1. Introduction

“Gap plasmon” waveguides emerged a decade ago [1] referring to surface plasmon polariton (SPP) propagation in a metal-insulator-metal channel waveguide–also called conductor-gap-conductor or CGC waveguide. Two metal strips were separated by a nano-scale gap filled or partly filled with dielectric, giving strong SPP localization in the direction perpendicular to the gap. Later, the definition of “gap plasmon” became broadened to include asymmetric structures of conductor-gap-dielectric (CGD) in which only one metal strip is present and where the “gap” G is a low-index dielectric layer, while “dielectric” D is a higher-index intrinsic semiconductor such as Si or Ge [2]. In CGD, the TM-polarized SPP propagates at the boundary between the conductor and the low-index dielectric [2]. This SPP mode guidance does not employ internal reflection at the low/high dielectric interface. The literature reports several examples of strip-shaped metal-gap-silicon CGD waveguides [3–5]. As early as 2007, a symmetric five-layer slab waveguide, similar in a generic way to the strip waveguides presented in this paper, was introduced by Adato and Guo [6] for the purpose of ultra-long-range SPP propagation. Their 5-layer had metal buried at mid-plane and their stack is characterized exactly as DGCGD where the long-range mode is symmetric with respect to the metal. There are two equivalent ways to describe the 5-layer SPP waveguide. One way is talk about the two plasmonic gaps that are present in the structure, and the other way is to describe the “metalized photonic slot” in the structure. There are two metal/low-dielectric interfaces in their 5-layer and hence two gaps, each having a 1.48 index [6]. The metal was an Au or Ag or Cu film with 20 nm minimum thickness. Dispersion curves and SPP mode cutoff conditions were presented there. The Si-based dual-gap waveguide was updated and optimized in the paper by Bian and Gong [7] who examined both horizontal and vertical geometries at λ = 1.55 μm. In lieu of gap terminology, they chose to describe the structures as a hybridized photonic slot –specifically a low-loss “silicon nano-slot-based symmetric hybrid plasmonic waveguide.” This hybrid category [8] has become an important research and development area [9,10].

The silicon photonic slot waveguide was discovered about a decade ago [11]. That structure has a low-index nano-scale dielectric slot between Si strips, a slot containing air or SiO₂ or polymer. The TM mode polarization was required for modal field enhancement in the slot, just as required in the present paper. In fact, we have modified this slot-in-Si. We have chosen to insert a thin metal ribbon in the mid-plane of this slot. The TM_o mode is a hybrid combination of plasmonic and photonic guidance.

For the 1.55-μm hybrid channel waveguides of [7], propagation lengths of ~1 mm were predicted for certain choices of “half slot” thickness and metal ribbon thickness. In this paper, we have made significant extensions of the prior work by: (1) investigating wavelengths of operation well into the mid infrared, from 1.55 μm out to 8 μm, (2) changing the waveguide body layers to germanium after analyzing silicon, (3) changing the SiO₂ gap material to silicon nitride which has a wider MIR transparency range, and (4) studying partial cladding of the waveguide core by air and comparing that performance to the results of all-around nitride cladding. In the prior Bian-Gong work [7] it was found that the thickness of the metal ribbon must be extremely small for good results on propagation length. That choice was carried through in this paper. We selected a copper ribbon thickness of 10 nm which is the minimum value that preserves electrical conductivity and avoids holes in the metal film. Copper is said to be compatible with CMOS processing, although there are some issues with Cu diffusion, and aluminum is perhaps a less controversial choice. The undoped intrinsic semiconductor body layers are termed “dielectrics” in the language of plasmonics, although N and P doping is surely feasible to transform the waveguide into an active electro-optical device.

This paper is organized as follows. Using the finite-element-modeling based software COMSOL^TM as the electro-magnetic mode solver, the fundamental TM-mode of Si and Ge nano-wire-ribbon waveguides is investigated over 1.55-8.0 μm. A systematic study of gap thickness and wire cross-section was performed and some unexpectedly long propagation lengths of several centimeters were found at mid infrared (MIR), unprecedented in the prior plasmonic art. Semiconductor “wires” of width W and overall thickness H were studied. Specifically, we predicted well-confined low-loss wires with cross-section W x H around 0.7λ/n x 0.7λ/n, with most mode energy in a λ²/400 area. By shrinking W and H further, very low loss was attained at the expense of considerable mode fringing into the cladding. As mentioned, the present mode is believed to be a new kind of hybrid mode [8] related to the photonic mode that propagates in a low-index nano-slot placed within an intrinsic semiconductor strip. The present mode combines metal-related surface-plasmon-polariton (SPP) guiding with photonic-slot concentration. The present hybrid slot/SPP waveguide (HSSPP) had lowest propagation loss when it was clad all-around by Si₃N₄, indicating that an “embedded wire” approach is best. Thus a “nano-circuit” network would be embedded. Examination of the Ge wire case indicated that the propagation lengths are “long” by SPP standards but are less than those in hybrid Si wires.

The significance of the present work is its examination and optimization of Si-based, CMOS-compatible, manufacturable, symmetric 5-layer Group IV channel waveguides (Si or Ge and Si₃N₄) with very long propagation lengths in the 1.55-8.0 μm wavelength range together with strong mode confinement. These waveguides are amenable to electro-modulation.

2. Results of numerical simulations on hybrid slot/SPP waveguides

Figures 1(a) and 1(b) illustrate perspective views of the silicon-based infrared channel waveguide: the air-clad silicon-on-nitride channel and the channel embedded completely in Si₃N₄. Both are expected to be compatible with CMOS processing. Figure 2 shows a cross-section view of the two structures investigated here, where the thickness of the “buried” metal ribbon is fixed at 10 nm, regardless of the wavelength chosen, meaning that this film thickness goes from λ/155 to λ/800 as the wavelength increases from 1.55 to 8.0 μm. There are two Si₃N₄ layers or “gaps” of thickness t that surround the Cu film. If we were to remove the metal, we would have a t + t photonic “slot”. There are two equivalent ways to describe the Fig. 2 waveguide: (1) a buried-ribbon dual-gap plasmonic waveguide, (2) a hybrid slot/surface-plasmon-polariton (HSSPP). The HSSPP descriptor is selected here.

 

Fig. 1 Two approaches to infrared channel waveguides built upon a “silicon-on-nitride” chip: left drawing (a) has the Si channel clad by air above the nitride layer; right drawing (b) has the channel embedded completely or clad-all-around by Si₃N₄. Here green represents silicon and yellow denotes Si₃N₄.

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Fig. 2 Cross-section view of the hybrid slot/surface-plasmon-polariton (HSSPP) waveguide built within a Si channel (structure at left) and in a Ge channel (structure at right). The thickness of the buried metal ribbon is fixed at 10 nm. There are two Si₃N₄ layers or “gaps” of thickness t that surround the Cu film. We can think of t as the “half slot” thickness.

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In the following simulations, Si₃N₄ is used for the gap layers, for the Si substrate “cover” and for the cladding. Its refractive index is constant at 1.98 over the wavelength range. We assume that the refractive index of the polycrystalline form is the same as that of the crystalline form for both Si and Ge. The refractive indices for crystalline silicon and germanium channel structures—including the λ-dispersion—are given in [12]. For the metal layer, the index of Cu can be expressed as$n_{Cu}=n_{Cu}^r+i{n}_{Cu}^i$ and both the real and the imaginary parts can be found in [13].

2.1 Effect of waveguide cladding materials and location

A realistic approach to understanding the effect of wavelength change is to scale both dimensions W and H with λ: in particular, we took from the silicon telecoms practice W = 0.4λ and H = 0.2λ for initial simulations. Then the TM_o effective index n + i k of the Si hybrid channel was determined at λ = 3 μm for the following four cases: (1) the Si₃N₄/Si substrate of Fig. 1(a) with air cladding, (2) the all-around Si₃N₄ cladding of Fig. 1(b), (3) air cladding all around, and (4) SiO₂ cladding all around. The results for cases (1) and (2) are presented in the simulation results of Fig. 3(a) and Fig. 3(b), respectively, for t = 20 nm and 10-nm Cu. Summarizing our four results, we found in four cases the following effective real indexes: (1) 2.27, (2) 2.47 (3) 2.04, and (4) 2.22. The imaginary effective index (the k extinction) was also determined and this can be expressed as the propagation loss in dB/cm as 4.34x10⁴ (4π/λ)k as follows for our four cases: (1)1062, (2)29.0, (3)44.6, and (4)37.5 dB/cm. A rather dramatic difference was found. The explanation for this comparative loss response is traced to the geometric symmetry of the waveguide cross-section structure. The symmetric structures of cases (2), (3) and (4) ensure low loss [6], whereas the “broken symmetry” of case (1) leads to high loss. In subsequent simulations, because of favorable symmetry, we limited ourselves to the Fig. 3(b) scenario. The metal thickness is 10 nm throughout the paper.

 

Fig. 3 Electric field distributions for HSSPP modes in Si channels with (a) air cladding and (b) all-around Si₃N₄ cladding. W x H = 0.4λ x 0.2λ, with λ = 3 μm, t = 20 nm, and Cu = 10 nm. The complex effective index is listed at the top. Field strength is shown in false color; the width dimension is x, and the height dimension is y.

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2.2 Relevant physics background for FEM modeling

We can set the stage for simulations by examining some waveguide physics that applies here. Waveguide physics results for the five-layer HSSPP slab system have been presented [6] and offer us guidelines for attaining long-range SPP propagation—the main features that we should “expect in advance” for the channel case. First, the waveguide layering must be symmetric in the vertical direction. Second, the half-slot dimension t must be on the nano scale. Third, there is an upper limit on t that must be avoided because at that dielectric height the mode “loses its attachment” to the metal. Specifically, when the TM mode attenuation is plotted against the “gap” thickness t (at a given gap index), then there is a definite critical thickness t_cut defined as the t-value at which the long -range SPP mode becomes cutoff. The interesting feature (see Fig. 3 of [6]) is that the attenuation becomes dramatically weaker (and the 1/e-intensity propagation length L becomes much longer) as t approaches t_cut. This behavior is seen clearly in our simulations as documented in section 2.4 below. What happens physically with increasing t is that the mode volume expands along y with a greater portion of its energy in the dielectric, producing lower loss. A second aspect of the cutoff thickness is its dependence upon the index contrast between the high- and low-index regions (between semiconductor and insulator) with [6] showing that the larger that index difference, the smaller is t_cut, and thus with a fixed t, the attenuation will be smaller due to smaller t_cut - t. We have verified the importance of this physics guideline in section 2.5 below.

Another relevant-physics aspect of the hybrid mode is related to the metal because mode loss is strongly linked to the metal. There are two aspects: the mode penetration into the metal and the wavelength dependence of the metal’s permittivity ε₁(λ) + i ε₂(λ). Plasmonic theory shows generally that mode loss is proportional to ε₂/(ε₁)². The physical ε₁(λ) and ε₂(λ) properties of Cu in the mid infrared predict a decreasing loss with increasing wavelength, and this is what we find. To be specific, we calculated the dispersion of mode loss for the structure with W = 0.4λ and H = 0.2λ at fixed t = 20nm and we found that the attenuation decreased from 0.02 dB/μm at 1.55 μm to 0.001 dB/μm at 8 μm. Therefore the known metal-linked dispersion relation supports our HSSPP mode. Tight confinement of light at/near the metal also produces here the expected penetration-linked effect as detailed below. The next sections of this paper examine the effect of changing the waveguide’s cross-section size, the effect of changing slot width t, the effect of changing the slot material, the effect of substituting Ge for Si, and the effect of size on mode profile.

2.3 Effects of W and H upon L

Keeping t again constant at 20 nm, we examined the effect of widening the channel W when keeping H fixed at a particular value. Both W and H were expressed in terms of λ. Results for L at λ = 3μm and λ = 6μm are presented in Fig. 4 for the silicon case. As shown in Fig. 4, the propagation length drops as W increases because the propagation loss is proportional to the width of metal “exposed” to the light field. Also, the L-decrease slows down with the increase of W because the modal field width will not increase linearly with the increasing waveguide width. A somewhat surprising Fig. 4 result is that the predicted L became several centimeters when both W and H were chosen in the sub-wavelength range of 0.1 to 0.2λ. The interpretation of these results is given in section 2.7 below. The next simulation was to hold W fixed and to vary H for the same devices, and the results are shown in Fig. 5 for both the λ = 3μm and λ = 6μm situations. Once again, the same dramatic increase of L when H moves below 0.2λ is found, with L rising into multiple-cm lengths.

 

Fig. 4 L versus W of the Si channel with different fixed heights at the wavelength of: (a) 3μm, (b) 6μm using t = 20nm.

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Fig. 5 L versus H for the Si channel with different fixed widths at (a) λ = 3μm, (b) λ = 6μm using t = 20nm.

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2.4 Influence of the gap-layer thickness upon L

We studied this t-dependence by choosing the 3 μm wavelength for the Si channel with W = 0.10λ and H = 0.15λ. The results are shown in Fig. 6 where it is found that an increased gap distance produces lower propagation loss. The propagation length increases 12x at t = 50 nm. As discussed in section 2.2, the increasing t in Fig. 6 brings t close to t_cut; consequently the mode loss decreases with t increase. However, at the larger t values in Fig. 6, the mode becomes easier to cut off for a given size of channel.

 

Fig. 6 L versus t in a silicon HSSPP structure with W = 0.1λ and H = 0.15λ at λ = 3 μm.

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2.5 Influence of the gap material’s refractive index upon L

Silicon-on-insulator, silicon-on-sapphire and silicon-on-nitride are three “mainstream” MIR photonic waveguides with relevance to plasmonics. That’s why we investigated the use of SiO₂, Al₂O₃ and Si₃N₄ as gap materials in the HSSPP structure. Again using λ = 3 μm as a test case, we took for Si the gap t = 30 nm with W = 0.1λ and H = 0.2λ, and we compared the resulting propagation length in the three cases. As shown in Fig. 7, we found that the lower the gap refractive index, the larger the L. The physical explanation for this result is related to the greater “bunching of light” that happens because the enhancement of the mode’s electric field within the photonic slot material is proportional to (n_s/n_g)² where n_s is the semiconductor index, and n_g the gap index.

 

Fig. 7 L versus λ for Si HSSPP structures having different gap materials: SiO₂, Al₂O₃ and Si₃N₄. We used λ = 3 μm, t = 30 nm, with W = 0.1λ and H = 0.2λ.

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2.6 Simulations of the germanium HSSPP waveguide

In order to connect the predicted behavior of a germanium HSSPP channel with the previously derived silicon results, we selected geometric conditions for Ge that were the same as those used in Figs. 4 and 5 above, namely: λ = 3 μm, t = 20 nm, with Si₃N₄ for both gaps and cladding. First H was fixed and the variation of W was looked at as shown in Fig. 8(a). Next W was fixed with H variable as presented in Fig. 8(b). For those cases, the calculated Ge propagation lengths are presented. As in silicon, when W is taken below 0.2λ, approaching 0.1λ in Fig. 8(a), and when H goes below 0.2λ towards 0.1λ in Fig. 8(b), then L rises evidently, even into the centimeter range. (The high-L interpretation of Fig. 8 is also offered in section 2.7.) Generally, the silicon L is about 50% larger than the germanium L. Because of Ge’s higher index, the semiconductor dielectric trapping is stronger in the Ge case of Fig. 8 than in the Fig. 3(b) Si case. The Ge slot has a greater percentage of the mode, which explains the lower-than-Si L.

 

Fig. 8 L versus (a) W of the Ge channel with different fixed H; (b) H of the Ge channel with different W at λ = 3μm with t = 20nm.

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2.7 Mode profile as a function of the W, H geometry

In this section, we interpret the long range HSSPP results revealed in Figs. 4, 5, and 8, and we highlight the tradeoff that long-range operation requires. Figures 4, 5, and 8 also show results of HSSPP propagation at millimeter lengths and those waveguide geometries of W and H are believed to be practical; they do not require much tradeoff. We begin by looking at the mode profile illustrated above in Fig. 3(b), and we consider the y direction. The y-distributed mode found in Fig. 3(b) is an eigen-solution of Maxwell’s equations with an associated eigenvalue (the mode effective index). It is not the superposition of three distinct modes. It is one mode with light “accumulated” in several regions. The mode has energy in the upper nitride cladding, the upper semiconductor strip, the double-gap, the lower semiconductor strip and the lower nitride cladding. When L is greater than 1 cm in Figs. 4, 5, and 8, we want to know what percentage of the mode power is present in each of the five regions. To answer the question, we performed several simulations, two of which are presented below.

A qualitative answer to the question is that the mode expands into the nitride cladding, when W and/or H approaches 0.10λ, which means the mode power in the double-gap is relatively small. This signals that the mode is approaching cutoff in the cm-L cases. For applications, it is usually not good to have the optical field fringing into the cladding and going outside of the semiconductor strips. For that reason, the cm-propagation may have limited utility. However, the mm-L structures discussed are found to be well-confined in the semiconductor channel like the result seen in Fig. 3(b). Figure 9(a) presents the mode profile under the same conditions as for Fig. 3(b) except that the height has been reduced from 0.2λ to 0.l25λ and the width was reduced from 0.4 λ to 0.1λ. Here we see that about 60% of the mode energy has been “forced” into the upper and lower “low loss” nitride claddings, with a resulting L of 4 cm. There is also some trapping in the upper and lower Si strips, but less energy resides there than in the claddings and double-gap area. While the propagation loss is very low in Fig. 9(a), most of the “fringy” mode resides away from the plasmonic region. In other words, the large L arrives at the expense of a large mode. Looking now at Figs. 4, 5, and 8, if we select W and/or H to be in the vicinity of 0.2λ rather than 0.1λ, then L is typically millimeters and here the mode has very little cladding component as in Fig. 3(b). Therefore, we judge those ~0.7λ/n choices of channel geometry to be practical. Generally L increases strongly with λ in Figs. 4, 5, 7, and 8.

 

Fig. 9 Mode distribution in Si channel at λ = 3μm, t = 20 nm with: (a) W = 0.1λ and H = 0.125λ; (b) W = 0.05λ and H = 0.8λ. In both cases, the five layers are: poly-Si, Si₃N₄, Cu, Si₃N₄ and Si.

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It is interesting to learn what happens in the high-aspect cases where H >> W. To illustrate this, we examined a “blade-shaped” channel waveguide, the Si channel with W = 0.05λ and H = 0.8λ at λ = 3μm having t = 20nm. The resulting mode is presented in Fig. 9(b). Once again, as in Fig. 9(a), the mode in Fig. 9(b) has expanded considerably into the surrounding nitride cladding, but now going in the lateral x-direction, and that mode enlargement is linked to the large L of 1.5 cm. The lateral fringing seen here may have value in the evanescent-wave side coupling of adjacent blade waveguides for directional coupling.

2.8 Effect of ribbon width in a wide gap

A question not answered thus far is whether the hybrid mode clings closely to the metal ribbon, or instead fills both gaps irrespective of ribbon size. To illuminate this issue, we returned to the Si waveguide at the 3μm wavelength, and fixed the W at 0.4λ with H at 0.2λ and t = 20 nm. The TM_o mode profile was derived for two cases of interest: one in which the ribbon width filled only 50% of the waveguide width, and the other with ribbon width of 0.25 W. These mode modeling results are given in Fig. 10 and can be contrasted with Fig. 3(b). It is found that the majority of mode energy does cling to the ribbon. However, there is noticeable leakage or fringing of light outside the ribbon “domain” and therefore an adjacent ribbon within the same gaps would experience some evanescent-wave side coupling.

 

Fig. 10 Electric field distributions for HSSPP modes in Si channels having different widths of the Cu ribbon: (a) 0.5W ribbon (b) 0.25W ribbon. Here W = 0.4λ, H = 0.2λ, t = 20 nm and λ = 3μm. The 10-nm ribbon is embedded in a 50-nm Si₃N₄ slot.

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3. Discussion and summary

The plasmonic literature indicates that the mid-infrared region is a favorable spectrum for plasmonics because generally the SPP waveguide losses decrease there with increasing wavelength. The work in this paper confirms this theme and indicates a strong, consistent increase in HSSPP propagation length as wavelength increases from 1.55 to 8 μm while the waveguide cross-section is also scaled up. For the hybrid slot/SPP TM_o mode, the 1/e mode-power propagation length L rises up into the multi-centimeter range when the Si or Ge channel width and/or the channel height is reduced down to the 0.3 to 0.4 λ/n range, where n is the semiconductor refractive index. This projected performance is really based upon the assumption of an ultra-thin metal film or “ribbon” buried within a low-index “gap” material of Si₃N₄ or Al₂O₃ or SiO₂, and it is found that the lower the gap index, the larger are the values of L attained. The metal thickness was set at 10 nm as the lowest feasible value for electrical conduction (continuity) and void-free integrity of that film. It was also essential for the HSSPP channel to be clad all around by a rather thick, multi-micron coating of dielectric such as Si₃N₄. When an air cladding covered the top three surfaces instead of this dielectric, the propagation length became significantly shorter. Each of two gaps had a thickness in the 20 nm range, and an increase of gap thickness to 50 nm gave longer HSSPP propagation lengths. The estimated values of L for Ge were slightly smaller than those for Si. If the width of the channel waveguide is set at a value of λ/n, for example, it is possible to reduce the width of the buried ribbon to λ/4n and thereby to have a majority of the mode energy cling to the localized ribbon, while the minority of energy resides in the metalless portion. In other words the ribbon is a local light-wire in the overall gap, although there exists infrared leakage around the ribbon that side-couples the narrow ribbon to nearby narrow ribbons.

The TM_o mode in Figs. 4, 5 and 8 closely approaches cutoff when the W, H dimensions of the Si or Ge waveguide are shrunk down to around 0.10 to 0.15 λ. The resulting expansion of the mode into the surrounding low-loss dielectric cladding thereby enables the propagation length to increase into the multi-cm domain. However, that mode size violates the “spirit” of mode confinement within the semiconductor channel. Hence the “smallest” waveguides may not be a good choice. Fortunately, there is a good compromise available. By selecting somewhat larger Si or Ge strip dimensions, W and/or H with values of 0.20 to 0.25 λ, the channel is expected to have good confinement (low fringing) as well as propagation at millimeter lengths. The W/H ratio influences the mode propagation loss and the ratio 0.5 has lower loss than the 2.0 ratio because the mode interacts with a smaller area of metal in the larger-H case. However, the aspect ratio alone does not determine reduction of the mode loss. That loss depends upon a combination of W and H values.

There are many future possibilities for these CMOS-compatible Si-based waveguides, for example, the creation of waveguided infrared resonant structures such as microrings and microdisks. With losses in the 5 dB/cm range, the Q value of such resonators could be fairly high and the bend radius could be on the wavelength scale. Regarding nonlinear optics, although the Si and Ge strips would have higher 3rd order nonlinear optical response than that of the lower-index gap material in which the light is concentrated, some significant four-wave mixing [14] could nevertheless be attained in the HSSPP by choosing the gaps to be As₂S₃ for example, with n_g around 2.4.

There are several ways in which to make this 5-layer channel into an active infrared device. The electrical path defined by the ribbon could, when connected to an electrical current supply, be used as a resistance heater to attain the thermo-optic effect (phase modulation) within the channel. Alternatively, the metal ribbon could be allowed to “float” electrically (no electrical connections to it), and then each of the two semiconductor strips would be doped, either P type or N type, to serve as electrical contacts for voltage to be applied to the channel waveguide. One electro-optical (EO) approach is to utilize an n ~1.8 organic polymer layer for each of the two gap layers. Then dc-poling would provide a strong second-order Pockels EO effect [15] in the double gap. For that device, an applied RF electric field across the two silicon strips would induce an additive and useful refractive index change in the two gaps; that is, EO phase modulation. In addition to this field effect, free-carrier electro-modulation appears feasible. The first case is charge accumulation or depletion in doped Si or doped Ge strips. The 5-layer is functionally back-to-back capacitors: SOM plus MOS. For the case of N and N doping, during bias a combination of electron accumulation and electron depletion in each Si at the Si/gap interface would happen during bias voltage applied to N and N contacts. For P and N doping, hole and electron accumulation in the top and bottom silicons should occur. Another possibility is to employ a doped transparent conductor—for example, indium tin oxide (ITO) — in each of two gap layers. The unbiased ITO index of 1.96 is appropriate for the slot, but the ITO complex index is very sensitive to charge injection [16]. The speculation here is that charge would accumulate during voltage application in the ITO gaps as well as in the Si and that could induce EO loss modulation in the channel.