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Pressure-Induced Enhancement of Superfluid Density in Transition Metal Dichalcogenides with and without Charge Density Wave

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Hydrostatic pressure raises the superfluid density of the layered superconductor 4H-NbSe2 by 75%, while leaving its superconducting gaps nearly unchanged.

desk verdict Solid pressure-muSR data on two TMDs, with a real caveat about the clean-limit conversion that inflates the headline 75% number; worth peer review. read the letter →

arxiv 2412.14381 v1 pith:OEF6R2VB submitted 2024-12-18 cond-mat.supr-con cond-mat.mtrl-sci

S.S. Islam , V. Sazgari , C. Witteveen , J.N. Graham , O. Gerguri , P. Král , M. Bartkowiak , H. Luetkens
show 3 more authors
R. Khasanov F.O. von Rohr Z. Guguchia
This is my paper · ORCID
classification cond-mat.supr-concond-mat.mtrl-sci
keywords superfluiddensitymuonspinrotationhydrostaticpressurechargewavetransitionmetaldichalcogenides4H-NbSe22H-NbS2unconventionalsuperconductivity
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 reports that compressing the layered superconductor 4H-NbSe$_2$ to about 2 GPa increases its superfluid density—the density of electrons that carry the superconducting current—by 75%, while the superconducting transition temperature rises only modestly. A smaller but robust 20% increase is seen in 2H-NbS$_2$, a compound with no charge-density-wave order. The authors argue that the large superfluid response cannot be attributed simply to pressure suppressing the competing charge-density-wave state, because the CDW is only mildly reduced and the increase is even larger in the compound without a CDW. The near-universal correlation between superfluid density and $T_c$ across these TMDs, cuprates, kagome, and iron-based superconductors suggests a common unconventional pairing mechanism. If correct, this points to pressure as a clean tuning knob for superfluid stiffness that is largely decoupled from $T_c$.

What carries the argument

The measurement engine is transverse-field muon spin rotation (TF-µSR): implanted muons precess in the vortex lattice of the superconductor, and the Gaussian relaxation rate $\sigma_{sc}$ is converted to an effective magnetic penetration depth $\lambda_{\rm eff}$ through the vortex-lattice relation $\sigma_{sc}/\gamma_\mu = 0.06091\,\Phi_0\, \lambda_{\rm eff}^{-2}$ (Brandt's result). Under the clean-limit assumption (coherence length $\xi$ much larger than mean free path $l$, so the factor $1/(1+\xi/l)\approx 1$), $\lambda_{\rm eff}^{-2}$ is proportional to the superfluid density $n_s/m^*$. The temperature dependence is analyzed with an $(s+s)$-wave two-gap model with pressure-independent weight factors, so the zero-temperature superfluid density can be tracked versus pressure. Magnetotransport (resistivity, Hall, magnetoresistance) independently locates $T_c$ and the CDW onset.

What would settle it

Measure the residual resistivity or use de Haas–van Alphen oscillations on the same crystals under pressure to estimate the mean free path $l$. If $l$ decreases by roughly a factor of two over 2 GPa, the $1/(1+\xi/l)$ correction would grow enough that the apparent 75% rise in $\lambda_{\rm eff}^{-2}$ could be largely a mean-free-path artifact rather than a true increase in $n_s/m^*$.

Watch

Extended reading notes

Core claim

The central claim is that hydrostatic pressure strongly enhances the zero-temperature superfluid density $n_s/m^*$ in niobium-based transition metal dichalcogenides, with the largest effect in 4H-NbSe$_2$: a 75% increase relative to ambient pressure at 2.05 GPa. In 2H-NbS$_2$, which has no charge-density-wave order, the enhancement is 20% at 1.8 GPa, and in 2H-NbSe$_2$ it is 32% at 2.2 GPa (from prior work). In all cases the superconducting gaps remain essentially pressure-independent, and $T_c$ changes only by about 0.2–0.9 K. Because the CDW onset temperature falls by only about 20% over the same pressure range in the selenides, the authors conclude that the superfluid-density enhancement is not a simple consequence of CDW suppression, and they highlight a pressure-independent slope in the $T_c$ versus $n_s/m^*$ scaling that resembles cuprate and iron-based superconductors.

Load-bearing premise

The reported superfluid-density increase assumes the clean limit, where the mean free path is long enough that the $\xi/l$ correction in the penetration-depth relation is negligible; the paper does not measure the mean free path.

Editorial extensions

If this is right

  • A 75% increase in superfluid density at only 2 GPa means pressure can nearly double the supercurrent-carrying capacity of 4H-NbSe$_2$ without a corresponding rise in $T_c$, a response that standard BCS theory does not predict.
  • The same slope in the $T_c$ vs $n_s/m^*$ scaling across 2H-NbSe$_2$, 4H-NbSe$_2$, and 2H-NbS$_2$ indicates a common microscopic mechanism for the pressure effect in the NbX$_2$ family.
  • Because 2H-NbS$_2$ shows a 20% enhancement with no CDW, a quantitative theory of these compounds must explain superfluid-density growth through a channel other than CDW suppression—such as pressure-driven changes in electron-phonon coupling, $p$-$d$ hybridization, or Fermi-surface topology.
  • The pressure independence of the two superconducting gaps constrains the pairing: pressure changes the condensate fraction without changing the gap magnitudes, which is consistent with phase-fluctuation or stiffness-driven pictures.
  • The addition of these TMDs to the Uemura-type scaling plot (with cuprates, iron-based, and kagome superconductors) suggests that a common mechanism may link superfluid density and $T_c$ across very different materials.

Reading between the lines

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

  • A testable extension not pursued here is to measure the mean free path (from residual resistivity or quantum oscillations) under the same pressures; this would test whether the clean-limit assumption holds and whether the 75% is entirely a superfluid-density effect.
  • The authors' explanation invokes pressure-driven changes in electron-phonon coupling, $p$-$d$ hybridization, and a saddle point near the Fermi level; first-principles band-structure calculations under pressure could verify which of these dominates.
  • The scaling plot places these TMDs on a Uemura-type correlation, which in cuprates has been read as BEC-like pairing; confirming that would require an independent measure of the effective mass $m^*$ to separate changes in $n_s$ from changes in $m^*$.
  • If the effect is a stiffness enhancement decoupled from $T_c$, uniaxial strain or chemical pressure in related TMDs might achieve similar or larger superfluid-density gains without raising $T_c$.
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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

4 major / 6 minor

Summary. This manuscript reports transverse-field muon spin rotation (TF-muSR) and magnetotransport experiments on 4H-NbSe2 and 2H-NbS2 under hydrostatic pressure. The authors extract the superconducting transition temperature, the vortex-lattice relaxation rate, and the effective penetration depth, converting the latter into superfluid density ns/m*. They report a 75% increase in superfluid density for 4H-NbSe2 at 2.05 GPa, a 20% increase for 2H-NbS2 at 1.8 GPa, and compare these with a 32% increase previously reported for 2H-NbSe2. Magnetotransport is used to map the CDW onset and a Hall sign-reversal temperature in 4H-NbSe2, showing only modest CDW suppression under pressure. The temperature dependence of the relaxation rate is analyzed with a two-gap (s+s)-wave model, yielding nearly pressure-independent gaps. The paper argues for an unconventional correlation between Tc and superfluid density across TMDs and draws analogies with cuprates, kagome, and iron-based superconductors.

Significance. If the clean-limit conversion is valid, the reported 75% enhancement of superfluid density in 4H-NbSe2 is a striking and quantitatively important result. The study is valuable because it uses a direct bulk probe (TF-muSR), covers both a CDW-bearing and a CDW-free compound, and provides a systematic pressure comparison with 2H-NbSe2. The finding that the superconducting gaps are essentially pressure-independent while the superfluid density changes substantially would impose strong constraints on models of superconductivity in TMDs. However, the headline numbers depend on an unmeasured mean free path through Eq. (1), and the cross-material slope claim in Fig. 5 is not quantified. These issues need to be resolved before the quantitative claims are fully supported.

major comments (4)
  1. [Eq. (1) and the following paragraph] The central quantitative claim, the 75% enhancement of ns/m* in 4H-NbSe2 reported in Fig. 4(c), is obtained from Eq. (1) by setting the clean-limit factor 1/(1+xi/l) to unity. The text correctly states that no accurate estimate of l is available and justifies this by the small coherence length xi ~ 3.5-3.8 nm. This justification is not sufficient: for a finite mean free path of order 10 nm, the factor is about 0.74, and if l grows from 10 nm to 25 nm under pressure, the conversion factor alone changes by about 18%, a substantial fraction of the reported 75% enhancement. Since the Hall and magnetoresistance data in Fig. 2 indicate Fermi-surface reconstruction and changing scattering under pressure, a pressure-dependent l is plausible. The authors should either provide a bound on l(p) from independent measurements or supply a sensitivity analysis showing how the reported relative changes of sigma_sc translate into ns/m* for a range of xi/l values. Without this, the 75% and 20% figures cannot be attributed purely to changes in superfluid density.
  2. [Fig. 5 and the discussion of it] The statement that "the slopes for 2H-NbSe2, 4H-NbSe2 and 2H-NbS2 are the same" is not supported by any quantitative fit or uncertainty estimate. Figure 5 shows only a few points without visible error bars, and the three TMD data sets cover different pressure ranges and different numbers of points. To claim a universal Tc-ns/m* correlation, the authors should report slopes with confidence intervals and goodness-of-fit measures; otherwise the comparison with cuprates and iron-based superconductors remains qualitative.
  3. [Methods, Eq. (6) and the following text] The pressure-cell background subtraction below Tc introduces a model-dependent linear coupling between sigma_pc and DeltaB_dia, with an unspecified function C(T). The value sigma_pc(T>Tc)=0.25 micros^-1 is given, but C(T) is not defined or constrained. Because roughly 60% of muons stop in the pressure cell rather than in the sample, the authors should demonstrate that the extracted sigma_sc(p) is robust to this modeling choice, or provide the form of C(T) and its uncertainty.
  4. [Fig. 4 and the paragraph citing extracted parameters] The 75% value is quoted from zero-temperature values obtained through the (s+s)-wave fits, which include two gaps and a weight factor, rather than directly from the low-temperature plateau of sigma_sc. Since the abstract and main text emphasize relative changes, the authors should show explicitly that the pressure-induced enhancement persists when the low-T data (for example, sigma_sc at 0.3 K) are ratioed directly, separated from the model-dependent two-gap extrapolation.
minor comments (6)
  1. [Methods, Eq. (3)] The functional form in Eq. (3) is described as a two-component fit, but the sum runs from i=0 to 2, while the moment formulas in Eqs. (4) and (5) use As,1+As,2 as the denominator. Please fix the index convention.
  2. [Reference [26]] Reference [26] is a placeholder with a 'LINK' for Supplemental Material; the actual link and the content of the Supplemental Material should be resolved before publication.
  3. [Discussion of the Hall effect] The phrase 'emergence of a secondary CDW order appears to induce Fermi-surface reconstruction' is speculative: the transport data show a Hall sign reversal but do not directly identify a secondary CDW order. Please soften this statement or provide supporting evidence.
  4. [Methods, pressure data analysis] In the pressure-cell analysis, the text introduces sigma_ns and sigma_pc but does not repeat the separation sigma_sc = sqrt(sigma^2 - sigma_ns^2) for the pressure runs. Clarify how the sample and cell relaxation rates are separated at pressure, especially given the temperature-dependent cell term below Tc.
  5. [Fig. 2 caption] Figure 2 uses many line colors and symbols without a panel-specific legend in the caption; a consistent legend or a table of Tc, TCDW, and T* values would improve readability.
  6. [Final discussion] The sentence 'the ratio Tc/TF is reduced by approximately 20 times compared to cuprates' lacks a definition of TF and a reference for the cuprate value; please specify both.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 75% superfluid-density enhancement is read from the zero-temperature muon relaxation-rate plateau, and Tc comes from independent transport and muSR data; the clean-limit factor is a stated assumption, not a derived input.

full rationale

The central claim (75% increase of ns/m* in 4H-NbSe2 at 2 GPa) is obtained from the pressure dependence of sigma_sc(0) in TF-muSR, which is an experimental observable read from the low-T plateau of the relaxation rate. Eq. (1) is the standard London penetration-depth relation; setting 1/(1+xi/l)=1 is an explicitly stated clean-limit assumption (the paper says 'reasonable to assume a very little effect from xi/l'), not an equation that encodes the claimed pressure enhancement. The (s+s)-wave alpha-model in Eq. (2) is only used to characterize the temperature dependence and to extract the gap values and weights; the relative superfluid-density change is carried by the measured sigma_sc(0,p)/sigma_sc(0,0) ratios. Tc is measured independently by resistivity and muSR. The comparison with 2H-NbSe2 uses previously published muSR data from the same group (Ref. [1]), but that is experimental data external to this paper's fitting, not a self-referential derivation. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is smuggled in via self-citation. The unmeasured mean free path is a genuine limitation (a pressure-dependent l could affect the size of the effect), but it is a caveat about an assumption, not circularity.

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

The central measurement (vortex-lattice relaxation rate) is direct, but converting it to superfluid density requires several domain assumptions, most notably the clean-limit condition with an unmeasured mean free path. The two-gap alpha-model introduces six fitted parameters (two gaps and one weight for each compound) that support the secondary claim that gaps are pressure-independent but do not affect the primary relative enhancement. No new physical entities are introduced.

free parameters (6)
  • Delta1 (small gap) for 4H-NbSe2
    Fitted from sigma_sc(T) using the (s+s)-wave alpha-model; value shown in Fig. 8 but not quoted in text. Used to support the claim that gaps are pressure-independent.
  • Delta2 (large gap) for 4H-NbSe2
    Second gap parameter in the alpha-model fit for 4H-NbSe2; value shown in Fig. 8.
  • Delta1 (small gap) for 2H-NbS2
    Fitted gap parameter for 2H-NbS2 from sigma_sc(T); value shown in Fig. 8.
  • Delta2 (large gap) for 2H-NbS2
    Second fitted gap parameter for 2H-NbS2; value shown in Fig. 8.
  • omega (weight factor) for 4H-NbSe2 = 0.54
    Relative weight of the two gap components in the (s+s)-wave fit; reported as approximately 0.54 and pressure-independent.
  • omega (weight factor) for 2H-NbS2 = 0.52
    Relative weight of the two gap components for 2H-NbS2; reported as approximately 0.52 and pressure-independent.
assumptions (6)
  • domain assumption Brandt vortex-lattice relation sigma_sc/gamma_mu = 0.06091 Phi0 lambda_eff^-2
    Used in Eq. (1) to convert the measured muon relaxation rate to penetration depth for an ideal triangular vortex lattice in a type-II superconductor.
  • domain assumption Local (London) approximation lambda >> xi
    Assumed in Eq. (7) and in the interpretation of lambda_eff. Justified by coherence lengths around 3.5-3.8 nm and penetration depths in the micrometer range.
  • domain assumption Clean-limit condition xi/l << 1
    The paper sets the factor 1/(1+xi/l) to unity without measuring the mean free path l. This is the weakest assumption, as stated in the text right after Eq. (1).
  • domain assumption Two-gap (s+s)-wave alpha-model
    The temperature-dependent sigma_sc is modeled with two independent s-wave gaps sharing a common Tc. This is a phenomenological fit model, not a derived electronic structure result.
  • domain assumption Polycrystalline powder-average relation for lambda_eff
    Samples are polycrystalline, and the effective penetration depth is treated with a powder average, which is standard in muSR but not explicitly derived in this paper.
  • ad hoc to paper Linear coupling model for pressure-cell background below Tc
    The pressure-cell relaxation rate sigma_pc(T) below Tc is modeled as sigma_pc(T>Tc) + C(T)(Delta B_dia), with an uncalibrated coupling coefficient C(T). This affects the extracted sigma_sc under pressure.

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Pith. "Pith review of Pressure-Induced Enhancement of Superfluid Density in Transition Metal Dichalcogenides with and without Charge Density Wave." pith.science (2026). https://pith.science/paper/OEF6R2VB

@misc{pith2026241214381,
  author       = {Pith},
  title        = {Pith review of: Pressure-Induced Enhancement of Superfluid Density in Transition Metal Dichalcogenides with and without Charge Density Wave},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEF6R2VB}},
  note         = {Machine review of arXiv:2412.14381}
}
abstract

Gaining a deeper understanding of the interplay between charge density wave (CDW) order and superconductivity in transition metal dichalcogenides (TMDs), particularly within the (4H/2H)-NbX$_{2}$ (X=Se,S) family, remains an open and intriguing challenge. A systematic microscopic study across various compounds in this family is therefore required to unravel this complex interplay. Here, we report on muon spin rotation and magnetotransport experiments investigating the effects of hydrostatic pressure on the superconducting transition temperature ($T_{\rm c}$), the temperature-dependent magnetic penetration depth ($\lambda_\mathrm{eff}$), and the charge density wave order (CDW) in two layered chalcogenide superconductors: 4H-NbSe$_{2}$, which exhibits CDW order, and 2H-NbS$_{2}$, which lacks such order. Our observations reveal a substantial 75$\%$ enhancement of the superfluid density ($n_{s}/m^{*}$) in 4H-NbSe$_{2}$ upon maximum applied pressure of 2 GPa, surpassing that of 2H-NbSe$_{2}$. Despite the absence of CDW order, a sizeable 20$\%$ growth in superfluid density is also observed for 2H-NbS$_{2}$ under an applied pressure of 1.8 GPa. Notably, the evaluated superconducting gaps in all these TMDs remain largely unaffected by changes in applied pressure, irrespective of pressure-induced partial suppression of CDW order in (4H/2H)-NbSe$_{2}$ or its general absence in 2H-NbS$_{2}$. These results underscore the complex nature of pressure-induced behaviors in these TMDs, challenging a simplistic view of competition solely between CDW order and superconductivity. Remarkably, the relationship between $n_{s}/m^{*}$ and $T_{\rm c}$ exhibits an unconventional correlation, indicating a noteworthy similarity with the behavior observed in cuprate, kagome, and iron-based superconductors.

Figures

Figures reproduced from arXiv: 2412.14381 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a): Top: Experimental pattern of 2H-NbS [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Pressure dependence of small superconducting gap [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

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