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REVIEW 3 major objections 4 minor 53 references

Geometric Scaling of Two-Level-System Loss in Superconducting Resonators

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the lithography resist HSQ, when left on finished NbN nanowire resonators, is the dominant source of two-level-system microwave loss.

desk verdict A genuinely useful paired measurement and a new Maxwell-London filling-factor method, but the headline HSQ loss tangent is model-dependent and needs a sensitivity analysis before I would trust the exact number. read the letter →

arxiv 1908.02606 v1 pith:TJ2LRBN4 submitted 2019-08-07 physics.app-ph cond-mat.supr-con

David Niepce , Jonathan Burnett , Martí Gutierrez Latorre , Jonas Bylander This is my paper · ORCID
classification physics.app-phcond-mat.supr-con
keywords superconductingresonatorstwo-levelsystemsdielectriclosshydrogensilsesquioxane(HSQ)NbNnanowireskineticinductancefillingfactorMaxwell-Londonsimulation
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 aims to identify and quantify where two-level-system (TLS) dielectric loss comes from in high-impedance superconducting nanowire resonators, and it points to the spin-on-glass resist hydrogen silsesquioxane (HSQ). The authors built nominally identical NbN coplanar waveguide resonators with and without a 30 nm HSQ layer on the center conductor, with gaps from 5 µm down to 500 nm, and found that the HSQ-covered resonators always lose more. Combining the measured loss with 3D finite-element simulations of the electric fields, they extracted an intrinsic loss tangent for HSQ of $8.0 \times 10^{-3}$, about four times the value typically used for silicon oxide, and showed that this one layer dominates the total loss in every geometry studied. If the paper is right, removing or avoiding HSQ after patterning would substantially reduce microwave loss in nanowire-based qubits, amplifiers, and detectors.

What carries the argument

The argument is carried by geometric filling factors, $F_k = U_k / U_{\mathrm{total}}$, the fraction of stored electric energy in each lossy region, computed from 3D finite-element solutions of the Maxwell–London equations. Because the disordered NbN film has a magnetic penetration depth of roughly $1\,\mu\mathrm{m}$, comparable to the device dimensions, the superconductor is modelled by a complex permittivity in the dirty local limit, and electrostatic simulations are shown to give the wrong filling factors. These $F_k$ feed the linear decomposition $1/Q_{\mathrm{TLS}} = \sum_k F_k \delta^i_k$, which converts measured total TLS loss into the intrinsic loss tangent of each region and singles out HSQ as the largest term.

What would settle it

Take finished resonators that still carry HSQ, remove the HSQ with a dedicated strip process, and remeasure the single-photon TLS loss; if the loss does not drop by the amount predicted from $F_{\mathrm{HSQ}} \times 8.0 \times 10^{-3}$, the dominance claim is wrong. Alternatively, measure the HSQ loss tangent directly in a resonator whose filling factor is fixed by geometry alone, such as a lumped-element capacitor with the HSQ as the known dielectric.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that HSQ is the dominant TLS loss channel in superconducting NbN coplanar waveguide resonators whenever it is left on the device, and that its intrinsic loss tangent is $\delta^i_{\mathrm{HSQ}} = 8.0 \times 10^{-3}$. This is roughly four times the loss tangent of the silicon-oxide interfaces that have historically been blamed for surface losses, so the standard assumption that a spin-on-glass resist behaves like silicon oxide would seriously underestimate its impact. The dominance is not because HSQ is extraordinarily lossy per volume, but because it sits directly above the center conductor, where the electric field is strongest, so its geometric filling factor is large and grows as the resonator gap shrinks. The paper establishes this by fabricating pairs of resonators that differ only in the HSQ layer, measuring the temperature dependence of the resonance frequency to isolate TLS loss, and using full-wave simulations to convert the measured total loss into per-material loss tangents.

Load-bearing premise

The fitted HSQ loss tangent assumes the device is exactly the modelled stack of uniform dielectric layers with the stated thicknesses and permittivities, with no niobium oxide under the HSQ; if those assumptions miss the real field distribution, the extracted HSQ loss changes even though the measured total loss is unchanged.

Editorial extensions

If this is right

  • Leaving HSQ on a finished nanowire resonator sets a floor on the achievable internal quality factor; stripping HSQ after e-beam patterning should raise $Q$ at single-photon powers.
  • At sub-micron gaps, the top dielectric's filling factor grows fastest, so high-impedance resonator design must treat the dielectric layer on top of the conductor as a first-order loss term, not a correction.
  • Electrostatic filling-factor calculations are not reliable for disordered-superconductor devices whose dimensions are comparable to the magnetic penetration depth; Maxwell–London simulation is needed to separate loss channels.
  • The extracted loss tangents of HSQ, substrate-metal interface, niobium oxide, and silicon oxide can be reused as material parameters to predict TLS loss in future nanowire circuits before fabrication.

Reading between the lines

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

  • The extracted HSQ loss tangent inherits the assumed layer stack; if the actual oxide thicknesses or the absence of $\mathrm{Nb_2O_5}$ under HSQ are wrong, the fitted value would compensate. A geometry with a single well-known lossy layer would separate those uncertainties.
  • Because developed HSQ is porous amorphous silica, its loss tangent likely depends on exposure dose, development, and baking; the reported value may represent one process condition, so a processing study could find lower-loss HSQ variants.
  • The same measurement-and-simulation workflow could rank other nanofabrication resists and hard masks by their predicted TLS loss, giving nanowire developers a screening tool before device fabrication.
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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 / 4 minor

Summary. The manuscript reports an experimental and numerical study of two-level-system (TLS) dielectric loss in NbN coplanar waveguide resonators with and without a hydrogen silsesquioxane (HSQ) capping layer on the center conductor. The authors measure the temperature-dependent frequency shift of four resonator geometries (gap widths 0.5–5 µm) at single-photon power and extract the product F_TLS·δ_TLS for each device. They then compute geometric filling factors using 3D finite-element simulations that solve the Maxwell–London equations for the disordered NbN film (penetration depth ≈1 µm), and fit the loss-budget model of Eq. (1) to obtain the intrinsic loss tangents of HSQ, the substrate–metal interface, niobium oxide, and silicon oxide. The central claim, quantified in Table II and stated in Section VI, is that HSQ has δ_i_HSQ = 8.0 × 10⁻³ and, because of its large filling factor, is the dominant TLS loss source in every measured geometry.

Significance. The qualitative result that HSQ increases microwave loss is convincingly established by the paired with/without-HSQ comparison: in each of the four geometries the measured F_TLS·δ_TLS is higher when HSQ is present (Table I), and the frequency-shift method isolates TLS loss from other mechanisms. If the quantitative value for δ_i_HSQ survives further scrutiny, it provides a useful design rule for high-impedance NbN nanowire circuits, where HSQ is commonly left in place. The methodological contribution—using 3D Maxwell–London simulations to account for the large magnetic penetration depth in disordered superconductors when computing participation ratios—is significant and clearly presented. The manuscript also gives credit to prior surface-loss literature and includes sufficient experimental detail for the measurement setup to be reproduced.

major comments (3)
  1. [Section IV, material-parameter paragraph; Table II] The quantitative extraction of δ_i_HSQ depends critically on the stated assumption: 'Because Nb2O5 requires several days to achieve any meaningful thickness, it is assumed that no Nb2O5 is present underneath the HSQ.' This assumption is load-bearing: the model removes the 5-nm Nb2O5 layer from the center conductor in the HSQ samples and replaces it with HSQ, while the measured with-HSQ loss is only modestly larger than the no-HSQ loss (e.g., 1.66 vs 1.36 × 10⁻⁵ at g_cpw = 5 µm). If a native oxide grows under the HSQ, the fitted δ_i_HSQ would be lower, and the claim of HSQ dominance in the wider geometries could fail. The fabrication sequence leaves the NbN exposed between RIE and HSQ spin with no time specified, and no direct oxide characterization or sensitivity analysis is provided. This issue directly affects the central quantitative claim and should be addressed with, e.g., XPS/TEM cross-section or a sensitivity study varying the under-HSQ oxide thickness.
  2. [Table II and Fig. 6] The four loss tangents in Table II are obtained by fitting Eq. (1) to the eight F_TLS·δ_TLS values in Table I, yet no uncertainties, covariance, or goodness-of-fit statistics are reported. With four free parameters and eight data points, the fit may not uniquely constrain the individual loss tangents; for instance, δ_i_SM and δ_i_SiO2 both affect the geometry dependence in similar ways. The two-significant-figure value δ_i_HSQ = 8.0 × 10⁻³ therefore conveys a precision that is not supported by the presented analysis. Please report parameter uncertainties (e.g., from a bootstrap or least-squares covariance) and a residual analysis, or explicitly discuss the confidence region for δ_i_HSQ.
  3. [Section VI and Fig. 7] The conclusion that HSQ is the dominant TLS loss source 'for all dimensions' goes beyond the direct measurements, which show only that HSQ increases loss; the dominance statement is an inference from the fitted parameters. Because the fitted parameters are sensitive to the assumed layer thicknesses, permittivities, and the no-Nb2O5-under-HSQ assumption (as noted above), the dominance claim for the widest geometry (g_cpw = 5 µm) is not yet robust. A sensitivity analysis showing how δ_i_HSQ and the per-region loss contributions in Fig. 7 change under plausible variations of the model parameters would be required to support this strong claim.
minor comments (4)
  1. [Abstract] The abstract contains an incomplete sentence: 'We emphasise that the loss caused by a spin-on-glass resist such as hydrogen silsesquioxane (HSQ), used for ultrahigh lithographic resolution relevant to the fabrication of nanowires, and find that, when used, HSQ is the dominant source of loss...' The phrase beginning 'the loss caused by...' lacks a main verb. Please rewrite.
  2. [Section II] The fixed ratio of gap to center conductor width is not stated; please give the ratio (or the center-conductor widths) to make the geometry table self-contained.
  3. [Section V] In Section V, 'These results confirms' should be 'These results confirm'.
  4. [Fig. 2 caption] The caption states that the curves are offset by 15 kHz, but does not explain which curves are offset; please clarify the offset convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the HSQ loss tangent is obtained by fitting Eq. (1) to independent frequency-shift data, and the dominance claim is a post-fit consequence rather than an input.

full rationale

The central quantity, δ_i_HSQ = 8.0 × 10^-3, is not defined in terms of the measured loss or of the paper's own prior results. It is extracted by fitting the linear decomposition 1/Q_TLS = Σ_k F_k δ_i_k, Eq. (1), to the independently measured F_TLS δ_i_TLS values listed in Table I. Those F_TLS δ_i_TLS values are themselves obtained from temperature-dependent frequency-shift measurements analyzed with the standard TLS model of Refs. [35,36]. The filling factors F_k come from 3D finite-element Maxwell–London simulations, which are separate from the loss measurements. The fit has eight measured data points (four geometries, with and without HSQ) and four fitted loss tangents, so the parameters are not forced by construction. The statement that HSQ dominates the loss is a post-fit interpretation of the fitted δ_i_HSQ together with the simulated filling factors, not a prediction that was assumed beforehand. The assumption that no Nb2O5 grows under HSQ is a model-input assumption; it affects the numerical value extracted, but this is a model-sensitivity limitation rather than circular reasoning. Self-citations such as Refs. [4] and [37] are used only for experimental setup details and are not load-bearing for the loss-tangent extraction. The paper also benchmarks its values against external literature, e.g., Refs. [26,28,34,43]. No step in the derivation reduces the claimed result to its own inputs, so the paper is not circular.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central numeric results are calibrated by a four-parameter fit of loss tangents to eight measured total-loss values, using simulated filling factors. The main unverified inputs are the geometric layer model, the no-Nb2O5-under-HSQ assumption, and the dirty-limit complex permittivity for NbN.

free parameters (4)
  • delta_i_HSQ (HSQ loss tangent) = 8.0 x 10^-3
    Fitted to total TLS loss versus gap width using Eq. (1) and the simulated filling factors; no uncertainty reported.
  • delta_i_SM (substrate-metal interface loss tangent) = 1.3 x 10^-3
    Fitted in the same four-parameter fit of Eq. (1) to the measured FTLS*delta_TLS values.
  • delta_i_Nb2O5 (niobium oxide loss tangent) = 4.7 x 10^-2
    Fitted in the same four-parameter fit; depends on the assumed 5 nm niobium-oxide layer.
  • delta_i_SiO2 (silicon oxide loss tangent) = 2.1 x 10^-3
    Fitted in the same four-parameter fit; depends on the assumed 5 nm silicon-oxide layer.
assumptions (5)
  • domain assumption The TLS tunneling model [35,36] relates the measured frequency shift to the product FTLS*delta_TLS via the digamma-function formula used in Fig. 2.
    The extraction of total TLS loss assumes this standard model and that the non-TLS kinetic-inductance contribution is correctly separated at low hfr/kBT.
  • domain assumption Total loss is a linear sum of independent lossy regions, Eq. (1).
    Standard participation-ratio decomposition; assumes independent regions with effective bulk-like loss tangents.
  • domain assumption Each TLS host is a homogeneous dielectric layer of fixed thickness and permittivity: 5 nm SiO2, 5 nm Nb2O5, 2 nm substrate layer, 30 nm HSQ.
    Layer parameters are taken from prior literature in Section IV; the true TLS distribution could be graded or have different thicknesses.
  • ad hoc to paper No Nb2O5 grows underneath the HSQ-covered center conductor.
    Assumed because Nb2O5 growth takes days; stated in Section IV but not directly verified on these samples.
  • domain assumption NbN's microwave response is described by the local dirty-limit complex permittivity, Eq. (5), with Mattis-Bardeen quasiparticle conductivity.
    Needed for the Maxwell-London finite-element simulation; relies on BCS dirty-limit formulas and measured Tc and sigma_n.

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Cite this review

Pith. "Pith review of Geometric Scaling of Two-Level-System Loss in Superconducting Resonators." pith.science (2026). https://pith.science/paper/TJ2LRBN4

@misc{pith2026190802606,
  author       = {Pith},
  title        = {Pith review of: Geometric Scaling of Two-Level-System Loss in Superconducting Resonators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJ2LRBN4}},
  note         = {Machine review of arXiv:1908.02606}
}
abstract

We perform an experimental and numerical study of dielectric loss in superconducting microwave resonators at low temperature. Dielectric loss, due to two-level systems, is a limiting factor in several applications, e.g. superconducting qubits, Josephson parametric amplifiers, microwave kinetic-inductance detectors, and superconducting single-photon detectors. Our devices are made of disordered NbN, which, due to magnetic-field penetration, necessitates 3D finite-element simulation of the Maxwell--London equations at microwave frequencies to accurately model the current density and electric field distribution. From the field distribution, we compute the geometric filling factors of the lossy regions in our resonator structures and fit the experimental data to determine the intrinsic loss tangents of its interfaces and dielectrics. We emphasise that the loss caused by a spin-on-glass resist such as hydrogen silsesquioxane (HSQ), used for ultrahigh lithographic resolution relevant to the fabrication of nanowires, and find that, when used, HSQ is the dominant source of loss, with a loss tangent of $\delta^i_{HSQ} = 8 \times 10^{-3}$.

Figures

Figures reproduced from arXiv: 1908.02606 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. shows that in our devices, the losses are domi￾nated by TLS, even for thin-film nanowires with widths down to 40 nm. In order to accurately account for the individual contributions of all TLS-containing regions of the circuit, we split the dielectric loss into a linear com￾bination of loss tangents each associated with a corre￾sponding filling factor [25–28, 40], 1 QT LS = FT LSδ i T LS = X k Fkδ i k (1) where δ i k… view at source ↗
Figure 3
Figure 3. FIG. 3. Simulated normalised current density inside the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Magnitude and field lines of the simulated elec [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Simulated filling factors [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Total TLS loss [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Contribution of each individual lossy region for resonators without HSQ [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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