**1. Introduction**

The market for gas sensors was estimated to be 2.23 billion USD in 2020 and is expected to reach 4.49 billion USD in 2028 [1]. The growing interest in gas sensors is driven by various field of applications, e.g., medicine [2], air quality [3], food processing [4], or security and defense [5], that address legislative (e.g., EU's air quality directives), National Ambient Air Quality Standards) and/or individual needs. Sensors commonly used in the market, according to the highest percentage contribution into the gas sensor market income, are electrochemical, semiconductor, and infrared sensors [1]. Electrochemical sensor principles are based on creation of an electrical signal after reaction with a target gas. Semiconductor sensors are made of heated metal oxides which in the presence of the gas change their resistivity. Infrared gas sensors are based on electromagnetic signal conversion into electrical signal [6]. Characteristics of these sensors are presented in Table 1 [7].

Gas sensors for real-life applications [7], e.g., air quality, toxic gasses, medicine, food processing, are required to be selective (perfectly distinguish one species among others), sensitive (able to detect few particles per million in volume (ppmv)), reliable (stable, suffer from small drift), and compact. Infrared gas sensors, like the ones based on tunable diode laser spectroscopy (TDLS), can perfectly discriminate the spectral signature of a gas species among others, thus providing an excellent selectivity, combined with a high sensitivity (sub-ppb detection) [8] (Table 1). The main drawbacks of infrared detection are: lack of absorption line in infrared spectrum for some gasses, poor selectivity for gasses with absorption line at the same wavelength, and lack of compactness.

Photoacoustic spectroscopy, an evolution of TDLS, permits reducing the size of the gas sensor while maintaining equivalent performances. In TDLS the detected signal is proportional to the length of the optical path while in photoacoustic spectroscopy, it is related to the laser emitted power, which allows keeping a high sensitivity even in a compact gas cells.

**Citation:** Trzpil, W.; Maurin, N.; Rousseau, R.; Ayache, D.; Vicet, A.; Bahriz, M. Analytic Optimization of Cantilevers for Photoacoustic Gas Sensor with Capacitive Transduction. *Sensors* **2021**, *21*, 1489. https://doi.org/10.3390/ s21041489

Academic Editor: Krzysztof M. Abramski

Received: 20 January 2021 Accepted: 18 February 2021 Published: 21 February 2021

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**Table 1.** Characteristics for various types of gas sensors based on [7].

e—excellent, g—good, p—poor, b—bad.

In photoacoustic spectroscopy, a modulated laser emitting at a wavelength corresponding to the absorption line of a targeted gas species is focused into a gas chamber. The measurement is performed by detecting the acoustic pressure generated by the local warming induced by molecular relaxation following optical absorption. The local temperature rise is a result of non-radiative vibrational–translational (V-T) relaxation processes occurring between excited molecules. At atmospheric pressure, the laser emission linewidth (∼MHz) is much smaller than the gas linewidth (∼GHz), which gives a perfect selectivity to this method.

The acoustic wave can be measured using a microphone [11] or a mechanical resonator such as a tuning fork [12]. The use of a mechanical resonator with high quality factor (around 10,000 for a quartz tuning fork (QTF)) improves the signal-to-noise (SNR) ratio and avoids the use of a resonant acoustic chamber.

Commercial QTF allows reaching very good sensing performances in Quartz Enhanced Photoacoustic Spectroscopy (QEPAS) [13] even if they were developed for the electronics market, and not for sensing purposes. As a consequence, the QTF is not optimized for photoacoustic spectroscopy and its potential integration in a compact system is limited compared to other mechanical resonators based on silicon materials. Silicon would offer several advantages such as its technological maturity, its design flexibility and its lower production costs. However, its best advantage lies in the feasibility of integration in complex CMOS electronics [14,15]. Recent progress in laser sources integration on silicon [16] makes it possible to consider fully integrated compact sensors. For these reasons, silicon-based micro-resonator seems to be the best choice for the future development of very compact gas sensors integrated on the same chip with electronics, a laser, and a mechanical resonator.

We study here the realization of a silicon-based micro resonator sensor, a cantilever, dedicated to photoacoustic sensing. This sensor, specifically designed for acoustic sensing purposes, would be an efficient transducer for sound wave detection.

The most common transduction methods in silicon-based micro-electromechanical systems (MEMS) are based on capacitive, piezoresistive, and piezoelectric effects. The capacitive transduction mechanism constitutes a more convenient method than piezoelectric [17] or piezoresistive [18] detection. It avoids any material deposition or implantation on the mechanical resonator, which may reduce the quality factor and make the fabrication process more complex. Capacitive detection employed in MEMS technology allows reaching high sensitivity. For example, the capacitive accuracy for accelerometers or position sensors is about a few ppm of their nominal capacitance [19], leading to a sub-femto-farrad resolution [20]. To improve a capacitive signal, it is advantageous to increase the capacitor surface which leads to a rise in viscous damping and abbreviates the devices performances. Undoubtedly, for parameters characterized by opposite trends, an optimization based on a theoretical model would be the first step towards sensor performance improvement.

The working principle of a gas sensor based on photoacoustic spectroscopy using a cantilever as a capacitive transducer is schematically presented in Figure 1. The acoustic pressure generated by laser light absorption applies a force on the cantilever and sets it in motion. To maximize the displacement, the acoustic wave is generated at the resonance frequency of the cantilever via laser wavelength modulation. The silicon cantilever is electrically insulated from the back silicon, forming a capacitor. One of the electrodes of the capacitor is the cantilever itself. The displacements of the cantilever cause the capacitance variations. Depending on the excitation frequency, the capacitance variations can be converted into a current or a voltage signal.

Performing solely a trial and error method for the sensor's performance optimization is not feasible due to economic and time constraints. Therefore, a computational method is the most reasonable choice. The sensing scheme imposes multi-physics problems in different domains and can be divided in four parts: (1) acoustic force, (2) damping mechanisms, (3) mechanical displacement, and (4) output signal. Many of these problems are not directly coupled and others are characterized by opposite trends in terms of geometry optimization. The main novelty in our approach is a simultaneous multi-physics optimization. This optimization aims to determine the geometrical parameters of the cantilever (length *L*, width *b*, thickness *h*, gap *d* (Figure 1)) and its resonance frequency, which would maximize the output electrical signal and the signal-to-noise ratio. For this, the cantilever has to be sized to maximize its displacement under acoustic wave exposition while exhibiting a strong capacitance variation.

**Figure 1.** Sensing scheme of a silicon cantilever-based sensor for photoacoustic gas detection with capacitive transduction mechanisms.

The paper is divided as follows:


described by *W*(*x*, *ω*) presented in Figure 1. Further terms describing the cantilever displacement refer to the fundamental vibration mode presented in Figure 1.


#### **2. Acoustic Force**

The purpose of this section is to study the cantilever dimensions (length *L*, width *b*, thickness *h*) and its resonance frequency in order to maximize the acoustic force. This part evaluates the photoacoustic pressure generation and the photoacoustic force applied to the cantilever. The source of the photoacoustic wave generation lies in periodic gas absorption induced by a modulated laser beam. This method is called wavelength modulation spectroscopy [21]. We consider a Gaussian laser beam propagating along the x-axis at an altitude *z* = *zL* and centered with respect to y-axis at *y* = *yL* (Figure 2).

The distribution of the light intensity *I*(*x*, *y*, *z*) is related to the laser power *PL*:

$$I(x, y, z) = P\_{L}g(x, y, z)\tag{1}$$

$$g(x, y, z) = \frac{2}{\pi w\_{L}(x)^{2}} \exp\left(-2\frac{(z - z\_{L})^{2} + (y - y\_{L})^{2}}{w\_{L}(x)^{2}}\right)$$

where *g*(*x*, *y*, *z*) is a normalized Gaussian profile and *wL*(*x*) = *wL*(*xL*) <sup>1</sup> + (*x*−*xL*)<sup>2</sup> *x*2 *R* is the laser radius which depends on the Rayleigh length *xR* <sup>=</sup> *<sup>π</sup>wL*(*xL*) *<sup>λ</sup><sup>L</sup>* , with *λ<sup>L</sup>* the laser emission wavelength.

**Figure 2.** Gaussian beam profile and its position on the axis in relation to the cantilever microbeam. *wL*(*xL*) = 100 μm, *λ<sup>L</sup>* = 1.65 μm, *xL* = 0.725L, *yL* = 0, *zL* = 150 mm.

The theoretical model used to describe the pressure and force of the acoustic wave generated by molecular absorption is based on the model developed by Petra et al. [22]. However, our model takes into account the variation of the laser beam radius *wL*(*x*) along the optical axis (x-axis) and the effects of the gas relaxation time constant. The assumptions used in the model are:


To fulfill the third assumption, the acoustic wave wavelength *λ<sup>a</sup>* must be at least one order of magnitude larger than the thickness and width of the cantilever: *λ<sup>a</sup>* ∼ 3.5 cm at *ν* = 10 kHz (*λ<sup>a</sup>* = *cs <sup>ν</sup>* , where *cs* is the speed of sound).

The absorption of the modulated light causes periodic heat changes and subsequently an acoustic wave. The heat production rate is given by:

$$H(x, y, z, t) = \frac{C\_f(\omega)g(x, y, z)}{\sqrt{1 + (\omega \tau)^2}} e^{i(\omega t - \arctan(\omega \tau))}\tag{2}$$

where *ω* is the laser modulation frequency, *τ* is the target gas relaxation time *Cf*(*ω*) is the effective absorption coefficient. The absorption and transmission line shapes can be ideally described with a Lorentzian line shape function. The laser emission wavelength scan the absorption line and is modulated around the central wavelength *λc*. The modulated wavelength can be expressed as: *λ*(*t*) = *λ<sup>c</sup>* + *λampsin*(*ωt*), where *λamp* is the modulation amplitude. When the laser wavelength is modulated the power remains constant and equal to *PL*, we can write *Cf*(*ω*) = 0.50*α*(*ω*)*PL*, where *α*(*ω*) is the absorption coefficient of the gas. The 0.5 factor is obtained by expansion of the absorption function in Fourier series. We consider only the first Fourier component (*a*<sup>1</sup> = 0.5) for the 1 *f* detection method. The second Fourier components would result in a coefficient of 0.35 (*a*<sup>2</sup> = 0.35) [22].

The photoacoustic wave generation is related to the heat production due to the light absorption. The expression of photoacoustic pressure *P*(*x*, *y*, *z*) is given by the wave equation:

$$\frac{\partial^2 P(\mathbf{x}, y, z, t)}{\partial t^2} - c\_s^2 \Delta P(\mathbf{x}, y, z, t) = (\gamma - 1) \frac{\partial H(\mathbf{x}, y, z, t)}{\partial t} \tag{3}$$

where *cs* <sup>=</sup> 347.276 m/s is the sound velocity in air and *<sup>γ</sup>* <sup>=</sup> *Cp Cv* the adiabatic gas coefficient or heat capacity ratio equal to the fraction ratio between heat capacities at constant pressure and volume.

Equation (3) is an inhomogeneous equation with time. By substituting *P*(*x*, *y*, *z*, *t*) = *p*(*x*, *y*, *z*)*eiω<sup>t</sup>* and *H*(*x*, *y*, *z*, *t*) = *h*(*x*, *y*, *z*)*eiω<sup>t</sup>* and imposing Sommerfeld radiation boundary conditions, Petra et al. [22] showed that the pressure equation takes the following form:

$$p(x,y,z) = -\frac{\pi A}{2c\_s^2 k\_s^2} (\mathcal{Y}\_0(k\_s r) + i l\_0(k\_s r)) \int\_0^{+\infty} u f\_0(u) \exp\left(\frac{-2u^2}{k\_s^2 w\_L(\mathbf{x})^2}\right) du\tag{4}$$

where *J*0,*Y*<sup>0</sup> are the zero-order Bessel functions of the first and the second kind, respectively. *<sup>A</sup>* <sup>=</sup> <sup>−</sup>(*<sup>γ</sup>* <sup>−</sup> <sup>1</sup>)*ωH*(*x*, *<sup>y</sup>*, *<sup>z</sup>*) <sup>2</sup> *<sup>π</sup>wL*(*x*)<sup>2</sup> represents the amplitude of photoacoustic pressure, *ks* <sup>=</sup> *<sup>ω</sup>*/*cs* is the wave number, and *<sup>r</sup>* <sup>=</sup> (*<sup>z</sup>* <sup>−</sup> *zL*)<sup>2</sup> + (*<sup>y</sup>* <sup>−</sup> *yL*)<sup>2</sup> is the distance between the laser beam and the cantilever.

The photoacoustic force *FPA* applied on the cantilever is defined as the difference of pressure between the top and bottom surfaces of the cantilever.

$$F\_{PA} = \int\_{0}^{1} \int\_{0}^{b/2} (p(\mathbf{x}, y, z) - p(\mathbf{x}, y, z - h)) \phi\_n(\mathbf{x}) \, d\mathbf{x} \, dy \tag{5}$$

*φn*(*x*) describes the one-dimension shape of the cantilever mechanical mode *n*. It gives the cantilever deflection and can be found analytically by solving an eigenvalue problem of the Euler–Bernoulli equation. The mode shape for a clamped-free cantilever is given by [23]:

$$\phi\_{\boldsymbol{n}}(\boldsymbol{x}) = \cosh\left(a\_{\boldsymbol{n}}\frac{\boldsymbol{x}}{L}\right) - \cos\left(a\_{\boldsymbol{n}}\frac{\boldsymbol{x}}{L}\right) - \frac{\sinh\left(a\_{\boldsymbol{n}}\right) - \sin\left(a\_{\boldsymbol{n}}\right)}{\cosh\left(a\_{\boldsymbol{n}}\right) + \cos\left(a\_{\boldsymbol{n}}\right)} \left(\sinh\left(a\_{\boldsymbol{n}}\frac{\boldsymbol{x}}{L}\right) - \sin\left(a\_{\boldsymbol{n}}\frac{\boldsymbol{x}}{L}\right)\right) \tag{6}$$

The acoustic force acting on the cantilever is frequency-modulated at the wavelength modulation frequency of the laser source. For a first harmonic detection (1f detection) it is adjusted to the cantilever mode frequency. In our model the cantilever vibrates at its fundamental mode-first harmonic *n* = 1 which corresponds to a mode constant *α*<sup>1</sup> = 1.875.
