## 2 Notations and Methodology
### Modulation Bandwidth  
The modulation bandwidth is also a figure of merit that must be taken into account in the conception of the MZM. Indeed, the longer the MZM electrode is, the higher its direct current (DC) modulation efficiency is but the smaller its radio-frequency (RF) bandwidth is. Figure 5 shows a simulation of the 3 dB RF bandwidth for an absolute bias voltage of 2.4 V of the electrical S21 parameter of the electrode itself as a function of its length for typical implantation and metallization conditions (detailed later). For these simulations we use an analytical equivalent circuit model for the traveling wave electrode which drives the carrier depletion silicon MZM. The model is composed of an elementary lossy dispersive transmission line (LDTL) model of length \(\partial l\) which is cascaded to fully represent the length of the electrode \(l_{\text{MZM}}\). The elementary LDTL model, depicted on Figure 6, includes frequency dependent resistors, capacitors and inductors. This model addresses two of the three factors that determine the operation speed of a traveling wave electrode MZM: (i) the microwave attenuation; (ii) the impedance matching; but do not take into account the velocity matching between the RF and the optical waves. However [PERSON] et al. proves that velocity mismatching does not limit the electrode bandwidth for electrode of reasonable length compared to the RF loss and the impedance mismatching effects [16]. The series resistor and inductor \(R_{\text{metal}}\) and \(L_{\text{metal}}\) are frequency dependent and their formulas are given in [17]. All the other elements except the PN junction capacitance (\(C_{\text{junction}}\)) and the access resistor (\(R_{\text{access}}\)) are computed similarly as in [16]. The junction capacitance is computed as in [18] and takes into account the fringing field effect. The access resistor is extracted from measurements on our fabrication platform.  
The simulation results depicted on Figure 5 have been obtained using a SPICE simulator (AMS 16.1, MENTOR Graphics, Wilsonville, OR, USA). We took a coplanar line with 200 \(\upmu\)m-width grounds, 10 \(\upmu\)m-width signal and 22.5 \(\upmu\)m-space between them. The substrate, the buried oxide and the optical guide (ribbon) heights are respectively 800 \(\upmu\)m, 2 \(\upmu\)m and 300 nm. The metallization thickness is 1.3 \(\upmu\)m. Doping concentrations for p and n part of the junction are both 2.5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\). As we expected, the shorter the length of the electrode is, the higher the bandwidth is. This is mainly due to the decrease of the RF loss. Ultimately the bandwidth will be limited by the doped-silicon access resistor and the junction capacitance. For very short electrode lengths, under 500 \(\upmu\)m, we can expect very high bandwidths up to 50 GHz. These results illustrate from another viewpoint the possible tradeoff between MZM parameters.  
Figure 4: Evolution of the doping concentrations and the position of the p-n junction in function of the targeted efficiency for the simulated active region. Symbols are the results of Lumerical simulations, lines are polynomial fits.