Impact of surface roughness on the stability of nanoelectromechanical pressure sensors in the Casimir regime

A.S.Korotkov; G.L.Klimchitskaya; V.M.Mostepanenko; V.V.Loboda

REVIEW 3 major objections 5 minor 28 references

Impact of surface roughness on the stability of nanoelectromechanical pressure sensors in the Casimir regime

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Surface roughness leaves a Casimir pressure sensor's stable position intact but shifts its collapse threshold outward.

desk verdict A useful niche extension of the authors' prior smooth-surface sensor work: roughness shifts the unstable equilibrium by a few nanometers while leaving the stable one nearly fixed, though the main quantitative claim rests on an unverified perturbative regime. read the letter →

arxiv 2506.00179 v1 pith:NO7RI5U6 submitted 2025-05-30 quant-ph

G.L.Klimchitskaya , A.S.Korotkov , V.V.Loboda , V.M.Mostepanenko This is my paper

classification quant-ph PACS 12.20.Fv12.20.Ds68.35.Ct

keywords CasimirforcesurfaceroughnesspressuresensorsnanoelectromechanicalsystemsLifshitztheorymembraneequilibriumpull-ininstability

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 asks whether surface roughness changes the allowed positions of the moving membrane in a nanoelectromechanical pressure sensor that operates in the Casimir regime. Balancing the measured pressure, elastic spring pressure, and the Casimir pressure computed with the Lifshitz theory, the authors find that the stable equilibrium—the position at which the sensor would normally rest—stays essentially unchanged when the membrane and ground plate are rough, while the unstable equilibrium, beyond which the membrane snaps into contact, moves to larger separations. For a silicon membrane above a gold-coated plate, the unstable point shifts by about 1 nm for 5 nm gold roughness and about 4 nm for 10 nm roughness. This matters for shrinking future sensors, which are expected to work with ordinary rougher surfaces rather than the specially smooth surfaces used in precision Casimir experiments.

What carries the argument

The load-bearing object is the equilibrium condition $f(z) \equiv k(h-z)/(LD)-P = P_R^C(z,T)+P_R^{el}(z)$, which equates the downward pressure available from the spring and the measured pressure with the upward sum of Casimir and electric pressures. Roughness enters through multiplicative corrections to both pressures: the Casimir pressure is multiplied by $1+10(\delta_1^2+\delta_2^2)/z^2+105(\delta_1^2+\delta_2^2)^2/z^4$ and the electric pressure by the analogous expression with coefficients $3$ and $5$. These corrections grow steeply as $z$ decreases, so they are strong near the small-separation unstable equilibrium and much weaker at the larger stable separation.

What would settle it

A direct measurement of the pull-in separation of a Si membrane above an Au-coated plate with r.m.s. roughness near 10 nm and independently characterized roughness features would decide the matter: the paper predicts a shift of about 4 nm to larger separation relative to a smooth plate, with the stable position unchanged.

Watch

Extended reading notes

Core claim

The paper's claim, stated on its own terms, is that surface roughness has a one-sided effect on pressure sensor stability: it does not perturb the stable equilibrium separation, but it does push the unstable equilibrium to larger separations, so the usable stability margin shrinks. In the elastic-plus-Casimir case with a Si membrane over an Au-coated plate, the smooth-surface unstable equilibrium near $z \approx 83$ nm becomes $z \approx 84$ nm at $\delta_{\rm Au}=5$ nm and $z \approx 88$ nm at $\delta_{\rm Au}=10$ nm, while the stable equilibrium stays near $z \approx 129.5$ nm. When the electric readout pressure is included, the unstable shift is about $0.5$ nm at $\delta_{\rm Au}=5$ nm and about $4$ nm at $\delta_{\rm Au}=10$ nm, with the stable equilibrium again unchanged near $164.5$ nm. The authors frame this as an extension of the earlier no-roughness model: the underlying Lifshitz/Casimir physics is unchanged, and the roughness correction acts mainly near the point where the pressure curves cross steeply.

Load-bearing premise

The result relies on the roughness correction formula staying accurate when the roughness height (up to 10 nm) is no longer very small compared with the membrane-plate separation (about 88 nm), and the paper does not give the lateral size of the roughness features.

Editorial extensions

If this is right

For the ultra-smooth surfaces used in precision Casimir experiments (about 0.1 nm Si roughness and 2 nm Au roughness), the rough and smooth equilibrium positions coincide, so previous smooth-surface sensor calculations remain valid.
At larger roughness, the stable separation of the membrane can still be predicted from the smooth-surface model; the calibration curve of the sensor does not need a roughness correction.
For a 10 nm rough gold coating, the collapse threshold moves approximately 4 nm outward, so a rough device has a smaller usable pressure range before snap-down than a smooth one.
Including the electric pressure used for capacitive readout preserves the qualitative result; the unstable shift is 0.5 nm for 5 nm gold roughness and 4 nm for 10 nm gold roughness, with no change in the stable equilibrium.

Reading between the lines

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

Extending the paper's logic, the rough-surface correction becomes larger as device separations shrink, so the asymmetry between stable and unstable equilibria should grow for next-generation miniaturized sensors even at fixed roughness amplitudes.
A testable extension is to measure the roughness power spectrum of the gold coating and recompute the Casimir pressure nonperturbatively; if the lateral size of the roughness features is comparable to the separation, the 4 nm shift predicted here would likely change.
The same one-sided stability effect should apply to other Casimir-actuated devices such as switches and accelerometers, whose switching thresholds are set by unstable equilibria.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

This is a straightforward, honest extension of the authors' earlier smooth-surface pressure sensor calculation. The new piece is inserting their 1995 perturbative roughness correction into the force balance and solving for the equilibrium. The result that catches the eye: the stable equilibrium position barely moves with roughness, but the unstable equilibrium shifts to larger separations by ~1 nm for δAu=5 nm and ~4 nm for δAu=10 nm. If correct, that is practically useful for sensor design.

The calculation is easy to follow and internally consistent. They use the Lifshitz pressure, measured roughness values for the low-roughness cases, and no fitted parameters. There is no circularity; Eq. (3) is an external perturbative result, and Eq. (4) is a standard force balance. In the smooth-surface limit they reproduce their previous numbers, which is a good check. I credit them for stating the validity conditions of Eq. (3) in Sec. 2, and for not overclaiming the magnitude of the effect.

The soft spot is the validity regime of Eq. (3) in the roughest cases. They apply it at δAu=10 nm with z≈88 nm (ratio 0.11) and never give the correlation length for the rough Au surface. The formula's coefficients 10 and 105 assume a pressure falling as z^{-4}. At these separations the Lifshitz pressure is indeed close to that for ideal conductors, so the coefficient is likely not wildly wrong, but the paper does not demonstrate that. If the effective exponent were 3 instead of 4, the 4 nm shift would drop to roughly 2.5 nm. The qualitative claim—roughness pushes the unstable point to larger gap—is robust because any positive roughness correction to an attractive pressure has that sign. The quantitative number is less secure. Also, Sec. 4 omits the value of U0 used, and the paragraph introducing Fig. 2 is duplicated. Both are fixable.

The paper is for readers designing or modeling Casimir-dominated MEMS/NEMS devices. It is a modest applied result, not a fundamental advance, but it is a legitimate one. I would send it to peer review, with a request that the authors (i) justify the roughness regime or soften the quantitative claim, and (ii) supply the missing voltage and an uncertainty estimate.

Best, [Your name]

Referee Report

3 major / 5 minor

Summary. The manuscript analyzes a nanoelectromechanical pressure sensor whose movable Si membrane is suspended by a spring above a Si or Au-coated Si ground plate, with equilibrium governed by Eq. (4) or Eq. (6). The Casimir pressure is computed from the Lifshitz formula (1) and multiplied by the perturbative roughness correction (3); the electrostatic pressure with roughness is given by Eq. (5). Solving the force-balance equations for a fixed measured pressure, the authors find two equilibrium separations: an unstable one at smaller z and a stable one at larger z. They report that surface roughness leaves the stable equilibrium essentially unchanged but shifts the unstable equilibrium to larger separations by about 1 nm for δAu = 5 nm and 4 nm for δAu = 10 nm, both with and without the electric pressure term.

Significance. The topic is practically relevant: quantifying how surface finish affects the collapse threshold of Casimir-regime pressure sensors is useful for device design. The Lifshitz calculation is standard, the numerical solution of Eq. (4) is straightforward, and the input roughness amplitudes are either measured values from Refs. 25 and 28 or explicit scenarios, so no parameter is fitted to the predicted shift. The main value of the paper is its identification of a robust qualitative effect: roughness pushes the unstable equilibrium to larger separations. The quantitative magnitudes, however, are only as reliable as the perturbative roughness correction (3) and the stated inputs, both of which need additional justification.

major comments (3)

[Sec. 2, Eq. (3)] The paper applies Eq. (3) to configurations with δAu = 10 nm at an unstable equilibrium z ≈ 88 nm (Fig. 1, right), i.e. δAu/z ≈ 0.11, and with δAu = 5 nm at z ≈ 84 nm. The text states that Eq. (3) assumes roughness amplitudes much smaller than z and correlation lengths much larger than z; no correlation length for Au is given, and δAu/z = 0.11 is not clearly in the asymptotic small-amplitude regime. Moreover, the coefficients 10 and 105 in Eq. (3) are those appropriate to a pressure scaling as z^{-4}; the Lifshitz pressure for Au at 80-130 nm does not necessarily follow this power law, so the leading correction should involve (1/2)P_C''/P_C times the roughness variance. Because the reported 4 nm shift rests on Eq. (3), the authors should justify its applicability for the roughest cases, for example by providing measured correlation lengths and checking sensitivity to the coefficient, or by using a numerical roughness treatment to support the quantitative claim.
[Sec. 4, Eq. (6)] The value of U0 is never stated in Sec. 4. The electric pressure in Eq. (5) depends on U0, and the equilibrium positions quoted in the text and in Fig. 2 (zunst ≈ 84.5 nm and 88 nm, zst ≈ 164.5 nm) cannot be reproduced or checked without this input. The authors should give U0 explicitly; if it is the same value as in Ref. 15, they should say so and cite the exact location.
[Sec. 4, Eq. (5)] Equation (5), the roughness correction to the electrostatic pressure, is asserted without derivation or citation. The coefficients 3 and 5 are plausible for a pressure scaling as z^{-2} with Gaussian roughness, but since Eq. (5) is used to compute the equilibrium shifts reported in Fig. 2, the authors should either derive it from an average over roughness heights or cite the source where it was established.

minor comments (5)

[Abstract] There is a typo in the abstract: "creatign" should be "creating", and "shrinked" should be "shrunken" or "further reduced in size".
[Sec. 2, Eq. (1)] The text below Eq. (1) refers to "Matrubara frequencies"; the standard spelling is "Matsubara frequencies".
[Fig. 1 caption] The caption contains a typo: "solid lne 2" should be "solid line 2".
[Sec. 4] The paragraph beginning "Now we consider the same pressure sensor with larger r.m.s. roughness amplitudes..." is duplicated verbatim in the text.
[Sec. 2, Eq. (3)] The paper would benefit from a brief statement of the regime of validity of Eq. (3) in terms of the correlation length, and from an indication of which measured correlation lengths are available for the Si and Au surfaces used in Refs. 25 and 28.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the roughness-induced shift of the unstable equilibrium follows from an external perturbative formula (Eq. 3) and an independent force-balance condition; no parameter is fitted to the claimed shift.

full rationale

The paper's quantitative claims (1 nm and 4 nm shifts of the unstable equilibrium) are obtained by solving Eq. (4), f(z) = P_C^R(z,T), with P_C^R given by Eq. (3). Eq. (3) is quoted from Ref. 23, an independent published perturbative result, and it carries stated assumptions (roughness amplitude much smaller than z and correlation length much larger than z); it does not contain the equilibrium shift as an input. The smooth-surface Lifshitz pressure Eq. (1) and the roughness amplitudes (0.1 nm, 1 nm, 2 nm, 5 nm, 10 nm) are taken from measurements or scenarios, not fitted to the final equilibrium shifts. Several cited works (Refs. 7, 15, 23, 25-28) share authors with this paper, but the load-bearing physical input, Eq. (3), is parameter-free and is not equivalent to the paper's conclusion. The only caveat is a validity concern rather than circularity: for delta_Au = 10 nm at z ≈ 88 nm (delta_Au/z ≈ 0.11) and with no correlation length specified, the conditions stated before Eq. (3) may not be fully met, so the numerical size of the shift is a robustness/correctness issue, not a circularity. The derivation is self-contained against external benchmarks; the central claim follows by calculation rather than by construction. Score 1 reflects the presence of multiple same-author citations in the supporting chain, none of which is load-bearing in a circular sense.

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

The central claim rests on standard Lifshitz theory, a published perturbative roughness expansion, and several design parameters taken from prior work. There are no new physical entities. The main burden is that the roughness expansion is used near its validity boundary and that one electric parameter is missing.

free parameters (5)

δAu (Au plate r.m.s. roughness) = 2 nm (measured ref.), 5 nm and 10 nm (scenarios)
Input values that drive the roughness correction in Eq. (3); the larger scenarios are chosen by hand and control the reported unstable shift.
δSi (Si membrane r.m.s. roughness) = 0.1 nm (measured ref.), 1 nm (scenario)
Roughness amplitude for the membrane; enters Eq. (3) and Eq. (5) symmetrically with δAu.
k (effective spring constant) = 30 N/m
Taken from Ref. 15 via kh = P0 LD with P0 = 3 kPa and h = 20 µm; sets the elastic pressure in Eqs. (4) and (6).
P (external measured pressure) = 2.979 kPa (Sec. 3), 2.973 kPa (Sec. 4)
Chosen just below the collapse pressure so two equilibria exist; the equilibrium positions and the existence of the unstable point depend on this value.
U0 (electric voltage) = not stated
Required by Eq. (5) for the electric pressure but no value is given, so the Sec. 4 numerical results are not reproducible from the manuscript.

assumptions (5)

domain assumption Lifshitz formula (Eq. 1) gives the Casimir pressure for smooth planar slabs at finite temperature.
Baseline pressure used throughout; accepted from Refs. 7, 16-18, not re-derived here.
domain assumption Perturbative roughness correction (Eq. 3) is valid for the roughness amplitudes and separations considered.
Quoted from Ref. 23; carries conditions δ << z and correlation length >> z that are not checked for δAu = 10 nm, z about 88 nm.
domain assumption Electric pressure with roughness is given by Eq. (5).
Stated without derivation or citation; assumes a parallel-plate capacitor with small roughness corrections.
domain assumption Free charge carriers in high-resistivity Si do not affect the Casimir pressure below 200 nm.
Invoked in Sec. 3 via Refs. 25-27 to justify using low-conductivity Si optical data at the Matsubara frequencies.
domain assumption The finite membrane can be modeled as an infinite plane.
L = 1000 µm, D = 200 µm, separations below 200 nm make edge effects negligible, but the approximation is not quantified.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Impact of surface roughness on the stability of nanoelectromechanical pressure sensors in the Casimir regime." pith.science (2026). https://pith.science/paper/NO7RI5U6

@misc{pith2026250600179,
  author       = {Pith},
  title        = {Pith review of: Impact of surface roughness on the stability of nanoelectromechanical pressure sensors in the Casimir regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NO7RI5U6}},
  note         = {Machine review of arXiv:2506.00179}
}

read the original abstract

The stability of nanoelectromechanical pressure sensors working in the Casimir regime is considered with account of surface roughness on both the sensor membrane and the ground plate. The equilibrium positions of the sensor membrane are found from the balance between the external measured, elastic, electric pressures, and the Casimir pressure computed by means of the Lifshitz theory. It is shown that the stable equilibrium position of the sensor membrane is nearly independent of the surface roughness, whereas its unstable equilibrium position is shifted to larger membrane-plate separations. The use of these results for creatign pressure sensors with further shrinked dimensions is discussed.

Figures

Figures reproduced from arXiv: 2506.00179 by the authors.

**Figure 2.** Unstable and stable equilibrium positions of the rou [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 24 canonical work pages

[1]

J. W. Gardner, V. K. Varadan and O. O. Awadelkarim, Microsensors, MEMS and Smart Devices (Wiley, Chishester, 2013)

work page 2013
[2]

R. B. Mishra and S. S. Kumar, J. Phys.: Conf. Ser. 1240, 012068 (2019). September 12, 2025 13:10 WSPC/INSTRUCTION FILE arxGs Impact of Surface Roughness on the Stability of Nanoelectro mechanical Pressure Sensors 7

work page 2019
[3]

A. T. Tutaev, A. S. Kozlov, J. V. Belyaev, V. V. Loboda, M. A. Bellavin and A. S. Ko- rotkov, IEEE Sensors J. 24, 7395 (2024)

work page 2024
[4]

R. P. Feynman, Eng. Sci. 23, 22 (1960)

work page 1960
[5]

H. B. G. Casimir, Proc. Kon. Ned. Akad. Wet. B 51, 793 (1948)

1948
[6]

V. A. Parsegian, van der Waals Forces: a Handbook for Biologists, Chemists, E ngi- neers, and Physicists (Cambridge University Press, Cambridge, 2005)

work page 2005
[7]

Bordag, G

M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Moste panenko, Advances in the Casimir Eﬀect (Oxford University Press, Oxford, 2015)

2015
[8]

L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-L opez, A. W. Rodriguez and R. Podgornik, Rev. Mod. Phys. 88, 045003 (2016)

work page 2016

Show all 28 references

[9]

G. L. Klimchitskaya and V. M. Mostepanenko, Int. J. Mod. Phys. A 37, 2241002 (2022)

2022
[10]

H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop and F. C apasso, Science 291, 1941 (2001)

2001
[11]

Esquivel-Sirvent and R

R. Esquivel-Sirvent and R. P´ erez-Pascual, Eur. Phys. J. B 86, 467 (2013)

2013
[12]

L. Tang, M. Wang, C. Y. Ng, M. Niclic, C. T. Chan, A. W. Rodri guez and H. B. Chan, Nature Photonics 11, 97 (2017)

2017
[13]

G. L. Klimchitskaya, V. M. Mostepanenko, V. M. Petrov and T. Tschudi, Phys. Rev. Appl. 10, 014010 (2018)

2018
[14]

Javor, Z

J. Javor, Z. Yao, M. Imboden, D. K. Campbell and D. J. Bisho p, Microsyst. & Nano- eng. 7, 73 (2021)

2021
[15]

G. L. Klimchitskaya, A. S. Korotkov, V. V. Loboda and V. M. Mostepanenko, Euro- phys. Lett. 146, 66004 (2024)

2024
[16]

E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 29, 94 (1955) [ Sov. Phys. JETP 2, 73 (1956)]

1955
[17]

I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii , Usp. Fiz. Nauk 73, 381 (1961) [Adv. Phys. 10, 165 (1961)]

1961
[18]

E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Pt. II (Pergamon, Oxford, 1980)

1980
[19]

Bordag, G

M. Bordag, G. L. Klimchitskaya and V. M. Mostepanenko, Int. J. Mod. Phys. A 10, 2661 (1995)

1995
[20]

P. J. van Zwol, G. Palasantzas and J. Th. M. De Hosson, Phys. Rev. B 77, 075412 (2008)

2008
[21]

Broer, G

W. Broer, G. Palasantzas, J. Knoester and V. B. Svetovoy, Phys. Rev. B 85, 155410 (2012)

2012
[22]

P. A. Maia Neto, A. Lambrecht and S. Reynaud, Phys. Rev. A 72, 012115 (2005)

2005
[23]

Bordag, G

M. Bordag, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Lett. A 200, 95 (1995)

1995
[24]

Handbook of Optical Constants of Solids , edited by E. D. Palik (Academic Press, New York, 1985)

1985
[25]

F. Chen, G. L. Klimchitskaya, V. M. Mostepanenko and U. Mo hideen, Phys. Rev. B 76, 035338 (2007)

2007
[26]

Chang, A

C.-C. Chang, A. A. Banishev, G. L. Klimchitskaya, V. M. Mo stepanenko and U. Mo- hideen, Phys. Rev. Lett. 107, 090403 (2011)

2011
[27]

A. A. Banishev, C.-C. Chang, R. Castillo-Garza, G. L. Kli mchitskaya, V. M. Mostepa- nenko and U. Mohideen, Phys. Rev. B 85, 045436 (2012)

2012
[28]

Chang, A

C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Kli mchitskaya, V. M. Mostepa- nenko and U. Mohideen, Phys. Rev. B 85, 165443 (2012)

2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

J. W. Gardner, V. K. Varadan and O. O. Awadelkarim, Microsensors, MEMS and Smart Devices (Wiley, Chishester, 2013)

work page 2013

[2] [2]

R. B. Mishra and S. S. Kumar, J. Phys.: Conf. Ser. 1240, 012068 (2019). September 12, 2025 13:10 WSPC/INSTRUCTION FILE arxGs Impact of Surface Roughness on the Stability of Nanoelectro mechanical Pressure Sensors 7

work page 2019

[3] [3]

A. T. Tutaev, A. S. Kozlov, J. V. Belyaev, V. V. Loboda, M. A. Bellavin and A. S. Ko- rotkov, IEEE Sensors J. 24, 7395 (2024)

work page 2024

[4] [4]

R. P. Feynman, Eng. Sci. 23, 22 (1960)

work page 1960

[5] [5]

H. B. G. Casimir, Proc. Kon. Ned. Akad. Wet. B 51, 793 (1948)

1948

[6] [6]

V. A. Parsegian, van der Waals Forces: a Handbook for Biologists, Chemists, E ngi- neers, and Physicists (Cambridge University Press, Cambridge, 2005)

work page 2005

[7] [7]

Bordag, G

M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Moste panenko, Advances in the Casimir Eﬀect (Oxford University Press, Oxford, 2015)

2015

[8] [8]

L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-L opez, A. W. Rodriguez and R. Podgornik, Rev. Mod. Phys. 88, 045003 (2016)

work page 2016