## 3 Optical Simulation Method  
Figure 3a shows the normalized electric field distribution in the plasmonic waveguide (Cu-SiO\({}_{2}\)-Si). Note that the plasmonic mode is dominant where the electric field distribution delays an exponential function beyond the SiO\({}_{2}\) layer. As the SiO\({}_{2}\) thickness increases, the mode splits both the plasmonic and the photonic ones gradually. Additionally, the normalized electric field of the photonic mode becomes stronger and the plasmonic one becomes weaker as the oxide layer becomes thicker. Moreover, a thicker SiO\({}_{2}\) layer causes the leaking energy for its poor mode limitation. Figure 3b shows the effective refractive index (n\({}_{\text{eff}}\)) and waveguide loss as a function for Cu-SiO\({}_{2}\)-Si plasmonic waveguide with SiO\({}_{2}\) thicknesses of 5, 10, 15, 20, 25, 30, 35, and 40 nm, respectively. It can be seen that the n\({}_{\text{eff}}\) decreases from 2.98 to 2.50, with the SiO\({}_{2}\) thickness increasing from 5 nm to 40 nm, which is explained by the leaking mode. The waveguide loss decreases from 0.5 dB/\(\upmu\)m to 0.08 dB/\(\upmu\)m, and this decrease is attributed to the mode splitting when more electric fields are in the photonic mode. Figure 1: Schematic drawing of the carrier-injection plasmonic micro-ring modulator (CIPMRM). Figure 2: The process flow of the CIPMRM is based on the standard CMOS process. The TM mode of the waveguide is coupled into CIPMRMs. For calculating the microring resonant peak intensity, the 3D finite-difference time-domain (FDTD) method is used to calculate the coupling efficiency. Figure 4a presents the coupling efficiency with different gap distances as the coupling length is 2.53 \(\upmu\)m. The coupling efficiency rises first and then falls as the gap decreases from 300 nm to 120 nm. When the gap is 150 nm, the coupling efficiency is at maximum (nearly 90%) because the narrower gap leads the mode to couple back with the waveguide and the wider one causes insufficient coupling, which is represented in Figure 4b. Moreover, the coupling efficiency curve slightly ascends with the wavelength on the insufficiency coupling condition; the curve descends with the wavelength on the over-coupled condition. The coupling curve is flat when the max coupling efficiency condition is achieved. It implies that the coupling efficiency is affected by the wavelength, especially when the maximum coupling efficiency is not reached. Here, the maximum coupling efficiency is discussed for when the loss of plasmonic micro-ring resonator is \(-9\) dB. To meet the critical coupling condition [19], where the coupling energy is equal to the micro-ring loss, maximum coupling efficiency (nearly 90%) is needed. Figure 5 is the transmission curve of the plasmonic micro-ring resonator with different SiO\({}_{2}\) thicknesses. To achieve an excellent extinction ratio (ER), the gap distance is adjusted to meet the requirements of the critical coupling condition. The loss of the plasmonic micro  
Figure 4: (**a**) The curve of coupling efficiency with the wavelength for different gap distances. (**b**) The power distribution from 3D-FDTD for 120, 150, and 180 nm gap distances. Figure 3: (**a**) The normalized electric field distribution of plasmonic waveguides with different SiO\({}_{2}\) thicknesses. (**b**) The refractive index and waveguide loss of plasmonic waveguides with different SiO\({}_{2}\) thicknesses. ring resonator with 15 nm SiO\({}_{2}\) is nearly 3.7 dB. To meet the critical coupling condition, the coupling gap is 210 nm and the coupling efficiency is 58%. The loss of the plasmonic microring resonator with 40 nm SiO\({}_{2}\) is nearly 1.85 dB. To meet the critical coupling condition, the coupling gap is 250 nm and the coupling efficiency is 36% according to the calculation. It can be seen that the resonance peak shifts for different SiO\({}_{2}\) thicknesses. This behavior can be attributed to the change in n\({}_{\text{eff}}\) for the plasmonic waveguide with different SiO\({}_{2}\) thicknesses.