## Materials and Methods  
Many approaches are available for designing BPFs, from which the inductor-capacitor (LC) resonant structure employed in this study is easy to implement, compared to other methods, while providing the required effectiveness. The schematic of the proposed BPF is shown in Figure 1a, where the resonant frequency is mostly determined by the inductor and capacitor, which have circular shapes to allow the capacitor to be located exactly in the space between the two divisions of the inductor. The performance of the inductor is enhanced by mutual inductance from the intertwined and overlapped circular lines with air bridges. An air bridge is depicted in the enlarged view at the upper right corner of Figure 1a in four vertical directions. Five similar air-bridge structures are present in the proposed BPF, one at the upper side, two symmetrically located at the left and right sides, and the remaining two at the lower side. As depicted in the enlarged view at the bottom right corner of Figure 1a, \(r_{1}\) and \(r_{2}\) are the radii of the two small circles in the innermost ring of the embedded capacitor. A sector structure with a diameter of 100 \(\upmu\)m in the middle can enhance coupling between the different parts. The line width and spacing of the outer ring are 20 \(\upmu\)m and 14 \(\upmu\)m, respectively, and both the line width and spacing of the inner ring are 10 \(\upmu\)m. The combination of the left and right transmission lines can improve matching with the terminal, and current flows without distortion or interference by the elimination of excessive sharp turns that mitigates impedance loss. The exact realization of an ideal element is difficult. Neglecting some small values of feedthrough capacitance, the equivalent circuit of the proposed LC-type BPF is shown in Figure 1b. The model mainly consists of the inductor, capacitor, and its associated parasitic lumped elements. Variables \(R_{L}\) and \(G\) denote the loss resistance and impedance in the inductor (\(L\)) and capacitor (\(C_{\text{O}}\)), respectively, and \(C_{\text{Nx}}\), \(C_{\text{S}}\), and \(R_{\text{S}}\) represent the capacitance and resistance associated with the SiN\({}_{\text{x}}\) layer and the ground. The above parameters can be estimated as [13; 14]:  
\[R_{L}=\frac{\rho l}{w\delta(1-e^{t/\delta})} \tag{1}\]  
\[C_{\text{S}}=\frac{1}{2}lwC_{\text{sub}} \tag{2}\]  
\[R_{\text{S}}=\frac{2}{lwC_{\text{sub}}} \tag{3}\]  
\[C_{\text{Nx}}=\frac{1}{2}lw\frac{\varepsilon_{\text{Nx}}}{t_{\text{Nx}}} \tag{4}\]  
where \(\rho\) is the resistance coefficient, \(l\) is the length of the circle, \(w\) is the metal width, \(\delta\) is the skin depth in the conductor, \(t\) is the thickness of the conductor line, \(C_{\text{sub}}\) and \(G_{\text{sub}}\) are the capacitance and conductivity per unit area of the GaAs substrate, respectively, and \(\varepsilon_{\text{Nx}}\) and \(t_{\text{Nx}}\) are the dielectric constant and thickness of the SiN\({}_{\text{x}}\) layer for passivation, respectively. Although many factors affect the performance of the BPF, the center frequency is mainly determined by the circular spiral inductor and capacitor in the proposed design. The inductance of the circular inductor can be derived as in [15]:  
\[L=\frac{\mu n^{2}d_{\text{A}}}{2}[\ln(2.46/\eta)+0.20\eta^{2}] \tag{5}\]  
with:  
\[d_{\text{A}}=\frac{(d_{\text{out}}+d_{\text{in}})}{2} \tag{6}\]  
\[\eta=\frac{d_{\text{out}}-d_{\text{in}}}{d_{\text{out}}+d_{\text{in}}} \tag{7}\]  
where \(\mu\) is the magnetic permeability, \(n\) is the number of turns of the inductor, \(d_{\text{A}}=862\)\(\upmu\)m is the average diameter of the outer and inner ring of the inductor, and the value of \(\eta\) is approximately 0.22. The linear function between the capacitance of the concentric pattern and radius of the innermost circle can be used to obtain the capacitance. The capacitance value is solely determined by the material properties for a fixed width of the annular ring and the spacing between two adjacent rings [16; 17].