## 2 Antenna Design Approach  
### Silicon-Based Antenna Design with Enhanced Bandwidth  
Using a high-permittivity substrate for microstrip antennas generally leads to penalties in performance [18], which is seen to be contrary to MMICs. For instance, the bandwidth of a microstrip patch for \(\epsilon_{\rm r}\) = 12.8 is less than 5% when d/\(\lambda_{0}\) < 0.04, and the surface wave power accounts for nearly half of the total input power when d/\(\lambda_{0}\) > 0.045 [19]. With high-permittivity silicon (\(\epsilon_{\rm r}\) = 11.9) showing fundamental issues and design challenges for microstrip antennas, it is necessary to enlarge the area of the radiation element or thicken the substrate for bandwidth enhancement [20]. Silicon-based rectangular patch antennas in four different dimensions are discussed with the aid of CMA, which describes the resonance behavior of each mode on the electromagnetic structure and the radiation behavior in the far-field zone [21]. Figure 1 shows the model used in the discussion. The rectangular patch is located on silicon with thickness h and is fed by the microstrip line through the H-shaped aperture coupling on the ground. Specifically, case 1 has the same thickness (h = 0.5 m) as case 2 and the same patch width (W = 5 mm) as case 3. The thickness discussed here is based on each silicon wafer being 250 \(\upmu\)m thick. Figure 2 shows the MS and MWC curves of the dominant mode in the four cases. The characteristic currents and far-field radiation of the dominant mode are shown in Figure 3. Note that the MWC curve for case 1 has two peaks, whose values are both larger than 0.15. It is supposed that other modes have an impact on the result. Figure 4 displays the MS and MWC results for case 1, from which we can see there are four resonant modes in the range and two of them are excited (J2 and J4). The characteristic currents and fields are shown in Figure 5. According to the current distributions, it can be determined that J2 refers to the TM\({}_{10}\) mode and J4 refers to the TM\({}_{10}\) mode. Therefore, the second peak in J2's MWC curve resulted from the onset of J4. The reflection coefficients of the antennas in the four cases (obtained using a full-wave simulation) are shown in Figure 6. It is seen that the best impedance matching in case 1 occurs in the middle of the two peaks in the MWC curve. Thanks to the joint effect of J2 and J4 on the input impedance, the antenna in case 1 has the widest bandwidth of 11.2%. Figure 1: Configuration of the microstrip patch antenna. (**a**) Top and side view; (**b**) exploded view. Figure 4: **(a)** MS and **(b)** MWC of the antenna with the dimensions of case 1. Figure 3: Characteristic currents and far-field radiation of the dominant mode. Figure 2: **(a)** MS and **(b)** MWC of the dominant mode. Case 1: W = 5 mm, L = 1.08 mm, h = 0.5 mm, Case 2: W = 2 mm, L = 1.11 mm, h = 0.5 mm; Case 3: W = 5 mm, L = 1.39 mm, h = 0.25 mm; Case 4: W = 3 mm, L = 1.39 mm, h = 0.25 mm. With the large ratio-aspect patch (W/L = 4.6) and two silicon layers (h = 500 \(\upmu\)m), the bandwidth is effectively extended. However, the large patch width imposes a penalty on the array design, considering the spacing requirement (typically 0.5\(\lambda_{0}\)) between the antenna elements at millimeter-wave frequencies. New microstrip configurations with a wider bandwidth have been exhaustively explored. In [22], a U-shaped slot was etched on a patch to generate an additional resonance. In [23], multiple stacked patches, with the upper patches acting as electromagnetically coupled parasitics, were designed for extending bandwidths. Based on the above works, the configuration of stacked patches with lower patch etched U-slots is proposed in this paper. The design approach is presented step by step using CMA, to reveal the radiation mechanism of the structure. An evolution of the proposed antenna is shown in Figure 7. Antenna I is a single patch with W/L = 3.4 and h = 500 \(\upmu\)m, and Antenna II adds a lower patch and maintains the same profile.