## 4 Metallic MEMS Cantilevers: Nonlinear Dynamics of the Pull-In Behavior
### Simulations on Pull-In and Actuation Time for Different Real Materials  
Since pull-in voltage and actuation time are strongly dependent on material properties in a MEMS cantilever, it is interesting to plot the dynamic pull-in voltage \(V_{\mathrm{pi^{\prime}}}\) as a function of the actuation time \(t_{\mathrm{a}}\) for MEMS cantilevers made of different metals, aluminum (Al), titanium (Ti), copper (Cu), nickel (Ni), silver (Ag), gold (Au), tungsten (W), tantalum (Ta), molybdenum (Mo) and chromium(Cr). The corresponding material properties are shown in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **Mass Density** & **Young’s Modulus** & **Poisson Ratio** \\ \hline Aluminum (Al) & 2700 kg/m\({}^{3}\) & 70 GPa & 0.35 \\ \hline Titanium (Ti) & 4506 kg/m\({}^{3}\) & 116 GPa & 0.32 \\ \hline Copper (Cu) & 8690 kg/m\({}^{3}\) & 120 GPa & 0.34 \\ \hline Nickel (Ni) & 8900 kg/m\({}^{3}\) & 219 GPa & 0.31 \\ \hline Silver (Ag) & 10,500 kg/m\({}^{3}\) & 83 GPa & 0.37 \\ \hline Gold (Au) & 19,300 kg/m\({}^{3}\) & 70 GPa & 0.44 \\ \hline Tungsten (W) & 19,350 kg/m\({}^{3}\) & 411 GPa & 0.28 \\ \hline Tantalum (Ta) & 16,690 kg/m\({}^{3}\) & 186 GPa & 0.34 \\ \hline Molybdenum (Mo) & 10,280 kg/m\({}^{3}\) & 329 GPa & 0.31 \\ \hline Chromium (Cr) & 7150 kg/m\({}^{3}\) & 279 GPa & 0.21 \\ \hline \end{tabular}
\end{table} 
Table 3: Properties of different metallic MEMS materials. Figure 15: Variation of the actuation time \(t_{\mathrm{a}}\) with mass density \(\rho\) and \(E\) for MEMS cantilevers (artificial materials). The results are shown in Figure 16, and it is evident that the \(V_{pi^{{}^{\prime}}}\) and \(t_{\text{a}}\) have opposite tendencies, which means one value increases while the other decreases. If the curve is close to the vertical axis, this shows better high-speed performance of the MEMS cantilevers. If the curves are close to the horizontal axis, this shows better low-voltage performance of the MEMS cantilevers. Considering both \(V_{pi^{{}^{\prime}}}\) and \(t_{\text{a}}\), it can be concluded that out of these considered metals, aluminum is the best for MEMS. For practical applications, low drive voltages and short actuation times are the most desirable goals, but it seems to be impossible to achieve both targets simultaneously. For application and research, it is important to consider the priorities and subsequently to make tailored optimum designs. Finally, the calculated actuation times are related to experimental MEMS closing times. Table 4 includes the type and dimensions of the MEMS device, the material of the moving part of the MEMS device, the closing time and the possible application. Although no quantitative comparison is possible, a qualitative agreement with the findings of this paper is obtained. For example, the shortest closing time of 1 \(\upmu\)s is obtained in our MEMS micro-shutters having \(l=40\)\(\upmu\)m and the longest closing time of 62.3 ms is obtained for a long cantilever with \(l=40\) mm. Due to the involved mechanical stress, the reported shutters and mirrors which have been fabricated are not easy to simulate. The stress in the thin layers modifies the elastic material properties. The material properties of very thin layers are often unknown and deviate from the values known for bulk materials. Deformations of thin stressed layers change the elastic properties additionally. This makes simulations of MEMS which include sub-layers (under stress) very challenging. Furthermore, an interdiffusion of material at interfaces, occurs and metallic surfaces are often covered by an unknown thin oxide layer. These altogether make a quantitative comparison between experiment and theory extremely challenging and sometimes nearly impossible. Thus, the qualitative agreement obtained in the last part is deemed acceptable. The metallic cantilever considered in this paper is a model structure to identify basic relations in nonlinear dynamics.