---
abstract: 'The operating principle of squeeze-film pressure sensors is based on the pressure dependence of a membrane’s resonance frequency, caused by the compression of the surrounding gas which changes the resonator stiffness. To realize such sensors, not only strong and flexible membranes are required, but also minimization of the membrane’s mass is essential to maximize responsivity. Here, we demonstrate the use of a few-layer graphene membrane as a squeeze-film pressure sensor. A clear pressure dependence of the membrane’s resonant frequency is observed, with a frequency shift of 4 MHz between 8 and 1000 mbar. The sensor shows a reproducible response and no hysteresis. The measured responsivity of the device is 9000 Hz/mbar, which is a factor 45 higher than state-of-the-art MEMS-based squeeze-film pressure sensors while using a 25 times smaller membrane area.'
author:
- 'Robin J. Dolleman'
- Dejan Davidovikj
- 'Santiago J. Cartamil-Bueno'
- 'Herre S.J. van der Zant'
- 'Peter G. Steeneken'
bibliography:
- 'squeezefilm.bib'
title: 'Graphene Squeeze-Film Pressure Sensors'
---

![image](abstractfigure)

Graphene, a single layer of $\mathrm{sp}^2$ bonded carbon atoms[@novoselov2005two], has exceptional mechanical properties. It has the highest Young’s modulus ($\sim$1 TPa) of all known materials [@lee2008measurement; @lee2013high]. Moreover, it has the lowest mass density and bending rigidity of all impermeable membranes[@bunch2008impermeable]. These properties make graphene a suitable material for nanomechanical sensors. Currently, pressure sensors are the most widespread membrane-based mechanical sensors and are present in most modern mobile handsets. Commercial microelectromechanical system (MEMS) based pressure sensors feature membranes of several hundreds of nanometers thickness. Replacing these by thin graphene membranes would allow an increase in responsivity and a size reduction by orders of magnitude. In order to exploit these advantages, several studies [@bunch2008impermeable; @smith2013electromechanical; @lee2014atomically; @lee2014air] have demonstrated the feasibility of sensing pressure changes with a graphene membrane suspended over a reference cavity at pressure $p_{\mathrm{ref}}$. When the ambient pressure ($p_{\mathrm{amb}}$) changes, the pressure difference ($p_{\mathrm{amb}}-p_{\mathrm{ref}}$) causes a deflection of the membrane. This has been directly detected by atomic force microscopy (AFM) and via a tension induced change in the membrane’s resonance frequency [@bunch2008impermeable]. Also the change in piezoresistance [@smith2013electromechanical] has been used to detect the change in pressure. However, the drawback of these pressure-difference based sensing methods is that they require a stable reference pressure $p_{\mathrm{ref}}$ over the $\sim$10 years lifetime of the sensor, posing extreme demands on the hermeticity of the reference cavity. Even though graphene sealed cavities were shown to have leak time constants of many hours [@bunch2008impermeable], at this stage it is unclear whether these can ever be increased to timescales of years. It is therefore of interest to develop pressure sensors that do not rely on the presence of an impermeable reference cavity.

In this work we demonstrate the feasibility of using graphene as a squeeze-film pressure sensor. The sensor consists of a membrane that covers a gas cavity, as is shown in Figure \[fig:fabrication\]. The main difference with conventional pressure sensors is the presence of an open venting channel that maintains the average pressure inside the cavity equal to the ambient pressure. Squeeze-film pressure sensors operate by compressing gas in the cavity that is at ambient pressure $p_{\mathrm{amb}}$. When the compression is performed at a high frequency, the gas fails to escape its effective position because of the viscous forces [@bao2007squeeze]. The added stiffness due to the compression of the gas is a function of pressure. For isothermal compression, this will change the resonance frequency ($f_{\mathrm{res}}$) of the resonator according to: $$\label{eq:squeezefilm}
f_{\mathrm{res}}^2 = f_0^2 + \frac{p_{\mathrm{amb}}}{4\pi^2 g_0 \rho h}.$$ Here, $f_{\mathrm{res}}$ is the membrane’s resonance frequency at pressure $p_{\mathrm{amb}}$, $f_0$ the resonance frequency in vacuum, $g_0$ the gap size between the membrane and the substrate that lies underneath the membrane and $\rho h$ the mass per unit square (see Supporting Information). Note, that the smaller the mass per unit square $\rho h$, the larger the frequency shift. The low mass density of graphene thus makes it a perfect material for this type of sensor.

As is shown in the Supporting Information, at high enough frequencies equation \[eq:squeezefilm\] is independent of mode-shape, thickness and boundary conditions of the membrane. The independence of the boundary conditions shows that the venting channel has no influence on the responsivity ($R={\rm d}f_{\mathrm{res}}/{{\rm d }p_{\mathrm{amb}}}$) of the device. Several works have demonstrated MEMS based squeeze-film pressure sensors with responsivities of up to 200 Hz/mbar [@andrews1993resonant; @southworth2009pressure; @andrews1993comparison; @kumar2015mems].

![a) Graphene flake transferred on a dumbbell shaped hole in a $\mathrm{SiO}_2$ substrate. Half of the dumbbell is covered, thus creating a drum with a venting channel. The diameter of the drum is 5 $\mu$m and the thickness of the oxide 400 nm. Lines A-A’ and B-B’ correspond to those in figures c) and b) respectively. b) Height profile from the atomic force microscopy (AFM) measurement, showing that the membrane is 10.5 $\pm$ 0.7 nm thick. c) Schematic cross-section of the squeeze-film sensor. d) Three-dimensional representation of the squeeze-film sensor design.[]{data-label="fig:fabrication"}](figure-fabrication-4.png)

We use an exfoliated few-layer graphene (FLG) flake that is suspended over dumbbell shaped holes using a dry stamping method [@castellanos2011atomically; @castellanos2013single]. The dumbbells have a diameter of 5 $\mu$m and are etched into a 400 nm $\mathrm{SiO}_2$ layer on a silicon substrate (Figure \[fig:fabrication\]a). The thickness of the flake after transfer is measured to be about 10.5 nm using atomic force microscopy (Figure \[fig:fabrication\]b). The stamping method allows accurate placement of the flake such that it covers half of the dumbbell shape, thus creating a graphene-based squeeze-film pressure sensor with a lateral venting channel (Figure \[fig:fabrication\]c–d). To demonstrate the importance of the venting channel for the sensor response, several sealed drums are created with the same flake. The resonance frequency of these sealed drums shows a strikingly different pressure dependence and undesired hysteresis as is discussed in the Supporting Information.

![Interferometry setup for detecting the resonance frequency of the graphene drum.[]{data-label="fig:setup"}](figure_setup.png)

Figure \[fig:setup\] shows the interferometry setup that is used to detect the sensor’s mechanical resonance modes as a function of gas pressure. The samples are mounted in a vacuum chamber with optical access. A dual-valve pressure controller connected to a nitrogen bottle controls the N$_{\rm 2}$ pressure in the chamber between 8 and 1000 mbar. An intensity modulated 405 nm blue laser drives the graphene membrane via optical absorption and thermomechanical force. A 632 nm red He-Ne laser beam targets the drum and cavity bottom and interference is detected at a photodiode. The intensity is modulated by the mechanical motion of the graphene drum. A vector network analyzer (VNA) modulates the blue diode intensity and detects the red laser light intensity on the photodiode to determine the frequency spectrum of the membrane [@castellanos2013single; @cartamil2015high].

The pressure-dependent resonance frequency of the sensor is studied by ramping the pressure upward and downward at a constant rate. During the pressure ramp the VNA continuously measures frequency spectra from 5-30 MHz at a rate of about 1 sweep every 2 seconds. Figure \[fig:datafits\] shows these frequency spectra at 4 different pressures. At 8 mbar 4 resonance modes are visible. At higher pressures the frequency of the fundamental mode increases while its Q-factor decreases. A damped harmonic oscillator model is fitted (red lines) to the data to extract the resonance frequency and quality factor as a function of pressure. The total frequency shift between 8 mbar and 1000 mbar is 4 MHz.

![\[fig:datafits\] Frequency spectra (blue) obtained from the VNA at different pressures. A damped harmonic oscillator model is fitted (red) to the fundamental mode to determine its resonance frequency and Q-factor.](newdatafits)

Figure \[fig:data\]a shows the frequency spectra taken during a pressure ramp in a contour plot. The frequencies of the first, third and fourth resonance modes increase as a function of pressure in close agreement (black dashed lines) with equation \[eq:squeezefilm\]. The intensity of the second mode vanishes above $\sim$50 mbar; therefore it is not possible to compare its response to equation \[eq:squeezefilm\]. For all modes the intensity decreases rapidly with pressure. The resonance frequency is plotted versus pressure in Figure \[fig:data\]b for a measurement at a ramp rate of 3.3 mbar/s. This measurement demonstrates the reproducibility of the sensor, showing no hysteresis as the pressure readings during upward and downward sweep are equal within the inaccuracy of the measurement. Equation \[eq:squeezefilm\] is plotted (dashed black line) in Figure \[fig:data\]b using the measured $f_0$ and no additional fit parameters. The theoretical curve is in close agreement with experimental data up to pressures of 200 mbar. We have also measured the pressure response with different gases as shown in the Supporting Information, this shows that the compression is isothermal and the use of equation \[eq:squeezefilm\] is valid. This also demonstrates that the pressure sensor operates independent of the type of gas. Above 200 mbar, the measured resonance frequency deviates from from equation \[eq:squeezefilm\]. This indicates that the assumptions underlying this equation cannot account anymore for the resonance frequency behavior at these higher pressures.

The quality factor is determined from harmonic oscillator fits (Figure \[fig:datafits\]) and plotted in Figure \[fig:data\]c. Three regimes can be distinguished: at pressures lower than 100 mbar, the quality factor drops as a function of pressure, approximately proportional to $1/p_{\mathrm{amb}}$. It is predicted by Bao et. al. [@bao2002energy] that the quality factor scales with $1/p_{\mathrm{amb}}$ in the free molecular flow regime. Between 100 and 500 mbar the quality factor appears to be more or less constant. Above 500 mbar the Q-factor reduces further approximately proportional to $1/p_{\mathrm{amb}}$. More sophisticated modeling is needed to explain the behavior of quality factor as a function of pressure.

From the data in Figure \[fig:data\]b it is possible to estimate the responsivity of the device: at low pressures the responsivity is approximately 9000 Hz/mbar while at atmospheric pressure it is 1000 Hz/mbar. The highest reported responsivity in squeeze-film MEMS pressure sensors is $200$Hz/mbar[@southworth2009pressure]. The responsivity of the graphene-based sensor is thus a factor of $\sim$5-45 larger than that of a MEMS sensor. At the same time the area of the graphene sensor is a factor 25 smaller. Based on equation \[eq:squeezefilm\], further improvement of the demonstrated sensor concept is possible by reducing the thickness of the membranes. It is estimated that using a single-layer graphene resonator will increase the responsivity by a factor of 5.6. A reduction of the gap size $g_0$ can enable a further increase of the responsivity.

In summary, a graphene-based squeeze-film pressure sensor has been demonstrated that does not need an impermeable reference cavity at stable reference pressure. Reproducible sensor response is demonstrated and a 4 MHz resonance frequency shift between 8 and 1000 mbar is measured. The resonance frequency closely follows the squeeze-film model up to 200 mbar, but at higher pressures deviations from the model are observed that require further theoretical study. In comparison with MEMS based squeeze-film sensors, the responsivity of the sensor is a factor 5-45 larger at an area of a factor 25 smaller. A further increase of the responsivity can be obtained using thinner membranes and reducing the gap size. In comparison to other graphene-based pressure sensing concept, the squeeze-film pressure sensor has the advantage that it does not rely on an impermeable reference cavity at constant pressure. It therefore provides a promising route towards size reduction and sensitivity improvements of pressure sensors.

Acknowledgements {#acknowledgements .unnumbered}
================

We thank Allard Katan, Ronald van Leeuwen and Warner J. Venstra for experimental support. The authors further thank the Dutch Technology Foundation (STW), which is part of the Netherlands Organisation for Scientific Research (NWO), and which is partly funded by the Ministry of Economic Affairs, for financially supporting this work. The research leading to these results has also received funding from the European Union Seventh Framework Programme under grant agreement no 604391 Graphene Flagship and this work was supported by the Netherlands Organisation for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program.

Methods {#methods .unnumbered}
=======

Substrates were fabricated using p-type silicon with thermally grown silicon dioxide on top. 400 nm cavities were etched into the silicon dioxide using reactive ion etching to obtain vertical etch profiles, using chromium as a mask and gases Ar at 2.7 sccm and $\mathrm{CHF}_3$ at 50 sccm with a power of 50 W and 7 $\mathrm{\mu}$bar pressure. The chromium mask was removed and samples were cleaned in nitric acid. For details on the graphene transfer process and the laser interferometer setup the reader is referred to Castellanos-Gomez et. al. [@castellanos2013single]. In the measurements the red laser power was 3 mW as measured before the objective entrance. The blue laser was kept at a power less than 1 mW, which was modulated using the VNA at -10 dB, resulting in a power modulation of 4.5%. Atomic force microscopy was performed on a Bruker Multimode 3 system to measure the thickness of the graphene flake. Pressure was controlled using a dual-valve pressure controller, calibrated for a pressure range between 0 and 1 bar. Dry nitrogen, carbon dioxide or argon was offered at the IN port of the controller and a (dry) scroll pump was connected to the vacuum port.
