## 2 Fabrication Process and Filter Design
### Filter Design and Fabrication  
A symmetrical stripline is shown in Figure 4. A stripline resembles a microstrip line and comprises a center conductor pattern symmetrically embedded completely within a dielectric, the top and bottom layers of which are conducting ground planes. The strip of width w is considered to have a thickness t that is very small so that the strip is a distance h from each of the ground planes and the ground planes are separated by b = 2 hD. The strip is completely surrounded by the dielectric and so this is a homogeneous medium and there is no need to introduce an effective permittivity. The characteristic impedance of a stripline is defined by Equation (1) [20]. \[Z_{0}=\frac{30\pi}{\sqrt{\varepsilon_{\mathrm{r}}}}\frac{(1-\mathrm{t}/ \mathrm{b})}{(\mathrm{W}_{\mathrm{eff}}/\mathrm{b})+\mathrm{C}_{\mathrm{f}}} \tag{1}\]  
where W is the conductor width, b is the substrate thickness, t is the conductor thickness, and \(\varepsilon_{\mathrm{r}}\) is the relative permittivity of the dielectric material. The effective width of the strip, \(\mathrm{W}_{\mathrm{eff}}\), is obtained from Equation (2). \[\frac{\mathrm{W}_{\mathrm{eff}}}{\mathrm{b}}=\left\{ \begin{array}{c}\frac{ \mathrm{W}}{\mathrm{b}}-\left[\frac{(0.35-\mathrm{W}/\mathrm{b})^{2}}{(1+12 \mathrm{t}/\mathrm{b})}\right]\frac{\mathrm{W}}{\mathrm{b}}<0.35\\ \frac{\mathrm{W}}{\mathrm{b}}\frac{\mathrm{W}}{\mathrm{b}}\geq 0.35\end{array} \right. \tag{2}\]  
Figure 3: MEMS filter with a stripline structure after wafer bonding (dimensions: 5680 \(\mathrm{\SIUnitSymbolMicro m}\times\) 1900 \(\mathrm{\SIUnitSymbolMicro m}\times\) 820 \(\mathrm{\SIUnitSymbolMicro m}\); stripline length: 5140 \(\mathrm{\SIUnitSymbolMicro m}\)). and C\({}_{\rm f}\) is defined by Equation (3). \[C_{\rm f}=\frac{2}{\pi}\ln\left[\frac{1}{1-\left(t/b\right)}+1\right]-\frac{t}{ \pi b}\ln\left[\frac{1}{1-\left(t/b\right)^{2}}-1\right] \tag{3}\]  
where C\({}_{\rm f}\) accounts for the fring capacitance at the edges of the strip and incorporates the effect of the strip thickness for t \(\ll b\). The fring capacitance per unit length (C\({}^{\prime}t\)) at each corner of the strip is defined by Equation (4). \[C_{\rm f}^{\prime}=\frac{2}{\pi}\ln\left[\frac{1}{1-\left(t/b\right)}+1\right] -\frac{t}{\pi b}\ln\left[\frac{1}{1-\left(t/b\right)^{2}}-1\right] \tag{4}\]  
The designed stripline structures follow the criteria of W/b \(\geq\) 0.35 and t/b \(<\) 0.1. The silicon substrate thickness (b) is 800 \(\upmu\)m, and the Through-Reflect-Line (TRL) metal thickness (t) is 5 \(\upmu\)m, resulting in t/b = 0.00625, which satisfied the theoretical condition for implementing RF characteristics. Based on these parameters, we designed the bottom wafer with a resonator width of 220 \(\upmu\)m (Figure 5) and modified the cap wafer design into three different configurations. Figure 6 illustrates three different cross-sectional designs for the MEMS filter structure, each representing a distinct relationship between the top (cap) and bottom resonator widths. The figure presents detailed schematics of the layered composition for each configuration, with color-coding to distinguish between materials (silicon wafers in light blue, copper plating in red, and gold plating as thin outer layers). Configuration (a) shows the L1 design, where the cap wafer contains no resonator structure (0% width relative to bottom), creating a simplified microstrip-like architecture with resonators only in the bottom wafer. Configuration (b) depicts the L2 design, where the cap resonator width is 60% of the bottom  
Figure 4: (a) Cross-sectional view of a stripline transmission line structure, and (b) fringe capacitance at the corners of the strip in stripline transmission lines. Figure 5: Bottom layout of the MEMS filter with a 220 \(\upmu\)m wide air gap (overall dimensions: 5680 \(\upmu\)m \(\times\) 190 \(\upmu\)m). resonator width (85 \(\upmu\)m compared to 220 \(\upmu\)m), providing an intermediate bonding area while reducing potential misalignment effects.