# Design of a Piezoelectrically Actuated Ultrananocrystalline Diamond (UNCD) Microantilever Biosensor
## 3 Results and Discussion
### Frequency Response Analysis of the Actuator  
The first step in designing the microcantilever involved determining the resonant frequency of the actuator. This frequency served as a reference for subsequent steps, where the goal is to match the eigenfrequency of the microcantilever beam to that of the actuator. This approach standardizes the design process, ensuring that any beam material--such as silicon (Si) or ultrananocrystalline diamond (UNCD)--is designed using the same initial parameters. Figure 7 shows the meshing and layer structure used for the simulations.  
It is important to note that at nanoscale thicknesses, the mechanical properties of materials, such as Young's modulus, residual stress, and fracture toughness, can vary significantly with film thickness because of surface effects, grain boundary evolution, and other size-dependent mechanical behaviors. This phenomenon has been well documented in the literature, particularly for piezoelectric and diamond-based thin films [29; 30]. In the present study, COMSOL Multiphysics assumes constant mechanical properties for each material layer and does not inherently account for thickness-dependent variations unless they are explicitly defined by the user. To partially address this limitation, a parametric analysis was conducted across a range of film thicknesses to examine their influence on resonant frequency and structural response. The results were interpreted with an understanding of the potential deviations introduced by this modeling simplification. Future work will focus on integrating experimentally measured thickness-dependent material data to enhance simulation accuracy. However, for the purposes of this study, a first-order approximation was deemed sufficient to identify the key design parameters required for the fabrication of UNCD-based microcantilevers.  
Table 5 shows the calculation of the resonance frequency for AlN layer thickness from 300 to 1000 nm. The width (20 \(\upmu\)m) and length (10 \(\upmu\)m) were kept fixed. As expected, the resonance frequency of the actuator increases as the thickness increases.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Order of BDF Solver** & **Displacement at Point 1 (nm)** & **Displacement at Point 2 (nm)** \\ \hline
2 & \(-383.16\) & \(-382.43\) \\
5 & \(-383.16\) & \(-382.34\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Displacement of the microcantilever at Points 1 and 2.  
Figure 7: (**a**) Mesh across the actuator (**b**) layers that compose the actuator.  
Based on the data presented in Table 5, the initial thickness of the AlN film was set to 300 nm, as this value provides a reasonable starting point for matching the eigenfrequency of the UNCD and Si microcantilever beams. This choice also offers greater flexibility in tuning the beam's eigenfrequency during the design process. Selecting a thicker AlN film from the outset would result in a significantly higher resonant frequency, which, in turn, would require a shorter beam length to achieve frequency matching, as indicated by Equation (2).