An improved 1D diode model for the accurate modeling of parasitics in silicon modulators

Silicon modulators paved the way for silicon photonics to take control of optical interconnects. Since its popularization, most works use the 1-D diode model approximation to design the horizontal PN junction, which estimates the modulator bandwidth and efficiency. Some works do not even consider the effects of fringe capacitance, alleging that the junction's dimensions are large. The 1-D model is suitable for vertically uniform PN junctions. However, there are essential deviations for the typical rib waveguide used in most horizontal-junction silicon modulators. Our work aims to quantify such deviations incorporating details from 2D model simulations and offer a corrected 1-D model for estimating modulation bandwidth. This study was carried out as follows: Firstly, we incorporated an improved scheme for phase shifting and loss for different junction locations and widely used doping concentrations. Next, we analyzed the generation-recombination effects and their impact on the depletion width at the top and bottom of the waveguide. We calculated the depletion width via the 1-D model and the two-dimensional Poisson's equation finite-element calculation for the rib and identified an important mismatch. Lastly, we propose and demonstrate an accurate equivalent circuit with our 1-D model corrections. Our model considers the total depletion capacitance, the fringe capacitance, the capacitance due to the wider depletion widths at the top and bottom surfaces of the diode, and other capacitive effects at the border of the rib as a result of high reverse bias. We found that although the 1-D model is well-suited for small reverse biases, higher voltages and extreme junction locations affect the bandwidth's estimation dramatically.