Open Access
Add to BibSonomyAbstract
Electrical, optical and electro-optical simulations are presented for a waveguided, resonant, bus-coupled, p-doped Si micro-donut MOS depletion modulator operating at the 1.55 μm wavelength. To minimize the switching voltage and energy, a high-K dielectric film of HfO2 or ZrO2 is chosen as the gate dielectric, while a narrow ring-shaped layer of p-doped poly-silicon is selected for the gate electrode, rather than metal, to minimize plasmonic loss loading of the fundamental TE mode. In a 6-μm-diam high-Q resonator, an infrared intensity extinction ratio of 6 dB is predicted for a modulation voltage of 2 V and a switching energy of 4 fJ/bit. A speed-of-response around 1 ps is anticipated. For a modulator scaled to operate at 1.3 μm, the estimated switching energy is 2.5 fJ/bit.
©2011 Optical Society of America
1. Introduction
Anderson and Fauchet [1] presented a stimulating study of MOS depletion modulators in photonic-crystal (PhC) three-point-defect (L3) resonators coupled to line defect waveguides. However, PhC integrated “circuits” have not yet been demonstrated experimentally and that’s why we decided to focus upon a demonstrated silicon-on-insulator (SOI) network-on-chip approach in which the network is constructed in part from arrays of SOI strip channel waveguides (nano-wires) that are side-coupled inplane to waveguided SOI micro-rings or micro-disks. Photonic integration of about a hundred rings has already been reported. We felt that the MOS-depleter modulation approach could be applied advantageously to this nanowire technology and we set-forth to design and simulate energy-efficient structures. Since the electromagnetic mode volume in a ring is larger than that in the PhC resonator, we had to trade off an increase in the switching energy for a greater ease of manufacture. The challenges we faced were to attain high-enough Q, low-enough drive voltage, and sufficient modal index change induced by carrier depletion.
This paper presents a modulator design optimized for adequate modulation depth at very low energy. The electrical, optical, and combined electro-optical response are simulated. Results are given on charge depletion, capacitance, voltage requirements, electromagnetic mode profile, polarization, gate dielectric choice, gate-electrode optical loading, waveguide doping effects, gate oxide thickness effects, resonator design, cavity Q, refractive index changes, mode overlap integral, and MOS-induced resonance shift.
2. Design discussion
The high Q resonator coupled to a readout nanowire has a FWHM spectral linewidth Δλq that is inversely proportional to Q. Depletion of carriers within the cavity increases the real index of the Si resonator material by an amount Δn which in turn changes the effective index of the resonant mode neff by an amount Δneff. Then Δneff produces a shift Δλ of the mode's initial resonance wavelength λ 0. For successful modulation the Q must be high enough and Δneff large enough so that Δλ is comparable to Δλq . High Q comes from low propagation loss in the waveguided resonator. There are several loss factors such as wall roughness loss, material loss, and waveguide bend loss. The principal loss here arises from gate-electrode loading since the optical mode-tail tunnels through the gate oxide and touches the electrode. In early simulations, we quickly found that a metal gate electrode was not acceptable because of its very high plasmonic loss. The solution to that problem is to employ doped polysilicon as the gate electrode (as is done in the CMOS industry) because the poly-Si loss is orders-of-magnitude smaller than that of metal. The silicon body of the resonator is uniformly doped p-type or n-type and there is material loss associated with free carrier absorption, although early studies show [2] that this loss can be kept below 1 cm−1 at the 1.55 μm communications wavelength by restricting the doping concentration to be less than 1017 cm−3.
We decided to introduce here the new silicon micro-donut resonator reported recently by the Georgia Tech group [3] because the donut has demonstrably higher Q than that of prior-art micro-rings. Donut wall losses are lower than ring losses because the donut’s whispering gallery mode uses internal reflection from only the outer sidewall, whereas the ring incorporates reflections from its inner and outer sidewalls. The MOS donut affords an opportunity to reduce the gate capacitance and the switching energy by employing a narrow-width ring-shaped gate electrode film that is positioned atop the whispering gallery region near the outer sidewall. For outer diameters in the 5 to 8 μm range, we determined the donut’s allowed modes and the low-loss polarization which is TE-like. Since it is a depletion device, this MOS donut does bear a generic resemblance to the vertical-pn-junction depletion modulators and switches that have been demonstrated in Si micro-disks [4,5].
The change in real Si index Δn arises from the free-carrier electro-refraction effect, where Δn nearly proportional to the change in free carrier concentration ΔN. From ionized impurities in the doped waveguide, there is an initial concentration N of free carriers that exists at zero bias. Carriers are swept out of the active region (rapidly) by the applied field, a removal of ΔN = N at full depletion. Thus a large N seems desireable to yield large Δn. However, at large N it becomes difficult to deplete the waveguide body since the depletion layer thickness depends upon N –0.5. Thus a compromise value of N such as 1016-1017 cm−3 must be chosen in order to deplete “most” of the donut height h—but this ceiling-on-N constrains Δn. The choice of electrons or holes as the free carriers plays a key role in attaining large Δn. Early studies [2] showed that free holes are about four times more effective than electrons for inducing Δn when sweeping out carrier densities of 1016-1017 cm−3. That prediction has been verified experimentally over the years; thus, we elected p-type doping of the resonator as discussed below.
The drive voltage V applied to the external terminals of the MOS capacitor resonator is divided internally between the Si waveguide region and the gate dielectric region—thus the terminal voltage is higher than the “useful voltage” that depletes the p-Si. This voltage division is governed by the low-frequency dielectric constant K of the gate dielectric with respect to the silicon K. By proper choice of gate dielectric material, the great majority of V will appear across the silicon as desired for minimum-energy switching. This freedom to choose the best dielectric is a valuable aspect of the MOS device. In its design, the MOS modulator has two more degrees-of-freedom than the pn-junction depletion device because the MOS device offers choices for the gate electrode and gate dielectric materials as well as the waveguide core-and-cladding materials. We have exploited this design freedom by selecting the high-K gate dielectric to be either Hafnium dioxide (HfO2) or Zirconium dioxide (ZrO2) instead of the conventional low-K gate of silicon dioxide. Hafnium dioxide has had great success in the modern MOSFET industry, and both HfO2 and ZrO2 have K ~25. As detailed below, when this thin high-K layer becomes gate material in the micro-donut, the resulting combination of low capacitance and low gate voltage enables ultralow-energy photonic switching.
3. Proposed designs
A 1 × 1 electrooptic modulator that has one input port and one output port is created when a resonator is coupled to one bus waveguide. A second bus waveguide coupled to the same resonator is optional but useful because a resonator coupled to two buses constitutes a 2 × 2 electrooptical spatial routing switch. Since the switch is wavelength selective, it also functions as a reconfigurable add-drop multiplexer within a wavelength-division-multiplexed (WDM) system. By interconnecting multiple switches, a multi-wavelength WDM switch or a multi-drop network can be formed.
A reconfigurable multi-passband optical filter can be created when resonators share two buses as shown in Fig. 1 for SOI micro-donut resonators in the MOS ring-gate structure. The purpose of Fig. 1 is to illustrate wavelength-multiplexed modulators (or switches) together with their electrical contact leads (the ground line and the independent signal-control lines) in a planar integrated SOI structure. It is often advantageous to introduce two resonators that are coupled to each other and to two buses [6] as shown in Fig. 2 for the 2 × 2 switch configuration. This important configuration allows the spectral passband shape of the switch - such as the side-skirt steepness - to be adjusted by design. Focusing now on the details of the one-donut, two-bus modulator device, we present in Figs. 3(a) and 3(b) the top view and the cross-section side view of the SOI device (The buried SiO2 is optically thick). As indicated, there is a thin (~50 nm high) doped Si “platform” situated in the central region of the p-doped silicon resonator body - an inner “disk extension” of the donut that offers a convenient means for electrically contacting the bottom of the annular MOS capacitor. We see in the side view the ring-shaped high- K gate insulator layer (of thickness tox) as well as the ring-shaped gate-electrode-material layer (of thickness te). The electrical bias source that impresses high-speed information on the guided light is also shown. There is charging current as the MOS device is switched, but the ac-coupled modulator does not have the dc currents that are present in the pn-junction depletion device, and the MOS device has some possibility of nonvolatile optical memory behavior in which the MOS capacitor holds its excited state without applied power.
Fig. 1 Perspective view of proposed multi-microdonut electro-optical spatial routing switch offering independent signal-gating of 2 x 2 MOS depletion devices.
Fig. 2 Perspective view of proposed double-microdonut 2 x 2 electrooptical MOS switch and wavelength-multiplexed ROADM.
Fig. 3 (a) top view of MOS-depletion microdonut modulator and switch, (b) cross-section side view of the device illustrating the ring-shaped high-K dielectric gate, the p-doped poly-silicon gate electrode, and the center-donut p-Si bottom contact region.
4. Electrical simulations
We can visualize that the cross-section view of Fig. 3(b) shows one section or “wedge” of the micro donut, and that the electrical response of this MOS capacitor segment is the same all around the donut circumference. For digital electrical inputs, the modulator has an initial state and an excited state - known as “off” and “on” states. In practice, the largest useable value of V is the threshold voltage Vth that begins to produce unwanted charge inversion (after depletion) at the gate/Si interface. The “off” and “on” states correspond to situations in which the silicon underneath the gate oxide is either in its original doped state or in its partially depleted state. It is well known that at zero bias a built-in field is sometimes induced in an MOS structure, resulting in a tilted energy band for the silicon near the oxide-silicon interface. In order to restore the flat band condition for silicon, a small flat-band voltage Vfb needs to be applied for the “off” state
where ϕms is the work function difference between the polycrystalline silicon gate and the silicon, Qox is the oxide effective charge per unit area at the oxide-silicon interface, Cox is the gate-oxide capacitance per unit areaHere ε 0 is the permittivity of vacuum, Kox is the dielectric constant of the oxide, and tox is the thickness of the oxide layer between the gate electrode and the silicon.In order for the MOS modulator to switch to the “on” state, a portion of the silicon height underneath the gate electrode should be depleted. The voltage that needs to be applied at the gate can be expressed as
where Vs is the part of the gate voltage that is actually dropped across the silicon. To prevent the MOS structure from reaching strong inversion at the oxide-silicon interface, we must limitwhere q is the electron charge, kB is the Boltzmann constant, and Ni is the intrinsic carrier concentration at the temperature T. The gate voltage Von for the “on” state should thus be limited by the threshold voltage Vth of the MOS structureUsing Eq. (5), our early simulations showed that an SiO2 gate with K = 3.9 required modulation voltages of 10 to 15 volts. We then examined a higher-K silicon-nitride gate dielectric. The Si3N4 offered lower-voltage control but was still not an optimum dielectric. As a solution to the “10-volt problem”, we turned to the high-K materials discussed above. Utilizing Eqs. (1)–(5), the “off” and “on” state voltages are shown in Figs. 4(a) and 4(b), respectively, for the effects of gate dielectric thickness in the range 100 nm < tox < 300 nm and the Si doping concentration for 6 × 1016 cm−3 < Np < 1 × 1017 cm−3. We have used Qox = 1010 cm–2 for the flat band voltage Eq. (1).Fig. 4 (a) MOS-modulator “off state” voltage versus tox, and (b) MOS modulator “on state” gate voltage versus tox.
To determine the switching energy for the MOS structure, we have estimated the capacitance for the ring-resonator shown in Fig. 3(b) where r1 and r2 are the outer and inner radii of the poly-Si electrode ring, respectively, and r 0 is the donut’s inner radius. The flat band capacitance for the “off” state can be calculated as
where the Debye length in silicon isand the “on” state capacitance iswhere the depletion depth in Si isSince the percentage of the silicon body height that can be depleted of charge depends upon the initial doping density Np, we have calculated with Eq. (9) the silicon height that could be fully depleted versus a given doping Np and the result is shown in Fig. 5 . We then took the single-mode waveguide height to be 230 nm at λ = 1550 nm as in the Georgia Tech experiments [3]. For the 8 × 1016 cm−3 doping, Fig. 5 indicates that 50% of the silicon waveguide can be depleted at the 2.2 V gate voltage, and this prediction is used in the mode overlap calculation of section 5 below.In our 1.55 μm simulations, we used h = 230 nm, r 1 = 3.0 μm, r 2 = 2.6 μm, r 0 = 2.2 μm and K = 25 for HfO2. The results obtained for “off” and “on” state capacitance using Eqs. (6)–(8) are shown in Figs. 6(a) and 6(b), respectively.
Fig. 6 (a) MOS modulator “off” state capacitance and (b) “on” state capacitance versus the HfO2 gate thickness tox for a range of p-doping concentration.
The switching energy Es of the MOS modulator can therefore be calculated
and the result is shown in Fig. 7 . Often, in practice, the applied modulation voltage V is a digital voltage that employs a return-to-zero line code in which the signal drops to zero within each clock period—implying that the bit’s energy occupies 50% of the time slot. For that reason, we shall assume that the modulator’s switching energy per bit is approximately one-half of Es.Fig. 7 The off-state to on-state switching energy of the MOS modulator versus the HfO2 gate thickness.
Examining Figs. 4, 6, and 7, the Von of 1.5 to 3.1 V is transistor-compatible (here Von = Vth) and the Voff of 0.097 to 0.125 V is quite small, while Coff of 4 to 12 fF is generally larger than Coff of 2.7 to 4.7 fF. The Es of 4 to 14 fJ is generally low. What are the “best” tox and Np parameters? We have selected from these graphs the values tox = 200 nm and Np = 8 × 1016 cm−3 as being optimum because they give a good solution to the minimum-versus-maximum problem of Es-vs-Δλ. Then Von = Vth = 2.2 volts, Voff = 0.11 volts, Con = 3.6 fF, and Coff = 6.9 fF. Assuming that the infrared extinction of the modulator is adequate at this gate voltage, Es = 8.5 fJ and the energy per bit is 4.3 fJ/bit. The MOS field effect modulator is inherently very fast and its modulation speeds are probably limited in practice by the device’s RC time constant, of the order of 1 ps.
5. Optical simulations
The TM-like mode in the donut had significantly higher loss than the fundamental TE-like mode; thus we elected the TE0 mode as the optimum modulation choice. This mode could be readily found when the donut’s outer radius r 1 was 3 μm but we had difficulty in finding stable TE modes for r 1 < 3 μm. However, at least for ungated donuts, smaller r1 are feasible because experiments [3] showed a practical TE0 mode at r 1 = 1.95 μm for a SiO2-clad Si donut on 1 μm of buried oxide, with h = 230 nm. In those experiments, the unloaded Q was 80,000 and a loaded Q of 30,000 was found when their micro-donut was side coupled across a 240 nm air gap to a Si strip waveguide.
Turning now to our gated microdonut case, using Lumerical software we calculated the unloaded Q versus HfO2 thickness. At λ 0 = 1558.92 nm, the following parameters were used in the electromagnetic simulations; a refractive index of 1.87 for HfO2 [7], an index of 3.4750 for both Si and poly-Si, and an index of 1.45 for the “thick” buried SiO2. The inner radius ro of the Si donut was 2.2 μm while the outer radius was 3.0 μm, with a Si height of 230 nm. For both the ring-shaped gate dielectric and gate electrode, the inner radius was 2.6 μm with a 3.0 μm outer radius so that the 400-nm radial width r 1 – r 2 matched the mode width. We used thicknesses of tox = 200 nm, te = 200 nm, with a gate electrode p-doping of 1 × 1019 cm−3 which produces an optical intensity attenuation factor of 80 cm−1 while the Si waveguide p-doping was 8 × 1016 cm−3 which gives a loss factor of 0.3 cm−1.
Utilizing all of these numbers, we then calculated the unloaded Q of the modulator as a function of tox, with the result presented in Fig. 8 . At the optimum tox = 200 nm discussed earlier, we see that the unloaded Q is 23,120 where we found λ 0 of 1558.92 nm and an effective mode index of 2.564. Figure 9 illustrates the spatial intensity distribution of this high-Q TE0 mode. When coupled to a bus channel waveguide for modulation use, the resonator Q decreases by an amount that depends upon the strength of evanescent-wave side coupling. We have chosen the critical-coupling condition as being useful for electro-modulation applications. For this choice, the loaded Q is 50% of the unloaded value; namely 11,560. This is equivalent to a mode propagation loss of about 39 dB per “cm of circumference” in the cavity.
Fig. 8 Unloaded Q versus tox for the MOS-depletion TE0-mode microdonut modulator whose dimensions are given in the text.
6. Electro-optical response
As mentioned, there is ample experimental evidence to support the Kramers-Kronig finding [2] that the change in real index Δn of bulk crystal silicon is larger for hole depletion than for electron depletion. The empirical relation that we have set forth [2,8] is a quantitative guide to the depletion response at 1559 nm, namely: Δn = 8.5 × 10–18(ΔNp)0.8 with the hole depletion ΔNp expressed in cm−3. We shall determine the Δneff that occurs when the resonator is partially depleted at V = Von. Our goal is to calculate the wavelength dependence of the bus waveguide’s output power in the Voff and Von states so as to determine the extinction ratio. Using the electromagnetic software, we first calculate the spectral dependence of bus intensity transmittance for the loaded-Q (11,540) device at Voff. Then we examine the spectral shift (the translation on the wavelength scale) of that curve.
We calculated the resonance shift Δλ using the approximations: Δλ/λ 0 = Δneff /neff (which may underestimate the actual Δλ). Further, we say that Δneff is given by the spatial overlap integral of the on-state E2 mode-intensity profile with the depleted region of the resonator. In the depleted volume of the Si waveguide ΔNp = 8 × 1016 cm−3 and from the above hole-depletion formula we get Δn = 4.5 × 10–4 in the upper half of the donut. Thus we use Δneff = MΔn, where M is the normalized overlap factor. To find M, we used the Gaussian mode profile shown in Fig. 10 where td = 0.5h. This gives M = 0.35 which implies that Δneff = 1.6 × 10–4. Using that, we then found Δλ = (Δneff /neff )λ0 = 0.097 nm when neff = 2.564. It is interesting to compare this shift to the FWHM linewidth Δλq = 0.135 nm of the loaded-Q modulator, namely Δλ=0.72Δλq. Also our Δλ compares to Δλ = 0.044 nm in Fig. 9 of [1].
Fig. 10 Mode-overlap calculation for the Fig-3 device with 2.2 V applied. The fundamental mode profile is approximated by the Gaussian shape illustrated here.
Our estimate of Δλ enabled us to plot the spectral transmittance of the modulator’s output waveguide for the off and on-states, as presented in Fig. 11 . Large Δλ/Δλq ratios generally make it easy to attain a large depth of modulation. Let us define Toff as the off-state transmittance, Ton as the on-state transmittance, and Ton/Toff as the extinction ration ER. We are going to operate the modulator in the “light-to-dark” mode. Thus, in Fig. 11 we have two choices of the operation-wavelength indicated by the vertical dashed lines; one approach gives the lowest insertion loss Toff for which ER ≥ 6 dB, the other wavelength selection gives the highest ER irrespective of Toff. For the first case, we estimate at λ = 1559.05 nm that Toff = −1.1 dB, Ton = - 7.1 dB and ER = 6.0 dB, while the second approach at the Ton valley where λ = 1550.02 nm, gives Toff = - 1.8 dB, Ton = - 25.5 dB and ER = 23.7 dB. Because Δλ scales with Von, there is a tradeoff between gate voltage magnitude and ER. For example, if we reduced the gate voltage Von to 1 volt, the 6 dB extinction would decrease to something like 2.7 dB.
Fig. 11 Infrared power transmission of the optimum MOS-depletion modulator in its off and on states.
The curves in Fig. 11 have a peak-to-valley ratio of about 25 dB; however, this theory does not take into account real-world issues that arise during microdonut construction—such as sidewall roughness and fabrication errors. Experiments on microdisks and microdonuts [3–5] show that a transmission dip in the range of −17 to −20 dB is feasible, and the type-2 ER estimated here would be adjusted in accordance with those findings.
7. Wavelength dependence of modulator performance
We investigated the modulator’s performance as a function of operation wavelength, going from 1.3 μm up to 5.0 μm. In wavelength-scaling the device, we want to keep the same Q, the same td/h, the same Δλ and the same ER found at 1.55 μm. This means (for a fixed HfO2 gate thickness) that we shall have to scale the p-type doping densities of both the waveguide body and the polysilicon gate electrode. For a given level of hole concentration in Si or poly-Si, we note that the free-carrier optical absorption loss of each region (parameters that enter into the mode Q calculation) is proportional to wavelength-squared [8], while at a given wavelength the optical intensity-attenuation factor of each material is roughly proportional to its individual free-hole concentration. Because of these scaling laws, it is necessary to reduce the doping levels of both regions as λ is increased beyond 1.55 μm in order to maintain the 1.55 μm performance. So the two p-dopings scale as 1/λ 2. We have found that this doping adjustment gives acceptable modulator performance over the 1.3 to 4.0 μm range. However, for λ > 4 μm, the p-doping of the gate electrode falls below 1018 cm−3, increasing the series resistance R of the device to a high value that is probably not acceptable.
We looked next at switching energy Es as a function of wavelength. The principle here is that Es is proportional to the fundamental mode volume of the donut. In order to attain single-TE0-mode operation over 1.3 < λ < 5.0 μm, we scaled r 0, r 1, r 2, and h in proportion to λ. This implies that the resonant-mode volume of our donut is scaling as λ 3. For that reason, we estimate—taking the best-case 1.55 μm result Es = 8.5 fJ as a reference—that Es = 5.0 fJ at λ = 1.3 μm (2.5 fJ/bit), with Es increasing to 146 fJ at λ = 4 μm. Clearly, we expect a low-energy technology at 1.3 μm.
We define the minimum switching energy Es,min will as the smallest Es that gives 6 dB of extinction. At a fixed gate-electrode doping, Es,min will depends upon the choice of Np, Von, and tox. As seen in Fig. 11 of their paper, Anderson and Fauchet [1] varied Np and Von at a fixed tox to minimize Es. They then repeated the procedure at other values of tox, thereby constructing a plot of Es,min vs tox. By analogy to their Fig. 11, we anticipate that our 1.55 μm Es,min will be less than 10 fJ over a range of tox centered at 200 nm.
8. Discussion and conclusion
One goal of silicon photonics today is to find a fast, compact, low-energy electro-optical switching-and-modulation technology for a single photonic layer of silicon (the top layer of an SOI chip). What is wanted is a technology that will enable active optical network-on-chip (NOC) applications using densely packed, interconnected components for medium-scale and large-scale integration. One candidate switching platform is the 2D silicon photonic-crystal (PhC) SOI membrane; another is the SOI nanowire waveguide platform. The PhC point-defect micro-resonators (L3, etc) offer some of the smallest possible mode volumes in silicon photonics, but the PhC technology has not yet proven practical for NOCs. By contrast, nanowires side-coupled to silicon micro-donut or micro-disk resonators appear significantly more viable in NOCs, although the micro-donut mode volume is larger than that of the L3 point defect. Thus the enhanced nanowire practicality comes at the expense of increased switching energies, but those energies are still very small as described here.
We have proposed and analyzed a novel HfO2-gated microdonut MOS-depletion electro-optical modulator with a p-doped poly Si gate electrode and a p-doped Si resonator body side-coupled through an evanescent-wave air-gap to one or two Si nanowire bus waveguides. At λ = 1.55 μm, useful depths of optical intensity modulation at the bus output are predicted and 2 × 2 electro-optical switching looks equally feasible. The modulator is anticipated to work well over the 1.3 to 4.0 μm wavelength range.
Our simulations have identified an optimum set of device parameters for the TE0 mode in a 6-μm-diam Si donut having a 8 × 1016 cm−3 p-type doping of the waveguide body, 230 nm waveguide height, 200 nm gate dielectric thickness, and 200 nm gate electrode thickness comprised of poly-Si doped p-type at 1 × 1019 cm−3. Electromagnetic mode modeling shows an unloaded Q of 23,120 with the side-loaded Q taken to be 11,560 giving a 0.135 nm FWHM linewidth. Then, with 2.2 volts applied to the gate, about half of the waveguide height is depleted of holes, resulting in a mode/charge overlap factor of 0.35 and a change in mode effective index of 1.6 × 10−4 which shifts the initial resonance by 0.097 nm, thereby yielding a modulation extinction ratio on the bus of at least 6 dB. The on- and off-state capacitances of the device are 3.6 and 6.9 fF, respectively, and the switching energy is estimated to be 8.5 fJ or 4.3 fJ/bit with a modulation response time of about 1 ps as limited by the RC time constant. For a modulator scaled to operate at the 1.3 μm wavelength, the switching energy is estimated to be 2.5 fJ/bit.
A final comment concerns alternative resonators for SOI modulator devices. Very recently, a new high-Q inline SOI waveguide resonator known as the silicon nanobeam has been reported [9–11]. This straight strip-waveguide cavity contains a 1D photonic crystal lattice that tapers in-and-out of the resonant region without a point defect. The nanobeam is equivalent in many respects to the circular resonators discussed in this paper, except that this cavity is inside the input/output waveguide instead of being external to it. To create an ultracompact one-waveguide MOS depletion modulator, we believe that the results obtained here can be applied immediately to an HfO2-gated nanobeam. Specifically, if the nanobeam has cross-section dimensions of 230 nm × 400 nm for TE0 guiding at λ = 1.55 μm, then the same p-dopings and the same gate dielectric-and-electrode layer thicknesses derived here for the micro-donut would be used in the nanobeam over an 8 μm active length. Our MOS structure also applies to the inline Fabry Perot cavity in Fig. 1(a) of [12].
Acknowledgments
This work was supported in part by the Air Force Office of Scientific Research, Dr. Gernot Pomrenke, Program Manager, under grant FA9550-10-1-0417.
References and links
1. S. P. Anderson and P. M. Fauchet, “Ultra-low power modulators using MOS depletion in a high-Q SiO₂-clad silicon 2-D photonic crystal resonator,” Opt. Express 18(18), 19129–19140 (2010). [CrossRef] [PubMed]
2. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
3. Z. Xia, A. A. Eftekhar, M. Soltani, B. Momeni, Q. Li, M. Chamanzar, S. Yegnanarayanan, and A. Adibi, “High resolution on-chip spectroscopy based on miniaturized microdonut resonators,” Opt. Express 19(13), 12356–12364 (2011). [CrossRef] [PubMed]
4. W. A. Zortman, M. R. Watts, D. C. Trotter, R. W. Young, and A. L. Lentine, “Low-power high-speed silicon microdisk modulators,” paper CThJ4, Conference on Lasers and Electro-Optics, San Jose, CA (2010).
5. M. R. Watts, D. C. Trotter, and R. W. Young, “Maximally confined high-speed second-order silicon microdisk switches,” paper PDP14, OFC/NFOEC, San Diego, CA (2008).
6. S. J. Emelett and R. A. Soref, “Analysis of dual-microring-resonator cross-connect switches and modulators,” Opt. Express 13(20), 7840–7853 (2005). [CrossRef] [PubMed]
7. M. Jerman, Z. Qiao, and D. Mergel, “Refractive index of thin films of SiO2, ZrO2, and HfO2 as a function of the films’ mass density,” Appl. Opt. 44(15), 3006–3012 (2005). [CrossRef] [PubMed]
8. M. Nedeljkovic, R. Soref, and G. Mashanovich, “Free-carrier electro-refraction and electro-absorption modulation predictions for silicon over the 1 – 14 μm infrared range,” submitted to Opt. Mater. Express (June 2011).
9. J. Hendrickson, R. Gibson, M. Gehl, J. D. Olitzky, S. Zandbergen, H. M. Gibbs, G. Khitrova, T. Alasaarela, A. Saynatjoki, S. Honkanen, A. Homyk, and A. Scherer, “One-dimensional photonic crystal nanobeam cavities,” Chapter in Quantum Optics with Semiconductor Nanostructures, F. Jahnke ed., (Woodhead Publishing), to be published (2011).
10. H.-C. Liu, C. Santis, and A. Yariv, “Coupled-resonator optical waveguide (CROWs) based on grating resonators with modulated bandgap,” paper SLTuB2 in OSA Advanced Photonics Conference, Slow Light, Toronto, Canada (12–15 June 2011).
11. B. Cluzel, K. Foubert, L. Lalouat, E. Picard, J. Dellinger, D. Peyrade, F. de Fornel, and E. Hadji, “Optical field molding within near-field coupled twinned nanobeam cavities,” paper IWB3 in OSA Advanced Photonics Conference, Integrated Photonics Research, Silicon, and Nanophotonics, Toronto, Canada (12–15 June 2011).
12. B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, “Compact electro-optic modulator on silicon-on-insulator substrates using cavities with ultra-small modal volumes,” Opt. Express 15(6), 3140–3148 (2007). [CrossRef] [PubMed]
