**1. Introduction**

In recent years, there has been a growing interest in the development and implementation of innovative solutions in the form of a miniaturized bio-sensors. In this regard, in the MEMS community emphasis has been given to design and fabricate highly sensitive, and precise biomass sensors. These bio-sensors are used for detection, identification and measurement of either single and/or multi-analyte/ s at lower cost, size, weight, and power consumption. Moreover, resonant devices are widely popular as a sensor for various chemical/biological applications [1]. In the context of biomass sensing, typical examples of resonant sensing include mass identification or detection [2–7]. A key attribute of these sensors is that the output signal is the variation/shift in the resonant frequency (*Δf*) of a vibrating structure that is subjected to small perturbations in the structural parameters i.e. effective mass/stiffness. Additional features of this method of detection are simple mechanical design, semi-digital nature of the signal (thus using simple frequency measurement system such as frequency counter and not requiring additional analog-todigital (A/D) conversion circuit), ultra-high resolution [8–10], (up to 10<sup>15</sup> grams scale [11–13] and up to 10<sup>18</sup> grams scale [14–17]). There are however also a challenge associated with the resonant sensor employing only one resonator; firstly, maximum theoretical frequency shift based parametric sensitivity, *Δf/f* is limited to 1/2 [18]. Sensors of these types are prone to environmental shifts such as pressure and/or temperature. Furthermore, these types of sensors, when used as a mass sensor are able to detect only one type of material (target analyte) at a time thus

**Figure 1.**

*A schematic representation of a 2-DoF coupled resonant mass-spring-damper system to be used as a highsensitivity biomass detector/identifier sensor. In a symmetric design, it is assumed that* M1 = M2 = M*,* K1 *=* K2 *=* K*. A mass perturbation,* Δm *is added onto the mass* M2*, causing system imbalance, leading to the energy localization/confinement and mode shape change. Mode-shape change is utilized as the output of the sensor.*

avoiding the possibility to rapidly detect and differentiate multiple biochemical analytes in parallel [19].

In the past few years, in the MEMS community, a paradigm shift is observed in the design and implementation of micromechanical resonating sensors. A new perspective is presented in using *1-d* chain of a coupled resonating proof masses, more familiarly refereed as multi degree-of-freedom (m-DoF) array, coupled resonator (CR) array, weakly coupled resonators (WCR) or mode-localized sensors [20–28]. **Figure 1** shows a representative schematic for such system. In this class of a sensors, coupling between the vibrating proof masses is constituted either electrostatically or mechanically. These sensors attribute an ultra-high parametric sensitivity (up to three to four orders high in magnitude) [29]. Such elevated levels of sensitivities are manifested via a novel transduction principle, i.e. sensing magnitude of vibrational energy exchange between the moving proof masses that are subject to a small disruption introduced into the system. This disruption serves to alter an effective mass, *Δm* of one of the proof mass element in a chain (see **Figure 1**). Primarily, due to relatively higher parametric sensitivities, m-DoF coupled resonators are emerging as an alternative and promising resonant sensing solution.

Other acknowledged advantages of weakly-coupled resonating sensors are linearity (attributed to high sensitivity, relative immunity against responding to common mode noise for example, ambient pressure and/or temperature [30–32] and the parallel detection capability in the context of the mass sensing applications [33–36]. These characteristics make mode-localized coupled resonators effective. For the obvious advantages as given, m-DoF coupled resonant sensors are being pursued over conventional method of resonant sensing, i.e. sensing the frequency shift, *Δf* of a single resonating device.

### **2. Theory**

A viable method to understand the operation of the mode-localized CR is its analysis through the transfer function model. The transfer function analysis enables to understand the system-level behavior of a biosensor unit. **Figure 1** shows a lumped parameter model of a 2 DoF mass-spring-damper system in the context of CR biosensor. It shows proof masses, *Mi*, mechanical spring constant, *Ki* and the damping coefficients, *ci*, (*i* = 1, 2). The *cc* models the damping force between the

#### *Ultra-Precise MEMS Based Bio-Sensors DOI: http://dx.doi.org/10.5772/intechopen.93931*

two resonating proof masses. Two proof masses are coupled through another spring, *Kc* as shown. A displacement of the proof mass, *xi* (*i* = 1, 2) in response to the applied force, *Fi* (*i* = 1, 2) is also indicated. Based on the free body diagram, a set of governing differential equations of motion for 2-DoF mass-spring-damper can be used to derive the theoretical transfer function. Subsequently, an expressions for mode-frequencies, *ω<sup>i</sup>* and modal amplitudes/amplitude ratio (AR) as a function of applied disorder in the mass, *δ<sup>m</sup>* can be obtained.

Following assumptions hold true for a symmetric device- *M1 = M2 = M*, *K1* = *K2* = *K* and *c1 = c2 = c*. Forcing vectors *Fi* acting on the proof masses are the harmonic excitation (drive) forces that cause displacements, *xi*, assumed to occur at one frequency. Quantities *δ<sup>m</sup>* and *κ* are normalized perturbation to the mass and normalized coupling factor, given as *δ<sup>m</sup>* = *Δm*/*M* and *κ* = *Kc*/*K*, respectively. When system experiences imbalance into the initial symmetry *i.e.*, *Δm* 6¼ 0, governing set of equations of motion for the two-coupled proof masses is given as follows:

$$\mathbf{M}\ddot{\mathbf{x}}\_1 + (c + c\_c)\dot{\mathbf{x}}\_1 + (K + K\_c)\mathbf{x}\_1 - c\dot{\mathbf{x}}\_2 - K\_c \mathbf{x}\_2 = F\_1(t) \tag{1}$$

$$(M + \Delta m)\ddot{\mathbf{x}}\_2 + (c + c\_c)\dot{\mathbf{x}}\_2 + (K + K\_c)\mathbf{x}\_2 - c\dot{\mathbf{x}}\_1 - K\_c\mathbf{x}\_1 = F\_2(t) \tag{2}$$

By operating the system in vacuum, the impact of the following can be reduced, i) damping force of individual proof mass and ii) damping force that occurs between two proof masses, hence *c*<sup>1</sup> ¼ *c*<sup>2</sup> ¼ *cc* ¼ 0 can be assumed for the simplified analysis. Therefore, (1) and (2) can be modified as below:

$$M\ddot{\mathbf{x}}\_1 + (K + K\_c)\mathbf{x}\_1 - K\_c\mathbf{x}\_2 = F\_1(t) \tag{3}$$

$$(M + \Delta m)\ddot{\mathbf{x}}\_2 + (K + K\_c)\mathbf{x}\_2 - K\_c \mathbf{x}\_1 = F\_2(t) \tag{4}$$

By applying a Laplace Transform to Eqs. (3) and (4), following expressions are obtained:

$$-G\_{11}(\mathfrak{s})X\_1(\mathfrak{s}) - G\_{12}(\mathfrak{s})X\_2(\mathfrak{s}) = F\_1(\mathfrak{s})\tag{5}$$

$$G\_{22}(\mathfrak{s})X\_2(\mathfrak{s}) - G\_{21}(\mathfrak{s})X\_1(\mathfrak{s}) = F\_2(\mathfrak{s}),\tag{6}$$

where

$$\mathcal{G}\_{11}(\mathfrak{s}) = \mathfrak{s}^2 \mathcal{M} + (\mathcal{K} + \mathcal{K}\_{\mathfrak{s}}) \tag{7}$$

$$\mathbf{G\_{12}(s) = G\_{21}(s) = K\_c} \tag{8}$$

$$G\_{22}(\mathfrak{s}) = \mathfrak{s}^2 (\mathcal{M} + \Delta m) + (\mathcal{K} + \mathcal{K}\_c) \tag{9}$$

In Eq. (6), set *F*2ðÞ¼ *s* 0, and derive an expression for *X*1ð Þ*s* and *X*2ð Þ*s* to use these values back in Eq. (5). An output transfer function is then obtained as follows:

$$G\_{1}(s) = \frac{X\_{1}(s)}{F\_{1}(s)} = \frac{G\_{22}(s)}{G\_{11}(s)G\_{22}(s) - G\_{12}(s)G\_{21}(s)}\tag{10}$$

$$\mathbf{G}\_{2}(s) = \frac{\mathbf{X}\_{2}(s)}{\mathbf{F}\_{1}(s)} = \frac{\mathbf{G}\_{21}(s)}{\mathbf{G}\_{11}(s)\mathbf{G}\_{22}(s) - \mathbf{G}\_{12}(s)\mathbf{G}\_{21}(s)}\tag{11}$$

Similar procedure can be applied to obtain an expression for *G*3ðÞ¼ *s X*1ð Þ*s <sup>F</sup>*2ð Þ*<sup>s</sup>* and *G*4ðÞ¼ *s X*2ð Þ*s <sup>F</sup>*2ð Þ*<sup>s</sup>* . Using the values of *<sup>G</sup>*11ð Þ*<sup>s</sup>* , *<sup>G</sup>*12ð Þ*<sup>s</sup> <sup>G</sup>*21ð Þ*<sup>s</sup>* and *<sup>G</sup>*22ð Þ*<sup>s</sup>* derived earlier in Eq. (7) through (9), following equations are obtained

$$G\_{1}(\varepsilon) = \frac{X\_{1}(\varepsilon)}{F\_{1}(\varepsilon)} = \frac{s^{2}(M + \Delta m) + K\_{a}}{s^{4}M(M + \Delta m) + s^{2}K\_{a}[M + (M + \Delta m)] + K\_{a}^{2} - K\_{\varepsilon}^{2}} \tag{12}$$

$$G\_{2}(\varepsilon) = \frac{X\_{2}(\varepsilon)}{F\_{1}(\varepsilon)} = \frac{K\_{\varepsilon}}{\varepsilon^{4}M(M + \Delta m) + \varepsilon^{2}K\_{a}[M + (M + \Delta m)] + K\_{a}^{2} - K\_{\varepsilon}^{2}} \tag{13}$$

Here *K<sup>α</sup>* ¼ ð Þ *K* þ *Kc* . Using *s=jω*, Eqs. (12) and (13) can be modified to attain

$$G\_{1}(jao) = \frac{X\_{1}(jao)}{F\_{1}(jao)} = \frac{-\alpha^{2}(M + \Delta m) + K\_{a}}{\alpha^{4}M(M + \Delta m) - \alpha^{2}K\_{a}(2M + \Delta m) + K\_{a}^{2} - K\_{c}^{2}} \tag{14}$$

$$G\_{2}(j\omega) = \frac{X\_{2}(j\omega)}{F\_{1}(j\omega)} = \frac{K\_{c}}{\alpha^{4}M(M+\Delta m) - \alpha^{2}K\_{a}(2M+\Delta m) + K\_{a}^{2} - K\_{c}^{2}} \tag{15}$$

A denominator of Eqs. (14) and (15) is given by

$$
\rho a^4 \mathcal{M}(\mathcal{M} + \Delta m) - \alpha^2 \mathcal{K}\_a(2\mathcal{M} + \Delta m) + K\_a^2 - K\_c^2 = \mathbf{0} \tag{16}
$$

Eq. (16) is the characteristic equation of 2 DoF coupled system. Roots of Eq. (16) can be given as

$$
\rho\_{ip}^2 \approx \frac{K\_a(2M + \Delta m) - \sqrt{\Delta m^2 K\_a^2 M^2 + 4K\_c^2 \left(M^2 + \Delta m\right)}}{2M^2} \tag{17}
$$

$$
\alpha\_{op}^2 \approx \frac{K\_a(2M + \Delta m) + \sqrt{\Delta m^2 K\_a^2 M^2 + 4K\_c^2 \left(M^2 + \Delta m\right)}}{2M^2} \tag{18}
$$

Here, *ω*<sup>2</sup> *ip* and *ω*<sup>2</sup> *op* are in-phase and out-of-phase natural mode frequencies of the device. With *Δm* = 0, Eqs. (17) and (18) take the form *ω*<sup>2</sup> *ip* <sup>¼</sup> *<sup>K</sup> <sup>M</sup>* and *ω*<sup>2</sup> *op* <sup>¼</sup> *<sup>K</sup>*þ2*Kc <sup>M</sup>* . Dividing (14) by (15), amplitude ratio (AR) is obtained as

$$\frac{H\_1(j\alpha)}{H\_2(j\alpha)} = \frac{-\alpha^2(M + \Delta m) + K\_a}{K\_c} \tag{19}$$

By substituting the values of *ω* in Eq. (19), the expression for mode AR as a function of mass perturbation, *Δm* is obtained as follows:

$$\frac{H\_1\left(\begin{smallmatrix}j\alpha\_{ip}\\j\alpha\_{ip}\end{smallmatrix}\right)}{H\_2\left(\begin{smallmatrix}j\alpha\_{ip}\end{smallmatrix}\right)} = \frac{-\left(\frac{\kappa\left(2M+\Delta m\right)-\sqrt{\Delta m^2 K\_a^2 M^2 + 4K\_c^2 \left(M^2 + \Delta m\right)}}{2M^2}\right) \left(M + \Delta m\right) + K\_a}{K\_c} \tag{20}$$

$$\frac{H\_1\left(\begin{array}{c} jao\_{ip} \\ H\_2\left(\begin{array}{c} jao\_{ip} \end{array}\right) = \end{array}\right) - \frac{\left(\frac{\kappa\left(2M + \Delta m\right) + \sqrt{\Delta m^2 K\_a^2 M^2 + 4K\_\varepsilon^2 \left(M^2 + \Delta m\right)}}{2M^2}\right) \left(M + \Delta m\right) + K\_a}{K\_\varepsilon} \tag{21}$$

With *Δm* = 0, Eq. (20) and (21) take the form as

$$\frac{H\_1(\
jo\_{ip})}{H\_2(\
jo\_{ip})} = \mathbf{1}; \frac{H\_1(\
jo\_{op})}{H\_2(\
jo\_{op})} = -\mathbf{1} \tag{22}$$

Eq. (22) represents initial balanced condition of a two CR.
