# Design of a Piezoelectrically Actuated Ultrananocrystalline Diamond (UNCD) Microantilever Biosensor
## 3 Results and Discussion
### AlN Actuator-Beam Integration
#### 3.4.1 Integration of the UNCD Beam  
Using the design illustrated in Figure 8, the AlN layer thickness was varied from 300 nm to 1000 nm. As shown in Table 7, increasing the AlN thickness resulted in only modest gains in the resonant frequency shift. However, when the length of the UNCD beam was reduced to 40 \(\upmu\)m--while keeping the AlN thickness at 1000 nm--the resonant frequency increased substantially from 1754.4 kHz to 3569.1 kHz. Although further shortening of the beam could enhance sensitivity by increasing the frequency shift, it would reduce the active area available for surface functionalization. This trade-off must be evaluated based on the specific requirements of the intended biosensing application.  
Based on the results in Table 7, the configuration with a 1000 nm thick AlN layer and a UNCD beam length of 40 \(\upmu\)m was selected for further analysis. To estimate the quality factor of this microcantilever, an electrical pulse was applied to its electrodes. The quality factor was then calculated using the amplitude decrement method, by measuring the displacement of the first two oscillation peaks--identified as Point 1 and Point 2 in Figure 9.  
In Figure 9, it is important to note that the actuator-microantilever system exhibits underdamped behavior, which is essential for accurately estimating the quality factor using this model. This is because Equations (8)-(10) are valid and reliable only for systems operating below the critical damping threshold.  
The value of the quality factor (\(Q\)) is relatively low, (see Table 8) which is desirable for microcantilever-based biosensors. A lower \(Q\) factor ensures that smaller shifts in resonant frequency can be detected more effectively, enhancing the sensor's sensitivity. Conversely, the amplitude of oscillations remains small (\(\sim\)-5-6 A), which is also advantageous for detecting low-mass biomolecules, as it minimizes noise and improves resolution.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**AIN Film** & **Beam Length** & **Number of** & **Resonant** & **Resonant** \\
**Thickness** & **(\(\upmu\)m)** & **Elements** & **(kHz)** & **Frequency** \\ \hline
300 & 53.3 & 101,348 & 1286.1 & 0.6 \\
500 & 53.3 & 110,218 & 1434.6 & 0.6 \\
750 & 53.3 & 128,271 & 1609.5 & 0.7 \\
1000 & 53.3 & 131,449 & 1754.4 & 0.8 \\
**1000** & **40.0** & **131,111** & **3569.1** & **2.4** \\ \hline \hline \end{tabular}
\end{table}
Table 7: Resonant frequency shift for UNCD microcantilevers. The added mass is 1 pg for each simulation. The bold values indicate the optimized value for the UNCD microcantilever.  
Figure 8: Free tetrahedral mesh across the cantilever. This mesh corresponds to an AlN film thickness of 1000 nm and a UNCD beam length of 40 \(\upmu\)m. The region of the actuator exhibits more elements, as necessary for piezoelectricity modeling.  
Afterward, Equation (7) was used to add a mass at different places at the center of the beam, varying the \(y_{0}\) coordinate while keeping the x\({}_{0}\) coordinate constant at 0 \(\upmu\)m (see Figure 1 for the coordinate reference system). The shift in resonant frequency was measured and is presented in Table 9.