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REVIEW 3 major objections 6 minor 51 references

Superconducting gap and its Little-Parks like oscillations with high-order harmonics in lithium intercalated 1T-TiSe$_2$

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Tunneling spectroscopy shows that the superconducting gap of lithium-intercalated TiSe2 closes well above the BCS transition temperature and oscillates with magnetic field.

desk verdict First tunneling-spectroscopy gap measurement in Li-intercalated TiSe2; the data are convincing, but the low 2Δ/Tg fluctuation claim needs tighter fitting and error analysis. read the letter →

arxiv 2506.23871 v1 pith:4UI5RI5L submitted 2025-06-30 cond-mat.supr-con

Jia-Yi Ji , Zongzheng Cao , Yi Hu , Haoyang Wu , Heng Wang , Yuying Zhu , Haiwen Liu , Lexian Yang
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Qi-Kun Xue Ding Zhang
This is my paper
classification cond-mat.supr-con
keywords superconductinggaptunnelingspectroscopylithiumintercalation1T-TiSe2Little-Parksoscillationanomalousmetalstateparticle-holesymmetryfluctuations
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

The paper reports tunneling spectroscopy of the superconducting gap in lithium intercalated 1T-TiSe$_2$, a quantity that had not been measured for this system. The authors find that the gap closes at $T_g = 0.42$ K and $0.76$ K for two doping levels, far above the BCS transition temperatures $0.22$ K and $0.50$ K derived from the zero-temperature gaps $\Delta(0) = 33\ \mu$eV and $76\ \mu$eV. They interpret the resulting ratios $2\Delta/T_g \approx 1.82$ and $2.31$ as strong superconducting fluctuations in a 70 nm film that is otherwise bulk-like. They also observe that the gap remains symmetric about zero bias in the anomalous metal state and that it oscillates in a perpendicular magnetic field with period $\Delta B = 9.01$ mT and five FFT harmonics. If correct, these measurements make the superconducting gap of doped TiSe$_2$ directly accessible and tie it to the particle-hole symmetry of failed superconductivity and to a regularly patterned superconducting network.

What carries the argument

The load-bearing object is the superconducting gap $\Delta(T,B)$ extracted from normal-insulator-superconductor tunneling spectra. The device stacks a Ti/Au electrode on a 3 nm MoS$_2$ barrier on 70 nm Li$_x$TiSe$_2$, with a solid ion conductor providing lateral lithium intercalation; the spectra are fitted by the Blonder-Tinkham-Klapwijk formula assuming a single $s$-wave gap, yielding $\Delta(T)$. The magneto-oscillation analysis uses the Little-Parks relation $S_1\Delta B = h/2e$, with the FFT of zero-bias conductance giving the fundamental frequency $f_1 = 111$ T$^{-1}$ and harmonics $f_2$--$f_5$, from which loop areas are inferred. A key interpretive mechanism is the assumed texture of commensurate CDW domains (non-superconducting) bounded by incommensurate CDW domain walls (superconducting), which supplies the loops needed for Little-Parks-like oscillations and the fluctuations that suppress $2\Delta/T_g$.

What would settle it

Tunneling into the same Li$_x$TiSe$_2$ material through a different barrier (for example, hexagonal boron nitride) and with a smaller-area junction should reproduce $\Delta(0) \approx 33/76\ \mu$eV, $T_g \approx 0.42/0.76$ K, and the symmetric zero-bias dip in the anomalous metal; if the apparent gap and its closing temperature instead track the barrier material or the junction area, the central claims are refuted.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the superconducting gap of Li$_x$TiSe$_2$ is real but unusual: $\Delta(0) = 33\ \mu$eV (state-2) and $76\ \mu$eV (state-3), yet the gap does not close at the BCS transition temperatures $0.22$ K and $0.50$ K derived from $2\Delta/T_c = 3.53$, but at $T_g = 0.42$ K and $0.76$ K. The small ratios $2\Delta/T_g \approx 1.82$ and $2.31$ are interpreted as strong superconducting fluctuations, not BCS-BEC crossover, because $E_F \approx 100$--$200$ meV makes $\Delta/E_F \sim 3\times 10^{-4}$. In the anomalous metal state under a magnetic field, the gap remains symmetric about zero bias, which the authors present as direct evidence of particle-hole symmetry and incoherent Cooper pairs. The zero-bias conductance also oscillates with perpendicular field at period $\Delta B = 9.01$ mT, and the FFT shows five harmonics $f_1$--$f_5$; Eq. (1) gives a fluxoid area $S_1 = 0.23\ \mu$m$^2$, and the harmonics are attributed to Cooper pairs circulating around combined non-superconducting domains. Together these observations are presented as evidence for a regularly structured superconducting network embedded in the charge-density-wave state, with the gap oscillations providing a local probe of that network.

Load-bearing premise

The argument collapses if the BTK $s$-wave single-gap fits return an apparent gap shaped by the MoS$_2$ barrier, the Ti/Au electrode, or the junction geometry rather than the intrinsic bulk pairing gap of Li$_x$TiSe$_2$, especially near $T_g$ where $k_B T \approx \Delta$ and broadening dominates the fits.

Editorial extensions

If this is right

  • If the ratios $2\Delta/T_g \approx 1.82$ and $2.31$ are intrinsic, then a 70 nm film of Li$_x$TiSe$_2$ belongs to the fluctuation-dominated class of superconductors, and bulk-like thickness alone does not guarantee BCS mean-field behavior.
  • If the symmetric gap in the anomalous metal state is a genuine density-of-states signature, then particle-hole symmetry in failed superconductors can be probed directly by tunneling rather than only by Hall measurements.
  • If the $9.01$ mT period and its harmonics reflect the superconducting network, then the network is regular enough to produce coherent multi-loop circulation, with an elementary fluxoid area $S_1 \approx 0.23\ \mu$m$^2$ (about 730 nm per side for a triangular lattice).
  • If the gap oscillations persist only to about 0.3 K and then shift to a larger area ($\approx 0.72\ \mu$m$^2$), the network itself is temperature dependent, with non-superconducting domains merging as the temperature rises.
  • The order-of-magnitude difference between the equilibrium gap ($33$--$76\ \mu$eV) and the light-induced 2 meV gap in TiSe$_2$ implies that the optically induced state is not the same equilibrium superconductor.

Reading between the lines

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

  • A testable extension the paper leaves implicit is to apply the same FFT harmonic analysis to other CDW superconductors (for example, Cu$_x$TiSe$_2$ or NbSe$_2$), using the harmonic content as a quantitative measure of how regularly the superconducting network is patterned.
  • If the small $2\Delta/T_g$ originates in phase fluctuations tied to the CDW domain network, then samples with larger or more disordered commensurate domains should show an even smaller ratio and weaker harmonics; this doping- or strain-dependent prediction could be checked without new theory.
  • The persistent symmetric zero-bias dip in the anomalous metal could be tested as a pseudogap by sweeping to magnetic fields high enough to suppress all superconducting correlations; the 30 mT data shown do not yet locate the upper critical field of the gap feature.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript reports tunneling spectroscopy measurements on lithium-intercalated 1T-TiSe2 using a solid-state ion-gated device that combines lateral lithium intercalation, resistance measurements, and planar tunneling through a 3-nm MoS2 barrier. The authors observe a superconducting gap in two doping states, with zero-temperature gap values Δ(0)=33 μeV (state-2) and 76 μeV (state-3), and gap closing temperatures Tg=0.42 K and 0.76 K, respectively. They note that 2Δ(0)/Tg ≈ 1.82 and 2.31, well below the BCS ratio of 3.53, and interpret this as evidence for pronounced superconducting fluctuations in a bulk-like 70-nm-thick flake. The paper further reports that the gap remains symmetric in the anomalous metal state, which they interpret as direct evidence of particle-hole symmetry, and that the gap oscillates as a function of perpendicular magnetic field with a fundamental period ΔB = 9.01 mT and higher harmonics up to the fifth order, from which they extract a loop area S1 = 0.23 μm² using the flux quantum relation. These findings are framed as the first direct observation of the superconducting gap in Li-intercalated TiSe2.

Significance. If the measurements are sound, the paper makes several notable contributions: it provides the first tunneling spectroscopy of the superconducting gap in Li-intercalated TiSe2, a system that has been studied primarily through transport; it demonstrates that a gap-like feature persists in the anomalous metal state, offering a direct DOS-based probe of particle-hole symmetry; and it reveals higher harmonics in the magneto-oscillations, supporting a regular superconducting network structure. The experimental setup includes thorough low-temperature filtering, which strengthens the anomalous-metal claim. However, the headline conclusion of strongly enhanced superconducting fluctuations depends critically on the reliability of the extracted Δ(0) and Tg from BTK fits. The current manuscript does not yet provide sufficient validation of these fits, and the relationship between Tg and the measured resistive transition temperature is not addressed. These points need to be resolved before the fluctuation claim can be accepted.

major comments (3)
  1. [§2, Fig. 1(b) and Fig. 2(b)] The gap closing temperature Tg=0.76 K for state-3 is lower than the resistive transition midpoint Tc=0.85 K reported in Fig. 1(b). This is opposite to the behavior expected for a superconducting gap that closes at or above the resistive transition, and it raises the possibility that the tunneling junction probes a region that is not representative of the bulk. The authors compare Tg only with Tc^BCS derived from the measured Δ via the BCS ratio, which is circular in the context of evaluating fluctuations. Please reconcile Tg with the transport Tc (including the zero-resistance temperature) and justify that the junction area is representative, or discuss how a local suppression of the gap would affect the 2Δ/Tg ratio.
  2. [§2, Figs. 2(c)(d) and Fig. S5] The extracted gap values and closing temperatures are obtained from BTK fits, but the manuscript does not show the normal-state conductance G_N(V) or demonstrate that it is flat over the bias window used for the fits, nor does it provide error bars on Δ(T). At Tg, k_BT equals 36 μeV (state-2) and 65 μeV (state-3), of the same order as Δ(0), so the fits near Tg are underdetermined unless thermal smearing and Dynes broadening are handled explicitly. The authors should provide a background-subtracted analysis with error bars, show representative fits at several temperatures (including near Tg), and test the sensitivity of Δ(0) and Tg to the choice of fit model (e.g., BTK vs. Dynes, and inclusion of an energy-dependent normal-state conductance). Without this, the quantitative claim that 2Δ/Tg is far below the BCS ratio is not established.
  3. [§3, Eq. (1) and Fig. 4(a)] The interpretation of the 9.01 mT period as a Little-Parks-like oscillation assumes that the period corresponds to the flux quantum through a single loop. The extracted area S1=0.23 μm² is one order of magnitude smaller than the junction area (3.4 μm²), and the higher harmonics are attributed to either multiple Cooper pairs (n>1) or multiple enclosed loops. The manuscript does not provide independent evidence for the real-space network area, and the deviation of the fifth harmonic from the linear trend in Fig. 4(a) is not quantified. A more direct comparison with the Little-Parks effect (e.g., measuring the oscillation amplitude versus temperature and its relationship to Δ(T)) would strengthen the interpretation.
minor comments (6)
  1. [Abstract and §2] The phrase 'the gap closing temperature well exceeds the transition temperature (Tc) expected from BCS theory' should specify that this Tc is Tc^BCS derived from the measured Δ, not the measured resistive Tc; otherwise it is misleading given the resistive Tc of 0.85 K for state-3.
  2. [§2, Fig. 2 caption] The definition of the gap closing temperature Tg should be stated explicitly, e.g., the temperature at which the fitted Δ extrapolates to zero or at which the coherence peaks vanish.
  3. [§2, Fig. 2(e)] The normal-state conductance at 1.4 K (black curve in Fig. 2(e)) should be shown over a wider bias range to demonstrate that it is flat and symmetric; this is important for the background subtraction of the BTK fits.
  4. [Methods and Fig. S5] The authors should report the fit parameters used in the BTK analysis (e.g., the barrier strength Z, the Dynes broadening Γ, and the normalization procedure) for each state.
  5. [Eq. (2)] The integer n is used for the harmonic order and also for the carrier density n in the text; consider using a different symbol (e.g., N) to avoid confusion.
  6. [§3, Fig. 4(a)] In Fig. 4(a), the error bars are the FWHM of the FFT peaks; please also provide the numerical values of the peak frequencies and the uncertainty of the period ΔB derived from the fundamental peak.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: gap extraction, BCS comparison, and flux-quantum area analysis rest on external benchmarks and original data.

full rationale

The derivation chain is self-contained. The superconducting gap Δ(T) is obtained by fitting measured tunneling spectra to the standard BTK formula (ref 35), an external model with s-wave pairing assumed from prior work on the related CuxTiSe2 system (refs 36,37); the extracted Δ(0) values (33 and 76 µeV) are then compared with the BCS gap-to-Tc ratio 3.53, an external benchmark, and with the independently fitted gap-closing temperature Tg (0.42 and 0.76 K). The parenthetical '2Δ/Tg = 3.53' is a typographical slip for the BCS ratio 2Δ/Tc; the numerical comparison shows Tc_BCS = 0.22/0.50 K and Tg = 0.42/0.76 K, so Tg is not set equal to Tc_BCS by construction. The magneto-oscillation analysis uses Eq. (1), S1 ΔB = h/2e, which is the standard flux-quantum condition applied to the measured FFT period f1 = 111 T^-1 (ΔB = 9.01 mT); the area S1 = 0.23 µm² follows from an external constant, not from a fitted parameter designed to reproduce the conclusion. The higher-harmonic interpretation through Eq. (2) is algebraically consistent with either larger loops (n=1, Sm = mS1) or higher-charge flux quantization (Sm = S1, n=m), and the paper argues against the latter using external data (ref 49). Self-citations (refs 13, 17, 28, 29) appear for the lateral intercalation technique, prior resistance oscillations, and the anomalous-metal background, but the central claims are supported by new tunneling data and external benchmarks, so no load-bearing step reduces to its own inputs or to an unverified self-citation chain.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on two fitted quantities (Δ(0) from BTK fits, Tg read from the Δ(T) curves) and two external benchmarks (BCS ratio 3.53, flux quantum h/2e). The interpretive load is carried by domain assumptions: s-wave BTK validity, bulk-like behavior at 70 nm, intrinsic anomalous metal, and the CCDW/ICDW network model. No invented entities.

free parameters (4)
  • Delta(0) from BTK fit, state-2 = 33 µeV
    Superconducting gap amplitude at zero temperature extracted from BTK fits to tunneling spectra; the central measured quantity and an input to the 2Δ/Tg fluctuation analysis.
  • Delta(0) from BTK fit, state-3 = 76 µeV
    Same as above for state-3; used to compute Tc_BCS = 0.50 K and 2Δ/Tg = 2.31.
  • Dynes broadening Gamma in BTK fit = not reported
    Broadening parameter in the BTK/Dynes fits (Fig. S5); influences the extracted Δ at high temperature where k_BT ≈ Δ, so Tg carries fit sensitivity.
  • Gap closing temperature Tg = 0.42 K (state-2), 0.76 K (state-3)
    Determined by inspection of the Δ(T) curves; a subjective endpoint choice that directly sets the 2Δ/Tg ratios.
assumptions (6)
  • standard math BCS weak-coupling ratio 2Δ/k_B T_c = 3.53 is the correct benchmark for judging the gap-closing temperature.
    Used to convert Δ(0) into Tc_BCS = 0.22 K and 0.50 K and to define the fluctuation deviation. Presumes weak-coupling BCS; a strong-coupling material would have a larger ratio, which would make the deviation even more unusual, so the direction of the conclusion is robust.
  • domain assumption The BTK model with s-wave single-gap pairing describes the NIS tunneling spectra.
    Assumed following refs [36,37] for Cu_xTiSe2; if the pairing were nodal or multigap, the extracted Δ and the 2Δ/Tg analysis would change. The paper states 'assuming s-wave pairing' without independent justification for LixTiSe2.
  • standard math Flux quantization in the network obeys S·ΔB = h/2e (Eq. 1) with the single-Cooper-pair flux quantum.
    Standard Little-Parks relation invoked without derivation; used to convert the measured period 9.01 mT into a loop area of 0.23 µm².
  • domain assumption A 70 nm thick TiSe2 flake can be treated as a bulk system.
    Stated without comparing 70 nm to the relevant superconducting length scales (coherence length, penetration depth, or the 730 nm texture period); the headline claim of fluctuation-dominated behavior even in a bulk-like system depends on this.
  • domain assumption The low-temperature resistance plateau under magnetic field is an intrinsic anomalous metal state rather than a measurement artifact.
    Supported by the filter chain, the FQHS electron-temperature calibration (80 mK), and prior literature [32-34], but the identification is inherited from transport and is load-bearing for the particle-hole symmetry claim.
  • ad hoc to paper The oscillatory network is intrinsic and consists of commensurate CDW (non-superconducting) domains separated by incommensurate CDW domain walls (superconducting).
    The summary says the structure 'likely consists' of this domain arrangement; it is an interpretive model drawn from refs [13,15,50] and is used to explain both the 730 nm scale and the doping dependence of the oscillations. An extrinsic origin (e.g., inhomogeneous lithium intercalation) is acknowledged for vortex pinning but not fully excluded for the network itself.

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Pith. "Pith review of Superconducting gap and its Little-Parks like oscillations with high-order harmonics in lithium intercalated 1T-TiSe$_2$." pith.science (2026). https://pith.science/paper/4UI5RI5L

@misc{pith2026250623871,
  author       = {Pith},
  title        = {Pith review of: Superconducting gap and its Little-Parks like oscillations with high-order harmonics in lithium intercalated 1T-TiSe$_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4UI5RI5L}},
  note         = {Machine review of arXiv:2506.23871}
}
abstract

The superconducting phase of doped 1T-TiSe$_2$ is a fruitful playground for exploring exotic quantum phenomena such as the anomalous metal state and spontaneously formed superconducting network. Here, we address these emergent states by studying the superconducting gap of lithium intercalated TiSe$_2$-a fundamental quantity that has remained unexplored so far. We fabricate a device that combines solid-state lateral lithium intercalation, resistance measurements and tunneling spectroscopy. We successfully probe the superconducting gap of TiSe$_2$ and reveal that the gap closing temperature well exceeds the transition temperature ($T_c$) expected from the Bardeen-Cooper-Schrieffer theory, indicating pronounced superconducting fluctuations even in a bulk-like system. Moreover, the symmetric gap persists even in the anomalous metal state, demonstrating the particle-hole symmetry of this exotic phase directly from the density of states. Finally, the superconducting gap shows magneto-oscillations with higher harmonics, attesting to a rather regular structure of the intrinsic superconducting network.

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Reviewed August 6, 2026 · model on record in the stance chip above.