Superheating field of clean superconductors near the type-I--type-II boundary: the low-temperature Meissner stability limit of niobium

Takayuki Kubo

Superheating field of clean superconductors near the type-I--type-II boundary: the low-temperature Meissner stability limit of niobium

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Clean niobium near the type-I/II boundary sustains a superheating field of about 290 mT at low temperatures.

desk verdict Kubo's Eilenberger calculation puts the low-T superheating field for Nb-like parameters at 290 mT, above the GL extrapolation. read the letter →

arxiv 2606.10420 v1 pith:CKVRP6YY submitted 2026-06-09 cond-mat.supr-con physics.acc-ph

Takayuki Kubo This is my paper · ORCID

classification cond-mat.supr-conphysics.acc-ph

keywords superheatingfieldniobiumEilenbergertheoryMeissnerstatetype-Itype-IIboundarycleansuperconductorsacceleratorcavities

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 calculates the superheating field in clean superconductors close to the boundary between type-I and type-II behavior, with emphasis on niobium. It applies self-consistent nonlinear nonlocal Eilenberger theory plus linear stability analysis of the Meissner state. For a material with Ginzburg-Landau parameter 0.7, the field reaches roughly 290 millitesla at one-fifth of the critical temperature when the thermodynamic critical field is 200 millitesla. This exceeds the result of simply extending the Ginzburg-Landau expression from near the critical temperature down to low temperature. The calculation translates to an intrinsic limit of about 67 MV per meter for a TESLA-shaped niobium cavity.

What carries the argument

Self-consistent nonlinear nonlocal Eilenberger theory combined with linear stability analysis of the Meissner state.

What would settle it

A measurement on clean niobium at T/Tc=0.2 that finds the superheating field significantly below 290 mT for B_c0 around 200 mT would falsify the central numerical result.

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Extended reading notes

Core claim

For a Nb-like material with κ_GL=0.7, we obtain B_sh ≃ 290 mT at T/Tc=0.2, using B_c0 ≃ 200 mT. This value is substantially higher than the value obtained by naively extrapolating the Ginzburg--Landau result near T_c to T ≪ T_c. For a TESLA-shaped Nb accelerator cavity, it corresponds to an intrinsic Meissner-stability limit of about 67 MV/m.

Load-bearing premise

The material is clean and the Eilenberger theory plus linear stability analysis accurately captures the Meissner stability limit at low temperature near the type-I/type-II boundary.

Editorial extensions

If this is right

The Meissner stability limit for clean niobium exceeds the value obtained by downward extrapolation of the Ginzburg-Landau result.
A TESLA-shaped niobium cavity has an intrinsic limit near 67 MV/m set by this low-temperature superheating field.
The result applies specifically to clean superconductors near the type-I/type-II boundary.

Reading between the lines

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

Maintaining cleanliness in niobium could allow higher accelerating gradients in radio-frequency cavities than current models based on Ginzburg-Landau extrapolation suggest.
The same Eilenberger-based approach could be applied to other materials with Ginzburg-Landau parameter near 0.7 to predict their low-temperature limits.
Comparison with existing experimental data on niobium cavities at low temperature would test whether real materials approach the calculated clean-limit value.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central result is a numerical value: for κ_GL = 0.7 and B_c0 = 200 mT, the superheating field reaches about 290 mT at T/Tc = 0.2. That number comes from solving the self-consistent nonlinear nonlocal Eilenberger equations and then checking linear stability of the Meissner state.

The work applies the right microscopic framework for clean material at low temperature near the type-I/II boundary. GL theory is known to be limited there, so replacing the extrapolation with the full theory is the natural step. The abstract states the method and the inputs clearly.

The result still rests on externally chosen values for κ_GL and B_c0. If those shift even modestly, the quoted 290 mT moves with them. Real niobium also has some scattering, so the clean-limit assumption needs checking against experiment. The abstract gives no convergence tests or error bars, which is the usual place where these calculations can hide problems.

The paper is aimed at the SRF cavity community that cares about the intrinsic Meissner limit for accelerator gradients. Anyone working on Nb cavity design or on microscopic limits in type-I/II materials will want to see the number.

I would send it to referees. The method is established, the question is well-posed, and the claimed improvement over GL is concrete enough to test.

Referee Report

0 major / 2 minor

Summary. The manuscript calculates the low-temperature superheating field B_sh of clean superconductors near the type-I--type-II boundary using the self-consistent nonlinear nonlocal Eilenberger theory together with linear stability analysis of the Meissner state. For parameters chosen to represent Nb (κ_GL=0.7, B_c0=200 mT) the authors report B_sh ≃ 290 mT at T/T_c=0.2; this exceeds the value obtained by naive extrapolation of the Ginzburg-Landau result and corresponds to an intrinsic cavity limit of ~67 MV/m for a TESLA-shaped Nb resonator.

Significance. If the numerical result holds, the work supplies a microscopic, low-T prediction for the Meissner stability limit in a regime directly relevant to superconducting radio-frequency cavities. The use of the established Eilenberger framework for clean-limit materials near the type-I/II boundary is a methodological strength; the reported elevation of B_sh relative to Ginzburg-Landau extrapolation is a concrete, falsifiable output that can be tested against experiment.

minor comments (2)

[Abstract] Abstract: the numerical value B_sh ≃ 290 mT is stated without accompanying error estimate, convergence criterion, or comparison to an independent method; adding one sentence on these points would improve clarity without altering the central claim.
The manuscript should explicitly state the temperature range over which the Eilenberger equations were solved and confirm that the reported point at T/T_c=0.2 lies within the regime where the nonlocal theory remains valid.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work, and recommendation for minor revision. The report correctly summarizes the central result: a self-consistent Eilenberger calculation yielding B_sh ≃ 290 mT at T/T_c = 0.2 for κ_GL = 0.7 and B_c0 = 200 mT, which lies above the naive Ginzburg-Landau extrapolation and implies an intrinsic cavity limit of ~67 MV/m. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper computes B_sh from the established self-consistent nonlinear nonlocal Eilenberger equations plus linear stability analysis of the Meissner state, taking material parameters κ_GL and B_c0 as external inputs. The reported numerical value is a direct output of that framework for the chosen parameters rather than a quantity fitted to itself or reduced by construction to the inputs. No self-citation chains, ansatz smuggling, or self-definitional steps are indicated in the abstract or description.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

Ledger constructed from abstract only; full paper may list additional parameters or assumptions.

free parameters (2)

κ_GL = 0.7
Input parameter chosen to represent Nb-like material near type-I/II boundary
B_c0 = 200 mT
Reference thermodynamic critical field value adopted for Nb

assumptions (2)

domain assumption Eilenberger theory in the clean limit accurately models the superheating field near the type-I/type-II boundary
Basis for the self-consistent nonlinear nonlocal calculation
domain assumption Linear stability analysis of the Meissner state identifies the superheating field
Method used to extract B_sh from the theory

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Cite this review

Pith. "Pith review of Superheating field of clean superconductors near the type-I--type-II boundary: the low-temperature Meissner stability limit of niobium." pith.science (2026). https://pith.science/paper/CKVRP6YY

@misc{pith2026260610420,
  author       = {Pith},
  title        = {Pith review of: Superheating field of clean superconductors near the type-I--type-II boundary: the low-temperature Meissner stability limit of niobium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKVRP6YY}},
  note         = {Machine review of arXiv:2606.10420}
}

abstract

We calculate the low-temperature superheating field $B_{\rm sh}$ of clean superconductors near the boundary between type-I and type-II superconductivity, with particular emphasis on Nb. The calculation is based on the self-consistent nonlinear nonlocal Eilenberger theory and the linear stability analysis of the Meissner state. For a Nb-like material with $\kappa_{\rm GL}=0.7$, we obtain $B_{\rm sh}\simeq 290\,{\rm mT}$ at $T/T_c=0.2$, using $B_{c0}\simeq 200\,{\rm mT}$. This value is substantially higher than the value obtained by naively extrapolating the Ginzburg--Landau result near $T_c$ to $T\ll T_c$. For a TESLA-shaped Nb accelerator cavity, it corresponds to an intrinsic Meissner-stability limit of about $67\,{\rm MV/m}$.

Figures

Figures reproduced from arXiv: 2606.10420 by the authors.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Microscopic theory of the field-induced instability of the vortex-free state in superconducting thin-film strips
cond-mat.supr-con 2026-08 accept novelty 8.0 of 10

A Usadel-theory stability analysis determines the vortex-free-state instability field of a superconducting thin-film strip without a core cutoff, yielding three width regimes and B_s proportional to 1/W in wide strips.
Microscopic equivalence of the vortex-entry current and the depairing current in a superconducting thin-film strip
cond-mat.supr-con 2026-07 conditional novelty 6.5 of 10

In an ideal homogeneous narrow thin-film strip at zero field with self-fields neglected, the microscopic vortex-entry current equals the depairing current at all temperatures.

Reference graph

Works this paper leans on

30 extracted references · 1 canonical work pages · cited by 2 Pith papers

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Superheating field of clean superconductors near the type-I--type-II boundary: the low-temperature Meissner stability limit of niobium

577 is the Euler constant. Numerically, κ GL ≃ 0. 958κ 0. We consider a clean semi-inﬁnite superconductor oc- cupying x > 0. The applied dc magnetic ﬁeld is paral- lel to the surface, Ba = Baˆz, and the Meissner screen- ing current ﬂows along ˆy. Energies are normalized by the zero-temperature BCS gap ∆ 0, and lengths by ξ0 = ℏvf /π ∆ 0. The magnetic ﬂux ...

work page Pith review arXiv 2026
[2]

Our 3 FIG

27Bc0, which has often been used as a rough estimate of the low-temperature superheating ﬁeld of Nb. Our 3 FIG. 1. (a) Temperature dependence of the superheating ﬁeld for a clean low- κ superconductor. Filled symbols show Eilenberger results at ﬁxed κ 0; the corresponding κ GL values are obtained from the near- Tc mapping. Open symbols at T /T c = 1 show ...
[3]

For a Nb-relevant value of κ GL, the estimated ﬁeld is even higher

34Bc0 for κ GL = 1. For a Nb-relevant value of κ GL, the estimated ﬁeld is even higher. Multiple experiments on high-purity Nb support its type-II/1 character. Small-angle neutron scattering measurements on Nb with RRR > 104 ob- served a well-ordered vortex lattice that transforms on cooling into an intermediate mixed state composed of dense vortex-lattic...
[4]

Using Bc(T /T c = 0

44Bc0. Using Bc(T /T c = 0 . 2) ≃ Bc0 ≃ 200 mT, this gives BNb sh ≃ 290 mT. For accelerator cavities, this sur- face magnetic-ﬁeld limit can be converted into a the- oretical limit on the accelerating gradient. Using the TESLA-shape conversion factor 4 . 26 mT/ (MV/ m), we obtain Emax acc ≃ 67 MV/ m. This value is substantially higher than both the conven...

2026
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Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Superheating field of clean superconductors near the type-I--type-II boundary: the low-temperature Meissner stability limit of niobium

577 is the Euler constant. Numerically, κ GL ≃ 0. 958κ 0. We consider a clean semi-inﬁnite superconductor oc- cupying x > 0. The applied dc magnetic ﬁeld is paral- lel to the surface, Ba = Baˆz, and the Meissner screen- ing current ﬂows along ˆy. Energies are normalized by the zero-temperature BCS gap ∆ 0, and lengths by ξ0 = ℏvf /π ∆ 0. The magnetic ﬂux ...

work page Pith review arXiv 2026

[2] [2]

Our 3 FIG

27Bc0, which has often been used as a rough estimate of the low-temperature superheating ﬁeld of Nb. Our 3 FIG. 1. (a) Temperature dependence of the superheating ﬁeld for a clean low- κ superconductor. Filled symbols show Eilenberger results at ﬁxed κ 0; the corresponding κ GL values are obtained from the near- Tc mapping. Open symbols at T /T c = 1 show ...

[3] [3]

For a Nb-relevant value of κ GL, the estimated ﬁeld is even higher

34Bc0 for κ GL = 1. For a Nb-relevant value of κ GL, the estimated ﬁeld is even higher. Multiple experiments on high-purity Nb support its type-II/1 character. Small-angle neutron scattering measurements on Nb with RRR > 104 ob- served a well-ordered vortex lattice that transforms on cooling into an intermediate mixed state composed of dense vortex-lattic...

[4] [4]

Using Bc(T /T c = 0

44Bc0. Using Bc(T /T c = 0 . 2) ≃ Bc0 ≃ 200 mT, this gives BNb sh ≃ 290 mT. For accelerator cavities, this sur- face magnetic-ﬁeld limit can be converted into a the- oretical limit on the accelerating gradient. Using the TESLA-shape conversion factor 4 . 26 mT/ (MV/ m), we obtain Emax acc ≃ 67 MV/ m. This value is substantially higher than both the conven...

2026

[5] [5]

Kramer, Stability limits of the Meissner state and the mechanism of spontaneous vortex nucleation in super- conductors, Phys

L. Kramer, Stability limits of the Meissner state and the mechanism of spontaneous vortex nucleation in super- conductors, Phys. Rev. 170, 475 (1968). 4

1968

[6] [6]

P. V. Christiansen, Magnetic superheating of high- κ su- perconductors, Solid State Commun. 7, 727 (1969)

1969

[7] [7]

A. J. Dolgert, S. J. Di. Bartolo, and A. T. Dorsey, Super- heating ﬁelds of superconductors: Asymptotic analysis and numerical results, Phys. Rev. B 53, 5650 (1996)

1996

[8] [8]

A. J. Dolgert, S. J. Di. Bartolo, and A. T. Dorsey, Er- ratum: Superheating ﬁelds of superconductors: Asymp- totic analysis and numerical results [Phys. Rev. B 53, 5650 (1996)], Phys. Rev. B 56, 2883 (1997)

1996