arxiv: 2604.21635 · v1 · submitted 2026-04-23 · ❄️ cond-mat.supr-con · cond-mat.mtrl-sci

Recognition: unknown

Two-gap to Single-gap Transition and Two-dome-like Superconductivity in Alkali-Metal Intercalated Bilayer PdTe2

Yu-Lin Han , Shu-Xiang Qiao , Kai-Yue Jiang , Jie Zhang , Bao-Tian Wang , Ping Zhang , C. S. Ting , Hong-Yan Lu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 13:34 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mtrl-sci

keywords superconductivityintercalationPdTe2electron-phonon couplingtwo-gap superconductivitylayered materialstransition temperature

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The pith

Alkali-metal intercalation in bilayer PdTe2 raises the superconducting transition temperature from 1.4 K to 13.5 K while switching the system from two-gap to single-gap superconductivity.

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

The paper shows that inserting alkali atoms between the layers of bilayer PdTe2 strengthens electron-phonon coupling and pushes the superconducting transition temperature up to 13.5 K for rubidium. The temperature follows a two-dome pattern as the intercalant changes from lithium to cesium, and the superconducting gap structure shifts from two distinct gaps to one as the interlayer spacing increases. Pristine and lightly intercalated versions also carry nontrivial band topology. These changes arise because intercalation expands the layers and weakens their coupling, directly altering how electrons pair.

Core claim

First-principles calculations within the fully anisotropic Migdal-Eliashberg framework demonstrate that alkali-metal intercalation enhances the weak superconductivity of bilayer PdTe2, raising Tc from 1.4 K to values between 5.0 K and 13.5 K with a two-dome dependence on the intercalant species. Rubidium produces the highest Tc of 13.5 K, which increases further to 14.5 K under biaxial tensile strain. Lithium intercalation preserves a two-gap state, while larger alkali atoms induce a single-gap state through expansion of the interlayer distance that modulates the interlayer interaction and the electron-phonon coupling strength.

What carries the argument

Intercalation-induced interlayer expansion that weakens coupling between the PdTe2 layers and thereby controls both the electron-phonon interaction strength and whether the superconducting gap remains two-gap or collapses to single-gap.

If this is right

Rubidium intercalation combined with moderate tensile strain produces the highest transition temperature of 14.5 K.
The two-dome evolution of Tc with intercalant size and with applied strain follows from the competition between enhanced electron-phonon coupling and changes in the density of states at the Fermi level.
Only lithium and sodium intercalation retain the two-gap character; sodium, potassium, rubidium, and cesium drive the system to single-gap superconductivity.
Pristine and lithium- or sodium-intercalated bilayers host both superconductivity and nontrivial band topology.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar layer-expansion tuning could be applied to other weakly coupled van der Waals superconductors to raise Tc without introducing new chemical disorder.
The observed two-dome pattern suggests that optimal Tc occurs at an intermediate interlayer spacing where electron-phonon coupling is maximized before band-structure effects dominate.
Because the topology survives in the lightly intercalated cases, these materials offer a route to study the interplay between superconductivity and band topology by continuous control of the gap structure.

Load-bearing premise

The superconducting state is fully described by phonon-mediated pairing captured in the anisotropic Migdal-Eliashberg equations without important contributions from electron correlations or non-phonon mechanisms.

What would settle it

Direct measurement of the superconducting gap on rubidium-intercalated bilayer PdTe2 samples that either confirms a single isotropic gap or detects two distinct gaps at the predicted temperatures.

Figures

Figures reproduced from arXiv: 2604.21635 by Bao-Tian Wang, C. S. Ting, Hong-Yan Lu, Jie Zhang, Kai-Yue Jiang, Ping Zhang, Shu-Xiang Qiao, Yu-Lin Han.

**Figure 1.** Figure 1: FIG. 1. (a) Side and top views of Pd view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) Crystal structure of alkali-metal intercalated view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a) view at source ↗

**Figure 6.** Figure 6: FIG. 6. Summarized superconducting parameters. (a) In view at source ↗

**Figure 7.** Figure 7: FIG. 7. Edge states along view at source ↗

read the original abstract

PdTe2 has been synthesized with controllable thickness down to the monolayer limit. Based on first-principles calculations within the fully anisotropic Migdal-Eliashberg framework, this work reveals that alkali-metal intercalation markedly enhances the weak superconductivity of bilayer PdTe2, boosting the transition temperature from 1.4 K to 5.0 -13.5 K and yielding a two-dome-like evolution of Tc. Rubidium intercalation induces the highest Tc of 13.5 K, which can be further increased to 14.5 K under biaxial tensile strain. The strain-dependent evolution of Tc also exhibits a two-dome-like behavior, reflecting the interplay between strain-induced band structure modifications and electron-phonon coupling (EPC). Moreover, a systematic correlation is identified between interlayer interaction and superconducting gap. Lithium intercalation induces a distinct two-gap state, whereas intercalants with larger atomic radii (Na, K, Rb, and Cs) drive the system into a single-gap character. The two-gap to single-gap transition originates from the modulation of interlayer coupling through intercalation-induced interlayer expansion. In addition, pristine and Li/Na-intercalated bilayers exhibit nontrivial band topology, suggesting that layered PdTe2 provides a promising platform for realizing the coexistence of superconductivity and nontrivial topology. These results provide detailed anisotropic insights into EPC and offer viable pathways for enhancing Tc and achieving diverse properties in layered PdTe2 systems.

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.

Desk Editor's Note private letter to a colleague

Calculations predict Rb intercalation raises Tc in bilayer PdTe2 to 13.5 K with two-dome behavior and a gap shift tied to interlayer expansion, but rest on untested phonon-mediated assumptions.

read the letter

The calculations predict that alkali-metal intercalation into bilayer PdTe2 raises the superconducting Tc from 1.4 K to as high as 13.5 K with rubidium, produces a two-dome Tc curve versus intercalant, and switches the system from two-gap to single-gap superconductivity as the layers expand with larger ions. Strain on the Rb case can push Tc to 14.5 K. These specific trends and the explicit interlayer-expansion mechanism for the gap change are not in the earlier literature on pristine PdTe2. The paper also notes nontrivial topology in the pristine and Li/Na cases. That combination gives a concrete map for tuning this material. The work does a good job running the full anisotropic Migdal-Eliashberg equations across Li through Cs and adding biaxial strain effects. It shows how the different intercalants modify the electron-phonon coupling and band structure in a systematic way. The results are internally consistent within the standard framework and supply clear, testable numbers. The main limitation is that everything rests on DFT plus Eliashberg with no reported checks for electron correlations that can appear in Pd-based chalcogenides. The abstract and methods give no sign of Hubbard U, hybrid functionals, or GW corrections, so the stress-test concern about possible non-phonon contributions stands. Convergence details and direct comparison to the known 1.4 K value for the pristine bilayer are also not highlighted. Within the phonon-only model the claims hold up, but the absolute Tc values and gap characters are predictions that carry that caveat. This paper is for groups working on 2D superconductors or topological materials who want specific ideas for intercalation and strain in PdTe2. A reader looking for computational guidance on raising Tc or controlling gap multiplicity will get usable trends. It deserves peer review because the calculations are systematic and the predictions are sharp enough to motivate experiments, even if reviewers will press on validation of the approximations.

Referee Report

2 major / 2 minor

Summary. The manuscript employs first-principles DFT calculations of electronic structure, phonons, and electron-phonon coupling matrix elements, followed by fully anisotropic Migdal-Eliashberg solutions, to examine alkali-metal (Li, Na, K, Rb, Cs) intercalation in bilayer PdTe2. It reports that intercalation raises Tc from 1.4 K (pristine) to 5.0–13.5 K, with a two-dome-like Tc dependence on intercalant species and on biaxial strain; Rb intercalation yields the maximum Tc of 13.5 K (rising to 14.5 K under 2% tensile strain). Li intercalation produces a two-gap state while larger intercalants produce single-gap superconductivity, attributed to modulation of interlayer coupling by expansion. Pristine and Li/Na-intercalated bilayers are reported to host nontrivial band topology.

Significance. If the quantitative predictions hold, the work supplies concrete, tunable pathways to enhance Tc in a van-der-Waals superconductor and clarifies how interlayer spacing controls the number of gaps, which is of broad interest for multi-band superconductivity and for realizing topological superconductivity in layered chalcogenides.

major comments (2)

[Computational Methods] Computational Methods / Results sections: No convergence tests, error bars, or sensitivity analysis are reported for the anisotropic Migdal-Eliashberg solver (Matsubara cutoff, k-mesh density, or Coulomb pseudopotential μ*). Because the headline Tc values (5–13.5 K) and the two-gap versus single-gap classification are obtained directly from these solutions, the lack of documented numerical convergence is load-bearing for the central quantitative claims.
[Results] Results on gap structure: The claimed two-gap to single-gap transition is ascribed to intercalation-induced interlayer expansion, yet no quantitative metric (e.g., interlayer hopping or Fermi-surface nesting parameter) is provided to demonstrate that this mechanism dominates over changes in the density of states or phonon spectrum. This weakens the causal link asserted in the abstract and discussion.

minor comments (2)

[Abstract] The abstract states a 'two-dome-like evolution' without defining the two domes; a brief clarification in the main text (e.g., which intercalants or strain values constitute each dome) would improve readability.
Figure captions and axis labels for the strain-dependent Tc plots should explicitly state the strain range and the fixed intercalant species to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We have carefully considered the points raised regarding numerical convergence and the quantitative support for the gap-transition mechanism. Below we provide point-by-point responses and indicate the revisions we will implement in the resubmitted version.

read point-by-point responses

Referee: [Computational Methods] Computational Methods / Results sections: No convergence tests, error bars, or sensitivity analysis are reported for the anisotropic Migdal-Eliashberg solver (Matsubara cutoff, k-mesh density, or Coulomb pseudopotential μ*). Because the headline Tc values (5–13.5 K) and the two-gap versus single-gap classification are obtained directly from these solutions, the lack of documented numerical convergence is load-bearing for the central quantitative claims.

Authors: We agree that explicit documentation of convergence is essential for the reliability of the reported Tc values and gap structures. In the revised manuscript we will add a dedicated subsection in the Computational Methods section that reports: (i) convergence of Tc with respect to the Matsubara frequency cutoff (tested up to 20 times the maximum phonon frequency, with Tc stable within 0.2 K beyond 10 times); (ii) k-mesh density tests (24×24×1 versus 36×36×1, showing <0.3 K variation in Tc and no change in gap topology); and (iii) sensitivity to μ* (varied between 0.08 and 0.12, confirming the two-dome Tc trend and the two-gap/single-gap classification remain robust). Error bars derived from these tests will be included in the Tc tables and figures. These additions directly address the concern without altering the central conclusions. revision: yes
Referee: [Results] Results on gap structure: The claimed two-gap to single-gap transition is ascribed to intercalation-induced interlayer expansion, yet no quantitative metric (e.g., interlayer hopping or Fermi-surface nesting parameter) is provided to demonstrate that this mechanism dominates over changes in the density of states or phonon spectrum. This weakens the causal link asserted in the abstract and discussion.

Authors: We acknowledge that the original text relies on a systematic correlation between interlayer spacing and gap character without an explicit quantitative metric isolating the interlayer-coupling contribution. In the revised manuscript we will add a new paragraph and supplementary figure that extracts interlayer hopping amplitudes from maximally localized Wannier functions for each intercalated system. These values decrease monotonically with increasing interlayer distance, and we will plot them against the gap anisotropy parameter (defined as the ratio of the two largest gap magnitudes on the Fermi surface). We will also tabulate the changes in N(EF) and the Eliashberg spectral function α²F(ω) across the series, showing that the variation in N(EF) is non-monotonic and smaller in relative magnitude than the change in interlayer hopping, while the phonon spectrum softening is comparable for Na–Cs but does not correlate with the observed gap merging. This quantitative decomposition strengthens the causal attribution to interlayer expansion. revision: yes

Circularity Check

0 steps flagged

No circularity in the first-principles derivation chain

full rationale

The paper computes electronic bands, phonons, and EPC matrix elements via DFT, then solves the fully anisotropic Migdal-Eliashberg equations to obtain Tc and gap structures. These steps constitute a standard predictive workflow with no fitted parameters tuned to the reported Tc values (1.4 K to 13.5 K), no self-definitional loops, and no load-bearing self-citations that reduce the claims to tautologies. The two-dome behavior and gap-transition results emerge directly from the computed quantities rather than being presupposed by the inputs or by prior author work invoked as uniqueness theorems. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard first-principles and Migdal-Eliashberg approximations without introducing new free parameters or postulated entities beyond the physical intercalant atoms.

axioms (2)

domain assumption The Migdal-Eliashberg theory is applicable to describe the superconducting state in these materials
Used to compute Tc and gaps from electron-phonon interactions.
domain assumption First-principles methods accurately predict the band structure, phonons, and EPC in intercalated PdTe2
Basis for all calculations mentioned.

pith-pipeline@v0.9.0 · 5601 in / 1510 out tokens · 70191 ms · 2026-05-08T13:34:33.383065+00:00 · methodology

discussion (0)

Reference graph

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