## Abstract

Metal-insulator-silicon-insulator-metal (MISIM) waveguides are proposed and investigated theoretically. They are hybrid plasmonic waveguides, and light is highly confined to the insulator between the metal and silicon. As compared to previous ones, they are advantageous since they may be realized in a simple way by using current standard CMOS technology and their insulator is easily replaceable without affecting the metal and silicon. First, their structure and fabrication process are explained, both of which are compatible with standard CMOS technology. Then, the characteristics of the single MISIM waveguide whose insulator has its original or an adjusted refractive index are analyzed. The analysis demonstrates that its characteristics are comparable to those of previous hybrid plasmonic waveguides and that they are very effectively tuned by changing the refractive index of the insulator. Finally, the characteristics of the two coupled MISIM waveguides are analyzed. Through the analysis, it is obtained how close or far apart they are for efficient power transfer or low crosstalk. MISIM-waveguide-based devices may play an important role in connecting Si-based photonic and electronic circuits.

© 2011 OSA

## 1. Introduction

Plasmonics has been developing rapidly for a recent decade since it is expected to become a bridge between size-limited photonics and speed-limited electronics [1]. As a key element of plasmonics, diverse plasmonic waveguides have been reported [2–8]. They include channel plasmon polariton waveguides [2], dielectric-loaded surface plasmon polariton waveguides [3–5], and metal slot or metal-insulator-metal waveguides [6–8]. The first two types of waveguides have a propagation length of tens of micrometers, but have a mode area of the order of 1 μm^{2}. In contrast, the metal slot waveguides have a propagation length of a few micrometers, but have a mode area smaller than 0.01 μm^{2}. Because of the Ohmic loss of metal, all the plasmonic waveguides have a limitation on providing both a long propagation length and a small mode area. In order to alleviate such a limitation, active research on hybrid plasmonic waveguides has been carried out recently [9–16]. Hybrid plasmonic waveguides are based on the principle that electric fields are highly enhanced in thin insulator with a low refractive index (RI) between metal and dielectric with a high RI. It has been shown that their propagation length and mode area can be made quite long and small simultaneously.

Most of the previous hybrid plasmonic waveguides have metal-insulator-semiconductor (MIS) structures [9–14]. For CMOS-compatibility and connection to silicon photonic waveguides, most of the MIS waveguides have the structure in which rectangular metal, insulator, and silicon lines with the same width are vertically stacked [10,12–14]. There are two things that need to be considered in the CMOS-compatible MIS waveguides: realization of them and tuning of their guiding characteristics. For the realization, a step of aligning the metal line with the insulator and silicon lines is required as reported in [12]. However, it has an inevitable alignment error that makes the realized MIS waveguides deviate from the ideal structure. For the tuning, the RI of the silicon line may be adjusted by using free carrier injection or depletion, but it requires an additional p-i-n structure. Instead of this, the RI of the insulator line may be adjusted to tune effectively the guiding characteristics since light is strongly confined to it. However, it is usually made of SiO_{2} so that it is not easy to change its RI. Moreover, after fabrication, it is impossible to replace it with a functional material since it is covered by and supports the metal or silicon line. Therefore, these two aspects of the CMOS-compatible MIS waveguides need to be improved.

This paper reports a metal-insulator-silicon-insulator-metal (MISIM) waveguide that is a different hybrid plasmonic waveguide. (Actually, the acronym of MISIM was introduced in [17], where S stood for semiconductor like InGaAs.) The structure of the MISIM waveguide is devised (1) to be realized in a simple way by using current standard CMOS fabrication technology and (2) to have a post-fabrication method of replacing its insulator with a functional material without influencing the structure of its metal and silicon. The dimensions of the MISIM waveguides are determined considering the limitations of standard CMOS technology. The designed MISIM waveguides are theoretically investigated. In Section 2, the structure and fabrication process of the MISIM waveguides are proposed and explained. The fabrication process is carried out with standard CMOS fabrication tools, and so it enables mass production of MISIM waveguide devices. For the metal of the MISIM waveguides, not only gold and silver but also copper and aluminum are considered since the latter are used in standard CMOS fabrication. In Section 3, the characteristics of the single MISIM waveguide are theoretically investigated. Also, changes of the effective index of its mode are handled, which are induced by respective changes of the real and imaginary parts of the RI of its insulator. Section 4 presents theoretical investigation of the coupled MISIM waveguides. Changes in the coupling of the two identical MISIM waveguides are analyzed with the distance between them varied. Finally, concluding remarks are given in Section 5.

## 2. Structure and fabrication process

The MISIM waveguide is based on a conventional silicon-on-insulator (SOI) wafer that is used for silicon photonics. The silicon, with a thickness *h _{S}*, of an SOI wafer is patterned into a rectangular line with a width

*w*. The Si line and the SiO

_{S}_{2}substrate are conformally covered by thin insulator with a thickness

*t*. On both sides of the Si line covered by the insulator, two rectangular metal lines with a width

_{I}*w*and a height

_{M}*h*are placed. The schematic diagram of its cross-section is shown in Fig. 1(a) . (Actually, this structure is somewhat similar to those in [15,16].) Using wet etching, we may remove easily its inverted U shaped insulator covering the Si line, and we obtain slots as shown in Fig. 1(b). These slots may be filled with a functional material. An example of such a functional material is electro-optic (EO) polymer with a very large EO coefficient [18,19]. If EO polymer fills the slots, the effective index of a MISIM waveguide mode may be effectively changed. Even a much larger change may be achieved if the slots are filled with liquid crystal. In these cases, the two metal lines not only play an important role in guiding a MISIM waveguide mode but also apply the electric field that controls the EO effect or the orientation of liquid crystal. In addition, if the slots are filled with a gain medium like polymer doped with quantum dots [20], the propagation length of a MISIM waveguide mode may be increased.

_{S}Its fabrication process is schematically shown in Figs. 1(c) to 1(g). First, the silicon is patterned by using 193-nm optical lithography and dry etching [21,22]. At this step, Si patterns for Si photonic waveguides and the MISIM waveguides are made. As shown in Fig. 1(c), for connection between the Si photonic waveguide and the MISIM waveguide, an about 450-nm-wide Si line for the former is tapered to have a width *w _{S}* for the latter. Then, silicon dioxide for the insulator of the MISIM waveguide is conformally deposited on the Si patterns by using a low pressure chemical vapor deposition (LPCVD) process with tetraethyl orthosilicate (TEOS) [23]. For the SiO

_{2}deposition, LPCVD is superior to thermal oxidation since the former is carried out at a lower temperature than the latter. Before deposition of metal on the silicon dioxide, silicon nitride (Si

_{3}N

_{4}) patterns are formed, as shown in Fig. 1(e), by using film deposition, optical lithography, and dry etching. At this step, the Si

_{3}N

_{4}patterns need to be aligned with the Si patterns. As inferred from Fig. 1(e), an alignment error along the

*x*-axis does not cause a realized MISIM waveguide to deviate from its ideal structure since the width of the opened area of the Si

_{3}N

_{4}patterns is much larger than

*w*. An alignment error along the

_{S}*z*-axis may affect the transformation of a Si photonic waveguide mode into a MISIM waveguide mode, but this may not be significant. This is because the alignment error expected from the 193-nm optical lithography is quite smaller than the length of the tapered region. The purposes of the Si

_{3}N

_{4}patterns are twofold. One purpose is to prevent the Si photonic waveguides from being covered by metal. If metal covers them, their propagation loss increases, which is undesirable. The other purpose is that the Si

_{3}N

_{4}patterns function as a mold for metal patterns. In other words, they are necessary for Damascene technology. After the formation of the Si

_{3}N

_{4}patterns, as shown in Fig. 1(f), metal is deposited to fill the empty space bounded by them. Finally, with chemical-mechanical polishing (CMP) employed, the surplus metal is removed until the top surfaces of the Si

_{3}N

_{4}patterns and the silicon dioxide deposited on the Si patterns are exposed as shown in Fig. 1(g). If the metal is copper, Cu CMP can be carried out without difficulty since it is an essential step for Cu interconnect formation in present industrial CMOS fabrication. Although CMP is not as commonly applied to gold, silver, and aluminum as it is applied to copper, CMP of them is also possible [24–26]. Figure 1(h) shows the cross-section of the finished MISIM waveguide in Fig. 1(g).

For the following theoretical investigation, it is assumed that the thickness of the deposited SiO_{2} film is equal to *t _{I}* on the SiO

_{2}substrate and on the top of the Si patterns. This assumption is reasonable since the LPCVD with TEOS gives very high conformality. In addition, the RI of the deposited SiO

_{2}film is assumed to be equal to that of the SiO

_{2}substrate. With regard to the dimensions of the Si line,

*h*is set to 250 nm since the Si thickness of a usual SOI wafer is around 250 nm. In the case of

_{S}*w*, the narrower the Si line is, the smaller the mode area of the MISIM waveguide is. Since the minimum line width that can be achieved from the 193-nm optical lithography is about 100 nm [21,22],

_{S}*w*is set to 100 nm. With regard to

_{S}*w*, it is assumed that the metal lines are so wide that the Si

_{M}_{3}N

_{4}patterns bounding them do not affect the characteristics of the MISIM waveguide. In other words, the structure in Fig. 1(a) is mainly analyzed rather than the one in Fig. 1(h). Table 1 summarizes the dielectric constants

*ε*

_{Au},

*ε*

_{Ag1},

*ε*

_{Cu}, and

*ε*

_{Al}of gold, silver, copper, and aluminum. They are obtained from Palik’s handbook [27]. However, Johnson and Christy’s paper [28] gives a quite different value for the dielectric constant of silver, which is denoted by

*ε*

_{Ag2}in Table 1. While the value of

*ε*

_{Ag1}was used in [13], that of

*ε*

_{Ag2}was used in [9–11,15,16]. Although the latter was used more frequently, it was reported that the former made simulation closer to experiment [29]. Therefore, both

*ε*

_{Ag1}and

*ε*

_{Ag2}are used in the following theoretical investigation of the MISIM waveguides. Finally, the wavelength

*λ*is set to 1.55 μm, and only the quasi-TE polarization is considered. This is because a hybrid plasmonic mode exists when its major electric field component is normal to the interface between its metal and insulator and the interface between its insulator and dielectric like silicon.

## 3. Characteristics of the single MISIM waveguide

The full-vectorial finite-element method provided by the commercial software FIMMWAVE was used to analyze the characteristics of the MISIM waveguides. In this section, the characteristics of the single MISIM waveguide are explained. Its insulator has the RI of SiO_{2} or the RI that is adjusted assuming the replacement of the insulator with EO polymer or a gain medium. In addition, two considerations related to a realized MISIM waveguide are discussed: the influence of a diffusion barrier for copper and that of a finite value of *w _{M}*.

#### 3.1 Insulator with the RI of SiO_{2}

The intensity profile, effective index, effective mode area, and normalized power of a MISIM waveguide mode were calculated when the RI of the insulator, *n _{I}* is equal to the RI of SiO

_{2}, which is 1.444. For every value of

*t*between 10 and 100 nm, the MISIM waveguide supports only one quasi-TE mode regardless of types of metal. Figure 2 shows the intensity (i.e. the magnitude of the Poynting vector ${P}_{z}\text{(}r\text{)}$) profiles of the MISIM waveguide mode for four different values of

_{I}*t*. In this calculation, the dielectric constant of the metal of the MISIM waveguide,

_{I}*ε*was set to

_{M}*ε*

_{Au}. As shown in Figs. 2(a) to 2(d), a large portion of the power carried by the MISIM waveguide mode is confined to the thin insulator layers between the metal lines and the Si line. The intensity in the insulator layers and the Si line decreases as

*t*increases, and it is almost inversely proportional to

_{I}*t*. This can be confirmed from Fig. 2(e), which shows the intensity distributions along a horizontal line

_{I}*y*= 125 nm. Figure 2(f) shows the intensity distribution along another horizontal line

*y*=

*t*/ 2. The larger the value of

_{I}*t*is, the further the intensity profile spreads over the insulator below the metal lines.

_{I}Figure 3
shows the relations of the effective index *n*
_{eff} of the MISIM waveguide mode to *t _{I}* for different values of

*ε*. As

_{M}*t*increases, the real part of

_{I}*n*

_{eff}, $\mathrm{Re}[{n}_{\text{eff}}]$ decreases, and the imaginary part of

*n*

_{eff}, $\mathrm{Im}[{n}_{\text{eff}}]$ increases. This is because the intensity in the metal lines decreases as

*t*increases, as shown in the inset of Fig. 2(e). Figure 3(b) also shows the propagation length

_{I}*L*of the MISIM waveguide mode, which is defined as the distance at which its field amplitude decays to 1/

_{p}*e*(i.e. ${L}_{p}=\lambda /(-2\pi \mathrm{Im}[{n}_{\text{eff}}])$). When

*ε*=

_{M}*ε*

_{Ag2},

*L*increases from 41 μm ( = 26.5

_{p}*λ*) to 136 μm ( = 87.7

*λ*) as

*t*increases from 10 nm to 100 nm.

_{I}*L*for

_{p}*ε*=

_{M}*ε*

_{Ag1},

*ε*

_{Au},

*ε*

_{Al}, and

*ε*

_{Cu}is on average, respectively, 19.7, 18.7, 18.3, and 13.8% of

*L*for

_{p}*ε*=

_{M}*ε*

_{Ag2}. The dependence of

*n*

_{eff}on the values of

*ε*is quite similar to that of the effective index

_{M}*n*

_{SPP}of a surface plasmon polariton propagating along the interface between semi-infinite metal with a dielectric constant equal to

*ε*and insulator with an RI equal to

_{M}*n*. It is well known that

_{I}*n*

_{SPP}is calculated by using the expression ${n}_{\text{SPP}}={[{\epsilon}_{M}{n}_{I}^{2}/({\epsilon}_{M}+{n}_{I}^{2})]}^{1/2}$. If

*ε*is equal to

_{M}*ε*

_{Cu},

*ε*

_{Ag1},

*ε*

_{Au},

*ε*

_{Ag2}, and

*ε*

_{Al},

*n*

_{SPP}is 1.463 –

*j*0.00265, 1.462 –

*j*0.00180, 1.460 –

*j*0.00186, 1.456 –

*j*0.000303, and 1.450 –

*j*0.00123, respectively. The magnitude orders of $\mathrm{Re}[{n}_{\text{SPP}}]$ and $\mathrm{Im}[{n}_{\text{SPP}}]$ depending on the values of

*ε*are almost the same as those of $\mathrm{Re}[{n}_{\text{eff}}]$ and $\mathrm{Im}[{n}_{\text{eff}}]$.

_{M}The effective mode area *A*
_{eff} of the MISIM waveguide mode was calculated by using the expression [30], where $W(\text{r})$ is the energy density given by Eq. (3) in [30]. The calculated relations of *A*
_{eff} to *t _{I}* are shown in Fig. 4(a)
. The real area

*A*of the thin insulator region denoted by

_{R}*R*in the inset of Fig. 4(b) is also shown as a function of

*t*, which is given by $2{t}_{I}({t}_{I}+{h}_{S})$. The ratio of

_{I}*A*to

_{R}*A*

_{eff}is between 0.6 and 0.7 for all the values of

*t*and

_{I}*ε*. This indicates that the region where the energy density of the MISIM waveguide mode becomes significant quite coincides with

_{M}*R*. As shown in Fig. 4(a),

*A*

_{eff}is smaller than the diffraction-limited area of silicon, ${(\lambda /2/{n}_{\text{Si}})}^{2}$, where

*n*

_{Si}is the RI of silicon, if

*t*< 55 nm. With regard to the power carried by the MISIM waveguide mode, Fig. 2 has shown that a large portion of the power is confined to

_{I}*R*. This confinement is quantitatively confirmed from the normalized power in

*R*, which is defined as ${\int}_{R}{P}_{z}(r)\text{d}A}{\scriptscriptstyle \raisebox{1ex}{$$}\!\left/ \!\raisebox{-1ex}{$$}\right.}{\displaystyle \int {P}_{z}(r)\text{d}A}$. Figure 4(b) shows the relations of the normalized power to

*t*. The normalized power reaches 0.6 around

_{I}*t*= 60 nm. As

_{I}*t*increases up to this value, it increases since the portion of the power, which is carried through the Si line, decreases. However, after this value, it decreases since the portion of the power, which is carried through the insulator below the metal lines, increases. The magnitude orders of the normalized power and

_{I}*A*

_{eff}depending on the values of

*ε*are the same as that of $\mathrm{Re}[{n}_{\text{eff}}]$ and the inverse of this order, respectively. The larger the intensity or energy density is in

_{M}*R*, the larger it is in the metal lines. Consequently, becomes larger as explained regarding Fig. 3.

The characteristics of the MISIM waveguide need to be compared with those of the CMOS-compatible MIS waveguides [10,12–14]. For comparison, the MIS waveguide was analyzed, which is a stack of the silver line (with *ε _{M}* =

*ε*

_{Ag2}), the insulator line with a thickness

*t*, and the 250-nm-high Si line on an SiO

_{I}_{2}substrate. The three lines have the same width as

*h*. When

_{S}*t*= 20 nm,

_{I}*L*of the MISIM waveguide mode is 56% of that of the MIS waveguide mode. This is because the MISIM waveguide mode is influenced by the two metal lines. In contrast, the normalized power in

_{p}*R*of the MISIM waveguide mode is 1.9 times larger than that in the insulator line of the MIS waveguide mode. This is because the geometrical area of the insulator line is slightly smaller than

*A*/ 2. Interestingly,

_{R}*A*

_{eff}of the MISIM waveguide mode is almost the same as that of the MIS waveguide. When the adjustment of the RI of the insulator induces an effective index change, the amount of such a change is almost proportional to the normalized power in the insulator as explained below. Consequently, the MISIM waveguide has about two times more effective tuning of its characteristics than the MIS waveguide at the cost of the decrease of

*L*by half without changing

_{p}*A*

_{eff}.

The results in Figs. 3 and 4 show that the characteristics of the Cu-based or Al-based MISIM waveguide are comparable to those of the Au-based one although *L _{p}* of the Cu-based one or the normalized power in

*R*of the Al-based one is a little smaller. Therefore, instead of gold, copper or aluminum may be used for the MISIM waveguides. In the case of the Ag-based MISIM waveguide, since

*L*changes significantly depending on whether

_{p}*ε*=

_{M}*ε*

_{Ag1}or

*ε*

_{Ag2}, the actual dielectric constant of an Ag film for a realized MISIM waveguide should be checked experimentally.

#### 3.2 Insulator with the adjusted RI

The MISIM waveguide was simulated with the slots in Fig. 1(b) filled with EO polymer or a gain medium. Under the assumption that the inverted U shaped region denoted by *U* in the inset of Fig. 5(a)
has an RI of *n _{I}* + Δ

*n*,

_{I}*n*

_{eff}was calculated as a function of

*t*for a fixed value of Δ

_{I}*n*. Figure 5(a) shows the relations of the change of $\mathrm{Re}[{n}_{\text{eff}}]$, $\Delta \mathrm{Re}[{n}_{\text{eff}}]$ to

_{I}*t*when Δ

_{I}*n*= 0.001. In this case, Δ

_{I}*n*means an electro-optically induced index change. If the EO coefficient of EO polymer is 100 pm/V, that amount of Δ

_{I}*n*is induced by applying ~1 V between the two metal lines. The value of the applied voltage is estimated from a simple capacitor model, which consists of parallel plates sandwiching the insulator-metal-insulator. (Actually, it is not easy to infiltrate EO polymer into a narrow and deep slot and pole it for the EO effect. However, this was successfully done in the case of the 75-nm-wide and 230-nm-deep slot in [19].) Since the power of the MISIM waveguide mode is well confined to

_{I}*R*as shown in Figs. 2 and 4, Δ

*n*effectively affects $\Delta \mathrm{Re}[{n}_{\text{eff}}]$. In other words, $\Delta \mathrm{Re}[{n}_{\text{eff}}]/\Delta {n}_{I}$ is close to 1, and it is even larger than 1 for some values of

_{I}*t*. When the relative change of $\mathrm{Re}[{n}_{\text{eff}}]$, denoted by ${\Delta}_{r}\mathrm{Re}[{n}_{\text{eff}}]$, is defined as $(\Delta \mathrm{Re}[{n}_{\text{eff}}]/\mathrm{Re}[{n}_{\text{eff}}])/(\Delta {n}_{I}/{n}_{I})$, the relations of to

_{I}*t*are shown in Fig. 5(b). Interestingly, these relations are quite similar to those of the normalized power in

_{I}*R*shown in Fig. 4(b). This similarity is approximately explained from the perturbation theory [31] for a planar MISIM waveguide that is uniform along the

*y*-axis. From the perturbation theory, it is obtained that

In this equation, *E _{x}* (

*H*) denotes the

_{y}*x*(

*y*) component of the electric (magnetic) field of a planar MISIM waveguide mode, and

*R*

_{∞}is the region that results from extending

*R*infinitely along the ±

*y*directions. The second approximate equality in Eq. (1) holds since $|\mathrm{Re}[{H}_{y}]|$ >> $|\mathrm{Im}[{H}_{y}]|$ for the planar MISIM waveguide mode so that ${H}_{y}$ ≈${H}_{y}^{*}$ and ${E}_{x}{H}_{y}$ ≈$2{P}_{z}$. The third term of Eq. (1) is the normalized power in

*R*

_{∞}. As shown in Fig. 3, $\mathrm{Re}[{n}_{\text{eff}}]$ >> $|\mathrm{Im}[{n}_{\text{eff}}]|$, and $\Delta \mathrm{Re}[{n}_{\text{eff}}]$ is much larger than the change of $\mathrm{Im}[{n}_{\text{eff}}]$, $\Delta \mathrm{Im}[{n}_{\text{eff}}]$ if Δ

*n*is real. Consequently, the first term of Eq. (1) becomes ${\Delta}_{r}\mathrm{Re}[{n}_{\text{eff}}]$, and it is proportional to the normalized power in

_{I}*R*

_{∞}for the planar MISIM waveguide mode. Since the MISIM waveguide mode approaches the planar MISIM waveguide mode as

*h*increases, Eq. (1) explains the aforementioned similarity to some extent.

_{S}When *U* is filled with a gain medium with a gain coefficient *g*, it is simply assumed that Δ*n _{I}* is given by $jg/(4\pi /\lambda )$. Δ

*n*=

_{I}*j*0.00617 if

*g*is 500 cm

^{–1}, which is somewhat larger than the gain coefficient of the polymer strip-loaded plasmonic waveguide doped with quantum dots in [20]. Figure 6(a) shows the relations of $\Delta \mathrm{Im}[{n}_{\text{eff}}]$ to

*t*in this case. $\Delta \mathrm{Im}[{n}_{\text{eff}}]$ changes with

_{I}*t*in almost the same way as $\Delta \mathrm{Re}[{n}_{\text{eff}}]$. When the relative change of $\mathrm{Im}[{n}_{\text{eff}}]$, denoted by ${\Delta}_{r}\mathrm{Im}[{n}_{\text{eff}}]$, is defined as $(\Delta \mathrm{Im}[{n}_{\text{eff}}]/\mathrm{Re}[{n}_{\text{eff}}])/(|\Delta {n}_{I}|/{n}_{I})$, the relations of ${\Delta}_{r}\mathrm{Im}[{n}_{\text{eff}}]$ to

_{I}*t*are shown in Fig. 6(a), and they also match almost perfectly those of ${\Delta}_{r}\mathrm{Re}[{n}_{\text{eff}}]$ in Fig. 5(b). In Eq. (1), if Δ

_{I}*n*is imaginary, the first term becomes ${\Delta}_{r}\mathrm{Im}[{n}_{\text{eff}}]$ since $\Delta \mathrm{Im}[{n}_{\text{eff}}]$ >> $\Delta \mathrm{Re}[{n}_{\text{eff}}]$. Hence, ${\Delta}_{r}\mathrm{Im}[{n}_{\text{eff}}]$ has almost the same dependence on

_{I}*t*as ${\Delta}_{r}\mathrm{Re}[{n}_{\text{eff}}]$. Since $\Delta \mathrm{Im}[{n}_{\text{eff}}]$ is positive,

_{I}*L*increases, and Fig. 6(b) shows the relations of the increase of

_{p}*L*, Δ

_{p}*L*to

_{p}*t*. The larger $\mathrm{Im}[{n}_{\text{eff}}]$ is, the larger Δ

_{I}*L*is. Except the case of

_{p}*ε*=

_{M}*ε*

_{Cu}, the gain coefficient of 500 cm

^{–1}makes

*L*increase by more than 100% for large values of

_{p}*t*. In the case of

_{I}*ε*=

_{M}*ε*

_{Ag2}, the gain of the gain medium compensates for the loss of the MISIM waveguide mode, and the mode has a gain coefficient reaching 280 cm

^{–1}.

#### 3.3 Two considerations related to realized MISIM waveguides

When copper is used for the metal of the MISIM waveguide, a diffusion barrier needs to be formed on the deposited SiO_{2} layer before the metal deposition in Fig. 1(f) since copper ions diffuse well into SiO_{2} and silicon. As a diffusion barrier, usually, a stack of tantalum nitride and tantalum (TaN/Ta) films is used. Since the TaN/Ta stack is in contact with the insulator to which the power of the MISIM waveguide mode is well confined, it may seriously affect the characteristics of the mode. Assuming that 5-nm-thick TaN and Ta films whose respective RIs are 4.388 – *j*2.822 and 4.865 – *j*5.471 [32] are conformally deposited on the insulator, the MISIM waveguide was analyzed. Because of the TaN/Ta stack, for *t _{I}* = 50 nm, its effective index changes from 1.920 –

*j*0.01988 to 1.914 –

*j*0.2374, and its propagation loss increases from 0.70 dB/μm to 8.4 dB/μm. Since its intensity profile changes slightly, e.g. the normalized power in

*R*decreases just by 0.01, the huge increase in the propagation loss is mainly attributed to the large absorption of the TaN/Ta stack. Therefore, instead of the TaN/Ta stack, a diffusion barrier with low absorption is required. Silicon carbide (SiC) is also used as a diffusion barrier [33], and it may be transparent at a wavelength of 1.55 μm [34]. If a 10-nm-thick SiC film with an RI of 3.096 [34] is used as a diffusion barrier, the MISIM waveguide mode has

*n*

_{eff}= 2.022 –

*j*0.02324, and its propagation loss is 0.82 dB/μm. Consequently, we had better use a SiC film rather than a TaN/Ta stack to realize the MISIM waveguides.

In realizing the MISIM waveguides, it is also necessary to consider the minimum value of *w _{M}*, which is required to maintain the explained characteristics of the MISIM waveguide mode. For this purpose, the structure in Fig. 1(h) was analyzed. The RI of silicon nitride was set to 1.97, and

*ε*was set to

_{M}*ε*

_{Cu}. As explained above, a small portion of the power of the MISIM waveguide mode is carried through the insulator below the metal lines, and the portion increases with

*t*. If

_{I}*w*is small, light in the insulator below the metal lines is coupled to the surface plasmon polaritons that exist along the boundaries between the metal lines and the Si

_{M}_{3}N

_{4}patterns. This coupling increases the mode area and propagation loss of the MISIM waveguide mode, and so

*w*should be large enough to make this coupling negligible. When ${n}_{\text{eff}}^{\text{d}}$ denotes the effective index of the deteriorated MISIM waveguide mode of the structure in Fig. 1(h), ${n}_{\text{eff}}^{\text{d}}$ was calculated with respect to

_{M}*w*for the four values of

_{M}*t*. Figure 7 shows the relations of $\mathrm{Re}[{n}_{\text{eff}}^{\text{d}}-{n}_{\text{eff}}]/\mathrm{Re}[{n}_{\text{eff}}]$ to

_{I}*w*and those of $\mathrm{Im}[{n}_{\text{eff}}^{\text{d}}-{n}_{\text{eff}}]/\mathrm{Im}[{n}_{\text{eff}}]$. The smaller

_{M}*t*is, the faster both $\mathrm{Re}[{n}_{\text{eff}}^{\text{d}}-{n}_{\text{eff}}]/\mathrm{Re}[{n}_{\text{eff}}]$ and $\mathrm{Im}[{n}_{\text{eff}}^{\text{d}}-{n}_{\text{eff}}]/\mathrm{Im}[{n}_{\text{eff}}]$ approach zero, i.e. the faster ${n}_{\text{eff}}^{\text{d}}$ converges to ${n}_{\text{eff}}$. If the minimum value of

_{I}*w*is defined as the value for which both their values become smaller than 0.001, the minimum values for

_{M}*t*= 20, 40, 60, and 80 nm are 290, 400, 510, and 670 nm, respectively.

_{I}## 4. Characteristics of the coupled MISIM waveguides

The analysis of the coupled MISIM waveguides is necessary, on the one hand, since it is necessary to know how closely the two MISIM waveguides should be placed to transfer efficiently the power of the one waveguide to the other. On the other hand, it is necessary to know how far apart they should be for negligible crosstalk between them. The coupled MISIM waveguides were analyzed by using a simple model of beating between the antisymmetric and symmetric modes of two coupled identical waveguides with loss [35]. For using the model, first, calculated were the effective indexes *N _{a}* and

*N*of the antisymmetric and symmetric modes of the structure in Fig. 8(a) . In the structure, the two MISIM waveguides share a metal line with a width

_{s}*s*between them. For

_{M}*ε*=

_{M}*ε*

_{Au}and

*t*= 40 nm, the real and imaginary parts of

_{I}*N*and

_{a}*N*are shown as functions of

_{s}*s*in Figs. 8(b) and 8(c), respectively. As

_{M}*s*increases,

_{M}*N*and

_{a}*N*approach

_{s}*n*

_{eff}. The upper (lower) insets of Fig. 8(b) show the distributions of the real part of the

*x*component of the electric field ${\text{E}}^{(a)}$ (${E}^{(s)}$) of the antisymmetric (symmetric) mode along the line

*y*= 125 nm for

*s*= 60 and 300 nm, respectively. In contrast to a coupling between two dielectric waveguides, $\mathrm{Re}[{N}_{a}]$ > $\mathrm{Re}[{N}_{s}]$.The reason for this is deduced from a coupling between two surface plasmon polaritons along the two sides of a metal film. It is well known that the coupling makes the metal film support antisymmetric and symmetric modes. Their effective indexes have the same relations as

_{M}*N*and

_{a}*N*.

_{s}By using *N _{a}* and

*N*, a coupling length

_{s}*L*was calculated. It is the distance at which the power of the left MISIM waveguide is maximally transferred to the right MISIM waveguide.

_{c}*P*

_{max}is defined as the maximally transferred power at

*L*, which is normalized with respect to the power that the left MISIM waveguide carries at the start of the structure in Fig. 8(a).

_{c}*L*and

_{c}*P*

_{max}are expressed by

*L*is a beating length given by $\lambda /2/\mathrm{Re}[{N}_{a}-{N}_{s}]$ and ${\overline{L}}_{p}$ is a sort of average propagation length given by $\lambda /\pi /\mathrm{Im}[-{N}_{a}-{N}_{s}]$ [35].

_{b}The relations of *L _{b}* to

*s*are shown in Fig. 9 . As shown in Fig. 8(b), with

_{M}*s*increasing, $\mathrm{Re}[{N}_{a}]$ decreases, $\mathrm{Re}[{N}_{s}]$ increases, and they approach asymptotically $\mathrm{Re}[{n}_{\text{eff}}]$. Thus

_{M}*L*increases almost exponentially with

_{b}*s*. Actually,

_{M}*L*is directly related to the degree of the coupling between the two MISIM waveguides: the stronger the coupling is, the smaller

_{b}*L*is. The coupled MISIM waveguides have two kinds of coupling: a coupling through the shared metal line and a coupling through the insulator below this metal line. The two couplings are affected by the intensity of the MISIM waveguide in

_{b}*R*. As it increases, the MISIM waveguide carries a larger portion of its power through the metal lines but a smaller portion of its power through the insulator below the metal lines. Consequently, with its intensity in

*R*increasing, the first coupling becomes strong, but the second coupling becomes weak. Figure 9(a), where

*ε*=

_{M}*ε*

_{Au}, shows that

*L*increases with

_{b}*t*when

_{I}*s*is small. This means that the first coupling is dominant for small

_{M}*s*since the intensity in

_{M}*R*decreases with

*t*increasing so that it becomes weak. In contrast, for large

_{I}*s*, the second coupling is dominant so that

_{M}*L*decreases with

_{b}*t*increasing. Figure 9(b), where

_{I}*t*= 40 nm, shows that the magnitude order of

_{I}*L*depending on the values of

_{b}*ε*is the inverse of that of $\mathrm{Re}[{n}_{\text{eff}}]$. This is because the intensity in

_{M}*R*increases with $\mathrm{Re}[{n}_{\text{eff}}]$ so that the first coupling becomes strong.

While *L _{b}* increases almost exponentially with

*s*, ${\overline{L}}_{p}$increases but approach

_{M}*L*, which is easily expected from its definition and Fig. 8(c). Since and $|\mathrm{Im}[{N}_{a}]|$ are much smaller than $\mathrm{Re}[{N}_{a}]$ and $\mathrm{Re}[{N}_{s}]$,

_{p}*L*is smaller than ${\overline{L}}_{p}$ for small

_{b}*s*. However,

_{M}*L*becomes much larger than ${\overline{L}}_{p}$ for large

_{b}*s*. This relation between

_{M}*L*and ${\overline{L}}_{p}$ makes

_{b}*L*bounded by

_{c}*L*for small

_{b}*s*and bounded by ${\overline{L}}_{p}$ for large

_{M}*s*, as deduced from Eq. (2). For sufficiently large

_{M}*s*,

_{M}*L*≈${\overline{L}}_{p}$ ≈

_{c}*L*<<

_{p}*L*, which means that the beating between the antisymmetric and symmetric modes is suppressed by the propagation loss of the MISIM waveguide mode.

_{b}Figures 10(a)
to 10(e) show the relations of *L _{c}* and

*P*

_{max}to

*s*for the four values of

_{M}*t*, respectively, for

_{I}*ε*=

_{M}*ε*

_{Au},

*ε*

_{Ag1},

*ε*

_{Al},

*ε*

_{Cu}, and

*ε*

_{Ag2}.

*L*and

_{c}*P*

_{max}for

*ε*=

_{M}*ε*

_{Au},

*ε*

_{Ag1},

*ε*

_{Al}, and

*ε*

_{Cu}change quite similarly with

*s*or

_{M}*t*, and there are no large differences between the values of

_{I}*L*and

_{c}*P*

_{max}for the different values of

*ε*. However, the relations of

_{M}*L*and

_{c}*P*

_{max}to

*s*for

_{M}*ε*=

_{M}*ε*

_{Ag2}are quite different from those in Figs. 10(a) to 10(d) since the MISIM waveguide mode for

*ε*=

_{M}*ε*

_{Ag2}has much smaller propagation loss than those for the other values of

*ε*. The explained change of

_{M}*L*with respect to

_{c}*s*is confirmed from Figs. 10(a) to 10(e). In Figs. 10(a) to 10(d), for small

_{M}*s*, although

_{M}*L*is bounded by

_{c}*L*,

_{b}*L*does not change exactly in the same way as

_{c}*L*shown in Fig. 9(a) (i.e.

_{b}*L*increases simply with

_{c}*t*regardless of

_{I}*s*) since ${\overline{L}}_{p}$ is not much larger than

_{M}*L*. However, if

_{b}*ε*=

_{M}*ε*

_{Ag2}, for small

*s*,

_{M}*L*changes almost in the same way as

_{c}*L*shown in Fig. 9(a) since ${\overline{L}}_{p}$ is much larger than

_{b}*L*in this case. With regard to

_{b}*P*

_{max}, it decreases as

*s*increases, and it decreases almost exponentially for large

_{M}*s*. The exponential decrease of

_{M}*P*

_{max}for large

*s*is explained from Eq. (3). If

_{M}*s*is sufficiently large,

_{M}*ϕ*≈$\pi {L}_{p}/2/{L}_{b}$ << 1, and so

*P*

_{max}is approximately given by $\mathrm{exp}(-2){(\pi {L}_{p}/2/{L}_{b})}^{2}$, which decreases almost exponentially. However,

*P*

_{max}increases with

*t*since the propagation loss of the MISIM waveguide mode decreases so that ${\overline{L}}_{p}$ increases with

_{I}*t*.

_{I}For the efficient power transfer between the two MISIM waveguides, *s _{M}* should be as small as possible. In Figs. 10(a) to 10(e), the smallest value of

*s*is 60 nm, and Fig. 10(f) shows

_{M}*P*

_{max}in the case of

*s*= 60 nm as functions of

_{M}*t*for the different values of

_{I}*ε*. The best power transfer is obtained when

_{M}*ε*=

_{M}*ε*

_{Ag2}. As

*t*increases from 20 nm to 80 nm,

_{I}*P*

_{max}increases from 0.86 to 0.89, and

*L*increases from 3.8 μm to 5.3 μm. The worst power transfer is obtained when

_{c}*ε*=

_{M}*ε*

_{Al}. As

*t*increases from 20 nm to 80 nm,

_{I}*P*

_{max}increases from 0.37 to 0.48, and

*L*increases from 4.2 μm to 6.3 μm. If

_{c}*s*is reduced below 60 nm,

_{M}*P*

_{max}increases a little, and

*L*decreases. However, the reduction of

_{c}*s*is limited by the following two factors. The one is that the distance between the two silicon lines in Fig. 8(a), which is

_{M}*s*+ 2

_{M}*t*, should be larger than the minimum line width of the 193-nm optical lithography. (If

_{I}*t*= 20 nm and the minimum line width is 100 nm,

_{I}*s*= 60 nm.) The other is whether the narrow channel for the shared metal line is well filled with metal.

_{M}For the low crosstalk between the two MISIM waveguides, *s _{M}* should be sufficiently large. The minimum value of

*s*, which is large enough for the low crosstalk, is determined as follows. First, the crosstalk is quantitatively defined as the ratio of

_{M}*P*

_{max}to the normalized power carried by the left MISIM waveguide at a distance of

*L*, which is given by $\mathrm{exp}(-2\varphi \mathrm{cot}\varphi ){\mathrm{cos}}^{2}\varphi $. Then, the crosstalk is given by $\mathrm{exp}(2){P}_{\mathrm{max}}$ for sufficiently large

_{c}*s*. If the desired value of the crosstalk is 0.01,

_{M}*P*

_{max}becomes 0.00135 for the minimum value of

*s*. This value is indicated as the dashed lines in Figs. 10(a) to 10(e), and the curves of

_{M}*P*

_{max}cross these dashed lines at the minimum values of

*s*. Figure 10(g) shows the minimum values of

_{M}*s*as functions of

_{M}*t*for the different values of

_{I}*ε*. On the one hand, the minimum value of

_{M}*s*is smallest for

_{M}*ε*=

_{M}*ε*

_{Cu}. It increases from 420 nm to 880 nm as

*t*increases from 20 nm to 80 nm. Actually, it is comparable to the distance between two coupled metal-insulator-metal (MIM) waveguides, for which the crosstalk between them is also 0.01. For example, in the case of the coupled Cu-based MIM waveguides (i.e. a 250-nm-thick Cu film with two 40-nm-wide slots, which is surround by SiO

_{I}_{2}), the crosstalk of 0.01 is obtained when the width of the metal line between the two slots is 430 nm. On the other hand, the minimum value of

*s*is largest for

_{M}*ε*=

_{M}*ε*

_{Ag2}. It increases from 660 nm to 1410 nm as

*t*increases from 20 nm to 80 nm. Interestingly, it is much smaller than the distance for the crosstalk of 0.01 between two coupled nanowire-based MIS waveguides, each of which has a very small mode area [36]. Reference [36] handled the case where two Si nanowires with the same diameter of 300 nm, each of which is surrounded by a 12-nm-thick SiO

_{I}_{2}shell, are placed on an Ag substrate whose dielectric constant is equal to

*ε*

_{Ag2}. Although the edge-to-edge distance between the SiO

_{2}–surrounded Si nanowires was 636 nm, the crosstalk was 0.74.

Lastly, there is one thing that is required to discuss with regard to the simple beating model. If ${\mathbf{\text{E}}}^{(l)}$ and ${\mathbf{\text{E}}}^{(r)}$denote the electric fields of the left and right MISIM waveguide modes, Eqs. (2) and (3) are derived under the assumption that ${\mathbf{E}}^{(l)}$ = ${\mathbf{\text{E}}}^{(s)}/\sqrt{2}$ + ${\mathbf{\text{E}}}^{(a)}/\sqrt{2}$ and ${E}^{(r)}$ = ${\mathbf{\text{E}}}^{(s)}/\sqrt{2}$ – ${\mathbf{\text{E}}}^{(a)}/\sqrt{2}$. However, as checked from the insets of Fig. 8(b), the assumption is not satisfied completely for small *s _{M}*. Therefore, the results based on Eqs. (2) and (3) may have some error for small

*s*, and it may be necessary to polish them by using a numerical method.

_{M}## 5. Conclusions

This paper has proposed and investigated theoretically the MISIM waveguides that are hybrid plasmonic waveguides with the two advantages: the simple realizability based on standard CMOS technology and the possession of the post-fabrication method of replacing the insulator with a functional material. First, their structure and fabrication process, both of which are compatible with standard CMOS technology, have been explained. Then, the following characteristics of the single MISIM waveguide have been analyzed: the effective index, propagation length, effective mode area, and normalized power of the MISIM waveguide mode. In the analysis, gold, silver, aluminum, and copper have been considered for its metal. It has been explained how its characteristics change depending on which metal is employed. In addition to this analysis, it has been demonstrated that the respective changes of the real and imaginary parts of the RI of its insulator tune the effective index very effectively. Moreover, with regard to a realized MISIM waveguide, it has been discussed that a diffusion barrier with low absorption should be used and that its metal lines should have a width larger than the determined value. Finally, the two coupled MISIM waveguides have been analyzed. On the one hand, when *ε _{M}* =

*ε*

_{Ag2},

*t*= 20 nm, and

_{I}*s*= 60 nm, 86% of the power carried by the one waveguide is transferred to the other at a distance of 3.8 μm. On the other hand, when

_{M}*ε*=

_{M}*ε*

_{Cu},

*t*= 20 nm, the crosstalk between them becomes very low for

_{I}*s*> 420 nm. The MISIM waveguides are expected to play an important role in connecting Si-based photonic and electronic circuits.

_{M}## Acknowledgments

This research was supported by the Basic Science Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022473).

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