Enhanced and modulable induced superconducting gap and effective Land\'e g-factor in Pb-InSb hybrid devices

Anqi Wang; Bingbing Tong; Erik P. A. M. Bakkers; Fanming Qu; Ghada Badawy; Guangtong Liu; Guang Yang; Guoan Li; Jiayu Shi; Jie Shen

REVIEW 3 major objections 5 minor 57 references

Enhanced and modulable induced superconducting gap and effective Land\'e g-factor in Pb-InSb hybrid devices

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that Pb-InSb hybrid devices offer a hard proximity-induced superconducting gap up to ~1.4 meV and effective g-factors up to 76, resolving a trade-off in aluminum-based Majorana platforms.

desk verdict Solid Pb-InSb gap experiment with a tunable hard gap; g-factor claims are suggestive but rest on visual line-slope fitting, so the paper needs a cautious referee. read the letter →

arxiv 2506.03563 v1 pith:JIG33O7C submitted 2025-06-04 cond-mat.mes-hall cond-mat.supr-conquant-ph

Guoan Li , Xiaofan Shi , Ziwei Dou , Guang Yang , Jiayu Shi , Marco Rossi , Ghada Badawy , Yuxiao Song

show 16 more authors

Ruixuan Zhang Yupeng Li Zhiyuan Zhang Anqi Wang Xingchen Guo Xiao Deng Bingbing Tong Peiling Li Zhaozheng Lyu Guangtong Liu Fanming Qu Erik P. A. M. Bakkers Micha{l} P. Nowak Pawe{l} Wójcik Li Lu Jie Shen

This is my paper

classification cond-mat.mes-hallcond-mat.supr-conquant-ph PACS 74.45.+c71.70.Ej73.23.-b

keywords Pb-InSbhybridnanowiresproximity-inducedsuperconductivityhardsuperconductinggapgate-tunablelargeeffectiveg-factorRashbaspin-orbitcouplingMajoranazeromodestunnelingspectroscopy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that lead (Pb), rather than aluminum (Al), is a superior parent superconductor for semiconductor-nanowire hybrid devices aimed at topological superconductivity. In Pb-InSb tunnel junctions the authors report a proximity-induced superconducting gap as large as ~1.4 meV, matching bulk Pb and roughly an order of magnitude above Al-based systems, and they find the gap can be tuned continuously from that maximum down to nearly zero by a back gate while staying hard. Magnetic-field spectroscopy exposes multiple conductance lines whose slopes yield effective g-factors up to 76, values that persist even in the strong-coupling limit where Al devices lose spin-orbit coupling and g-factor. If correct, this removes a known trade-off and widens the parameter space in which Majorana zero modes could be stabilized.

What carries the argument

The load-bearing objects are the measured conductance resonances in the magnetic-field maps and the one-dimensional model used to interpret them. The Hamiltonian combines kinetic energy, chemical potential, an induced pairing term $\Delta(x)$, Rashba spin-orbit coupling $\alpha$, and Zeeman energy $E_Z = \frac{1}{2} g \mu_B B$; from it the authors derive an analytic positive-energy spectrum $E_\pm$ whose bending and anti-crossings reproduce the gap evolution. In the multimode version each transverse mode has its own $\alpha_i$ and $g_i$ with off-diagonal mixing $\gamma$, which accounts for multiple radial lines with different slopes that do not all share a common high-field slope. These modes are what connect the raw conductance maps to the extracted effective g-factors.

What would settle it

Follow one of the tracked conductance lines to the highest accessible magnetic field: the paper's own analytic dispersion predicts the slope, hence the extracted g-factor, should bend at low field and saturate by roughly 5 T. If the slopes instead remain linear from zero field or fail to show the predicted bending, the spin-orbit renormalization explanation would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that a Pb superconductor on an InSb nanowire simultaneously solves the two bottlenecks of Al-based hybrids: the induced superconducting gap is large and hard, and the effective spin-orbit coupling and Landé g-factor remain large in the same coupling regime. In the measured devices the zero-field gap reaches ~1.4 meV, the above-gap to subgap conductance ratio reaches two orders of magnitude, and the gap can be gate-tuned from roughly zero back to its maximum without becoming soft. Out-of-plane magnetic-field maps show radial resonance lines, anti-crossings, and gap reopening, and the authors interpret the slopes of these lines as effective g-factors of about 7 to 76. They attribute the persistence of strong spin-orbit coupling in the strong-coupling limit to Pb's Rashba spin-orbit coupling, and they reproduce the qualitative structure with a one-dimensional multimode model in which each transverse mode carries its own g-factor and spin-orbit strength.

Load-bearing premise

The claim that the effective g-factors reach 76 rests on identifying the tracked conductance lines as spin-split bound states of nanowire modes; if those lines instead come from disorder states or interference effects, the extracted g-factors and the strong-spin-orbit conclusion would not be supported.

Editorial extensions

If this is right

A Pb-based induced gap near 1.4 meV raises the upper bound on any topological gap roughly sevenfold relative to Al, making Majorana experiments less sensitive to thermal occupation and disorder.
Continuous gate tuning of a hard gap from maximum to near zero gives a single device a wide range of hybridization strengths, which is useful for mapping the topological phase diagram.
Effective g-factors up to 76, sustained in the strong-coupling regime, lower the Zeeman field needed for a trivial-to-topological transition and enlarge the safe field window below the parent superconductor's critical field.
The same platform can be used for Coulomb-blockaded Cooper-pair transport and artificial Kitaev chain geometries, where a larger gap and g-factor improve coherence and operating temperature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural experiment beyond the present data is to measure the high-field slope of the same resonance lines past 5 T; the analytical model predicts the extracted g-factor saturates there, which would distinguish spin-orbit renormalization from other line-drawing artifacts.
If Pb's Rashba coupling is indeed the source of the preserved g-factor, varying the Pb film thickness or the Ti wetting-layer details should systematically change the extracted effective g-factor, a knob the paper does not sweep.
The results suggest Pb-InSb islands should also show robust 2e-periodic Coulomb-blockaded Cooper-pair transport; demonstrating that would connect these spectroscopy findings to qubit-oriented island experiments.
Because the extracted g-factors come from line slopes rather than direct spin measurements, a spin-resolved or weak-antilocalization measurement of the same nanowires would give an independent estimate of the spin-orbit strength.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Peter,

You should know two things about arXiv:2506.03563. First, the central experimental result—a large, hard, gate-tunable induced gap in Pb-InSb hybrids—looks real and is a useful advance over Al-based devices. Second, the claimed enhancement of the effective g-factor up to 76 is plausible but not yet solid; it rests on visual slope-fitting of conductance lines whose identity is not uniquely established.

What's genuinely new: quantitative tunneling spectroscopy on Pb-InSb, a material combination not covered by the earlier Pb-InAs (Kanne) or Pb-PbTe (Zhang) papers. The data in Fig. 1 show a hard gap reaching ~1.4 meV in the island device, with subgap conductance suppressed by two orders of magnitude. The gate tunability from ~0 to 1.4 meV while keeping the gap hard is impressive; Al devices typically get soft at small gaps. The Appendix B argument excluding Coulomb blockade is clean and convincing.

The soft spot is the g-factor extraction. The Supplemental describes Savitzky-Golay smoothing and peak-finding, but the slopes are then visually evaluated. Nothing proves that a fitted straight segment follows a single physical eigenstate across the whole field range. In a multimode wire, the conductance maximum can jump between modes, and the slope then mixes unrelated states. The authors note that lines bend at low field and fit only the high-field slopes, but they don't quantitatively model that bending beyond pointing to their model. The multimode model has enough free parameters (alpha_i, g_i, mixing gamma, barrier V, Delta, mu) that its agreement with the slopes is not a strong constraint. The abstract also overstates the gap as an "order of magnitude" larger than Al when the text says six-fold; that's an easy fix but should not be left as is.

If I had to bet, the large-gap result will survive scrutiny, and the g-factor observation is interesting enough to warrant a follow-up with better line-tracking. The paper deserves a serious referee; it should not be desk-rejected. A careful referee should ask for automated peak-tracking across the 2D maps, explicit criteria for line identity, and error bars that come from something other than visual inspection. The theory section should be repositioned as illustrative rather than as confirmation.

Overall: a solid experimental contribution with a soft interpretive overlay. Worth citing for the gap, and worth a reading-group discussion on how much slope-based g-factor extraction can be trusted.

Referee Report

3 major / 5 minor

Summary. The manuscript reports tunneling spectroscopy measurements on Pb-InSb hybrid nanowire devices. The authors demonstrate a proximity-induced superconducting gap up to ~1.4 meV (matching bulk Pb), with hardness reaching two orders of magnitude, and show that the gap can be continuously tuned by the back gate from near zero up to its maximum while remaining hard. In out-of-plane magnetic fields, the conductance maps show multiple finite-bias radial lines with different slopes; these are interpreted as spin-split resonance lines, and their slopes are converted into effective g-factors up to 76. A single-mode and a multi-mode one-dimensional model are used to argue that spin-orbit coupling from Pb renormalizes the g-factor and produces the observed bending, anti-crossings, and mode-dependent slopes. The central claims are (i) a large, hard, tunable induced gap and (ii) preserved strong spin-orbit coupling and enhanced g-factors in the strong-coupling regime.

Significance. If both claims hold, this work would be an important step for Majorana-based quantum computing: Pb provides a ~7x larger induced gap than Al, tunable without loss of hardness, and the effective g-factors up to 76 exceed those of Al-based hybrids, potentially enlarging the topological parameter space. The large, hard, gate-tunable gap is directly supported by conductance data with strong subgap suppression and is the most robust contribution. The g-factor and strong-SOC conclusions are plausible but rest on a less rigorous extraction of slopes from noisy conductance maps and on an illustrative rather than quantitative model. The paper also ships a useful multimode simulation framework, but its parameters are not fitted to data.

major comments (3)

[Supplemental Material, "Extraction of the g-factor" and Fig. 2] The extraction pipeline (Savitzky-Golay smoothing, peak finding, symmetrization of positive and negative bias, and then visual evaluation of slopes) does not demonstrate that a fitted straight segment follows the same physical eigenstate over the entire magnetic-field range. In a multimode nanowire, the conductance maximum can shift from one mode to another as B and the induced gap evolve, and the resulting slope would then be an artifact of stitching unrelated states. This is load-bearing for the headline g-factor values up to 76 and for the conclusion that Pb preserves strong SOC in the strong-coupling regime. Please provide a global peak-tracking algorithm that enforces continuity of a line in the 2D map, report how many identified peaks are discarded, and show that the extracted slopes are stable with respect to smoothing parameters and peak-finding thresholds.
[Appendix C and Eq. (S3)] The multimode model, which uses independent α_i, g_i, γ, μ, Δ, and barrier parameters, is used only for qualitative illustration; no quantitative comparison with the experimental conductance maps is made. The claim that the observed slopes arise from g-factor renormalization by Pb's spin-orbit coupling is therefore not tested against alternatives. The model also keeps Δ constant and omits the B-dependence of the parent Pb gap, and although the authors argue that orbital effects are isotropic, the parent-gap suppression could influence the line slopes. Please fit at least representative experimental traces, or provide a sensitivity analysis showing how robust the extracted g-factors are to variations in the model parameters.
[Abstract and page 2] The abstract states that the induced gap exceeds that of Al by "an order of magnitude", but the text (page 2, around the discussion of [24]) and later statements give a factor of approximately six to seven. The measured maximum of ~1.4 meV compared to Al's ~0.2 meV is roughly a factor of seven, not an order of magnitude. This overstatement should be corrected to avoid misleading readers about the quantitative improvement.

minor comments (5)

[Fig. 2] The colored dashed lines used to indicate the radial resonance lines are difficult to distinguish in black-and-white print; please use distinct line styles or add labels directly on the panels.
[Appendix B] The second exclusion argument (the maximum induced gap is nearly identical to the parent Pb gap) is not by itself decisive against a Coulomb gap, since a charging energy of order 1 meV is possible; however, when combined with the gate-voltage range and absence of excited states it is reasonable. Please state explicitly that the large gate range over which the gap persists is the primary discriminator.
[Fig. 3(e)] The error bars on the extracted g-factors are described only as being obtained from visual evaluation in the Supplemental Material. Please specify whether these error bars include systematic uncertainty from the ambiguity of line identification, and how the error bars were calculated.
[Eq. (2)] The sign convention for E± and the branch of the square root should be stated explicitly, along with the condition under which the gap closes; this would help readers connect the analytical formula to the numerical spectra in Fig. 4.
[Fig. 1 and main text] The statement that the gap remains hard while being tuned "down to nearly zero" is ambiguous, as the hardness ratio (above-gap vs subgap conductance) is ill-defined when the induced gap vanishes. Please clarify the gate-voltage range over which the hardness is quantified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: large gap and g-factors are direct measurements; the theory is illustrative with hand-picked parameters.

full rationale

The paper's central claims rest on direct transport measurements rather than on a fitted parameter renamed as a prediction. The induced superconducting gap and its hardness are read directly from differential-conductance maps (Fig. 1), and the gate tunability is the measured variation of those gap edges with V_BG. The effective g-factor is defined operationally from the slope of finite-bias conductance lines versus magnetic field using g = (2/mu_B)(dE/dB); the extraction pipeline described in the Supplemental Material (Savitzky-Golay smoothing plus peak finding and visual line fitting, Fig. S5) is a standard peak-tracking procedure and does not fit any model parameter to force the reported slopes. The theoretical model (Eq. 1 and Eqs. S1, S3) is used after the fact to illustrate how Rashba spin-orbit coupling bends and renormalizes the resonance lines (Fig. 3); its alpha_i, g_i, gamma, and mu are hand-picked rather than least-squares fitted to the data, so the comparison is illustrative rather than a self-referential prediction. The self-citations present in the paper ([41] for fabrication details and [50] for spin-orbit-induced g-factor enhancement) are supporting external publications and are not the sole justification for the central observations; they are not load-bearing in the sense of forcing a conclusion by definition or by an unverified uniqueness claim. The skeptic's concern that the tracked conductance lines might not be spin-split Andreev levels, and that alternative origins such as disorder-localized states or Fabry-Perot interference are not explicitly excluded, is a validity/correctness concern about the extraction, not a circularity: if the line assignment is invalid the g-factor claim would be wrong, but the derivation would not become equivalent to its inputs. No step in the paper reduces a prediction to a fitted input or imports a conclusion from a self-citation chain, so the paper is self-contained against circularity.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central experimental observations (large gap, tunability) do not depend on the model parameters. The free parameters listed are inputs to the interpretive model used to explain the magnetic-field dependence and the g-factor renormalization. The model is not a fit to the data, and no new physical entities are introduced.

free parameters (7)

induced gap Δ in model = 1 meV (chosen)
Used in the numerical and analytical calculations to represent the proximity-induced gap; not fitted to each gate voltage.
chemical potential μ = 10 meV (chosen)
Fermi energy in the 1D model; not directly measured in the experiment.
Rashba spin-orbit strength α = 10, 20, 30, 45 meV·nm (chosen)
Values chosen to show the effect of SOC on the resonance line slopes; no unique extraction from data.
bare g-factors of modes = -40, -60, -80 (chosen)
Assumed in the multimode model; the experimental effective g-factors are extracted separately.
mode-mixing parameter γ = not specified
Off-diagonal coupling between modes in Eq. (S3); chosen to achieve qualitative match.
tunnel barrier height = 0.2 μeV (chosen)
Sets the junction in the tunneling regime in the model.
effective mass m* = 0.014 m (literature)
InSb effective mass taken from literature; the model results depend on this choice.

assumptions (5)

standard math Bogoliubov-de Gennes formalism and scattering matrix theory
Used to compute tunnel conductance from the Hamiltonian in Eq. (1) via the Kwant package; this is a standard method in mesoscopic physics.
domain assumption Rashba spin-orbit coupling model with Zeeman field
Assumes the nanowire and Pb interface are described by a 1D Rashba Hamiltonian with an out-of-plane Zeeman field (Eq. 1 and Supplementary Eq. S1).
domain assumption Proximity effect induces a uniform pairing potential Δ in the nanowire
The model treats Δ as constant in the proximitized region, neglecting disorder and interface details.
domain assumption Effective g-factor extraction from the slope assumes linear Zeeman splitting
The formula g = (2/μ_B)(∂E/∂B) is applied to the slopes of conductance lines, assuming they originate from spin-split states.
domain assumption Lead provides strong Rashba spin-orbit coupling to the nanowire
The paper cites monolayer Pb experiments to argue that Pb induces strong SOC, but does not directly measure the SOC strength in these devices.

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Cite this review

Pith. "Pith review of Enhanced and modulable induced superconducting gap and effective Land\'e g-factor in Pb-InSb hybrid devices." pith.science (2026). https://pith.science/paper/JIG33O7C

@misc{pith2026250603563,
  author       = {Pith},
  title        = {Pith review of: Enhanced and modulable induced superconducting gap and effective Land\'e g-factor in Pb-InSb hybrid devices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JIG33O7C}},
  note         = {Machine review of arXiv:2506.03563}
}

read the original abstract

The hybrid system of a conventional superconductor (SC) on a semiconductor (SM) nanowire with strong spin-orbit coupling (SOC) represents a promising platform for achieving topological superconductivity and Majorana zero modes (MZMs) towards topological quantum computation. While aluminum (Al)-based hybrid nanowire devices have been widely utilized, their limited superconducting gap and intrinsic weak SOC as well as small Land\'e g-factor may hinder future experimental advancements. In contrast, we demonstrate that lead (Pb)-based hybrid quantum devices exhibit a remarkably large and hard proximity-induced superconducting gap, exceeding that of Al by an order of magnitude. By exploiting electrostatic gating to modulate wavefunction distribution and SC-SM interfacial coupling, this gap can be continuously tuned from its maximum value (~1.4 meV, matching the bulk Pb gap) down to nearly zero while maintaining the hardness. Furthermore, magnetic-field-dependent measurements reveal a radial evolution of the gap structure with anti-crossing feature, indicative of strong SOC and huge effective g-factors up to 76. These findings underscore the superior functionality of Pb-based hybrid systems, significantly advancing their potential for realizing and stabilizing MZMs and the further scalable topological quantum architectures.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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