Membrane sandwich squeeze film pressure sensors Free

Large-area, low thickness drums acting as deformable diaphragms are commonly used for pressure sensing. In conventional miniaturized sensors, the pressure difference with a reference cavity is determined by measuring the deflection of the suspended drum or by dynamically monitoring the change of the drum resonance frequencies.^1,2 As illustrated in Fig. 1, the pressure readout can be achieved by piezo-resistive, piezoelectric, capacitive, or optical means. Typically, such sensors operate around atmospheric pressure or in low vacuum (millibar). For high and ultrahigh vacuum pressure measurements, ionization gauge sensors are routinely used due to their very high sensitivity, but they intrinsically require the knowledge of the involved species and their ionization cross section in order to determine an absolute pressure.

Squeeze film pressure sensors—which exploit the dynamical modification of the mechanical properties of oscillating elements due to the compression of a fluid in a small gap region³—offer a solution to this problem. By measuring the compressive force rather than the force due to the pressure difference with a reference cavity (Fig. 1), such sensors allow for direct and absolute pressure measurements and bypass the challenge of fabricating stable, sealed, and non-outgassing reference cavities. Displacement measurements allowing for determining the mechanical resonance frequencies and linewidths of the drums can be conveniently performed electrically in MEMs systems, although they require the use of electrodes/lossy materials, which may deteriorate the mechanical properties of the resonators, and electronic noise ultimately limits the achievable sensitivity [Figs. 1(c)–1(e)]. Optical interferometric displacement measurements [Fig. 1(f)] may, in contrast, be more sensitive and allow for the use of purely dielectric thin films, such as, e.g., silicon nitride, with very low mass and excellent mechanical properties.

Since Blech’s proposal to use squeeze film effects to control the response of seismic interferometers,⁴ the squeeze film sensors have been realized with various materials, geometries, and detection techniques^5–22 Progress in fabricating and controlling micro- and nanostructures has, in particular, allowed for the application of high-frequency and high mechanical quality resonators to the exploration of a broad range of hydrodynamics regimes.^23–27 

This Tutorial article focuses on the working principles of squeeze film pressure sensors based on “membrane sandwiches,” i.e., pressure sensors consisting of a parallel arrangement of two large-area, ultrathin suspended films forming a few-micrometer gap and immersed in a fluid. In the first part, various pressure regimes and relevant characteristic numbers are introduced in order to discuss the nature of the interaction of the fluid with the oscillating membranes. This article focuses mostly on the free molecular flow regime, in which the fluid can be considered a gas, which is isothermally compressed in the gap region. The basic equations describing the dynamics of the membrane vibrations are derived and the squeeze film effect consequences, in particular, the gas-induced coupling between the membranes, are generally discussed. In the second part of the article, recent experimental implementations^21,22 and illustrations of these effects using high-tensile stress silicon nitride membrane resonators are presented.

The previous analysis holds for a plate oscillating in a fluid far from other structures/boundaries. However, in the presence of a nearby structure, e.g., a fixed parallel plate defining a small gap $d$ as illustrated in Fig. 2(b), the dynamics of the molecules inside the gap may be substantially modified.³ The compression of the fluid inside the gap results in an additional squeeze film force on the moving plate.

Figure 5 illustrates the variations of the bright and dark mode frequencies, $ω ±$⁠, as the gas-induced coupling $η$ is increased. When the membranes’ intrinsic degenerate frequencies are perfectly degenerate (⁠ $δ=0$⁠), the dark mode frequency remains unchanged, while that of the bright mode experiences a linear shift given by $2η$⁠. For nondegenerate intrinsic frequencies, positive linear frequency shifts given by $η$ are first observed as long as $η≪δ$⁠. When $η∼δ$⁠, the dark mode becomes decoupled from the pressure and its frequency converges toward $ω 0$⁠, while the bright mode experiences a doubled linear shift given by $2η$⁠. When $η$ becomes so large that the frequency shift is no longer small with respect to the bare frequency, the frequency shift no longer scales linearly with the coupling, but as $2 η$⁠, as illustrated in Fig. 5(b).

The membrane resonators used in the examples discussed below are commercial (Norcada Inc.), high tensile stress (⁠ $∼0.7−0.9 GPa$⁠), $500μ m$-square, and $∼100 nm$-thick stoichiometric silicon nitride films deposited on a 5 mm-square and $500μ m$-thick silicon frame by low pressure chemical vapor deposition. The membrane sandwiches used in this work were realized using the method described in Ref. 31: a spacer was deposited on one of the SiN films, and the membranes were positioned parallel to each other using actuators and monitoring the transmission of broadband light through the Fabry–Pérot interferometer constituted by both membranes. A picture of an assembled sandwich with a gap of $d=2.95μ m$ and its normalized transmission spectrum is shown in Fig. 6. The membrane sandwiches are then placed in a vacuum chamber and positioned parallel with a 50:50 beam splitter mirror to form a 7 mm-long Fabry–Pérot interferometer, whose length can be controlled by means of a piezoelectric transducer (Fig. 7). Monochromatic light from a tunable external cavity diode laser (890–940 nm) is injected into the interferometer and the transmitted light is detected by a fast photodiode and its fluctuations recorded on a narrow resolution bandwidth spectrum analyzer. The length of the interferometer is chosen so as to maximize the displacement sensitivity. The recorded thermal noise spectra around the mechanical resonance frequencies allow for their precise determination, as well as that of the mechanical quality factors. The pressure is varied by letting air at room temperature in the chamber.

The fundamental drummodes of the membranes typically have intrinsic (in vacuum) resonance frequencies between 700 and 850 kHz and display quality factors of the order $10 5$ after assembly. Typical variations of the Knudsen number with pressure in the range investigated are shown in Fig. 3, illustrating that the transition regime occurs in the few to few tens of millibars range. For sandwiches with gaps between 2 and $8.5μ m$ and the frequencies considered, the Weissenberg number is $∼80−340$ so that the resonators operate well in the high-frequency regime, in which the squeeze film force is expected to be essentially elastic. This behavior is also expected in the transition and viscous regimes, since estimates of the squeeze parameter also give high values, as can be seen in Fig. 3. Figure 4 additionally shows that, for the high mechanical frequencies considered, squeeze film damping is expected to be negligible with respect to kinetic damping.

First, investigations were carried out with membrane sandwiches having gaps in $8−9μ m$⁠, provided by a predeposited dielectric spacer on one of the commercial chips.³¹ Thermal noise spectra around the fundamental drummode frequencies were recorded at various air pressure at room temperature and for different sandwiches.²¹  Figure 8(a) shows examples of such spectra in the few millibar range for non-degenerate sandwich for which the intrinsic frequencies of the fundamental modes were 712 and 733 kHz, corresponding to $δ=0.014 ω 0$⁠. Positive linear resonance frequency shifts of 0.84 and 0.82 kHz/mbar, respectively, are observed in good agreement with the elastic squeeze film predictions. In contrast, measurements performed with a single membrane resonator from the same fabrication batch showed comparatively much smaller, negative resonance frequency shifts [the inset of Fig. 8(a)].

The near-degenerate sandwich case is shown in Fig. 8(b), where a clear air-induced hybridization of the fundamental drummodes with intrinsic frequencies of 721 and 721.2 kHz (⁠ $δ= 10 − 4 ω 0$⁠) can be observed. There, positive resonance frequency shifts of $≃0$ and 1.66 kHz/mbar are observed for the bright and dark modes, respectively, again in good agreement with the theoretical predictions. The solid lines in Fig. 8(b) also show the results of fits to the theoretically expected thermal noise spectra of coupled harmonic oscillators, in which $γ kin$ and $κ s q$ are given by the theoretical predictions based on the independently determined membrane thickness and gap. The only free fitting parameters are a global scaling factor for the thermal noise spectrum amplitude and the relative weight of the measured membrane displacements, which non-trivially depend on a number of parameters (interferometer length, intermembrane separation, optical/mechanical mode overlap, membrane parallelism, etc.). A very good agreement is observed between the measured spectra and the coupled oscillator model predictions.

To further ascertain that the squeeze film damping contribution was indeed negligible with respect to that of the kinetic damping, the variations with the pressure of the quality factor of the fundamental mode of a single free-standing membrane and of a membrane in an $8.5μ m$-gap sandwich were accurately measured in the range $10 − 6−10 mbar$ and are displayed in Fig. 9. The same linear increase of the quality factor is observed as the pressure is reduced, until air damping becomes negligible with respect to the intrinsic damping at pressures below $10 − 3 mbar$⁠. The measured quality factors were also found to be in very good agreement with the kinetic damping predictions of Eq. (2).

To increase the magnitude of the squeeze film effects, sandwiches with shorter gaps (⁠ $2−3μ m$⁠) were assembled using similar membranes, albeit with slightly lower (⁠ $∼87 nm$⁠) thickness and higher (⁠ $∼0.9 GPa$⁠) tensile stress, and various spacers made of aluminum or UV-light cured resist. Figure 10 shows the effects of pressure on the fundamental drummode resonance frequency of a $2.95μ m$-gap sandwich in the range $0.1−50 mbar$⁠. Very large squeeze film-induced shifts of up to 175 kHz—representing a relative variation of $∼20%$ of the fundamental drummode frequency—are observed at 50 mbar. In the range 2–20 mbar a pressure responsivity of about 4 kHz/mbar is achieved, which is a factor approximately 2 lower than the largest squeeze film sensor responsitivities reported with graphene microdrums.²⁰ The crossovers from the linear, independent membrane shift to the doubled linear, coupled membrane shift—and even the nonlinear strong coupling shift discussed in Sec. II B—are clearly observed. The measured shifts also match very well the theoretical expectations based on the independently determined intermembrane gap and membrane thickness. The variations of frequency shifts and damping rates were furthermore measured for higher order modes and found to evolve as well as expected from the squeeze film model predictions.²² 

To evaluate the sensitivity of these sensors at low pressures, systematic measurements of the resonance frequencies referenced to that in vacuum were performed by sequential rapid increase and decrease of the pressure in order to minimize the thermal drifts during the measurements. For these measurements, a $2.1μ m$-gap sandwich with membranes having intrinsic fundamental drummode resonance frequencies of 820.1 and 831.2 kHz with quality factors 86 000 and 46 000, respectively, was used. At low pressure, an equal pressure responsitivity of 3.1 kHz/mbar at low pressures is observed for each mode with a sensitivity at the $μ$bar level, essentially limited by the thermal drifts of the vacuum chamber used in these measurements (Fig. 11).

The workings of membrane sandwich pressure sensors have been introduced with particular focus on the free molecular flow regime, and recent experimental implementations using silicon nitride resonators operating in the high frequency regime have been presented.

The use of a membrane sandwich in which a fluid is isothermally compressed in the small gap region between the membranes makes it possible to exploit squeeze film effects in order to determine in a direct and species-independent way the pressure exerted on the membranes. High pressure responsivity/sensitivity can then be obtained, as small gaps (few micrometers) and thin membranes (⁠ $<100 nm$⁠) make for strong squeeze film effects and high tensile stress make for high frequency (⁠ $∼ MHz$⁠) and high mechanical quality factor modes, which, combined with the resonators’ large area, ensures effective trapping of the gas. The gas-induced coupling between the modes is also shown to enhance the pressure responsivity. In addition, the “canonical” large, unperforated, thin, parallel membrane geometry allows for a direct comparison with the theoretical predictions.

Let us note that, while this article primarily focused on the free molecular flow regime, similar concepts would also be applicable to the viscous regime, which could be explored, e.g., with higher frequency modes. Even more exciting perhaps is to extend the sensitivity of such membrane sandwich pressure sensors further into the high/ultrahigh vacuum regimes, which would be relevant for a number of applications, ranging from absolute high vacuum pressure calibration^32,33 to the direct determination of the vapor pressure of chemically or environmentally relevant low volatile substances.³⁴ Interesting prospects for improving the sensitivity of these devices at low pressures exist, e.g., via dissipation dilution and nanostructuring in order to dramatically improve the mechanical quality factors of specific modes³⁵ [Fig. 12(a)]. Improving the optical reflectivity of the suspended films by the patterning of subwavelength nanostructures such as photonic crystals or gratings^36–41 [Fig. 12(b)], or exploiting the squeeze film-induced intermembrane coupling via, e.g., electrical tuning of the mechanics^42,43 [Fig. 12(c)] would also represent interesting avenues to explore for improving the performances of squeeze film sensors.

The author is grateful to Andreas Naesby and Sepideh Naserbakht for their contributions to this work and acknowledges financial support from Villumfonden and from Independent Research Fund Denmark.

The data that support the findings of this study are available from the corresponding author upon request.