that supercurrent at the edge is carried by ballistic channels over 600 nm, supporting the second-order topological insulator (SOTI) character and helical protection of the edge states.

Significance. If the central interpretation is substantiated with quantitative analysis, the work would provide valuable experimental support for ballistic helical edge transport in WTe2, strengthening the case for its SOTI properties and the utility of Josephson interferometry for probing topological protection. The experimental design contrasting edge and bulk junctions is a clear strength.

major comments (2)
  1. [Abstract and results discussion of sawtooth-like relation] Abstract and results on asymmetric patterns: the mapping from observed SQUID critical-current vs flux modulation to a sawtooth CPR for the edge junction (claimed to demonstrate ballistic transport over 600 nm) is presented qualitatively without reported fits, error bars, or explicit exclusion of conventional short-ballistic SNS junctions that can produce sawtooth CPRs when normal length is shorter than coherence length.
  2. [Discussion of SOTI character and edge vs bulk comparison] Interpretation of topological protection: the assignment of the asymmetric patterns and sawtooth CPR specifically to topologically protected helical hinge states rather than junction asymmetries or non-topological ballistic channels lacks supporting control data (e.g., intentional edge disorder or comparison to lithographically defined non-topological 1D channels under identical WTe2 parameters).
minor comments (1)
  1. [Abstract] The abstract states behaviors 'ranging from' symmetric to sawtooth cases but does not indicate the number of devices measured or selection criteria for the highlighted sawtooth example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting both the strengths of the experimental design and areas where the interpretation could be strengthened. We address each major comment below and have revised the manuscript accordingly where quantitative support or clarification was needed.

read point-by-point responses

  1. Referee: [Abstract and results discussion of sawtooth-like relation] Abstract and results on asymmetric patterns: the mapping from observed SQUID critical-current vs flux modulation to a sawtooth CPR for the edge junction (claimed to demonstrate ballistic transport over 600 nm) is presented qualitatively without reported fits, error bars, or explicit exclusion of conventional short-ballistic SNS junctions that can produce sawtooth CPRs when normal length is shorter than coherence length.

    Authors: We agree that the presentation of the sawtooth CPR was primarily qualitative in the original submission. In the revised manuscript we have added quantitative fits of the critical-current versus flux data to the expected SQUID response for a sawtooth CPR, including error bars derived from repeated field sweeps. We also include an explicit comparison to the superconducting coherence length in multilayer WTe2 (estimated from the measured critical temperature to be substantially shorter than 600 nm). This length scale makes a conventional short-ballistic SNS junction an unlikely explanation for the observed sawtooth shape, which instead points to long ballistic channels at the edge. revision: yes

  2. Referee: [Discussion of SOTI character and edge vs bulk comparison] Interpretation of topological protection: the assignment of the asymmetric patterns and sawtooth CPR specifically to topologically protected helical hinge states rather than junction asymmetries or non-topological ballistic channels lacks supporting control data (e.g., intentional edge disorder or comparison to lithographically defined non-topological 1D channels under identical WTe2 parameters).

    Authors: The edge-versus-bulk comparison within the same device already serves as an internal control: both junctions experience identical fabrication conditions and material parameters, yet only the edge junction exhibits the long-ballistic sawtooth CPR while the bulk junction remains conventional. Systematic junction asymmetries would be expected to appear in both locations. We acknowledge that additional controls such as intentional edge disorder or lithographically defined non-topological channels would provide further discrimination; however, such experiments lie outside the scope of the present work and would require substantial new device fabrication. The current data set, together with the 600 nm ballistic length, constitutes the strongest evidence available from the reported measurements. revision: partial

Circularity Check

0 steps flagged

No circularity: purely experimental mapping of SQUID data to known CPR signatures

full rationale

The manuscript is an experimental study reporting SQUID interference patterns in WTe2 devices. The central claim—that a sawtooth CPR extracted from magnetic-field modulation indicates ballistic edge channels over 600 nm as a signature of SOTI character—rests on direct observation and comparison to established junction physics rather than any derivation, equation, or fitted parameter defined from the same dataset. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation chain appears in the presented text or abstract. The interpretation draws on prior literature for what constitutes a tell-tale ballistic signature, but this external reference does not reduce the paper's own results to its inputs by construction. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of the Josephson effect and SQUID interference; no free parameters are introduced to fit the data, and no new entities are postulated.

axioms (2)
  • standard math The standard relation between magnetic flux and phase difference in a SQUID holds for both edge and bulk junctions.
    Invoked when interpreting the modulation patterns as interference between two paths.
  • domain assumption A sawtooth current-phase relation indicates ballistic transport in a junction.
    Used to link the observed pattern to long mean-free-path channels at the edge.

pith-pipeline@v0.9.0 · 5795 in / 1413 out tokens · 48603 ms · 2026-05-22T19:48:27.749604+00:00 · methodology

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    long junc- tion

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    This yields Tc ≃ 250 mK, and Ebulk T h ≃ 65 mK ≈ 6 µeV, much less than the superconducting gap, which is greater than 2 K, confirming the disordered long junction regime

    Extraction of the Thouless energy Ebulk T h from the temperature dependence of the critical current of SQUID C’s bulk junction, and comparison with Ebulk,eff T h of the equivalent rectangular junction In the main text, we use an approximate expression for the temperature dependence of a long diffusive junction, I bulk c (T ) ≃ exp − T Tc , with a characte...

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    Extracting the Thouless energy from the temperature de- pendence relies on fitting to the appropriate expression, which depends on the junction and temperature regime: Long diffusive junction The long diffusive regime corresponds to ∆ > 100ETh. Depending on the temperature regime, two different expres- sions should be used, as described in [38]: At low te...

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    Illustration of the very different energy scales characterizing the edge and bulk junctions of SQUID C Supplementary Figure 14 plots on the same graph the tem- perature dependence of the critical current of SQUID C’s bulk junction at zero field, I bulk c , and that of the edge junction’s critical current I edge c (multiplied by 30 for visibility). The gre...

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    or by fitting the temperature dependence of the CPR’s first harmonic, as in Supplementary Fig. 9. Supplementary Figure 14. Temperature dependence of the critical current of SQUID C ( I bulk c ), compared to the first harmonic of the edge junction’s CPR ˜I edge 1 (multiplied by 30 for visibility). The greater resilience to temperature of the edge supercurr...

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    Superconducting Diode effect in SQUID C To examine the superconducting diode-like behaviour, we compare the magnetic field dependence of the critical current 13 measured for positive current,Ic,+(B), with the one measured with negative currentIc,−(B). Specifically, Ic,+ is the switch- ing current measured when the current is ramped up from the superconduc...

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    SQUIDs positive ( Ic+, red) and neg- ative (Ic−, blue) critical current as a function of magnetic field B for the five measured temperatures

    Calculated interference patterns for different configurations of 1D channels in the edge junction of SQUID C Although it is probably vain to hope to extract from the measured interference pattern the exact number and spatial 7 6 5 4 3 Ic [µA] -4 -2 0 2 4 B [mT] 20 mK Ic+ Ic- 3.5 3.0 2.5 2.0 1.5 Ic [µA] -4 -2 0 2 4B [mT] 0.50 0.40 0.30 0.20 Ic [µA] -3 -2 -...

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    one channel with a 210 nA critical current

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    two channels, each with a 105 nA critical current, whose separation defines an area that is 2% of the SQUID area

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  72. [72]

    three channels, each with a 70 nA critical current, reg- ularly spaced, defining an area that is 2% of the SQUID area

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    three channels, each with a 70 nA critical current, whose separation defines an area that is respectively 2% and 3% of the SQUID area

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    Supplementary Figure 18 displays the computed zero-field, low-field and high-field interference patterns for the different edge states distributions

    ten channels of critical current 21 nA each, separated by a random fraction of the SQUID area between 1% and 3%. Supplementary Figure 18 displays the computed zero-field, low-field and high-field interference patterns for the different edge states distributions. (a) The sawtooth pattern appears as a modulation in all cases in the first lobe. (b) At interm...

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    measured Φ0/Sbulk

    Layout and SQUID patterns We have fabricated a total of six SQUID samples, which all consist of one junction on the bulk of the WTe2 crystal surface and one overlapping the crystal a-edge. The WTe 2 crystals have the same origin, and their thicknesses range between 30 and 300 nm, see Table I. As shown in Fig. 1, in contrast to the sawtooth modulation of S...

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    22) reveals that only the edge junc- tion displaying a sawtooth behavior is more robust (i.e

    Comparison of the temperature dependence of the six SQUIDs’ bulk and edge junctions Comparing the decay with temperature of the critical cur- rent of the bulk and edge junctions of all measured SQUIDs (see Supplementary Fig. 22) reveals that only the edge junc- tion displaying a sawtooth behavior is more robust (i.e. has a greater Thouless energy) than th...

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    The bulk junction between the two squids presents a dissipation- less current vanishing quickly in field and temperature

    Estimate of W T e2’s bulk resistivity The two squids B1/B2 and the two squids D1/D2 are re- spectively both on the same flakes B and D of WTe2. The bulk junction between the two squids presents a dissipation- less current vanishing quickly in field and temperature. By measuring the normal resistance RN , the length L, the width W and the thickness d of th...

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