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REVIEW 4 major objections 5 minor 91 references

Microwave characterization of superconducting coplanar resonators made out of granular aluminium

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that the kinetic inductance fraction of granular aluminium microwave resonators can be extracted from the temperature dependence of the resonance frequency, with Mattis-Bardeen fits giving values from 0.137 to 0.356…

desk verdict A useful grAl dataset paper whose quantitative claims are undercut by the authors' own caveat about Mattis-Bardeen and by an internal table/text inconsistency. read the letter →

arxiv 2508.03873 v1 pith:456E4BVI submitted 2025-08-05 cond-mat.supr-con cond-mat.mtrl-sci

Kelvin J. Ramos , Ivana Curci , Erick Potosí , Ignacio Lobato , Leonardo Salazar Alarcón , Hernán Pastoriza , Leandro Tosi This is my paper · ORCID
classification cond-mat.supr-concond-mat.mtrl-sci
keywords granularaluminiumkineticinductancefractioncoplanarwaveguideresonatorsMattis-Bardeentheorytwo-levelsystemssuperconductingmicrowavedevicesdisorderedsuperconductors
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 microwave measurements of coplanar waveguide resonators fabricated from granular aluminium films with four different oxygen levels, and argues that the temperature dependence of the resonance frequency can be used to extract the kinetic inductance fraction. By fitting the relative frequency shift to Mattis-Bardeen theory, the authors obtain kinetic inductance fractions from 0.137 for pure aluminium up to 0.356 for the most oxidized film, and sheet kinetic inductances between 1.8 and 6.9 pH per square. The same measurements give the two-level-system contribution to microwave loss, about $1.5\times10^{-4}$ in $F\delta_{\mathrm{TLS}}^0$, with no clear trend versus oxygen content. The central point is that this characterization route, combining temperature sweeps with Mattis-Bardeen fits and conformal-mapping or simulated geometric frequencies, is simple enough to be useful for designing MKIDs, high-impedance resonators, and superinductors.

What carries the argument

The load-bearing object is the Mattis-Bardeen complex conductivity in the local limit, $\sigma = \sigma_1 - i\sigma_2$, used to compute the surface impedance $X_s(T)$ of a thin film, together with the perturbation relation $\Delta f/f_0 = -(\alpha/2)(\Delta X_s/X_s(0))$. The kinetic inductance fraction $\alpha$ enters as the only free parameter in the frequency-shift fit, while $T_c$ is fixed to the measured value of each film. Geometric resonance frequencies are obtained independently from finite-element simulations and from conformal-mapping expressions for a coplanar waveguide, including the kinetic-inductance geometric factor used to convert $\alpha_1$ into sheet inductance. For two-level systems, the machinery is the resonant TLS model relating $1/Q_i$ to photon number through $F\delta_{\mathrm{TLS}}^0$, $n_c$, $\beta$, and a loss floor $\delta_1$.

What would settle it

Measure the kinetic inductance of the same four films by a method independent of Mattis-Bardeen, such as the magnetic-field dependence of the resonance frequency or a lumped-element tank-circuit inductance measurement, and compare with the values derived here; disagreement outside the quoted errors, for example $L^\square_k$ differing by more than about 20% for grAl-3, would falsify the claim that the temperature-shift fit returns the true kinetic inductance.

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Extended reading notes

Core claim

The authors establish that the kinetic-inductance fraction $\alpha$ of a granular aluminium film can be estimated from the measured temperature dependence of the resonance frequency, without needing an absolute calibration of the geometric resonance frequency. The relation $\Delta f/f_0 = -(\alpha/2)(\Delta X_s/X_s(0))$, with $X_s$ computed from Mattis-Bardeen complex conductivity in the local limit, reproduces the measured shift from base temperature up to $T_c$. From that fit, $\alpha_1 = 0.137 \pm 0.005$ for aluminium, $0.201 \pm 0.001$ for grAl-1, $0.291 \pm 0.007$ for grAl-2, and $0.356 \pm 0.008$ for grAl-3, giving sheet kinetic inductances $L^\square_k$ of 1.8 to 6.9 pH per square after matching conformal-mapping expressions to $\alpha_1$. The authors report that these Mattis-Bardeen values are systematically about 25% larger than rigid-shift estimates using simulated geometric frequencies and about 9% larger than those using conformal mapping, and they adopt $\alpha_1$ as the estimate because the frequency shift is more reliable than the quality-factor shift. The two-level-system analysis yields $F\delta_{\mathrm{TLS}}^0 \approx 1.5\times10^{-4}$, $\beta$ close to 1, and non-TLS losses limiting the internal quality factor to about $5\times10^4$.

Load-bearing premise

The extraction assumes that Mattis-Bardeen local-limit surface impedance with a single BCS gap of $\Delta_0 = 1.76 k_B T_c$ describes strongly disordered granular aluminium films, even though the authors themselves note that for 1 pH per square the effective penetration depth reaches about 800 nm, where the theoretical expression is not justified.

Editorial extensions

If this is right

  • The kinetic inductance fraction can be read from a single temperature sweep, avoiding the need for an absolute geometric-frequency calibration, which makes the method applicable to films whose geometric resonance frequency is hard to simulate.
  • Controlling oxygen content tunes sheet kinetic inductance from 1.8 to 6.9 pH per square, and growing thinner films would push these values higher.
  • The Mattis-Bardeen-derived $\alpha$ is systematically larger than the rigid-shift $\alpha$, so designs based on rigid-shift estimates may underestimate kinetic inductance by roughly 25%.
  • Two-level-system losses in granular aluminium are approximately $F\delta_{\mathrm{TLS}}^0 = 1.5\times10^{-4}$, independent of oxidation level, so further improvement in quality factor will have to target non-TLS losses, which currently cap $Q_i$ near $5\times10^4$.
  • The same characterization flow can be reused for other disordered superconducting films, since it only needs $S_{21}(T)$ and a film thickness.

Reading between the lines

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

  • Beyond the paper, the systematic offset between Mattis-Bardeen and rigid-shift values of $\alpha$ suggests a temperature-independent bias in one of the two methods; a decisive test would be to extract $\alpha$ from the magnetic-field dependence of the same resonators, since that route does not rely on Mattis-Bardeen local-limit assumptions.
  • Beyond the paper, the lack of correlation between TLS loss and oxygen content hints that the loss floor is set by the granular microstructure, such as grain boundaries and the aluminium-oxide matrix, rather than by the oxide concentration; if so, annealing or altered growth protocols could reduce TLS without changing $\alpha$.
  • Beyond the paper, the reported sheet inductances up to 6.9 pH per square and quality factors near $10^4$ could be fed directly into MKID sensitivity estimates; a concrete extension would be to fabricate MKID pixels from these same films and compare photon-noise-limited performance with predictions based on the extracted parameters.
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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 / 5 minor

Summary. The paper reports microwave characterization of coplanar-waveguide quarter-wave resonators fabricated from four aluminum films with different oxygen content (Al, grAl-1, grAl-2, grAl-3). The authors extract the kinetic-inductance fraction by fitting the temperature dependence of the resonance frequency and quality factor to Mattis-Bardeen theory (α1 from Eq. (7), α2 from Eq. (8)), compare these values with rigid-shift estimates based on finite-element simulations and conformal mapping, derive sheet kinetic inductances by inversion of the conformal-mapping formula (Table II), and characterize two-level-system losses from the power dependence of the internal quality factor (Eq. (10)). The main reported results are α1 between 0.137 and 0.356, sheet kinetic inductances between 1.8 and 6.9 pH/□, and an average TLS loss factor F·δ_TLS0 of about 1.5×10⁻⁴. The paper is framed as a characterization study useful for MKIDs, high-impedance resonators, and superinductors.

Significance. If the quantitative extraction were validated, this would be a useful contribution: it provides a consistent set of kinetic-inductance and TLS parameters for granular aluminum films with different disorder levels, along with detailed fabrication and design information. Strengths include the careful multi-probe film characterization (TEM, XRD, profilometry), the explicit reporting of discrepancies among extraction methods, and the TLS power-dependence analysis yielding β close to 1, consistent with the standard TLS model. However, the central quantitative claims are currently contingent on an unvalidated Mattis-Bardeen local-limit, single-gap assumption that the authors themselves flag as not justified; the paper would be considerably strengthened by an independent validation route or a clear statement of model uncertainty on every derived quantity.

major comments (4)
  1. [§V.1, Eq. (7), Table II] The central extraction assumes that the Mattis-Bardeen local-limit, single-gap surface impedance (Eqs. (5)–(7)) describes these strongly disordered granular films, but this assumption is neither tested nor independently supported. The authors themselves state in §V.1 that λ_eff ≈ 800 nm for 1 pH/□ and that 'the use of the theoretical expressions to extract α is not justified and we take these values only as indicative.' Since Eq. (7) multiplies the theoretically prescribed ΔX_s(T)/X_s(0) by a single scale factor α, any error in the gap ratio, gap distribution, or local-limit kernel is absorbed into α1 and propagates through Table II into L□_k. The paper should validate the Mattis-Bardeen assumption for these films (for example, by comparing against a direct low-temperature measurement of the geometric resonance frequency from the same geometry, or by modeling the disordered-film conductivity) or explicitly present α1 and all derived quantities as indicative values with a model uncertainty.
  2. [§V.2, Fig. 4, Table II] The unresolved discrepancy between α1 and α2 (1–22%) is attributed to unsaturated two-level systems, but the frequency-shift data used for α1 are not corrected for the TLS frequency shift described by Eq. (9). If TLS also shifts the resonance frequency at the operating photon number, then α1 is biased as well and cannot be considered more reliable merely because it comes from the frequency shift rather than the quality factor. A joint fit of Δf/f0(T) and 1/Qi(T) that includes both the Mattis-Bardeen term and the TLS term, or an explicit numerical estimate of the TLS-induced frequency shift at the measurement power, is needed before α1 is adopted as the reported kinetic-inductance fraction.
  3. [§V.1, Fig. 3, Table II] The sheet inductances in Table II are not an independent check of the Mattis-Bardeen extraction: they are obtained by solving α(L□_k) = α1 with the conformal-mapping expression of Eq. (B9), so any systematic error in α1 is transferred directly to L□_k. Moreover, the rigid-shift estimates α_SIM and α_CM are systematically 9–25% lower than α1, and the origin of this offset is not explained; the spread is comparable to or larger than the quoted statistical errors. The paper should either reconcile these estimates (including the effect of the coupling capacitance and finite film thickness on the geometric frequency f_g) or quote a model uncertainty that covers the full spread of values from the different extraction methods.
  4. [§V.1 and §VI] The internal inconsistency between the presentation of α1 and L□_k as extracted values and the statement that the theoretical expressions used for the extraction are 'not justified' should be resolved in the text. As written, the conclusion states that the authors 'were able to extract the kinetic inductance contribution' and provide the sheet kinetic inductance, while the body of the paper simultaneously warns that these values are only indicative. The authors should either strengthen the validation to justify the numbers or consistently frame all quantitative kinetic-inductance results as estimates with explicit error bars that include the model uncertainty.
minor comments (5)
  1. [§I] The theory name is spelled 'Mathis-Bardeen' at the end of the introduction; it should be 'Mattis-Bardeen' as used elsewhere.
  2. [Eqs. (5), (6), (10)] The typeset equations contain garbled exponential factors and bracket expressions in the provided text; please ensure the final formulas are correct and consistent with the standard Mattis-Bardeen and TLS expressions.
  3. [Fig. 4 caption] The caption says the extracted α values are indicated in the legend, but the numerical values are not visible in the current figure; adding the fitted values to the caption would improve readability.
  4. [§V.2] The phrase 'with two outliers resonators which show a larger participation ratio' is unclear; please specify which resonators are outliers and quantify the deviation from the average F·δ_TLS0.
  5. [General] No data or code availability statement is provided; archiving the S21 datasets, the fitted resonance parameters, and the fit residuals would substantially improve reproducibility, since the central claims are based on fits to temperature and power sweeps.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: alpha1 is an explicitly fitted parameter, the sheet inductance is a transparent model inversion, and independent rigid-shift and sheet-resistance comparisons anchor the central claim.

full rationale

The derivation chain is self-contained rather than circular. The central quantity alpha1 is obtained by fitting Eq. (7) to the measured temperature dependence of the relative frequency shift; the paper explicitly describes this as an extraction, not as a prediction from the fit. The sheet kinetic inductances in Table II are computed by solving alpha(Lk_sheet)=alpha1 using the defining relation Eq. (2) and the conformal-mapping expression Eq. (B9); this is transparently labeled 'matching the theoretical alpha(Lk_sheet) to the experimental value alpha1' and is a model inversion rather than an independent check, so it does not constitute a fitted-input-called-prediction step. The central claim is independently anchored by the rigid-shift estimates alphaSIM and alphaCM (Fig. 3), which use simulated or CMT geometric frequencies and are independent of Mattis-Bardeen theory, and by the sheet-resistance estimate of Eq. (1). The paper's own caveat that 'the use of the theoretical expressions to extract alpha is not justified' is a validity limitation for strongly disordered films, not a circular step. No load-bearing self-citation, imported uniqueness theorem, or ansatz-smuggling-via-citation was found; the two self-cited theses support only minor fabrication details.

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

The central extraction depends on standard BCS and TLS models and geometric formulas rather than on new physics. The most important load-bearing items are the Mattis-Bardeen local-limit conductivity, the thin-film surface impedance relations, the Watanabe/Clem conformal-mapping kinetic-inductance formula, and the TLS saturation model. The fitted parameters carry the quantitative content. No new entities are introduced.

free parameters (4)
  • alpha1 (Mattis-Bardeen frequency-shift kinetic inductance fraction) = 0.131-0.361 per resonator (Table II)
    Obtained by fitting Eq. (7) to the temperature-dependent frequency shift with Tc fixed; the central extracted quantity.
  • alpha2 (Mattis-Bardeen quality-factor-shift kinetic inductance fraction) = 0.133-0.353 per resonator (Table II)
    Obtained by fitting Eq. (8); disagrees with alpha1 by 1-22 percent, so it is used only as a cross-check.
  • Lk_sheet (sheet kinetic inductance from conformal-mapping inversion) = 1.8-6.9 pH/sq (Table II)
    Solved by matching theoretical alpha(Lk_sheet) to fitted alpha1; not an independent measurement and it differs from the Eq. (1) estimate.
  • TLS parameters (F*delta_TLS0, beta, nc, delta1) = F*delta_TLS0 avg 1.5e-4; beta close to 1; nc below one photon; delta1 around 2e-5
    Fitted to the power dependence of Qi using Eq. (10); reported without error bars in Fig. 6.
assumptions (4)
  • domain assumption Mattis-Bardeen local-limit complex conductivity (Eqs. 5-6) with Delta0 = 1.76 kBTc applies to granular Al films.
    The central alpha1 extraction fits frequency shifts with this model; the local-limit condition xi0 << lambda is stated but not quantitatively verified for the grAl films.
  • domain assumption Thin-film surface impedance Zs = t/sigma and resonator relations (Eqs. 4, 7-8) connect measured shifts to kinetic inductance.
    These relations convert measured Delta f/f0 into alpha1; they assume uniform film thickness and negligible geometric or parasitic contributions.
  • domain assumption CPW conformal-mapping formulas including the kinetic-inductance correction (Eq. B9 from Watanabe and Clem) yield the sheet inductance.
    The paper uses this formula to solve Lk_sheet from alpha1 but later says the theoretical expression is not justified for the large effective penetration depth.
  • domain assumption TLS saturation model (Eq. 10) with empirical beta describes the power dependence of the internal quality factor.
    Used to extract F*delta_TLS0, beta, nc, and delta1; the derivation is cited rather than repeated in the paper.

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Pith. "Pith review of Microwave characterization of superconducting coplanar resonators made out of granular aluminium." pith.science (2026). https://pith.science/paper/456E4BVI

@misc{pith2026250803873,
  author       = {Pith},
  title        = {Pith review of: Microwave characterization of superconducting coplanar resonators made out of granular aluminium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/456E4BVI}},
  note         = {Machine review of arXiv:2508.03873}
}
read the original abstract

We have designed, fabricated and characterized microwave resonators made out of thin films of granular aluminium (grAl) with different oxygen content. We extract the contribution of the large kinetic-inductance of this disordered superconductor from the frequency shift of the resonators as a function of temperature, and discuss the correlation with the resistivity of the films. At low temperatures, measurements of the internal quality factor as a function of microwave power indicate the presence of two-level systems, which are inherent to the growth process of the granular aluminium. The characterization methods presented here may be useful for the design of MKIDs, high-impedance resonators and superinductors.

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Pith tools

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