REVIEW 2 major objections 4 minor
Dynamics of amorphous membranes in the two-dimensional limit
T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Suspended monolayer amorphous carbon membranes act as nanomechanical resonators with pretension one to two orders lower than graphene, placing their nonlinear dynamics in a low-drive regime.
desk verdict A credible first demonstration of MAC nanomechanical resonators with an honest but unresolved density calibration that keeps the headline low-tension claim provisional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the suspended MAC nanodrum itself (a monolayer over a circular through-hole) read out through a Fabry–Pérot cavity formed with the underlying silicon. The quantitative argument rests on the tension-dominated circular membrane model, which relates the fundamental frequency to pretension by $f_0 = \frac{\alpha_{01}}{2\pi R}\sqrt{n_0/\rho_{2D}}$ with $\alpha_{01} = 2.4048$, the first zero of the Bessel function $J_0$. This single equation converts measured frequencies into pretension estimates, and the same model supplies the mode-sequence ratios against which the observed spectra deviate. The transduction scheme—above-bandgap (488 nm) optothermal actuation and below-bandgap (633 nm) interferometric detection—provides the measurement access; the membrane model supplies the physical interpretation.
What would settle it
Measure the fundamental frequency of a single MAC drum, then add a known mass (for example, by depositing a small gold particle) and remeasure; the frequency shift determines the modal mass and hence the areal mass density without assuming $\rho$ or $h$. If the resulting pretension falls in the graphene-like range ($\approx 0.1$–$1$ N/m), the paper's central regime claim is wrong. Alternatively, measure the frequency of at least two radial modes of the same drum and require them to match the tension-dominated membrane ratios; if the ratio deviates substantially from the ideal Bessel-function ratios in a way that cannot be explained by tension inhomogeneity, the model converting frequency to pretension is not adequate.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that monolayer amorphous carbon membranes are not merely transferable and robust enough to form suspended nanodrums, but that they sit in a previously hard-to-reach mechanical regime: their effective two-dimensional pretension is one to two orders of magnitude lower than that of graphene, with fundamental frequencies between 2.4 and 19 MHz. In the tension-dominated circular membrane model, the frequency–pretension relation $f_0 = \frac{2.4048}{2\pi R}\sqrt{n_0/\rho_{2D}}$ converts those frequencies into $n_0 \approx 10^{-4}$–$10^{-3}$ N/m. The authors present this low-pretension regime as the common root of the membrane's unusual dynamics: geometric nonlinearities, stress heterogeneity, and intermodal coupling become significant at low drive powers. They observe hardening, softening, and mixed Duffing traces, nonlinear damping that scales quadratically with drive amplitude, parametric excitation at $2f_0$, and a dip in the driven response that suggests internal resonance. The central claim is that these phenomena are not artefacts but direct consequences of the amorphous network and the low tension it permits.
Load-bearing premise
The pretension numbers hinge on an assumed areal mass density ($1300\,\mathrm{kg/m^3} \times 0.6\,\mathrm{nm}$) that has never been measured for suspended MAC; if the true mass density is larger or smaller by a factor, all inferred pretensions (and the central low-pretension regime) shift by that same factor.
Editorial extensions
If this is right
- Monolayer amorphous carbon drums become a new experimental system for 2D nanomechanics in which disorder, rather than crystallinity, sets the elastic response.
- Because the nonlinear onset displacement scales as $x_{nl} \propto R\sqrt{n_0/(Eh)}$, the measured one-to-two-orders lower pretension implies nonlinear effects appear at drive amplitudes roughly 3–10 times smaller than in comparable graphene drums.
- The observation of both hardening and softening, sometimes in the same device, indicates that competing nonlinear mechanisms (static sag, stress inhomogeneity, motion-dependent heating) are active; mapping which mechanism wins as a function of frequency and drive would test the low-tension picture.
- The 5-nm-thick MAC drums show more regular membrane spectra and a pretension around 0.1 N/m, suggesting that thickness averaging restores near-crystalline behavior; the paper cautions that this is not yet isolated from differences in pretension and morphology.
- The dip in the driven response, interpreted as 2:1 internal resonance between the fundamental and a nearby mode, implies that mode-counting and tunability are available in amorphous nanodrums, with implications for signal processing and sensing.
Reading between the lines
- If the low-pretension regime is confirmed by an independent mass-density measurement, MAC drums should be extraordinarily sensitive to external force or mass loads: the compliance is high, so small added masses or pressure changes would produce large frequency shifts. A testable extension is to compare the responsivity of MAC and graphene drums of identical geometry.
- The paper's reliance on an assumed areal mass density suggests a direct calibration route: measure the frequency shift after depositing a known mass (or measure two mode orders), which yields $\rho_{2D}$ without assuming a density or thickness. This would collapse the main uncertainty in the pretension claim.
- The correlation between pure softening and lower fundamental frequencies hints that the sign of the cubic nonlinearity could serve as a read-out of local stress heterogeneity or static sag in amorphous membranes. This is an inference the paper does not state explicitly.
- Because the amorphous network is electrically insulating, the same resonators could probe dynamics free of electronic damping, potentially reaching higher quality factors in a cleaner regime if the clamping and material losses are minimized.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper reports an experimental study of suspended monolayer amorphous carbon (MAC) nanodrums. The authors synthesize and transfer MAC membranes over circular holes, actuate them optothermally with a 488 nm laser, and read out their motion interferometrically with a 633 nm laser. They resolve thermomechanical Brownian motion, driven linear resonances, multimode spectra, and strongly nonlinear responses. Fundamental frequencies range from 2.4 to 19 MHz with quality factors 235 to 1750. Using the standard tension-dominated circular membrane model (Eq. 6) and an assumed areal mass density ρ2D = ρh with ρ = 1300 kg m−3 and h = 0.6 nm, they convert the measured frequencies into effective pretensions n0 in the range 10−4 to 10−3 N/m, which they state are one to two orders of magnitude lower than pretensions commonly reported for graphene nanodrums. At higher drive, they observe hardening, softening, and mixed Duffing nonlinearities, nonlinear damping, possible parametric excitation, and signatures of internal resonance. They also find that monolayer mode-frequency ratios deviate from ideal tensioned-membrane ratios, while 5 nm thick MAC samples show more regular spectra and higher effective pretension.
Significance. If the central claim is robust, this work would be a valuable first demonstration of monolayer amorphous carbon as a nanoelectromechanical resonator platform, extending nanomechanics of 2D materials from crystalline to amorphous systems. The paper has clear strengths: the experimental setup is described in detail, the analyses follow standard practice for 2D membrane resonators, the optical transduction is modelled with a transfer-matrix approach, and the nonlinear observations are qualitatively consistent with a low-pretension regime. The data availability statement is a positive feature. However, the headline quantitative claim, namely that MAC pretensions are one to two orders of magnitude below those of graphene, rests on an unmeasured areal mass density and on an ideal tension-dominated mode-shape assumption that is in tension with the paper's own observations of sagging and non-ideal mode ratios. The qualitative phenomenology is credible, but the central quantitative comparison is not yet secured.
major comments (2)
- [Results, Eq. (6) and following paragraph] The inferred pretension n0 is directly proportional to the assumed areal mass density ρ2D = ρh, with ρ = 1300 kg m−3 and h = 0.6 nm, which the paper explicitly states has not been directly measured. Because n0 scales linearly with ρ2D, any error in ρ2D translates directly into the claimed pretension range in Fig. 2(d). Residual PMMA, adsorbed water or hydrocarbons, or an effective thickness different from 0.6 nm could plausibly change ρ2D by a factor of several; a factor of 10 increase would move the inferred n0 range to 10−3–10−2 N/m and erase the headline one-to-two-order difference from graphene. The AFM step of approximately 1 nm does not constrain the suspended areal density, since it includes interface-related offsets and possible contamination. The displacement calibration in Eqs. (4) and (5) inherits the same assumption, so all absolute amplitude statements and the comparison of nonlinear-onset scaling with Eq. (8) are affected. Please state this sensitivity explicitly in the main text, and either provide an independent calibration of ρ2D or present the low-pretension comparison with graphene as conditional on the assumed density.
- [Results, Fig. 5 and Eq. (6)] The conversion of measured fundamental frequencies into pretension assumes an ideal, tension-dominated, uniform circular membrane mode shape. The paper, however, reports that membranes are recessed 40–100 nm below the SiNx surface and that monolayer mode-frequency ratios deviate substantially from the ideal tensioned-membrane ratios (Fig. 5(a)). Under these conditions, Eq. (6) yields an effective parameter whose physical interpretation as a uniform pretension is not established. In particular, as the paper itself lists in the discussion of Fig. 5, local mass loading can lower resonance frequencies without any reduction in tension; a low fundamental frequency is therefore not, by itself, evidence of low pretension. Please justify the mode-shape assumption for the specific devices used for the n0 extraction, or reframe the central claim as a model-dependent effective value and state what additional measurements (e.g., mode-shape imaging or frequency-shift mass calibration) would be needed to confirm the low-pretension regime.
minor comments (4)
- [Fig. 1 caption and text near Fig. 1(d)] The sentence 'AFM of MAC transferred onto SiO2 yielded an step height of approximately 1 nm' contains a grammatical error ('an step' should be 'a step'). Also, in the fabrication description, 'SiNx/Si (0.1/200µm)' should have a space before the unit, and '10−6 mbar' should be written with a space and a proper minus sign.
- [Fig. 2(c) and (d)] The quality-factor and pretension distributions are presented without stating the number of devices measured for each radius. Reporting the sample size and, where possible, the statistical uncertainty in the median and interquartile ranges would make the comparison between R = 1.25 μm and R = 2.5 μm more informative.
- [Fig. 3(f) and related text] The categories 'purely softening behavior' and 'mixed hardening–softening behavior' are not defined quantitatively. Please state the criterion used to assign a device to each category and give the number of devices in each group, since this classification supports the claim that pure softening correlates with lower fundamental frequencies.
- [Eq. (8)] The nonlinear-onset scaling xnl ∝ R sqrt(n0/(Eh)) omits the proportionality constant and does not specify whether E is the three-dimensional Young's modulus or a two-dimensional modulus. Specifying the factors and definitions would allow readers to reproduce the estimated factor-of-3–10 reduction in onset displacement.
Circularity Check
No circularity: n0 follows from measured f0 by a standard, openly stated membrane-model inversion; the unmeasured ρ2D input is disclosed and merely rescales the estimate, and self-citations provide material parameters rather than the low-pretension result.
full rationale
The derivation of the headline low-pretension claim is a transparent forward calculation, not a fit. Equation (6), f0 = (α01/2πR)√(n0/ρ2D), inverts the textbook tension-dominated circular-membrane model (Bessel zero α01 = 2.4048, Ref. [27]) to obtain n0 = ρ2D(2πRf0/α01)² from the measured fundamental frequencies (2.4–19 MHz, Fig. 2(c)); the comparison with graphene uses independent literature pretensions [28]. Nothing in this chain is fitted to the claimed output, and the observed nonlinearity is not fed back to adjust n0: Eq. (8) predicts xnl ∝ R√(n0/Eh), and the hardening/softening, nonlinear damping, and 2:1 mode features are reported as 'consistent with the low-pretension scaling' as an independent check. The one sensitive input is ρ2D = ρh with ρ = 1300 kg m⁻³ and h = 0.6 nm, which the paper openly flags: 'the areal mass density of suspended monolayer MAC has not been directly measured' (paragraph after Eq. (5)); the resulting n0 values are labelled 'order-of-magnitude estimates.' I flag this disclosed limitation explicitly: because n0 ∝ ρ2D, a factor-of-several mass error (e.g., residual PMMA or adsorbed contaminants) would linearly rescale all inferred pretensions and could erode the one-to-two-order comparison with graphene. That is a robustness/correctness risk, not circularity, because f0 is measured independently of ρ2D and the model inversion is standard. Self-citations (Refs. [16] and [19], with overlapping authorship) supply material parameters — synthesis, optical constants, and the nominal thickness/density — from published prior work that does not itself assert low pretension; these citations are real external evidence and not load-bearing for the central result. No Eq. X = Eq. Y by construction, no fitted parameter renamed as prediction, and no imported uniqueness theorem were found. Score 2 reflects only the minor disclosed self-citation feeding an ancillary assumption.
Assumptions & free parameters
free parameters (3)
- Effective areal mass density rho_2D =
7.8e-7 kg/m^2 (rho=1300 kg/m^3, h=0.6 nm)
- Modal mass prefactor alpha =
0.269
- Nominal monolayer thickness h =
0.6 nm
assumptions (4)
- domain assumption The fundamental mode of a suspended circular membrane obeys f0 = alpha01/(2*pi*R) * sqrt(n0/rho_2D) for a tension-dominated membrane (Eq. 6).
- domain assumption The large-amplitude motion is described by a Duffing oscillator with nonlinear damping (Eq. 7).
- domain assumption The nonlinear onset displacement scales as x_nl proportional to R * sqrt(n0/(E*h)) (Eq. 8).
- domain assumption Temperature of the membrane is 293 K (room temperature) for equipartition calibration.
Cite this review
Pith. "Pith review of Dynamics of amorphous membranes in the two-dimensional limit." pith.science (2026). https://pith.science/paper/UAAJKCF3
@misc{pith2026260810486,
author = {Pith},
title = {Pith review of: Dynamics of amorphous membranes in the two-dimensional limit},
year = {2026},
howpublished = {\url{https://pith.science/paper/UAAJKCF3}},
note = {Machine review of arXiv:2608.10486}
}
read the original abstract
Atomically thin mechanical resonators have been realized predominantly in crystalline two-dimensional (2D) materials, such as graphene, where long-range crystalline order sets their elastic properties and defines their nonlinear resonant behavior. Extending these concepts to the amorphous 2D limit has remained largely unexplored. Here, we demonstrate that monolayer amorphous carbon (MAC) forms suspended membranes that support optothermal actuation and sensitive interferometric readout across both linear and nonlinear regimes of its resonant motion. We resolve thermomechanical motion, driven resonances, and multimode spectra in MAC nanodrums. The frequencies of fundamental vibration modes correspond to unusually low pretensions, placing monolayer MAC nanodrums in a regime where geometric nonlinearities, stress heterogeneity, and mode coupling emerge at comparatively low drive powers. Consistently, we observe pronounced nonlinear dynamics, including hardening, softening, and mixed Duffing responses, nonlinear damping, parametrically excited modes, and signatures of intermodal coupling. These results establish MAC as a robust nanoelectromechanical platform and open an experimental route to disorder-governed nanomechanics in the 2D amorphous limit.
Figures
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Reviewed August 15, 2026 · model on record in the stance chip above.
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