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REVIEW 3 major objections 7 minor 59 references

Origin of the superconductor-insulator transition in disordered two-dimensional films

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Phase fluctuations of the superconducting order parameter, not the destruction of pairing, drive the superconductor-insulator transition in disordered two-dimensional NbN films.

desk verdict Careful NbN stiffness and transport study; the BKT analysis is strong, but the key contrast between finite Tc0 and suppressed Js(0) near the SIT depends on an inflection-point Tc0 that lacks an independent cross-check. read the letter →

arxiv 2608.07101 v1 pith:FFUM6HNH submitted 2026-08-07 cond-mat.supr-con

Alexander Weitzel , Lea Pfaffinger , Animesh Panda , Arpan Das , Ilaria Maccari , Simon Reinhardt , Sven Linzen , Evgenii Il'ichev
show 3 more authors
Nicola Paradiso Ferdinand Evers Christoph Strunk
This is my paper · ORCID
classification cond-mat.supr-con PACS 74.25.Dw74.40.-n74.62.En
keywords superconductor-insulatortransitionquantumphasefluctuationsBKTsuperfluidstiffnessdisorderedNbNfilmstwo-dimensionalsuperconductivitykineticinductancepairingtemperature
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 measurements of resistance and superfluid stiffness on fifteen superconducting and three insulating disordered NbN films spanning a wide range of disorder. It establishes that the finite-temperature transition from superconductor to resistive state remains of Berezinskii-Kosterlitz-Thouless (BKT) type all the way to the superconductor-insulator transition (SIT), with Halperin-Nelson fits holding over three to four decades of resistance. The central finding is that very close to the SIT the zero-temperature superfluid stiffness and the BKT temperature drop sharply below the mean-field expectation while the pairing temperature stays finite and continues to follow the fermionic suppression law. The authors conclude that the SIT in these films is a second-order quantum phase transition driven by quantum fluctuations of the superconducting phase, not by disorder destroying the pairing amplitude.

What carries the argument

The argument is carried by three objects: the superfluid phase stiffness $J_s(T)$, extracted from the sheet kinetic inductance in a resonant circuit; the mean-field pairing temperature $T_{c0}$, assigned to the inflection point of $R(T)$ where phase- and amplitude-fluctuation regimes meet; and the BKT universal line $J_s(T_{BKT}) = 2T_{BKT}/\pi$. Resistance in the phase-fluctuation window is fitted to the Halperin-Nelson formula $R(T) = C \exp(-b/\sqrt{T/T_{BKT}-1})$, and $T_{c0}$ is compared with the fermionic suppression formula and with the Ginzburg-Levanyuk correction. Vortex core energy $\mu$ extracted from numerical solution of the Kosterlitz renormalization-group equations links the thermally activated depletion of $J_s$ to vortex-antivortex pairs, and a mapping between normal-state resistance and Anderson disorder strength $W$ ties the experiment to mean-field Bogoliubov-de Gennes simulations of the pairing landscape.

What would settle it

A direct falsifier would be tunneling spectroscopy on the same NbN films for $R_N$ between roughly 12 and 16 kΩ: if the single-particle gap $E_g$ closes as $J_s(0)$ vanishes near the critical resistance, the fermionic (pairing-destruction) scenario is correct; if the gap stays open on the insulating side while the stiffness approaches zero, the phase-fluctuation claim survives. A sharper version is to measure $E_g$ and $J_s(0)$ on the same device and check whether $J_s(0)$ reaches zero at a disorder strength where $E_g$ is still a finite fraction of its clean value.

Watch

Extended reading notes

Core claim

On the authors' own terms, the paper establishes that the superconductor-insulator transition in homogeneously disordered two-dimensional NbN films is driven by quantum phase fluctuations. The evidence: as disorder increases toward the critical value, the mean-field pairing temperature $T_{c0}$ extracted from the inflection point of $R(T)$ remains finite and follows the fermionic suppression law, while the zero-temperature superfluid stiffness $J_s(0)$ and the BKT temperature $T_{BKT}$ drop sharply below their mean-field and Ginzburg-Levanyuk expectations. Near the transition $J_s(0)$ and $T_{BKT}$ vanish together, consistent with the universal BKT relation and with $J_s(0) \propto (\delta g)^{\nu}$ and $T_{BKT} \propto (\delta g)^{z\nu}$, where $\delta g = (R_N - R_c)/R_c^2$, $\nu \simeq 0.67$, and $z \simeq 1$. The finite-temperature transition remains of BKT type throughout, so the zero-temperature SIT is a continuous quantum phase transition of the phase variable.

Load-bearing premise

The load-bearing assumption is that the inflection point of the resistance curve continues to mark the mean-field pairing temperature $T_{c0}$ in the high-disorder regime, where the independent quasiparticle cross-check used at lower disorder is no longer available; if the inflection point stops tracking $T_{c0}$ near the SIT, the contrast between a finite $T_{c0}$ and a sharply suppressed $J_s(0)$ collapses.

Editorial extensions

If this is right

  • The zero-temperature superconductor-insulator transition in disordered NbN films is a continuous (second-order) quantum phase transition driven by phase fluctuations, not by destruction of the pairing amplitude.
  • The finite-temperature transition remains of BKT type up to the critical disorder; the zero-resistance state is always bounded by the BKT line $J_s = 2T/\pi$.
  • Very close to the SIT, $J_s(0) \propto (\delta g)^{\nu}$ and $T_{BKT} \propto (\delta g)^{z\nu}$ with $\nu \simeq 0.67$ and $z \simeq 1$, following the bosonic quantum-phase-fluctuation scenario.
  • Thermal depletion of superfluid stiffness crosses over from quasiparticle-dominated to vortex-like excitations near the transition, which matters for quantum circuits built from disordered superconductors.
  • The SIT in NbN is distinct from the first-order, glassy transition reported in InO$_x$; the difference is attributed to material parameters such as thickness, disorder strength, and screening.

Reading between the lines

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

  • If pairing indeed survives into the insulating phase, one expects a regime of preformed or localized Cooper pairs on the insulating side; a testable prediction is that diamagnetic or kinetic-inductance signatures should continue tracking the gap rather than the stiffness.
  • Because the inflection-point $T_{c0}$ is the only anchor for the pairing scale at high disorder, the paper's central contrast would be strengthened by a concurrent tunneling measurement of the single-particle gap on the same films used for stiffness data; the authors note this cross-check is currently missing.
  • The $\mu \simeq (\pi^2/2)J_s(0)$ result and the near-power-law stiffness depletion reported in other disordered superconductors suggest that a single vortex-core-energy scale may govern low-temperature electrodynamics of high-kinetic-inductance films; if so, the design rules for microwave circuits should account for vortex excitations rather than only quasiparticles.
  • A direct extension of the comparison to other materials with the same combined resistance-plus-stiffness protocol could test whether the order of the transition (continuous versus first-order) is controlled by screening strength and dimensionality, as the authors conjecture.
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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 / 7 minor

Summary. The manuscript reports a systematic study of the disorder-driven superconductor-insulator transition (SIT) in ultrathin NbN films, measuring both sheet resistance R□(T) and superfluid stiffness Js(T) on the same meander devices for fifteen superconducting and three insulating samples with normal-state resistance up to the critical range 16-18 kΩ. The finite-temperature transition is found to remain of Berezinskii-Kosterlitz-Thouless (BKT) type across the whole superconducting range: Halperin-Nelson fits describe R□(T) over 3-4 decades, and the vortex-unbinding temperature TBKT from resistance agrees with that from the universal stiffness jump Js(TBKT) = 2TBKT/π. The central claim is that close to the SIT the zero-temperature stiffness Js(0) and TBKT drop sharply below the mean-field expectation, quantified through a one-parameter Finkel'stein fit for Tc0 and the Ginzburg-Levanyuk formula, while the pairing temperature Tc0, assigned to the inflection point of R(T), remains finite and follows the Finkel'stein curve. This is interpreted as evidence that the SIT is a second-order quantum phase transition driven by quantum phase fluctuations, with pairing surviving into the insulator, in contrast to the first-order scenario reported for InOx [26]. The paper also reports the vortex-core energy µ and the activation energy E* extracted from the low-temperature stiffness, and comparisons with self-consistent BdG simulations of the pairing amplitude distribution.

Significance. If correct, the main payoff is a clean discrimination between fermionic (pairing-destruction) and bosonic (phase-fluctuation) mechanisms for the SIT in a canonical disordered two-dimensional superconductor, together with the demonstration that the BKT character of the finite-T transition survives up to critical disorder. The manuscript has genuine strengths: simultaneous R□(T) and Js(T) measurements on identical devices; two fully independent determinations of TBKT (Halperin-Nelson resistance fits and the universal stiffness jump) that agree across the entire disorder range, which solidly grounds the BKT-character claim; a cross-check of Js from nonlinear IV characteristics near the SIT (Extended Data Fig. 2); a one-parameter Finkel'stein baseline for Tc0; and falsifiable predictions (power-law suppression of Js(0) and TBKT with z ≃ 1) that connect to quantum-critical scaling. The persistence of the tunneling gap Eg across the SIT (Fig. 3g) gives partial independent support for surviving pairing at strong disorder.

major comments (3)
  1. [Supplementary Sec. I.E.a and Fig. S7; Fig. 3f; Fig. 2a] The central contrast of the paper—a finite Tc0 (light blue diamonds in Fig. 3f) coexisting with a sharply suppressed Js(0) (black circles) in regime III—rests on identifying Tc0 with the center of the linear segment / inflection point of R(T) at disorder levels where tc0 = (Tc0 - TBKT)/Tc0 reaches 0.65-0.8, i.e., where the phase-fluctuation window is about 80% of Tc0. Supplementary Sec. I.E.a states explicitly that the previous independent sanity check (quasiparticle suppression of Js(T) consistent with Eq. 3) is no longer possible for strongly disordered samples because Js(T) is almost rectangular (Fig. 2a). The near-SIT validation shown in Fig. S7 (red line) is a fit of the same amplitude-fluctuation theory to the same R(T) data that motivates the inflection-point convention, so it does not constitute an independent determination. In the regime tc0 = 0.65-0.8, the curvature crossover that defines the inflection point is itself shaped by the vortex-unbinding regime that spans most of the transition; if the true pairing temperature actually tracks Js(0) and TBKT near Wc, the apparent drop of Js(0) below the mean-field expectation would be an artifact of an overestimated Tc0, and the quantum-phase-fluctuation conclusion would lose its experimental foundation. I request (i) a sensitivity analysis of the Fig. 3f comparison under alternative Tc0 assignments (e.g., extrapolating the low-disorder amplitude-fluctuation fits without anchoring to the inflection point, or bounding Tc0 by the tunneling-gap scale), and (ii) an explicit statement of how much the regime-III Tc0 values would have to change for the claimed drop of Js(0) and TBKT relative to the Finkel'stein baseline to disappear. The zero-temperature tunneling gap Eg persisting into the insulator (Fig. 3g) supports the survival of pairing in the ground state, but it does not by itself pin the temperature scale Tc0, so it does not remove this concern.
  2. [Methods (Finkel'stein fit, Eq. 6); Fig. 3f; Extended Data Fig. 4] The mean-field baselines against which the regime-III drop is measured are not independent of the disputed Tc0 convention. The black line in Fig. 3f is obtained by inserting the Finkel'stein fit for Tc0(RN)—a one-parameter fit (γc = 5.312, Extended Data Fig. 4) to the very same inflection-point-based Tc0 data—into Eq. 4, and the red line by inserting that fit into Eq. 5. Any systematic error in the inflection-point assignment in regime III therefore enters both the data points and the baselines, and the manuscript does not quantify how robust the regime-III deviation is to this shared input. Please (i) report the residuals and confidence interval of the one-parameter Finkel'stein fit, and (ii) re-examine the deviation of Js(0) and TBKT from the baselines when the Finkel'stein fit is performed on regime I-II data only (RN ≲ 10-12 kΩ) and extrapolated into regime III, where the inflection-point assignment is least constrained. This test would separate the claimed drop from the calibration of the baseline.
  3. [Fig. 3f inset] The quantum-critical part of the conclusion (second-order transition, z ≃ 1) leans on the power-law analysis in the inset of Fig. 3f, where Js(0) ∝ (δg)^ν and TBKT ∝ (δg)^(zν) are stated to be 'consistent with' ν = 0.67. No fitting procedure, goodness-of-fit, or uncertainty is given, and the number of superconducting samples in regime III is small (of order four). Moreover δg = (Rc - RN)/Rc² depends on Rc, which is only bracketed as 16 kΩ < Rc < 18 kΩ; the resulting spread in δg can be large for near-critical points, and the implied uncertainty in (ν, z) is not propagated. Please provide the fit range, confidence intervals for ν and z, the number of points, and the sensitivity of the exponents to the choice of Rc within the stated bracket. If the exponent is too weakly constrained by the present data, the claim should be softened accordingly, since the first-order-versus-second-order distinction drawn from the InOx comparison in the Summary partly rests on this scaling analysis.
minor comments (7)
  1. [Abstract] The closing sentence of the abstract invokes the experimental discovery by Haviland, Liu, and Goldman, but the original paper (Phys. Rev. Lett. 62, 2180 (1989)) is not among the references; it should be cited there and where the SIT is introduced in the opening paragraph.
  2. [Eq. (6) vs. Supplementary Eq. S.11] The definition of the Finkel'stein parameter is inconsistent between the main text (γc = ln[ħ/(kBTc00τ)] in Eq. 6) and the Supplement (γc = 1/log(Tc00τ) in Eq. S.11), and the supplementary form is dimensionally inconsistent as written. Since γc = 5.312 is the single free parameter of the central baseline fit, the two definitions must be reconciled.
  3. [Main text, 'Similarly, Eq. 5 ...' and 'Close to the SIT ...' paragraphs] The universal BKT relation appears with an inverted factor in two places: 'TBKT = 2Js(TBKT)/π' and 'Js(TBKT) = πTBKT/2' should both read Js(TBKT) = 2TBKT/π (equivalently TBKT = (π/2)Js(TBKT)); elsewhere the paper uses the correct form (Fig. 2 caption), so this is presumably a typographical slip, but it should be corrected because the 2/π factor is invoked in the inset of Fig. 3f.
  4. [Methods; Supplementary Sec. I.D.c; Extended Data Fig. 4 caption] Several typographical errors should be cleaned up: 'One film film with lower disorder' (Methods), 'previous tudies' (Supplementary Sec. I.D.c), 'the the measured Tc0' (Methods, analysis of Tc0), and 'Rychaudhuri' (Extended Data Fig. 4 caption, for Raychaudhuri).
  5. [References [10], [12], [56]] The Kosterlitz 1974 paper is cited twice with different page numbers (J. Phys. C 7, 1060 in [10] and 1046 in [56]); reference [12] also contains a garbled author string ('V. Ambegaokar, D. R. N., B. I. Halperin & Siggia, E. D.'). These should be normalized.
  6. [Methods, two-coil comparison] The claim that broadening in two-coil measurements 'occurs for the two-coil method only' and is caused by roughly 10% edge-to-center variations of Js(0) and RN is supported only by an 'in preparation' citation; since this claim is used to reinterpret previously published BKT broadening results [27], it should either be presented with its supporting data or clearly flagged as preliminary.
  7. [Abstract and Summary] Given the sensitivity of the Tc0 assignment discussed in the major comments, the phrases 'unambiguous evidence for quantum phase fluctuations' (Abstract) and 'strongly suggest' (Summary) overstate the present support; I recommend qualifying these statements until the near-SIT Tc0 determination is independently corroborated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central deviation is an observed quantity compared against semi-empirical baselines, and the acknowledged missing cross-check is a validation gap, not a reduction of the conclusion to its inputs.

full rationale

Walking the paper's derivation chain, the central claim is an observed comparison: the measured superfluid stiffness Js(0) and BKT temperature TBKT deviate from semi-empirical baselines near the SIT, while Tc0 does not. The baselines are constructed from a one-parameter Finkel'stein fit to Tc0(RN) together with standard BCS and BKT relations (Eqs. 4 and 5); the measured Js(0) and TBKT are not inputs to that fit, so the reported sharp drop is not statistically forced. The extraction of Tc0 from the R(T) inflection point (Supplementary Sec. I.E.a) is a definitional convention, and the authors explicitly state that the independent quasiparticle suppression cross-check is no longer possible for strongly disordered samples. This is a real validation gap for the finite-Tc0 claim, but it is not a circular reduction: no equation equates the conclusion to the definition of Tc0, and the paper does not claim to derive Tc0 from the phase-fluctuation scenario. Self-citations ([19], [20], [21], [25]) are used for methods, fit forms, and computational procedures, not as the sole justification of the central conclusion. The BKT character is tested against standard Halperin-Nelson fits over 3–4 decades of resistance, which could have failed, and TBKT is also cross-checked independently via Js(TBKT)=2TBKT/pi. The RN-to-W mapping via tc0 is a coordinate calibration with explicitly listed caveats, not a hidden prediction. No fitted parameter is renamed as the claimed result, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in solely through self-citation. The paper is therefore self-contained in the sense required for a circularity finding.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central measurement is empirical, but the interpretation leans on several theoretical baselines: Finkel'stein theory with one fitted parameter gamma_c, BCS/Mattis-Bardeen formulas, BKT universality, and a BdG model whose disorder scale is mapped to experiment through the dimensionless ratio (Tc0 - TBKT)/Tc0. No new particles or mediators are introduced. The most load-bearing inputs are gamma_c and the inflection-point rule for Tc0.

free parameters (6)
  • gamma_c in Finkel'stein fit = 5.312
    Single free parameter in Eq. S.11 fitted to Tc0(R300) data; sets the theoretical Tc0 curve and therefore the BCS Js(0) and TBKT baselines used to define the anomaly near the SIT.
  • Numerical conversion factor for Js = 42.0 K/t
    Multiplies raw BdG superfluid stiffness values, given in units of hopping t, to match experimental Js in Kelvin in Fig. 3f.
  • Numerical conversion factor for Eg and E* = 113.4 K/t
    Multiplies raw BdG quasiparticle energy and activation energy to match experimental values in Fig. 3g.
  • TLS subtraction parameters f0(0) and F*delta0_TLS = F*delta0_TLS = 0.191
    Fit of Eq. S.9 to the low-temperature resonance frequency data; subtracted from f0(T) before extracting Js(T).
  • Power-law exponent nu = 0.67
    Inferred by fitting Js(0) and TBKT versus reduced conductance in Fig. 3f inset; reported as consistency with dotted lines, without confidence intervals.
  • Halperin-Nelson fit parameters C, b, TBKT per sample = reported per sample
    Three-parameter fits of Eq. 1 to R(T) supply the TBKT values used throughout; Fig. S8 shows the sensitivity to +/-30 mK.
assumptions (7)
  • standard math The zero-resistance transition in a 2D superconductor is governed by the BKT mechanism, including the Halperin-Nelson resistance form and the universal jump Js(TBKT) = 2 TBKT / pi.
    Used throughout to extract TBKT from R(T) and to interpret Js(T) (main text Eq. 1 and Fig. 2).
  • standard math The superfluid stiffness follows the Mattis-Bardeen/BCS form Eq. 3, with Eg and Delta allowed to differ.
    Used to extrapolate Js(0) from kinetic inductance data (Eq. 3 and Fig. 2b,c).
  • domain assumption Finkel'stein theory describes the fermionic suppression of Tc0 with a single parameter gamma_c.
    Eq. S.11 with gamma_c = 5.312 fitted to Tc0 data provides the mean-field baseline for Js(0) and TBKT.
  • ad hoc to paper The inflection point of R(T) marks the mean-field pairing temperature Tc0 even at high disorder where the phase fluctuation window spans 80% of Tc0.
    Supplementary Sec. I.E.a and Fig. S7; the independent cross-check via quasiparticle suppression is unavailable near the SIT, which the paper acknowledges.
  • domain assumption The level of disorder is represented by the normal-state resistance RN = R(Tmax), and RN maps to Anderson disorder W through the dimensionless ratio (Tc0 - TBKT)/Tc0 common to experiment and BdG numerics.
    Methods section 'Mapping between RN and W' and Fig. S11; caveats are listed in Supplementary Sec. II.B.b.
  • domain assumption The attractive-U BdG mean-field model on a 96x96 lattice captures the relevant physics of the NbN films.
    Theoretical Methods and Fig. 3a-d; used for pair-potential maps and comparison lines in Fig. 3f,g.
  • standard math Two-level system contributions to the resonance frequency are described by Eq. S.9 and can be subtracted using a fit.
    Supplementary Sec. I.D.c; affects the Js(T) data near millikelvin temperatures.

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Cite this review

Pith. "Pith review of Origin of the superconductor-insulator transition in disordered two-dimensional films." pith.science (2026). https://pith.science/paper/FFUM6HNH

@misc{pith2026260807101,
  author       = {Pith},
  title        = {Pith review of: Origin of the superconductor-insulator transition in disordered two-dimensional films},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFUM6HNH}},
  note         = {Machine review of arXiv:2608.07101}
}
abstract

Theory predicts the superconductor-to-insulator transition (SIT) to emerge from the competition between Anderson localization, which tends to localize single-particle wavefunctions, and superconductivity, which establishes long-range correlations in the superconducting order parameter. In two-dimensional (2D) superconducting films, the transition temperature $T_\text{c}$ at which resistance vanishes, $R_\Box(T_\text{BKT}){=}0$, is set by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism and satisfies $T_\text{BKT}< T_{c0}$, where $T_{c0}$ is the mean-field transition temperature. In weakly disordered samples $T_\text{BKT}\lesssim T_{c0}$, whereas increasing disorder drives $T_\text{BKT}\ll T_{c0}$ near the SIT. Whether the finite-temperature transition retains its BKT character throughout this crossover remains an open question. Here, we investigate the evolution of both sheet resistance $R_\Box(T)$ and superfluid stiffness $J_s(T)$ over a wide range of disorder strength $W$. We establish that even near the SIT, the finite-temperature transition from the superconducting to the resistive state remains of BKT type. However, as disorder approaches the critical value, the zero temperature superfluid phase stiffness, $J_s(0)$, is found to vanish rapidly while $T_{c0}$ remains finite, which we attribute to quantum phase fluctuations as the drive for the zero-temperature transition. Three decades after its experimental discovery by Haviland, Liu, and Goldman, our measurements clarify the origin of the SIT in 2D films.

Figures

Figures reproduced from arXiv: 2608.07101 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. e we plot 1 − Js(T)/Js(0) for five samples with dif￾ferent RN in an Arrhenius fashion, i.e., vs. 1/T. Using the activation energy E∗ and Js(0), we fit the data in Fig. 2e to (i) Eq. 3 (grey lines). Here, we use Eq. 3 in a heuristic way, without assuming E∗ = Eg. and (ii) using the NRG-procedure [27, 28] with Eq. 3 as starting point (red lines). The values of µ are determined by adjusting the position of the universa… 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 (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 1
Figure 1. Figure 1: Additionally, we show in Fig. S8 the fits on linear and logarithmic scale for two exemplary datasets. The [PITH_FULL_IMAGE:figures/full_fig_p019_1.png]
Figure 3
Figure 3. Figure 3: Thirdly, numerically obtained Tc0 does not reflect the experimentally observed suppression of Tc0 due to the insensitivity of the numerical procedure towards Coulomb interactions. The expected inaccuracies of the first two effects, based on standard BKT RG treatment an…

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    One film film with lower disorder (RN ≃880 Ω) was prepared by magnetron sputtering on a MgO substrate and is similar to the films presented in Ref

    orientation). One film film with lower disorder (RN ≃880 Ω) was prepared by magnetron sputtering on a MgO substrate and is similar to the films presented in Ref. [46]. For the ALD-grown films, the normal state re- sistance can be controlled in two ways: (i) variation of theN 2...

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