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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13425–13428
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Design of High Efficiency Multi-GHz SiGe HBT Electro-Optic Modulator

Shengling Deng, Z. Rena Huang, and J. F. McDonald  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13425-13428 (2009)
http://dx.doi.org/10.1364/OE.17.013425


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Abstract

We design and theoretically analyze a heterojunction bipolar transistor (HBT) electro-optic (EO) modulator with a composition graded SiGe base. The waveguide has a large cross-section of 1µm for ease of fiber alignment. At a base-emitter bias of VBE =2.5V, a π-phase shift requires 74.5µm interaction length for TM polarization at λ=1.55µm. The total optical attenuation is 3.9dB to achieve a π-phase shift in this condition. This device is expected to operate at a switching speed of 2.4GHz.

© 2009 OSA

1. Introduction

Development of integrated electro-optic (EO) modulators with high speed, small footprint, low power, and compatibility with silicon fabrication processes is one of the challenges for building silicon-based integrated optical networks. In the past few years, extensive research has been performed in novel silicon EO modulators. For example, strained Ge/SiGe multiple quantum wells (MQW) [1

1. Y. H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437(7063), 1334–1336 (2005). [CrossRef] [PubMed]

] using quantum-confined Stark effect (QCSE), and strained Silicon with the Pockels effect [2

2. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

] have been explored. However, these modulators are either immature in processing or of extremely low efficiency. Free carrier dispersion and the thermo-optic effect are possible mechanisms to create index modulation in silicon. The thermo-optic effect is rather slow with a modulation frequency only up to 1MHz [3

3. C. Cocorullo, M. Iodice, I. Rendina, and P. M. Sarro, “Silicon thermo-optical micro-modulator with 700 kHz - 3 dB bandwidth,” IEEE Photon. Technol. Lett. 7(4), 363–365 (1995). [CrossRef]

]. Therefore, free carrier dispersion is the primary mechanism utilized in silicon EO modulators with gigahertz bandwidth and above.

Various electrical structures of free carrier dispersion based EO modulator have been explored in the past two decades. If possible, a device that is already in production for other applications is preferable to relying upon a totally new device due to the cost of introduction of new technology. According to how the free carrier plasma is generated, elementary structures can be categorized into three types: forward biased PIN diodes [4

4. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

], reverse biased PN diodes [5

5. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15, 660–668 (2007). [CrossRef] [PubMed]

,6

6. F. Y. Gardes, G. T. Reed, A. P. Knights, and G. Mashanovich, “Evolution of optical modulation using majority carrier plasma dispersion effect in SOI”, Proc. of SPIE 6898, 68980C-68980C-10 (2008). [CrossRef]

], and Metal-Oxide-Semiconductor (MOS) capacitors [7

7. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]

]. For forward PIN diodes, the injection of minority carriers in the intrinsic region gives rise to a large refractive index modulation. This in turn leads to high modulation efficiency and small device footprint. However, forward PIN is typically slow because the injection of minority carriers is a slow process. For reverse biased PN diodes, the depletion of the majority carrier contributes to the refractive index modulation. It is a fast process with a delay characterized by the majority carrier relaxation time. The major drawback of the reverse biased EO modulator is its low modulation efficiency, since the refractive index modulation depth is limited by the background doping. For the MOS capacitor based EO modulators, the refractive index modulation is achieved through majority carrier accumulation, and it is thus an intrinsically fast device. However, the modulation efficiency is even lower because the thickness of the accumulated carriers is extremely thin. As a result, the MOS based EO modulator requires an active waveguide of several millimeters long for a phase shift of π. Traveling wave electrode design can greatly improve modulation efficiency. The fastest traveling wave EO modulator based on reverse biased PN structure has a demonstrated bit rate at 40Gbps [8

8. A. Liu, “Announcing the world’s first 40G silicon laser modulator”, http://blogs.intel.com/research/2007/07/40g_modulator.php

]. The propagation of RF signal and optical wave along the long dimension of this type of device must be such that the wavefronts of these two waves move at the same speed in tandem. This poses a design challenge since the speed synchronization is very difficult for microwave propagation in the transmission line and optical wave propagation in the dielectric waveguide.

In addition to exploring new electrical structures, the design of optical structures is another research focus. Photonic crystal (PC) waveguide [9

9. Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005). [CrossRef]

] utilizing slow light effect, and resonant structures, such as distributed Bragg reflector [10

10. A. Cutolo, M. Iodice, A. Irace, P. Spirito, and L. Zeni, “An electrically controlled Bragg reflector integrated in a rib silicon on insulator waveguide,” Appl. Phys. Lett. 71(2), 199 (1997). [CrossRef]

12

12. C. A. Barrios, V. Rosa de Almeida, and M. Lipson, “Low-power-consumption short-length and high-modulation-depth silicon electrooptic modulator,” J. Lightwave Technol. 21(4), 1089–1098 (2003). [CrossRef]

], Fabry-Perot (FP) resonators [13

13. X. Xiao, J. C. Sturm, K. K. Goel, and P. V. Schwartz, “Fabry-Perot optical intensity modulator at 1.3µm in silicon,” IEEE Photon. Technol. Lett. 3(3), 230–231 (1991). [CrossRef]

15

15. C. A. Barrios, V. R. Almeida, R. Panepucci, and M. Lipson, “Electrooptic Modulation of Silicon-on-Insulator Submicrometer-Size Waveguide Devices,” J. Lightwave Technol. 21(10), 2332–2339 (2003). [CrossRef]

], and ring resonator [4

4. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

] are attractive in certain applications. An EO modulator incorporating ring resonator with diameter of 12µm has demonstrated a date rate of 18Gb/s [16

16. S. Manipatruni, Q. Xu, B. Schmidt, J. Shakya, and M. Lipson, “High Speed Carrier Injection 18Gb/s Silicon Micro-ring Electro-optic Modulator,” in Proceedings of Lasers and Electro-Optics Society (IEEE, 2007), pp.537–538.

]. The high bit rate was achieved because the transmission spectrum in this structure is very sensitive to carrier density change. However, for the same reason, this type of EO modulator is ultra sensitive to variations including temperature, bias voltage, and working wavelength. Additionally, one desires a low-Q structure to maintain a wider bandwidth.

In this paper, we propose a novel EO modulator that consists of a heterojunction bipolar transistor (HBT) with SixGe1-x graded-base. The SiGe HBT EO modulator was first suggested by R.D.Lareau, et al [17

17. R. D. Lareau, L. Friedman, and R. A. Soref, “Waveguided electro-optical intensity modulation in a Si/GexSi1-x/Si heterojunction bipolar transistor,” Electron. Lett. 26(20), 1653–1655 (1990). [CrossRef]

] in 1990, in which the germanium composition was constant. Due to limitations in SiGe processing, it was not until 1992 that the composition graded SiGe HBT became available in commercial BiCMOS production line. The graded base SiGe HBT device benefits from a built-in electric field induced by the varying bandgap difference between the germanium and the silicon. An adjustable bandgap of SixGe1-x can greatly enhance the device speed and offer many new design options for performance improvement in a HBT device. Of particular interest is the ability to trade HBT current gain enhancements for higher base doping, and lower base resistance.

The operation of an EO modulator with three terminals was first explored on bipolar mode FET (BMFET) [18

18. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,” J. Lightwave Technol. 15(3), 505–518 (1997). [CrossRef]

], in which the lightwave was weakly guided by small index contrast between the intrinsic silicon and the heavily doped silicon. The BMFET EO modulator was long in active region and slow in speed. However, by principle the third terminal in an EO modulator can provide an additional freedom in manipulation of the electric field to maximize the overlap between the carrier plasma and the optical field in the waveguide. In this paper, we will show through our theoretical investigation that a phase shift of π can be achieved within an interaction length of 74.5µm. To our knowledge, the proposed HBT EO modulator is the smallest silicon-based EO modulator that works at GHz bandwidth without using any resonant structure. Among demonstrated GHz EO modulators either with PN or PIN structure, the cross-section of the rib waveguide is exclusively in submicron scale. A large waveguide cross-section (>1µm) designed in this work can alleviate the complexity of light coupling from a fiber to the chip [19

19. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). [CrossRef] [PubMed]

]. The trade-off of large cross-section is reduced switching speed due to longer carrier transit time across the base. In MOS EO modulator, the index modulation is obtained only in a thin layer consisting of inversion carriers; whereas the index modulation in a HBT is extended from the base to the collector and emitter region due to large carrier injection, leading to an increased overlap between the optical wave and the free carrier plasma. This thickness can be several 10’s of nanometers to 100 nanometers.

2. Device structure

The EO modulator of this research incorporates a SiGe HBT with graded base as its electrical structure, and the Mach-Zehnder interferometer (MZI) as its optical structure to convert phase modulation into intensity modulation. One of the MZI arms is controlled by the HBT and the other one is for reference. The cross-section of a simplified device model used in the theoretical calculation is sketched in Fig. 1. The optical beam is coupled into the HBT from one facet and propagates along the z-axis. The designed HBT EO modulator has a vertical structure, which is prevalent in modern technology. The significance of vertical HBT in modern technology is that the emitter and the base widths are free from lithography limit, making ultra-high speed possible.

Fig. 1. 3D Schematic of the HBT EO modulator (not drawn to scale)

In our model, the emitter region is made of heavily doped n-type poly-silicon with a peak doping concentration of ~2×1020cm-3. The emitter contact is placed on the top of the device. The base region consists of a p-type Si1-xGex epi-layer with x varying from 0% at the base-emitter junction to 20% at the base-collector junction. The active area of the base region is underneath the emitter, which is termed as intrinsic base. The intrinsic base has a peak doping concentration of 2×1019cm-3 and is electrically connected through heavier doped extrinsic base regions with p-type doping (~1020cm-3) from both sides. This arrangement in a bipolar device targets at minimizing the base contact resistance as well as series resistance. The collector region is n-type doped at ~5×1017cm-3. An additional n+ region, or the sub-collector, is buried at the bottom. The sub-collector is to reduce the contact resistivity to the collector terminal. In a real HBT device, the collector region is brought to the top metal layer via a reach-through structure. For the analysis of the electrical characteristics of the HBT, we place the collector contact at the bottom for simplicity. We assume that the waveguide is covered by an overglass SiO2 (nSiO2=1.45) layer and the bottom is confined by a SiO2 layer as well. The bottom SiO2 can be introduced by substrate thinning followed by oxidization, or using Silicon on Insulator (SOI) technology. The large refractive index contrast between the silicon layer and the SiO2 layer provides a strong optical confinement. The refractive index of intrinsic silicon is taken as nSi=3.42 in this work, while the Si1-xGex refractive index is calculated from nSi(1-x)Gex=3.42+0.37x+0.22×2. The change of the refractive index in silicon as a result of free carrier plasma effect follows the well-known equations derived from Kramers-Kronig relation by R.A.Soref and B.R.Bennett [20

20. R. A. Soref and B. R. Bennett, “Kramers-Kronig analysis of electro-optical switching in silicon,” SPIE Integr. Opt. Circuit Eng. 704, 32–37 (1986).

]:

{Δn=Δne+Δnh=[8.8×1022ΔNe+8.5×1018(ΔNh)0.8]Δα=Δαe+Δαh=8.5×1018ΔNe+6.0×1018ΔNh
(1a)
{Δn=Δne+Δnh=[6.2×1022ΔNe+6.0×1018(ΔNh)0.8]Δα=Δαe+Δαh=6.0×1018ΔNe+4.0×1018ΔNh
(1b)

where Δn is the change of refractive index, and ΔN is the change of carrier density. The subscript “e” and “h” denotes the property change caused by electrons and holes, respectively. Equation (1)(a) and 1(b) corresponds to λ=1.55µm and λ=1.3µm, accordingly.

The complete doping profile at the device center along the y-direction is shown in Fig. 2.

Fig. 2. Doping profile through the device center

Similar to PIN based EO modulator, the free carrier plasma effect in a HBT EO modulator is also produced by carrier injection. The injection can take place at emitter-base junction and/or the base-collector junction, depending on bias conditions. Elaboration on carrier injection behavior is presented in section 3.

In the PIN EO modulator, carrier injection is solely governed by minority carrier diffusion, which is a slow process; whereas for the HBT EO modulator, the built-in electric field in the base effectively reduces the electrons transit time. Therefore, HBT EO modulator is expected to work much faster. The built-in electric field also permits heavy doping in the base, which is a design not possible in homojunction bipolar device because the electron mobility greatly suffers from strong scattering. In addition, high emitter injection ratio enabled by the bandgap discontinuity ΔEg at the emitter-base heterojunction is another reason why the base of HBT can have a high doping. Advantages of high doping in the base include less current crowding effect and smaller RC delay. It is worth noting that the base thickness favors high speed when small, but favors phase modulation efficiency when thick.

The HBT EO modulator is analyzed according to the following procedure: we first compute the carrier distribution throughout the device using a 2D device simulation package MEDICI; we then convert the 2D carrier distribution into refractive index profile through Eq. (1); next, we verify the single mode condition using both effective index method (EIF) and RSoft BeamProp CAD tool based on beam propagation method (BPM); finally, we import the refractive index data into BeamProp for mode computation. The EO modulation efficiency and the mode loss are calculated based on the effective index information. A MEDICI transient analysis is also performed to evaluate device switching performance.

3. Electrical and optical analysis

The proposed HBT EO modulator, as a three-terminal device, allows more options to control the carrier distribution in the device. In this research, we chose the common-emitter configuration, or more specifically, we maintain a reverse bias VCE between the collector and the emitter, and apply a voltage at the base terminal. In the following DC characteristics investigation, we sweep the base bias VBE from 0V up to 3.0V with a 0.5V step, while keeping a constant VCE=1.5V.

3.1 Carrier distribution

A bipolar transistor consists of two PN junctions, namely, the emitter-base junction and the base-collector junction. The injected electron density in the vicinity of the emitter-base junction can be expressed as [21

21. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (John Wiley & Sons, Inc, 2007), Chap. 5.

]

n(0)=np0exp[q(VBEΔVBE)(nkT)]
(2)

where np0 is the electron density in the base at equilibrium, VBE is the applied emitter-base bias, ΔVBE is the voltage dropped on the series resistances in the quasi-neutral regions, and n is an ideality factor. The hole injection in the emitter follows a similar relation, except that np0 is replaced by pn0. The equilibrium carrier concentration np0 and pn0 in the heavily doped emitter and the base are larger than their nominal values due to bandgap narrowing effect [21

21. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (John Wiley & Sons, Inc, 2007), Chap. 5.

]. As a result, increased minority carrier injections in the emitter (~50 times higher) and the base (~7 times higher) need to be taken account in the simulation.

With VBE ramping up from 0V to 3.0V, the HBT operation mode shifts from cutoff to forward-active, and eventually to saturation. The energy band diagrams for thermal equilibrium condition, VBE=0V, 1.0V, and 2.5V are shown in Fig. 3. The Fermi level at thermal equilibrium condition is chosen as the reference point of the potential energy. The gradual energy bandgap narrowing in the base is the result of monotonic increase of germanium composition from the emitter side to the collector side. The conduction band bending establishes a built-in electric field that is favorable to the transport of electrons in the base. At VBE=0V, no carrier injection takes place at the emitter-base junction, though a constant bias VCE=1.5V set the device off the equilibrium. Thus, the device is in its “OFF” state. At moderate to high VBE bias, as shown in Fig. 3 (c) and (d), the potential barrier at the emitter-base junction is reduced considerably, which allows the electrons from the emitter injected to the collector through the thin base region. This corresponds to the “ON” state. At high VBE bias, the slope of the energy band bending is reduced at the base-collector interface due to Kirk effect.

Fig. 3. Energy band diagram along y-axis at the device center under (a) Equilibrium condition, (b) No injection (VBE=0V, VCE=1.5V) from emitter to base, (c) Low injection (VBE=1.0V, VCE=1.5V), and (d) high injection (VBE=2.5V, VCE=1.5V) condition.

The carrier distribution at the middle of the device (along y-direction) is plotted in Fig. 4. When VBE is below the threshold (~0.7V), carrier injection throughout the device is negligible, thus the free carrier plasma effect is minimal. Once VBE is biased beyond 0.7V, obious electron injection from the emitter to the base is observed. The injected electrons move across the base and are finally collected in the collector. Similar injection takes place for the holes from the base side to the emitter side; therefore, the free carrier plasma also forms in the emitter. However, due to the previously mentioned bandgap offset ΔEg, the hole injection level is much lower than the electron injection with a difference proportional to exp(-ΔEg/kT).

As we can see from Eq. (2), the carrier injection process follows an exponential relation to VBE. Therefore, higher VBE is the key to higher carrier density change, and thus higher phase modulation efficiency. To achieve the best efficiency, the HBT device needs to work in saturation mode, which requires VBE>VCE=1.5V. In this condition, we find from Fig. 4 (a) that the injected electron density is above the base doping level. As the consequence of charge neutrality requirement in the base, a significant hole density increase is observed. On the collector side, both electron and hole density increase slightly in the region immediately close to the base-collector junction. The increase in electron density indicates the amount of electrons entering the collector is greater than collector background doping. Similarly, the increase of hole density in the collector is also required by charge neutrality. This hole density increase in collector is equivalent to a widened base. In summary, when the HBT device is biased into saturation mode, strong electron and hole plasmas will generate in the base and its nearby regions. Thinner bases enhance speed, but reduce the volume of the carriers and their contribution to the index change. The extension of the free carrier plasma region beyond the base of the HBT is favorable, though weak, for the enhancement of modulation efficiency. But this also comes at the cost of speed.

Fig. 4. One-dimensional carrier density plot along y-axis at device center for (a) electrons, and (b) holes (insets for locations between 0.5µm to 0.7µm at VBE=1.5V to 3.0V)

In the case of large VBE, high-level injection effect and the parasitic series resistances effect have to be taken into consideration in choosing the working point. These two effects lead to a clamped injection level, because a significant portion of the increased voltage drops in the quasi-natural regions in this condition, instead of across the emitter-base junction. The carrier density saturation can be clearly observed in Fig. 4. In other words, at high VBE, a further increase in voltage only bring small extra increase to free carrier density; thereby an upper limit in the EO modulation efficiency is anticipated.

As we will elaborate in the following sections, the strong carrier density change at high VBE also has an impact on waveguide properties, including mode distortion and possible cutoff of the fundamental mode.

3.2 Two-dimensional carrier distribution and refractive index contour

In section 3.1, a 1D carrier distribution at the center of the HBT device is plotted and discussed. However, the carrier distribution in the HBT is highly non-uniform in the lateral direction. This is because the series resistance in the intrinsic base is proportional to current path length. In the middle of the device where the current path is the longest, the resistance is the largest; near the intrinsic base edge, the resistance has the smallest value. As a result of the position dependent resistance, the current crowds at the edge of the intrinsic base and the emitter. We convert the 2D carrier distribution into refractive index map according to Eq. (1), and plot it in Fig. 5 for λ=1.55µm at a few bias points. The current crowding effect is evident in this 2D refractive index contour. For the equilibrium condition, as is shown in Fig. 5 (a), the refractive index in the intrinsic base is higher than that in the extrinsic base, which is an outcome of the doping difference. A vertical index gradient is also observed. In the emitter, the gradient is a consequence of non-uniform doping profile; while in the base, it is primarily due to the germanium composition gradient. Although the doping profile in the HBT causes a higher refractive index in the intrinsic base than the extrinsic base at equilibrium, the free carrier plasma effect at high voltage can result in substantial refractive index decrease, and as a result, makes the intrinsic base index lower than that of the neighboring extrinsic base areas. This effect can be found in Fig. 5 (c). The non-uniform index distribution with reduced index in the waveguide center region could strongly alter the mode profile compared to the equilibrium condition, and lead to optical field expansion towards the slab regions. This phenomenon will be discussed next.

Fig. 5. Refractive index profile for λ=1.55µm with VBE=(a) 0V (b) 1.5V (c) 2.5V

3.3 Optical mode and modulation efficiency

The eigenmode in the HBT EO modulator waveguide is solved using Rsoft BeamPROP. Since this is not a mode coupling waveguide, we chose to use the semi-vector option to analyze the TE and the TM mode separately.

3.3.1 Verification of single mode condition

We first verify that the waveguide in this work satisfies single mode condition. For SOI wafers, a single mode rib waveguide usually needs to be as small as a few hundred nanometers in dimension. However, Soref et al. [22

22. R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si-on-SiO2,” IEEE J. Quantum Electron. 27(8), 1971–1974 (1991). [CrossRef]

] provided a guideline for designing a single mode rib waveguide whose dimensions can be up to a few microns. Although the waveguide explored here is not a strict rib/ridge waveguide - due to the finite dimension of SB in the base - we can still employ Soref’s rule to check two extreme conditions as a start. Figure 6 (a) shows the sketch of the cross-section of a rib waveguide with the parameters a, b, and r defined as the same as those in [22

22. R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si-on-SiO2,” IEEE J. Quantum Electron. 27(8), 1971–1974 (1991). [CrossRef]

]. The notation λ0 refers to the free-space wavelength. In Fig. 6 (b), the simplified HBT structure of our design in analogy to a standard rib waveguide is plotted. When SB=1µm as the base is aligned with the emitter width, or SB is practically infinite, the HBT waveguide retrogrades to a standard rib waveguide with b=0.484 for λ=1.55µm (or b=0.577 for λ=1.3µm) and a/b=0.667 according to our HBT parameters. When SB=1µm, the rib waveguide parameter r is 0.533, while when SB approaches infinite, r is 0.667. The single mode regimes are calculated and plotted in Fig. 6 (c). The calculation suggests that single mode condition holds for all cases.

Fig. 6. (a) Cross-section of a standard rib waveguide, (b) Cross-section of a HBT waveguide structure, and (c) Single mode region for SiO2/Si/SiO2 with r=0.667 and 0.533.

In the above analysis, we neglected the index deviation caused by the SiGe base layer in the rib waveguide. This should not be a problem considering the small index difference between SiGe and silicon compared to the index contrast between silicon and SiO2. However, if SB does not fit into the above two situations, chances are higher order modes can be supported in the waveguide. Due to the requirement of placing base electrodes, SB needs to be much larger than 1µm. Therefore, we need to determine the proper SB range in which the waveguide exhibits similar properties as the second case (large SB). We searched for the first order mode cutoff condition by scanning SB in the range of 5µm to 13µm with a 0.5µm step. The simulation results for λ=1.55µm indicate that the first order is cut off at SB=7.5µm for TE mode and SB=8.5µm for TM mode. At λ=1.3µm, SB is greater than 7.5µm for TE mode and 8µm for TM mode. Thus, the proposed HBT waveguide is of single mode.

3.3.2 Mode profile and interaction length for π-phase shift (Lπ)

The eigenmode profiles for λ=1.55µm are plotted in Fig. 7 at three different VBE biases. The TM mode generally exhibits a better confined profile than the TE mode. Since the free carrier plasma is formed in the center of the device, a more compact mode profile of the TM mode implies a larger overlap integral of the electric field with the modified refractive index area. For TM mode, the Ey field contour maintains its general shape as VBE increases, although spatial expansion is shown along the x-direction. For TE mode, the Ex field shows a more complicated behavior with the increase of VBE, in which the peak of the electric field splits and drifts from the rib region to the nearby slab regions. In the refractive index discussion in section 3.2, we pointed out the drastic refractive index decrease in the rib region at high VBE due to strong carrier injection. As a result, the rib region loses confinement of the electric field in the lateral direction, which gives rise to what is shown in Fig. 7 (c). At VBE=2.5V, a dumbbell shaped mode profile appears in the TE mode profile. Since the electric field does not have a node in the waveguide center, the dumbbell shaped electric field should still be the fundamental mode of the waveguide.

Fig. 7. Mode profiles at λ=1.55µm, TE mode (a) VBE=0V (b) VBE=1.5V (c) VBE=2.5V; TM mode (d) VBE=0V (e) VBE=1.5V (f) VBE=2.5V

Fig. 8. Δneff as a function of VBE

In a length-fixed EO modulator, the performance is usually represented by the voltage required for π phase shift (Vπ). In our work, since we investigate device behavior by sweeping the voltage, we use the interaction length required for π phase shift (Lπ) as an alternative to characterize the performance. Lπ is calculated using Eq. (3) [7

7. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]

]:

Lπ=λ02Δneff
(3)

Table 1. Comparison of Reported Silicon EO Modulator in Recent Years (ts is the total switching time, length refers to the active component length, and FCAM is the acronym for free carrier absorption modulator)

table-icon
View This Table

We also examined how the carrier plasmas in the emitter and collector affect the EO modulation efficiency. This is done by comparing Lπ calculated above with conditions in which the refractive index of the emitter and collector is fixed at its VBE=0V value. For example, for TM mode at λ=1.55µm under the bias of VBE=2.5V, we obtain =76.7µm without considering carrier plasma in the emitter. A similar investigation not considering index variation in collector produces Lπ=75.2µm. In this condition, the modulation efficiency has 3% contribution from the emitter due to the hole injection, and 0.9% from the collector due to the base widening. These results are consistent with the 1D carrier distribution plot in Fig. 4 and the 2D refractive index map in Fig. 5. Therefore, we can conclude the performance of the HBT EO modulator heavily relies on the carrier plasmas generated in the base region. In the device design, consequently, we can minimize the base widening effect and further compress the hole injection from base into the emitter in order to get faster switching speed.

We calculated the attenuation of the HBT EO modulator according to Eq. (1). Though high attenuation is held for device with heavy doping, the significant reduction in Lπ allows the total attenuation of the designed HBT EO modulator to be comparable to other types of silicon EO modulators. For example, at λ=1.55µm and VBE=2.5V, the optical loss for TM mode is 3.9dB for an active waveguide length of Lπ=74.5µm. This value is even smaller than reported results from many other work [18

18. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,” J. Lightwave Technol. 15(3), 505–518 (1997). [CrossRef]

,30

30. W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]

].

4. Dynamic response considerations

The transient response of the EO modulator refers to switching the device from “ON” to “OFF” or verse vice. It is affected collectively by several mechanisms, namely the carrier diffusion, depletion capacitances charging/discharging time, and the RC delay from the parasitic capacitances and resistances. In general, carrier redistribution is the dominant delay component, especially when the voltage swing is large. This delay can be represented by the forward transit time tF as expressed in Eq. (4). tF is the sum of the emitter diffusion time tE, the base transit time tB, the emitter-base depletion region transit time tBE, and the base-collector depletion region transit time tBC, and is expressed as [31

31. Y. Taur and T. H. Ning, Fundamentals of Moerdern VLSI Devices (Cambridge University Press, 1998), Chap. 6.

]

tF=tE+tB+tBE+tBCtE+tB+tBC=wB2ηDn+wEwBθDn+wdBC2vs
(4)

η2[1+(εbiε0)32]
(5)

where ε0=2kT/qWB, and εbi is the built-in field. In this design, η, as a result sorely from SiGe composition gradient is calculated to be 6. It is evident, according to Eq. (4), that tF is strongly affected by the vertical dimensions of the device. However, tF only accounts for part of the total delay. Thus, a numerical analysis is necessary to estimate the switching speed of the proposed device.

The transient analysis is performed with MEDICI by switching the base-emitter bias VBE between -1.5V and 2.5V with a 1ps ramp time. The collector current density IC verse time is assessed to characterize the switching speed, and is plotted in Fig. 9. The rise time that measures the carrier building up process, is 0.21ns. The fall time, or the carrier removal time, is 0.15ns. Evaluated according to the longer delay process, i.e., the turn-on delay, this device supports up to 2.4Gbps data rate for rectangular non-return to zero (NRZ) pulse train. Although carrier injection type of EO modulator is typically considered slow unless an optical resonant structure is used [4

4. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

], we demonstrated in this work that Multi-GHz switching speed is possible due to the thin base and the built-in electric field.

Fig. 9. Transient analysis: (a) input pulse VBE, and (b) IC response.

5. Discussion of device design strategy

In this work, the device geometry parameters are chosen to maximize the modulation efficiency, and the switch speed is secondary in the design consideration. We first investigated how the mode profile varies with rib waveguide cross-section dimensions, emitter thickness WE, and the base region width SB. The geometry parameters that lead to single mode operation of the rib waveguide are determined. Next, we slightly adjusted the emitter thickness as well as the base thickness WB to maximize the carrier plasma overlap with the optical field. It turns out that the base thickness is the most influential parameter in determining the interaction length Lπ and the switching speed. A trade-off between the modulation efficiency and the switching speed is observed.

There are several aspects in this design that distinguish the HBT EO modulator from other approaches. First, the composition graded SiGe base induces a built-in electric field that facilitates carrier transport across the thin base, making the HBT device inherently fast. Second, the proposed HBT EO modulator attains plasma manipulation by carrier injection. Thus, it exhibits much higher modulation efficiency compared to PN junction or MOS based EO modulators. Third, the proposed device provides an extra degree of freedom in device operation because of the use of the third terminal compared to two-terminal devices such as PN and PIN EO modulators. The third terminal can be used to manipulate the electric field profile in the device in order to maximize the overlap between the optical mode and the plasma cloud [18

18. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,” J. Lightwave Technol. 15(3), 505–518 (1997). [CrossRef]

].

6. Conclusions

In this paper, we discussed the optical and electrical characteristics of a SiGe HBT EO modulator with composition graded SiGe base. The strong carrier injection in the base provides very high modulation efficiency. It also has a considerable impact on the mode profile of the HBT waveguide. In this device, the interaction length for a phase shift of π is Lπ=74.5µm for TM mode at λ=1.55µm. The propagation loss for Lπ=74.5µm is 3.9dB. This device is expected to operate at a switching speed of 2.4GHz or higher. The results are obtained in a device with a waveguide cross-section larger than 1µm. We expect enhanced modulation efficiency and ultra-fast dynamic response in an optimized submicron HBT EO modulator.

References and links

1.

Y. H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437(7063), 1334–1336 (2005). [CrossRef] [PubMed]

2.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

3.

C. Cocorullo, M. Iodice, I. Rendina, and P. M. Sarro, “Silicon thermo-optical micro-modulator with 700 kHz - 3 dB bandwidth,” IEEE Photon. Technol. Lett. 7(4), 363–365 (1995). [CrossRef]

4.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

5.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15, 660–668 (2007). [CrossRef] [PubMed]

6.

F. Y. Gardes, G. T. Reed, A. P. Knights, and G. Mashanovich, “Evolution of optical modulation using majority carrier plasma dispersion effect in SOI”, Proc. of SPIE 6898, 68980C-68980C-10 (2008). [CrossRef]

7.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]

8.

A. Liu, “Announcing the world’s first 40G silicon laser modulator”, http://blogs.intel.com/research/2007/07/40g_modulator.php

9.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005). [CrossRef]

10.

A. Cutolo, M. Iodice, A. Irace, P. Spirito, and L. Zeni, “An electrically controlled Bragg reflector integrated in a rib silicon on insulator waveguide,” Appl. Phys. Lett. 71(2), 199 (1997). [CrossRef]

11.

G. Coppola, A. Irace, M. Iodice, and A. Cutolo, “Simulation and analysis of a high-efficiency silicon optoelectronic modulator based on a Bragg mirror,” Opt. Eng. 40(6), 1076–1081 (2001). [CrossRef]

12.

C. A. Barrios, V. Rosa de Almeida, and M. Lipson, “Low-power-consumption short-length and high-modulation-depth silicon electrooptic modulator,” J. Lightwave Technol. 21(4), 1089–1098 (2003). [CrossRef]

13.

X. Xiao, J. C. Sturm, K. K. Goel, and P. V. Schwartz, “Fabry-Perot optical intensity modulator at 1.3µm in silicon,” IEEE Photon. Technol. Lett. 3(3), 230–231 (1991). [CrossRef]

14.

M. Y. Liu and S. Chou, “High-modulation-depth and short-cavity-length silicon Fabry-Perot modulator with two grating Bragg reflectors,” Appl. Phys. Lett. 68(2), 170 (1996). [CrossRef]

15.

C. A. Barrios, V. R. Almeida, R. Panepucci, and M. Lipson, “Electrooptic Modulation of Silicon-on-Insulator Submicrometer-Size Waveguide Devices,” J. Lightwave Technol. 21(10), 2332–2339 (2003). [CrossRef]

16.

S. Manipatruni, Q. Xu, B. Schmidt, J. Shakya, and M. Lipson, “High Speed Carrier Injection 18Gb/s Silicon Micro-ring Electro-optic Modulator,” in Proceedings of Lasers and Electro-Optics Society (IEEE, 2007), pp.537–538.

17.

R. D. Lareau, L. Friedman, and R. A. Soref, “Waveguided electro-optical intensity modulation in a Si/GexSi1-x/Si heterojunction bipolar transistor,” Electron. Lett. 26(20), 1653–1655 (1990). [CrossRef]

18.

A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,” J. Lightwave Technol. 15(3), 505–518 (1997). [CrossRef]

19.

S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). [CrossRef] [PubMed]

20.

R. A. Soref and B. R. Bennett, “Kramers-Kronig analysis of electro-optical switching in silicon,” SPIE Integr. Opt. Circuit Eng. 704, 32–37 (1986).

21.

S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (John Wiley & Sons, Inc, 2007), Chap. 5.

22.

R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si-on-SiO2,” IEEE J. Quantum Electron. 27(8), 1971–1974 (1991). [CrossRef]

23.

A. Sciuto, S. Libertino, A. Alessandria, S. Coffa, and G. Coppola, “Design, Fabrication, and Testing of an Interated Si-Based Light Modulator,” J. Lightwave Technol. 21(1), 228–235 (2003). [CrossRef]

24.

C. E. Png, G. T. Reed, R. M. H. Atta, G. Ensell, and A. G. R. Evans, “Development of Small Silicon Modulators in SOI,” Proc. SPIE 4997, 190–197(2003). [CrossRef]

25.

L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High Speed Silicon Mach-Zehnder Modulator,” Opt. Express 13(8), 3129–3135 (2005). [CrossRef] [PubMed]

26.

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s Carrier-injection-based silicon microring silicon modulators,” Opt. Express 15, 430–436 (2007). [CrossRef] [PubMed]

27.

S. Manipatruni, Q. Xu, and M. Lipson, “PINIP based high-speed high-extinction ratio micron-size silicon electro-optic modulator,” Opt. Express 15(20), 13035–13042 (2007). [CrossRef] [PubMed]

28.

F. Y. Gardes, K. L. Tsakmakidis, D. Thomson, G. T. Reed, G. Z. Mashanovich, O. Hess, and D. Avitabile, “Micrometer size polarisation independent depletion-type photonic modulator in Silicon On Insulator,” Opt. Express 15(9), 5879–5884 (2007). [CrossRef] [PubMed]

29.

S. Deng, Z. R. Huang, J.-R. Guo, J. F. McDonald, and R. P. Kraft, “Numerical Investigation of a SiGe HBT Electro-optic Modulator,” in proceedings of IEEE/LEOS Winter Topicals Meeting Series, 14–15, (2009) [CrossRef]

30.

W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]

31.

Y. Taur and T. H. Ning, Fundamentals of Moerdern VLSI Devices (Cambridge University Press, 1998), Chap. 6.

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.2090) Optical devices : Electro-optical devices
(130.4110) Integrated optics : Modulators

ToC Category:
Integrated Optics

History
Original Manuscript: April 29, 2009
Revised Manuscript: July 10, 2009
Manuscript Accepted: July 13, 2009
Published: July 20, 2009

Citation
Shengling Deng, Z. Rena Huang, and J. F. McDonald, "Design of High Efficiency Multi-GHz SiGe HBT Electro-Optic Modulator," Opt. Express 17, 13425-13428 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13425


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References

  1. Y. H. Kuo, Y. K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437(7063), 1334–1336 (2005). [CrossRef] [PubMed]
  2. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]
  3. C. Cocorullo, M. Iodice, I. Rendina, and P. M. Sarro, “Silicon thermo-optical micro-modulator with 700 kHz – 3 dB bandwidth,” IEEE Photon. Technol. Lett. 7(4), 363–365 (1995). [CrossRef]
  4. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
  5. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15, 660–668 (2007). [CrossRef] [PubMed]
  6. F. Y. Gardes, G. T. Reed, A. P. Knights, and G. Mashanovich, “Evolution of optical modulation using majority carrier plasma dispersion effect in SOI”, Proc. of SPIE 6898, 68980C–68980C–10 (2008). [CrossRef]
  7. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]
  8. A. Liu, “Announcing the world's first 40G silicon laser modulator”, http://blogs.intel.com/research/2007/07/40g_modulator.php
  9. Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005). [CrossRef]
  10. A. Cutolo, M. Iodice, A. Irace, P. Spirito, and L. Zeni, “An electrically controlled Bragg reflector integrated in a rib silicon on insulator waveguide,” Appl. Phys. Lett. 71(2), 199 (1997). [CrossRef]
  11. G. Coppola, A. Irace, M. Iodice, and A. Cutolo, “Simulation and analysis of a high-efficiency silicon optoelectronic modulator based on a Bragg mirror,” Opt. Eng. 40(6), 1076–1081 (2001). [CrossRef]
  12. C. A. Barrios, V. Rosa de Almeida, and M. Lipson, “Low-power-consumption short-length and high-modulation-depth silicon electrooptic modulator,” J. Lightwave Technol. 21(4), 1089–1098 (2003). [CrossRef]
  13. X. Xiao, J. C. Sturm, K. K. Goel, and P. V. Schwartz, “Fabry-Perot optical intensity modulator at 1.3μm in silicon,” IEEE Photon. Technol. Lett. 3(3), 230–231 (1991). [CrossRef]
  14. M. Y. Liu and S. Chou, “High-modulation-depth and short-cavity-length silicon Fabry-Perot modulator with two grating Bragg reflectors,” Appl. Phys. Lett. 68(2), 170 (1996). [CrossRef]
  15. C. A. Barrios, V. R. Almeida, R. Panepucci, and M. Lipson, “Electrooptic Modulation of Silicon-on-Insulator Submicrometer-Size Waveguide Devices,” J. Lightwave Technol. 21(10), 2332–2339 (2003). [CrossRef]
  16. S. Manipatruni, Q. Xu, B. Schmidt, J. Shakya, and M. Lipson, “High Speed Carrier Injection 18Gb/s Silicon Micro-ring Electro-optic Modulator,” in Proceedings of Lasers and Electro-Optics Society (IEEE, 2007), pp.537–538.
  17. R. D. Lareau, L. Friedman, and R. A. Soref, “Waveguided electro-optical intensity modulation in a Si/GexSi1-x/Si heterojunction bipolar transistor,” Electron. Lett. 26(20), 1653–1655 (1990). [CrossRef]
  18. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,” J. Lightwave Technol. 15(3), 505–518 (1997). [CrossRef]
  19. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). [CrossRef] [PubMed]
  20. R. A. Soref and B. R. Bennett, “Kramers-Kronig analysis of electro-optical switching in silicon,” SPIE Integr. Opt. Circuit Eng. 704, 32–37 (1986).
  21. S. M. Sze, and K. K. Ng, Physics of Semiconductor Devices (John Wiley & Sons, Inc, 2007), Chap. 5.
  22. R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si-on-SiO2,” IEEE J. Quantum Electron. 27(8), 1971–1974 (1991). [CrossRef]
  23. A. Sciuto, S. Libertino, A. Alessandria, S. Coffa, and G. Coppola, “Design, Fabrication, and Testing of an Interated Si-Based Light Modulator,” J. Lightwave Technol. 21(1), 228–235 (2003). [CrossRef]
  24. C. E. Png, G. T. Reed, R. M. H. Atta, G. Ensell, and A. G. R. Evans, “Development of Small Silicon Modulators in SOI,” Proc. SPIE 4997, 190–197 (2003). [CrossRef]
  25. L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High Speed Silicon Mach-Zehnder Modulator,” Opt. Express 13(8), 3129–3135 (2005). [CrossRef] [PubMed]
  26. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s Carrier-injection-based silicon microring silicon modulators,” Opt. Express 15, 430–436 (2007). [CrossRef] [PubMed]
  27. S. Manipatruni, Q. Xu, and M. Lipson, “PINIP based high-speed high-extinction ratio micron-size silicon electro-optic modulator,” Opt. Express 15(20), 13035–13042 (2007). [CrossRef] [PubMed]
  28. F. Y. Gardes, K. L. Tsakmakidis, D. Thomson, G. T. Reed, G. Z. Mashanovich, O. Hess, and D. Avitabile, “Micrometer size polarisation independent depletion-type photonic modulator in Silicon On Insulator,” Opt. Express 15(9), 5879–5884 (2007). [CrossRef] [PubMed]
  29. S. Deng, Z. R. Huang, J.-R. Guo, J. F. McDonald, and R. P. Kraft, “Numerical Investigation of a SiGe HBT Electro-optic Modulator,” in proceedings of IEEE/LEOS Winter Topicals Meeting Series, 14–15, (2009) [CrossRef]
  30. W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10Gb/s silicon Mach-Zehnder modulator,” Opt. Express 15(25), 17106–17113 (2007). [CrossRef] [PubMed]
  31. Y. Taur, and T. H. Ning, Fundamentals of Moerdern VLSI Devices (Cambridge University Press, 1998), Chap. 6.

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