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

#### **Figure 5.**

*The output response of a 2-DoF coupled resonator mode-localized design used for biomass sensing applications. Due to the mass addition or removal process, initial eigenmodes and eigen-frequencies of the* j th *resonator (*j *= 1, 2) change due to the mode-localization. Changes in the amplitudes,* Δ<sup>a</sup> *are found to be 2 to 3 orders high in magnitude than that of the corresponding changes into the frequencies,* Δf *at the* i th *mode (*i *= 1, 2) of the output response.*

### **2.2 Frequency response and a finite element model (FEM)**

**Figure 6(a)** shows a frequency response (bode diagram) of a 2-DoF CR system. **Figure 6(b)** shows a COMSOL mode shape simulation for a designed geometry of a two mechanically coupled resonators. A structural mechanics module of COMSOL Multiphysics [37] can be used to design CR sensor and simulate for the mode shape

#### **Figure 6.**

*Output frequency response (bode diagram) of the CR sensor showing the two modes of a 2-DoF sensor. A finite element model is also depicted to determine the mode shape and resonant frequencies of the design. Note the swapping of the two modes as per the coupling used in the sensor.*

and eigen-frequencies of the design. As seen from the FEM results, for mode 1, vibrating elements (cantilevers) move in the same direction (in-phase mode) and the same amplitudes [1, 1]. For mode 2, both the cantilevers move in opposite direction (out-of-phase mode) and the same amplitudes [1,-1]. The two simulated mode frequencies are *f1* and *f2* for mode 1 (in-phase) and mode 2 (out-of-phase), respectively. As the FEM illustrates, the coupling used in the design is mechanical. If the coupling used in the design is electrical, out-of-phase mode precedes the inphase mode as seen in the bode diagram (note the phase difference of the two resonators).

### **3. Case studies in m-DoF resonant mass sensing**

In this section, different types of MEMS ultra-precise sensors based on the m-DoF CR architecture are discussed. In MEMS resonant biosensors, a surface of the micromechanical resonator is coated with a sensitive thin film. A resonant frequency shift is monitored as a result of adsorption/absorption of the target analyte/s [19]. In the same framework, CR structures are used as a mass sensors owing to the enhanced mass sensitivity and parallel monitoring of multiple analyte/s.

#### **3.1 Study I**

For the first time, it was proposed that a vibration mode localization can be used to demonstrate an elevated mass sensitivity [36]. A fabricated prototype is shown in **Figure 7(a)**. In this work, two nearly identical mechanically coupled gold-foil microcantilevers were used. For the experimentation, borosilicate microspheres (mean diameter of 4.9 *μm* with a mass of ≈154 *pg* were added on cantilever 2. Piezoelectric shaker was used for the driving scheme for the sensor. A laser doppler vibrometer was used to capture the tip velocities at different locations of individual cantilevers. An output plot as seen in **Figure 7(b)** was obtained to show eigenstate variation as a function of normalized mass perturbation, *δm*. With *Δm* = 0, vibration amplitudes of cantilevers 1 and 2 at two mode frequencies (i.e. two distinct modes as in-phase and out-of-phase) are seen. Uneven amplitudes for both the cantilevers at both the modes are result of fabrication mismatch. With *Δm* 6¼ 0, vibration amplitudes of both the cantilevers change at both the modes. With mass added to cantilever 2, amplitudes of both the resonators at both the modes are seen to be increased. Amplitude of resonator 1 is relatively higher than amplitude of resonator 2 at the first mode (at lower frequency in the response). Amplitude of resonator 2 is relatively higher than amplitude of resonator 1 at the second mode (at the higher frequency in the response). A relatively larger shift (either in amplitude or resonant frequency) indicates vibration energy is localized to that particular cantilever at the mode of operation. With a mass differential (*Δm*) in the system, resonant frequencies of both the cantilevers at both the modes also change. However, relative changes in the amplitudes are orders of magnitude higher than the changes in frequencies. This work experimentally demonstrated about two orders higher in magnitude relative changes into the eigenstates (5–7%) than relative changes in the frequencies (0.01%). Enhanced sensitivity of eigenstates to the added mass was attributed to the decreased scaled coupling strength, *κ* between the two cantilevers. Each eigenstate is the normalized vector formed by the amplitudes of the two vibrating elements (cantilevers) at a corresponding resonance frequency. In the same work, mass removal from the cantilever surface resulted in return of eigenstates to their original values.

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

**Figure 7.**

*Ultrasensitive mass sensor using a mode localization in coupled microcantilevers (a) fabricated prototype and (b) amplitude-frequency response of a fabricated prototype before and after the mass imbalance introduced into the system [33]. Reprinted from [33] with the permission of AIP Publishing.*

#### **3.2 Study II**

Moreover, using an array of polysilicon microcantilevers (up to 15) it is possible to record up to 3 orders higher changes in eigenstate based output of the sensor [35]. In an array of cantilevers, each pattern of eigenmode shifts is unique. Therefore, by examining an experimentally measured pattern of eigenmode shifts it is possible to determine to which cantilever a target analyte particle has adhered. A mass sensitivity of up to two orders higher was found as opposed to the previous work [36] reported by the same group. A mass sensitivity of up to three orders higher was found as opposed to relative frequency shifts. It is therefore feasible to design coupled resonant (CR) microstructures and use eigenmode as an output metric for enhanced parametric sensitivity over resonant sensors that use frequency shift output. However, it is also evident that merely adding the number of resonators in a 1-dimensional (1-*d*) chain does not necessarily increase the parametric sensitivity in proportion.
