## 2 Optical Coupling Structures  
We focused on the optical coupling between the fundamental optical mode of the 1 \(\upmu\)m high and 1 \(\upmu\)m wide Si\({}_{3}\)N\({}_{4}\) waveguide (Figure 1a) and the fundamental mode of the 200 nm high and 590 nm wide Ge-based FKE optical modulator (Figure 1c). The optical mode from Si\({}_{3}\)N\({}_{4}\) had an effective index of \(\sim\)1.79, which was significantly smaller than the effective index of \(\sim\)3.55 obtained from the Ge-based on Si structure. Moreover, the optical mode size of the former was much larger than that of the latter, which was well-confined in the Ge-based material due to its larger refractive index than that of the top and bottom Si (n \(\sim\) 3.45). It should be noted that decreasing the Si\({}_{3}\)N\({}_{4}\) dimension lowered its effective index and also eventually made the optical mode size larger as the optical power spread out of the Si\({}_{3}\)N\({}_{4}\) core, which was not beneficial for direct optical coupling with the compact Ge-based FKE optical modulator. Modulators are typically required to have a relatively smaller dimensions in order to allow high electric field at low bias voltage, which is critical for low-voltage operation. In this paper, the lateral amorphous GeSi taper was optimized for optical coupling with the 200 nm high and 590 nm wide Ge-based FKE optical modulator; this coupling structure should be able to be adapted to other compact Ge-based devices with comparable dimensions with corresponding adjustments in taper width, height, and length. Additionally, it should be noted that although a butt coupling structure was proposed at the telecommunication wavelength of 1.31 \(\upmu\)m for optical integration between a Si\({}_{3}\)N\({}_{4}\) waveguide and Ge/SiGe multiple quantum wells (MQWs) on Ge-rich SiGe buffer devices [32], their optical mode areas were relatively comparable, which was not the case for the Ge-on-Si device, in which a large index difference between Ge-based modulator region and Si resulted in a highly confined and much smaller optical mode in the Ge-based region, as shown Figure 1c. Regarding the amorphous taper's dimensions, its length and height were 15 \(\upmu\)m and 200 nm, respectively, while the width at the beginning and at the end of the taper were 30 nm and 290 nm, respectively. The aspect ratio of \(\sim\)6.7 (200/30) at the beginning of the taper could be considered critical; nevertheless, it was within the capability of the advanced CMOS deep etching process. Regarding the possibility of practical fabrication for 1 \(\upmu\)m thick SiN, it was proposed by Ref. [23] that physical vapor deposition (PVD) had the potential to achieve a low-loss SiN material with maximum thickness beyond that normally obtainable with chemical vapor deposition (CVD) due to its lower built-in stress. Performance variation with respect to the change in taper tip width is discussed in the next section. The top cladding employed in simulation was SiO\({}_{2}\); therefore, the etching was stopped on the SiO\({}_{2}\) surface used to support the amorphous taper. The SiO\({}_{2}\) top cladding is not included in Figure 1d to allow a clear demonstration of the entire integrated devices. As in Figure 1e for the top view of optical propagation, the optical mode was transferred from the 1 \(\upmu\)m high and 1 \(\upmu\)m wide Si\({}_{3}\)N\({}_{4}\) waveguide to the 200 nm high and 590 nm wide Ge-based modulator region via the amorphous Ge\({}_{0.83}\)Si\({}_{0.17}\) taper (Figure 1b). For the Ge-based FKE p-i-n structure, the top Si n-layer and the bottom Si p-layer were 200 nm and 400 nm high, respectively. The bottom Si p-layer had a 200 nm etching depth to accommodate the bottom contact metallization. Regarding the optical parameters employed in simulation, the absorption coefficient data of the Ge-based material were as previously reported from Ge\({}_{0.9925}\)Si\({}_{0.0075}\) FKE modulator by [PERSON] et al.