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Abstract
3D doping structure has significant advantages in modulation efficiency and loss compared with 2D modulator doping profiles. However, to the best of our knowledge, previous work on 3D simulation methods for interdigitated doping designs applied simplified models, which prohibited complex 3D doping. In this work, innovative omni junctions, based on the effective 3D Monte-Carlo method, are believed to be the first proposed for high-performance modulators. Simulation results show that the modulation efficiency reaches 0.88 V·cm, while the loss is only 16 dB/cm, with capacitance below 0.42 pF/mm. This work provides a modulator design with superior modulation efficiency and serviceability for high-speed datacom.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the benefit of complementary metal-oxide-semiconductor (CMOS) compatibility, low cost and strong optical confinement, silicon photonics has been vastly applied in optoelectronic communication. As the core component, silicon modulators play an important role in realizing high-speed data transmission over short and long distances. Concerning higher modulation speeds and lower power consumption, the silicon carrier-depletion modulators are generally applied in data transceivers, owing to major carrier movement at reverse biases and infinitesimal leakage current [1]. In terms of structure design, the Mach-Zehnder modulator (MZM) provides higher bandwidth and fabrication robustness compared to the microring modulator (MRM) [2]. The promotion in the MZM working performance such as modulation efficiency, insertion loss, and electro-optic (EO) bandwidth is strongly related to the doping profile in the rib waveguide of the silicon carrier-depletion modulator.
Methods to modulate the effective refractive index more efficiently without greatly increasing capacitance have been extensively studied. Previously, the solution for higher modulation efficiency and lower loss is mostly to optimize the doping profile in the cross-section of the rib waveguide, namely, 2D optimization. Modulators based on various kinds of junctions have been reported, such as lateral junctions, vertical junctions, L-shaped junctions, U-shaped junctions and wrapped junctions [3]. Apart from 2D optimization of doping profile, the non-uniform implantation in the direction of light propagation has a beneficial effect on improving modulation efficiency and decreasing modulator insertion loss, which is considered 3D doping optimization. Due to the capability of simple modeling and fabrication, interdigitated pure P-type and pure N-type MZMs [2,4–8] and MRMs [9–12] were mostly introduced, typically with reasonable modulation efficiency and low insertion loss due to depletion region generated at the interface. However, the capacitance of those designs tends to be large, which limits modulator bandwidth. Thus, the specific zigzag-type modulator has been adopted in the MRM with high bandwidth in [13], whose ratio of P-type to N-type doping along the propagation direction changes at a certain slope for enhanced carrier-light interaction to overcome the cavity lifetime limitation, yet the driving voltage and the power consumption become problems. To optimize for low-voltage application, an MRM based on predominantly vertical and horizontal p-n junctions was demonstrated with high modulation efficiency and splendid power-efficiency [14], with bit rates up to 40 Gbps. In addition, modulators with subwavelength grating waveguides, based on interdigitated capacitors [15] were currently proposed with extremely high modulation efficiency and loss, however, also with high capacitance. The interdigitated junctions were also utilized in photonic crystal modulators, allowing data transmission with low loss and high efficiency [16–21]. Nevertheless, previous work on 3D doping structure mainly generated junctions in the propagation direction while most did not simultaneously form junctions at the cross-sections. The doping profiles in these studies result from simplified models with uniform or Gauss distribution [22–26]. While in 2D doping profile designs, the Monte-Carlo method has been proved to become a reliable solution for more realistic doping profiles [27–29], this method is also applicable in building a 3D doping model closer to the real distribution. With the capability to precisely generate complex omni junctions in three dimensions, high-performance modulators based on novel doping profiles are likely to be found.
In this work, the effective 3D Monte-Carlo method is first proposed to generate complex 3D junctions in modulator designing. This effective 3D Monte-Carlo method is implemented by utilizing 2D Monte-Carlo results to save computation time and resources. For demonstration, we propose a novel Mach-Zehnder modulator based on the well-designed omni junctions. The modulator provides excellent performance in modulation efficiency and insertion loss, with wide EO bandwidth. The fabrication tolerance is considered to prove the serviceability. A comparison of the results is also presented. The effectiveness and ease of the effective 3D Monte-Carlo method enable efficient optimization of modulators.
2. Methods
In comparison with other doping distribution models, the Monte-Carlo method is the iteration of photons histories as they are scattered and absorbed [30], thus doping profiles are close to the real situation. However, 3D modeling requires high computation complexity and makes 3D Monte-Carlo simulation very difficult. To solve the computing-resource-consuming and time-cost problems, the 3D Monte-Carlo simulation in this work is realized by mapping 2D Monte-Carlo results into the 3D structure. Next, we introduce the procedure on how to carry out the doping modeling.
In the top view of the modulator waveguide, Fig. 1 illustrates the whole 3D Monte-Carlo procedure for single implantation in two steps: inside and outside the direct implantation region respectively. Each step utilizes 2D Monte-Carlo distribution. The details are described in the following.
Fig. 1. (a) Top view, (b) cross-section of the implanted modulator waveguide to show doping profile generation.
Firstly, the direct implantation region is doped as denoted by Fig. 2. Figure 2(a) shows the implantation region for doping. Given the implantation dose and energy, and the specific fabrication procedure like rapid annealing temperature and time, the accurate cross-section doping profile by 2D Monte-Carlo simulation is depicted in Fig. 2(b). Since the doping gradient in the z direction at the certain depth of perpendicular implantation is small, each cross-section inside direct implantation is assumed the same doping profile. The example here defines the implantation totally on the rib. For direct implantation both on the rib and slab, the doping profiles need to be obtained from the 2D Monte-Carlo simulation.
Fig. 2. Schematic diagram of doping profile generation for xy cross-sections of 3D modulator inside implantation region. (a) 3D structure for doping, (b) 2D Monte-Carlo simulation profile of the rib waveguide.
Secondly, Fig. 3 demonstrates the implementation of carrier diffusion. The 3D doping structure is denoted by Fig. 3(a). For the region on the silicon at a certain depth, Fig. 3(b) shows that the diffusion of different directions outside the direct implantation region applies the same diffusion profile from the 2D Monte-Carlo result, as shown in Fig. 3(c). At the corner of the diffusion region, Fig. 3(d) implies that carriers share the same distribution when diverging from the center of the concentric circular arcs. Each center is the vertex of the direct implantation region. The divergence paths for diffusion at the corner follow the distribution profile in Fig. 3(c).
Fig. 3. Schematic diagram of doping profile for carrier diffusion outside direct implantation region. (a) 3D structure for doping, (b) diffusion directions of implantation, (c) diffusion profile in the silicon at a certain depth, (d) diffusion corner of implantation.
With those steps accomplished at each implantation dose and implantation energy, the doping profile of a 3D modulator is generated by the effective 3D Monte-Carlo method. The overall performance of the whole 3D structure such as modulation efficiency (VpiL) and insertion loss is the synthesis of the performance of each cross-section.
As for overall VpiL for 3D structure, the detailed formulas are deduced as the following:
The phase shift for the xy cross-section located at $z$ at $V=V_{bias}$ is
where $n(z)$ is the effective index of the specific cross-section, $\lambda$ is the light wavelength and $L_z$ is the interval between two cross-sections and is assumed equal in the following deduction.Then the whole phase shift is
To reach $\pi$ shift, the required length of the periodic modulator is calculated as
where $L_{tot}$ is the total length of the period.Thus the VpiL for this period is calculated as
Equivalent loss for this period is simply the average of loss of each xy cross-section:
The resistance and capacitance of the 3D modulator regarding various frequencies are directly calculated via optoelectronic simulation by TCAD tools.
3. Discussion
The basic idea of this design is to look for optimized periodic junctions for silicon modulators. The period of the designed structure is composed of two kinds of junctions. Each kind is set to have an implantation length of 0.6 $\mu$m in the propagation direction. The rib width and height of the modulator waveguide are 410 nm and 220 nm, respectively, while the slab height is 60 nm. The implantation parameters to optimize are implantation doses, energies and locations of each junction, and the implantation conditions are all within ordinary two-step doping process limits. After optimizing the implantation parameters according to the need of high modulation efficiency, low loss and high bandwidth, the appropriate implantation conditions for the vertical junction and the lateral junction are listed in the Table 1 and Table 2, respectively. The whole optimization is realized by particle swarm optimization (PSO) algorithm. The obtained doping profile is displayed in the Fig. 4. The composition of vertical and lateral junctions forms omnidirectional PN junctions in the three-dimensional space, which are simply called omni junctions, referring to the PN junctions generated both in the cross-section and in the propagation direction. The doping levels are around $10^{18}/cm^{-3}$ to reach high modulation efficiency. At this doping level, with the absorption expression at 1550 nm [31]
where $\Delta n_e$ and $\Delta n_h$ are changes in the free-electron and free-hole carrier concentrations, the free carrier loss is over 10 dB/cm, much higher than the propagation loss in the silicon waveguide [32]. Thus the loss of the modulator mainly considers the free carrier loss and the validity of this model is examined with comparison to other experimentally demonstrated results [2,4,11]. The optimized vertical junction locates 0.17 $\mu$m above the top of the buried oxide (BOX). Figure 4(c) shows that the vertical junction has no current path for voltages to be added to the electron region in the upper part. The electron current path is provided by the lateral junction from the propagation direction, which is identified as red arrows in Fig. 4. The optimized lateral junction locates at exact the middle of the rib.Fig. 4. (a) Schematic diagram of the omni-junction modulator based on vertical and lateral junctions, (b) lateral and (c) vertical junctions of a single period.
Table 1. Implantation steps for the vertical junction (the tilt angle is 0$^{\circ }$ and rapid thermal annealing at 1030$^{\circ }$C within 10s)
Table 2. Implantation steps for the lateral junction (the tilt angle is 0$^{\circ }$ and rapid thermal annealing at 1030$^{\circ }$C within 10s)
Based on the optoelectronic simulation, the VpiL and Loss for each cross-section are illustrated in Fig. 5. The overall VpiL of the whole period at -2V bias is calculated as 0.88 V$\cdot$cm, shown as the dashed line in the Fig. 5(a) and the insertion loss is calculated as 16 dB/cm at 0V bias while at -2V the loss is as small as 11.9 dB/cm, as shown in the Fig. 5(b). The low VpiL and Loss are mainly induced by the vertical junction and the depletion zone in between vertical and lateral junctions. The results demonstrate a significant advantage of the omni-junction modulator that the well-designed 3D doping profile enables greater depletion region variation in both the cross-section and the propagation direction, which enhances the effective index change, hence modulation is stronger. The larger depletion region expanded at the interface of the vertical and lateral junctions in the propagation direction results in lower loss. Another important reason is that the 3D doping structure can be conductive in the propagation direction. Thus the design of one junction can focus on decreasing VpiL and loss to drag down the overall VpiL and loss while the other junction keeps the resistance and capacitance at a low level. As for the designed structure, the current path is provided in the propagation direction, hence the vertical junction is designed to stretch across the rib for extremely low VpiL and low loss. Figure 6 illustrates that the capacitance of this modulator is calculated as 0.42 pF/mm at the bias of -2 V, which proves the possibility for high-speed application.
Fig. 5. (a) VpiL and (b) loss of the modulator along the propagation direction under reverse biases (solid lines for attributes of each cross-section along the z direction, dashed lines for equivalent attributes of the whole period).
The RF performance is simulated when the modulator is driven through the coplanar strip transmission line which can be fabricated in the CMOS foundry. The cross-section of the carrier-depletion-based optical modulator is shown in the Fig. 7. The parameters of the transmission line is that $Wgnd\_s=155.2 \mu m$, $Ws=10 \mu m$, $Wgnd\_c=94.4 \mu m$, $G=6.4 \mu m$, $Hm2=1.3 \mu m$, $Hvia=0.8 \mu m$, $Wvia=2.4 \mu m$, $Hm1=0.7 \mu m$, $Wm1=5 \mu m$, $Hct=1.25 \mu m$ and $Wct=2.4 \mu m$. The attenuation, effective index, and characteristic impedance of the designed coplanar line are simulated in Fig. 8.
Fig. 8. (a) Microwave attenuation, (b) microwave effective index, and (c) characteristic impedance of the designed transmission line.
With the velocity match, impedance match and the microwave loss taken into account, the normalized frequency response of the modulator is calculated as [33]
As fabrication variation affects the performance of modulators, an analysis of fabrication tolerance is investigated. For misalignment of vertical and lateral junctions within 0.1 $\mu$m, the variations of performance are demonstrated in the Fig. 10. The value of the upward vertical junction misalignment is defined as positive and the value of the rightward lateral junction misalignment is defined as positive in this tolerance analysis. The (a)-(c) shows the influence of misalignment of the vertical junction that VpiL has a fluctuation of over 31% when biased at -3 V and Loss has a fluctuation of around 33% for negative misalignment and a fluctuation inferior to 5% for positive misalignment. Besides, the resistance and capacitance variation lead to bandwidth variation of over 25%. Deduced from performance fluctuation, this modulator tends to work better with an appropriate margin design of positive vertical junction misalignment. The (d)-(f) suggests lateral junction misalignment has a weaker influence on modulator performance, as VpiL of positive misalignment varies within 24% while VpiL of negative misalignment varies within 7%, and loss of positive misalignment varies around 35% while loss of negative misalignment varies within 7%. The bandwidth has a 22% reduction for negative misalignment and 1% reduction for positive misalignment. The analysis of lateral junction misalignment implies that lateral junction design has better robustness and the fabrication tolerance is better with a margin design of negative lateral junction misalignment. The fabrication tolerance analysis demonstrates that this modulator design is appropriate for realistic application, as the margin designs for junctions can be introduced to reduce fabrication errors. Within the conventional two-step manufacturing process flow, this modulation structure will be fabricated as our future work.
Fig. 10. Influence of vertical and lateral junction misalignment on modulator performance: (a) VpiL fluctuation, (b) loss fluctuation, and (c) bandwidth fluctuation of vertical junction misalignment; (d) VpiL fluctuation, (e) loss fluctuation, and (f) bandwidth fluctuation of lateral junction misalignment.
The comparison of this design with other reported interdigitated modulators aimed at high-speed communication is displayed in Table 3, showing the superiority in VpiL$\times$Loss and bandwidth. As our design is capable of working at different biases to cater to the requirement of high bandwidth or high extinction ratio, the VpiL at -2V bias and the bandwidth at -3V bias are presented for direct comparison. Compared with modulators based on interdigitated pure P and pure N, it is demonstrated that the design of omni junctions based on vertical and lateral types has a great advantage on VpiL. While sharing a similar structure with [14], the better results of this design are ascribed to the current path for the vertical junction provided from the Z direction rather than from the X direction, enabling a wider vertical P-N junction and thus higher carrier modulation at different biases. Furthermore, this design enables a wider junction in the Z direction to overlap with mode profile in the rib, which also greatly decreases VpiL and loss. Hence, the 3D design of the omni-junction modulator provides the advantage of carrier variation along the propagation direction and this mechanism has great potential for high-efficiency design. In addition, the omni-junction modulator design enables current paths provided in the Z direction rather than ordinarily provided in the xy cross-section which makes it possible to decrease VpiL and loss without increasing resistance and capacitance of the modulator, demonstrating the possibility of high-baud application with high modulation efficiency and low loss.
Table 3. Comparison for this work with reported C-band high-speed interdigitated modulators
4. Conclusion
In conclusion, we propose the effective 3D Monte-Carlo method to precisely generate complex omni junctions, enabling higher design diversity of doping profiles for better modulation. A novel modulator based on the effective 3D Monte-Carlo method is optimized to have the low VpiL of 0.88 V$\cdot$cm while the loss is below 16 dB/cm, which requires a short active region to reach high-baud transmission. This design enables compact size, catering to the need of large-scale integration. EO bandwidth reaches over 31.6 GHz under -3 V bias. These significant results demonstrate the advantage of complex doping profile designs which greatly decrease VpiL and loss, providing new insights and intuition into modulator design. With the capability to implement the effective 3D Monte-Carlo method, more promising omni junctions with excellent performance are to be studied. The doping structure consisting of unequal proportions of junctions or multiple junctions has great potential for low-voltage and high-speed applications, which remains as our future work. Moreover, the designs of segmented traveling-wave electrodes and drivers are beneficial for impedance and velocity matching and improving high-frequency performance of this designed modulator.
Funding
National Key Research and Development Program of China (2022YFB2803100); National major scientific research instrument development project (22127901); Shanghai Sailing Program (22YF1456700).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010). [CrossRef]
2. I. Goykhman, B. Desiatov, S. Ben-Ezra, J. Shappir, and U. Levy, “Optimization of efficiency-loss figure of merit in carrier-depletion silicon mach-zehnder optical modulator,” Opt. Express 21(17), 19518–19529 (2013). [CrossRef]
3. A. Rahim, A. Hermans, B. Wohlfeil, D. Petousi, B. Kuyken, D. V. Thourhout, and R. G. Baets, “Taking silicon photonics modulators to a higher performance level: state-of-the-art and a review of new technologies,” Adv. Photonics 3(2), 024003 (2021). [CrossRef]
4. H. Xu, X. Xiao, X. Li, Y. Hu, Z. Li, T. Chu, Y. Yu, and J. Yu, “High speed silicon mach-zehnder modulator based on interleaved pn junctions,” Opt. Express 20(14), 15093–15099 (2012). [CrossRef]
5. H. Zhu, L. Zhou, X. Sun, Y. Zhou, X. Li, and J. Chen, “On-chip optical power monitor using periodically interleaved p-n junctions integrated on a silicon waveguide,” IEEE J. Sel. Top. Quantum Electron. 20(4), 56–63 (2014). [CrossRef]
6. Y. R. R. Bustamante, G. B. de Farias, D. A. Motta, A. P. Freitas, L. Grabielli, and H. E. Hernandez-Figueroa, “Comparison between lateral and interleaved junctions for high-speed o-band silicon mach-zehnder modulator,” in Latin America Optics and Photonics Conference (Optica Publishing Group, 2018), paper Th3A.6.
7. Z. Hu, L. Jin, J. Guo, and J. Feng, “High bandwidth silicon mach-zehnder modulator based on interleaved pn junctions,” in Asia Communications and Photonics Conference (2018), pp. 1–3.
8. W. Wang, Z. Zhang, X. Guo, J. Zhou, S. J. X. Brian, M. S. Rouifed, C. Liu, C. Littlejohns, G. T. Reed, and H. Wang, “Mid-infrared (mir) mach-zehnder silicon modulator at 2µm wavelength based on interleaved pn junction,” in Conference on Lasers and Electro-Optics (Optica Publishing Group, 2018), paper STh1B.1.
9. M. Ziebell, D. Marris-Morini, G. Rasigade, P. Crozat, J.-M. Fédéli, P. Grosse, E. Cassan, and L. Vivien, “Ten gbit/s ring resonator silicon modulator based on interdigitated pn junctions,” Opt. Express 19(15), 14690–14695 (2011). [CrossRef]
10. X. Xiao, H. Xu, X. Li, Y. Hu, K. Xiong, Z. Li, T. Chu, Y. Yu, and J. Yu, “25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions,” Opt. Express 20(3), 2507–2515 (2012). [CrossRef]
11. D. Marris-Morini, C. Baudot, J.-M. Fédéli, G. Rasigade, N. Vulliet, A. Souhaité, M. Ziebell, P. Rivallin, S. Olivier, P. Crozat, X. L. Roux, D. Bouville, S. Menezo, F. Bœuf, and L. Vivien, “Low loss 40 gbit/s silicon modulator based on interleaved junctions and fabricated on 300 mm soi wafers,” Opt. Express 21(19), 22471–22475 (2013). [CrossRef]
12. L. Alloatti, D. Cheian, and R. J. Ram, “High-speed modulator with interleaved junctions in zero-change cmos photonics,” Appl. Phys. Lett. 108(13), 131101 (2016). [CrossRef]
13. X. Xiao, X. Li, H. Xu, Y. Hu, K. Xiong, Z. Li, T. Chu, J. Yu, and Y. Yu, “44-gb/s silicon microring modulators based on zigzag pn junctions,” IEEE Photonics Technol. Lett. 24(19), 1712–1714 (2012). [CrossRef]
14. J. C. Rosenberg, W. M. J. Green, S. Assefa, D. M. Gill, T. Barwicz, M. Yang, S. M. Shank, and Y. A. Vlasov, “A 25 gbps silicon microring modulator based on an interleaved junction,” Opt. Express 20(24), 26411–26423 (2012). [CrossRef]
15. D. Perez-Galacho, A. Abraham, S. Olivier, L. Vivien, and D. Marris-Morini, “Silicon modulator based on interleaved capacitors in subwavelength grating waveguides,” Proc. SPIE 9891, 989112 (2016). [CrossRef]
16. Y. Terada, H. Ito, H. C. Nguyen, and T. Baba, “Theoretical and experimental investigation of low-volgage and low-loss 25-gbps si photonic crystal slow light mach–zehnder modulators with interleaved p/n junction,” Front. Phys. 2, 61 (2014). [CrossRef]
17. Y. Hinakura, Y. Terada, T. Tamura, and T. Baba, “Wide spectral characteristics of si photonic crystal mach-zehnder modulator fabricated by complementary metal-oxide-semiconductor process,” Photonics 3(2), 17 (2016). [CrossRef]
18. Y. Terada, T. Tatebe, Y. Hinakura, and T. Baba, “Si photonic crystal slow-light modulators with periodic p–n junctions,” J. Lightwave Technol. 35(9), 1684–1692 (2017). [CrossRef]
19. M. Passoni, D. Gerace, L. O’Faolain, and L. C. Andreani, “Slow light with interleaved p-n junction to enhance performance of integrated mach-zehnder silicon modulators,” Nanophotonics 8(9), 1485–1494 (2019). [CrossRef]
20. M. Passoni, D. Gerace, L. O’Faolain, and L. C. Andreani, “Optimizing an interleaved p-n junction to reduce energy dissipation in silicon slow-light modulators,” Photonics Res. 8(4), 457–467 (2020). [CrossRef]
21. O. Jafari, S. Zhalehpour, W. Shi, and S. LaRochelle, “Mode-conversion-based silicon photonic modulator loaded by a combination of lateral and interleaved p-n junctions,” Photonics Res. 9(4), 471–476 (2021). [CrossRef]
22. H. Yu, M. Pantouvaki, J. V. Campenhout, D. Korn, K. Komorowska, P. Dumon, Y. Li, P. Verheyen, P. Absil, L. Alloatti, D. Hillerkuss, J. Leuthold, R. Baets, and W. Bogaerts, “Performance tradeoff between lateral and interdigitated doping patterns for high speed carrier-depletion based silicon modulators,” Opt. Express 20(12), 12926–12938 (2012). [CrossRef]
23. H. Xu, X. Li, X. Xiao, Z. Li, Y. Yu, and J. Yu, “Demonstration and characterization of high-speed silicon depletion-mode mach–zehnder modulators,” IEEE J. Sel. Top. Quantum Electron. 20(4), 23–32 (2014). [CrossRef]
24. D. Perez-Galacho, D. Marris-Morini, E. Cassan, and L. Vivien, “Modeling of PN interleaved phase shifters for high speed silicon modulators,” Proc. SPIE 9133, 913312 (2014). [CrossRef]
25. D. Pérez-Galacho, D. Marris-Morini, R. Stoffer, E. Cassan, C. Baudot, T. Korthorst, F. Boeuf, and L. Vivien, “Simplified modeling and optimization of silicon modulators based on free-carrier plasma dispersion effect,” Opt. Express 24(23), 26332–26337 (2016). [CrossRef]
26. A. J. Shaikh, F. Packeer, M. Baig, and O. Sidek, “A full 3d model of the modulation efficiency of a submicron complementary metal-oxide-semiconductor (cmos)-compatible interleaved-junction optical phase shifter,” J. Comput. Electron. 18(4), 1379–1387 (2019). [CrossRef]
27. G. Hobler and S. Selberherr, “Monte carlo simulation of ion implantation into two- and three-dimensional structures,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 8(5), 450–459 (1989). [CrossRef]
28. G. Hobler, “Monte carlo simulation of two-dimensional implanted dopant distributions at mask edges,” Nucl. Instrum. Methods Phys. Res., Sect. B 96(1-2), 155–162 (1995). Selected Papers of the Tenth International Conference on Ion Implantation Technology (IIT’94). [CrossRef]
29. A. K. W. Chee, R. F. Broom, C. J. Humphreys, and E. G. T. Bosch, “A quantitative model for doping contrast in the scanning electron microscope using calculated potential distributions and monte carlo simulations,” J. Appl. Phys. 109(1), 013109 (2011). [CrossRef]
30. S. Prahl, M. Keijzer, S. Jacques, and A. Welch, “A monte carlo model of light propagation in tissue,” SPIE Inst. Ser. IS 5 (1989).
31. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
32. T. Horikawa, D. Shimura, and T. Mogami, “Low-loss silicon wire waveguides for optical integrated circuits,” MRS Commun. 6(1), 9–15 (2016). [CrossRef]
33. H. Yu and W. Bogaerts, “An equivalent circuit model of the traveling wave electrode for carrier-depletion-based silicon optical modulators,” J. Lightwave Technol. 30(11), 1602–1609 (2012). [CrossRef]
