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 μm². In contrast, the metal slot waveguides have a propagation length of a few micrometers, but have a mode area smaller than 0.01 μm². 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 SiO₂ 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_S. The Si line and the SiO₂ substrate are conformally covered by thin insulator with a thickness t_I. On both sides of the Si line covered by the insulator, two rectangular metal lines with a width w_M and a height h_S 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.

 

Fig. 1 (a) Cross-sectional structure of the MISIM waveguide. This structure is mainly analyzed in Section 3. (b) Cross-sectional structure of the MISIM waveguide whose insulator is partially removed by using wet-etching. The fabrication process of the MISIM waveguide consists of (c) formation of silicon patterns, (d) deposition of oxide, (e) formation of Si₃N₄ patterns, (f) deposition of metal, and (g) chemical-mechanical polishing. The cross-section along the line AB is shown in (h).

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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 SiO₂ 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 (Si₃N₄) patterns are formed, as shown in Fig. 1(e), by using film deposition, optical lithography, and dry etching. At this step, the Si₃N₄ 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 Si₃N₄ patterns is much larger than w_S. An alignment error along the 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 Si₃N₄ 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 Si₃N₄ patterns function as a mold for metal patterns. In other words, they are necessary for Damascene technology. After the formation of the Si₃N₄ 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 Si₃N₄ 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 SiO₂ film is equal to t_I on the SiO₂ 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 SiO₂ film is assumed to be equal to that of the SiO₂ substrate. With regard to the dimensions of the Si line, h_S is set to 250 nm since the Si thickness of a usual SOI wafer is around 250 nm. In the case of w_S, 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], w_S is set to 100 nm. With regard to w_M, it is assumed that the metal lines are so wide that the Si₃N₄ 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.

Table 1. Dielectric Constants of Gold, Silver, Copper, and Aluminum at a Wavelength of 1.55 μm

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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 SiO₂ 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 SiO₂

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 SiO₂, which is 1.444. For every value of t_I 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 t_I. In this calculation, the dielectric constant of the metal of the MISIM waveguide, ε_M was set to 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_I increases, and it is almost inversely proportional to t_I. This can be confirmed from Fig. 2(e), which shows the intensity distributions along a horizontal line y = 125 nm. Figure 2(f) shows the intensity distribution along another horizontal line y = t_I / 2. The larger the value of t_I is, the further the intensity profile spreads over the insulator below the metal lines.

 

Fig. 2 Intensity profiles of the MISIM waveguide mode for ε_M = ε _Au. For t_I = 20, 40, 60, and 80 nm, they are shown in (a) to (d), respectively. (e) Intensity distributions along a horizontal line y = 125 nm for the different values of t_I. An enlarged part is shown in the inset. (f) Intensity distributions along another horizontal line y = t_I / 2 for the different values of t_I. In (e) and (f), the black, red, green, and cyan lines correspond to the cases of t_I = 20, 40, 60, and 80 nm, respectively. The correspondence between the line colors and the values of t_I is used in the following figures.

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Figure 3 shows the relations of the effective index n _eff of the MISIM waveguide mode to t_I for different values of ε_M. As t_I increases, the real part of 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_I increases, as shown in the inset of Fig. 2(e). Figure 3(b) also shows the propagation length L_p of the MISIM waveguide mode, which is defined as the distance at which its field amplitude decays to 1/e (i.e. $L_p=\lambda /(-2\pi \mathrm{Im}[n_{\text{eff}}])$). When ε_M = ε _Ag2, L_p increases from 41 μm ( = 26.5λ) to 136 μm ( = 87.7λ) as t_I increases from 10 nm to 100 nm. L_p for 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_p for ε_M = ε _Ag2. The dependence of n _eff on the values of ε_M is quite similar to that of the effective index n _SPP of a surface plasmon polariton propagating along the interface between semi-infinite metal with a dielectric constant equal to ε_M and insulator with an RI equal to n_I. It is well known that 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_M is equal to ε _Cu, ε _Ag1, ε _Au, ε _Ag2, and ε _Al, n _SPP is 1.463 – j0.00265, 1.462 – j0.00180, 1.460 – j0.00186, 1.456 – j0.000303, and 1.450 – j0.00123, respectively. The magnitude orders of $\mathrm{Re}[n_{\text{SPP}}]$ and $\mathrm{Im}[n_{\text{SPP}}]$ depending on the values of ε_M are almost the same as those of $\mathrm{Re}[n_{\text{eff}}]$ and $\mathrm{Im}[n_{\text{eff}}]$.

 

Fig. 3 Relations of the effective index n _eff to t_I for different values of ε_M. The real and imaginary parts of n _eff, $\mathrm{Re}[n_{\text{eff}}]$ and $\mathrm{Im}[n_{\text{eff}}]$ are shown in (a) and (b), respectively. (b) also shows the relations of the propagation length L_p to t_I, which are represented by the dashed lines. In the inset of (a), an enlarged part is shown. The square, circle, triangle, inverted triangle, and diamond symbols correspond to the cases of ε_M = ε _Au, ε _Ag1, ε _Ag2, ε _Al, and ε _Cu, respectively. This correspondence between the symbol shapes and the values of ε_M is used in the following figures.

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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_R of the thin insulator region denoted by R in the inset of Fig. 4(b) is also shown as a function of t_I, which is given by $2t_I(t_I+h_S)$. The ratio of A_R to A _eff is between 0.6 and 0.7 for all the values of t_I and ε_M. This indicates that the region where the energy density of the MISIM waveguide mode becomes significant quite coincides with 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_I < 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 R. This confinement is quantitatively confirmed from the normalized power in R, which is defined as ${\displaystyle {\int }_R{P}_z(r)\text{d}A}\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_I. The normalized power reaches 0.6 around t_I = 60 nm. As t_I 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 A _eff depending on the values of ε_M 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 R, the larger it is in the metal lines. Consequently, becomes larger as explained regarding Fig. 3.

 

Fig. 4 (a) Relations of the effective mode area A _eff to t_I for the different values of ε_M. An enlarged part is shown in the inset. The dotted curve represents the real area A_R of the thin insulator region R. In addition, the thin horizontal line represents the diffraction-limited area of silicon. (b) Relations of the normalized power in R to t_I for the different values of ε_M. An enlarged part is shown in the upper middle inset. The lower right inset shows the definition of the thin insulator region R.

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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_I, and the 250-nm-high Si line on an SiO₂ substrate. The three lines have the same width as h_S. When t_I = 20 nm, L_p 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 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_R / 2. Interestingly, 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_p by half without changing 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_p changes significantly depending on whether 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 + Δn_I, n _eff was calculated as a function of t_I for a fixed value of Δn_I. Figure 5(a) shows the relations of the change of $\mathrm{Re}[n_{\text{eff}}]$, $\Delta \mathrm{Re}[n_{\text{eff}}]$ to t_I when Δn_I = 0.001. In this case, Δn_I means an electro-optically induced index change. If the EO coefficient of EO polymer is 100 pm/V, that amount of Δn_I 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 R as shown in Figs. 2 and 4, Δn_I 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 t_I. 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 t_I are shown in Fig. 5(b). Interestingly, these relations are quite similar to those of the normalized power in 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

 

Fig. 5 (a) Relations of the change of $\mathrm{Re}[n_{\text{eff}}]$, $\Delta \mathrm{Re}[n_{\text{eff}}]$ to t_I for the different values of ε_M. The dotted line represents the case of $\Delta \mathrm{Re}[n_{\text{eff}}]$ = 0.001. The inset shows the inverted U shaped sub-region of the insulator, U, whose RI is n_I + Δn_I. In this figure, Δn_I = 0.001. (b) Relations of ${\Delta }_r\mathrm{Re}[n_{\text{eff}}]$ to t_I for the different values of ε_M. ${\Delta }_r\mathrm{Re}[n_{\text{eff}}]$ is the relative change of $\Delta \mathrm{Re}[n_{\text{eff}}]$, which is given by $(\Delta \mathrm{Re}[n_{\text{eff}}]/\mathrm{Re}[n_{\text{eff}}])/(\Delta n_I/n_I)$.

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In this equation, E_x (H_y) denotes the 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 $E_x{H}_y$ ≈$2P_z$. 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 Δn_I 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 R _∞ for the planar MISIM waveguide mode. Since the MISIM waveguide mode approaches the planar MISIM waveguide mode as h_S increases, Eq. (1) explains the aforementioned similarity to some extent.

When U is filled with a gain medium with a gain coefficient g, it is simply assumed that Δn_I is given by $jg/(4\pi /\lambda )$. Δn_I = j0.00617 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_I in this case. $\Delta \mathrm{Im}[n_{\text{eff}}]$ changes with t_I 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 t_I 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 Δn_I 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 t_I as ${\Delta }_r\mathrm{Re}[n_{\text{eff}}]$. Since $\Delta \mathrm{Im}[n_{\text{eff}}]$ is positive, L_p increases, and Fig. 6(b) shows the relations of the increase of L_p, ΔL_p to t_I. The larger $\mathrm{Im}[n_{\text{eff}}]$ is, the larger ΔL_p is. Except the case of ε_M = ε _Cu, the gain coefficient of 500 cm^–1 makes L_p increase by more than 100% for large values of t_I. In the case of 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.

 

Fig. 6 (a) Relations of the change of $\mathrm{Im}[n_{\text{eff}}]$, $\Delta \mathrm{Im}[n_{\text{eff}}]$ to t_I for the different values of ε_M when Δn_I = j0.00617, which means that the gain coefficient of U is 500 cm^–1. The dotted line represents the case of $\Delta \mathrm{Im}[n_{\text{eff}}]$ = 0.00617. Also, the relations of ${\Delta }_r\mathrm{Im}[n_{\text{eff}}]$ to t_I are shown together. ${\Delta }_r\mathrm{Im}[n_{\text{eff}}]$ is the relative change of $\Delta \mathrm{Im}[n_{\text{eff}}]$, which is given by $(\Delta \mathrm{Im}[n_{\text{eff}}]/\mathrm{Re}[n_{\text{eff}}])/$ $(|\Delta n_I|/n_I)$. (b) Relations of the increase of L_p, ΔL_p to t_I. Also, the gain coefficient of the MISIM waveguide mode for ε_M = ε _Ag2 is shown as a function of t_I.

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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 SiO₂ layer before the metal deposition in Fig. 1(f) since copper ions diffuse well into SiO₂ 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 – j2.822 and 4.865 – j5.471 [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 – j0.01988 to 1.914 – j0.2374, 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 – j0.02324, 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_M was set to 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_I. If w_M 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 Si₃N₄ patterns. This coupling increases the mode area and propagation loss of the MISIM waveguide mode, and so w_M 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 w_M for the four values of t_I. Figure 7 shows the relations of $\mathrm{Re}[n_{\text{eff}}^{\text{d}}-n_{\text{eff}}]/\mathrm{Re}[n_{\text{eff}}]$ to w_M and those of $\mathrm{Im}[n_{\text{eff}}^{\text{d}}-n_{\text{eff}}]/\mathrm{Im}[n_{\text{eff}}]$. The smaller t_I 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 w_M is defined as the value for which both their values become smaller than 0.001, the minimum values for t_I = 20, 40, 60, and 80 nm are 290, 400, 510, and 670 nm, respectively.

 

Fig. 7 (a) Relations of $\mathrm{Re}[n_{\text{eff}}^{\text{d}}-n_{\text{eff}}]/\mathrm{Re}[n_{\text{eff}}]$ to w_M for the different values of t_I. $n_{\text{eff}}^{\text{d}}$ denotes the effective index of the deteriorated MISIM waveguide mode of the structure in Fig. 1(h). (b) Relations of $\mathrm{Im}[n_{\text{eff}}^{\text{d}}-n_{\text{eff}}]/\mathrm{Im}[n_{\text{eff}}]$ to w_M for the different values of t_I.

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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_s 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_M between them. For ε_M = ε _Au and t_I = 40 nm, the real and imaginary parts of N_a and N_s are shown as functions of s_M in Figs. 8(b) and 8(c), respectively. As s_M increases, N_a and N_s approach 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_M = 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 N_a and N_s.

 

Fig. 8 (a) Cross-sectional structure of the coupled MISIM waveguides. (b) Relations of $\mathrm{Re}[N_a]$ (black) and $\mathrm{Re}[N_s]$ (red) to s_M. N_a and N_s are the effective indexes of the antisymmetric and symmetric modes of the structure in (a). The upper (lower) insets show the distributions of the real part of the x component of the electric field of the antisymmetric (symmetric) mode, $\mathrm{Re}[E_x^{(a)}]$ ($\mathrm{Re}[E_x^{(s)}]$) along the line y = 125 nm for s_M = 60 and 300 nm, respectively. (c) Relations of $\mathrm{Im}[N_a]$ (black) and $\mathrm{Im}[N_s]$ (red) to s_M. In the calculation for this figure, ε_M = ε _Au, and t_I = 40 nm.

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By using N_a and N_s, a coupling length L_c was calculated. It is the distance at which the power of the left MISIM waveguide is maximally transferred to the right MISIM waveguide. P _max is defined as the maximally transferred power at L_c, which is normalized with respect to the power that the left MISIM waveguide carries at the start of the structure in Fig. 8(a). L_c and P _max are expressed by

The relations of L_b to s_M are shown in Fig. 9 . As shown in Fig. 8(b), with s_M increasing, $\mathrm{Re}[N_a]$ decreases, $\mathrm{Re}[N_s]$ increases, and they approach asymptotically $\mathrm{Re}[n_{\text{eff}}]$. Thus L_b increases almost exponentially with s_M. Actually, L_b is directly related to the degree of the coupling between the two MISIM waveguides: the stronger the coupling is, the smaller L_b 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 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_b increases with t_I when s_M is small. This means that the first coupling is dominant for small s_M since the intensity in R decreases with t_I increasing so that it becomes weak. In contrast, for large s_M, the second coupling is dominant so that L_b decreases with t_I increasing. Figure 9(b), where t_I = 40 nm, shows that the magnitude order of L_b depending on the values of ε_M is the inverse of that of $\mathrm{Re}[n_{\text{eff}}]$. This is because the intensity in R increases with $\mathrm{Re}[n_{\text{eff}}]$ so that the first coupling becomes strong.

 

Fig. 9 Relations of the beating length L_b to s_M (a) for the different values of t_I and (b) for the different values of ε_M. In the calculation for (a), ε_M = ε _Au. In the calculation for (b), t_I = 40 nm. Enlarged parts are shown in the insets.

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While L_b increases almost exponentially with s_M, ${\overline{L}}_p$increases but approach L_p, 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 $\mathrm{Re}[N_s]$, L_b is smaller than ${\overline{L}}_p$ for small s_M. However, L_b becomes much larger than ${\overline{L}}_p$ for large s_M. This relation between L_b and ${\overline{L}}_p$ makes L_c bounded by L_b for small s_M and bounded by ${\overline{L}}_p$ for large s_M, as deduced from Eq. (2). For sufficiently large s_M, L_c ≈${\overline{L}}_p$ ≈L_p << L_b, which means that the beating between the antisymmetric and symmetric modes is suppressed by the propagation loss of the MISIM waveguide mode.

Figures 10(a) to 10(e) show the relations of L_c and P _max to s_M for the four values of t_I, respectively, for ε_M = ε _Au, ε _Ag1, ε _Al, ε _Cu, and ε _Ag2. L_c and P _max for ε_M = ε _Au, ε _Ag1, ε _Al, and ε _Cu change quite similarly with s_M or t_I, and there are no large differences between the values of L_c and P _max for the different values of ε_M. However, the relations of L_c and P _max to s_M for 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_M. The explained change of L_c with respect to s_M is confirmed from Figs. 10(a) to 10(e). In Figs. 10(a) to 10(d), for small s_M, although L_c is bounded by L_b, L_c does not change exactly in the same way as L_b shown in Fig. 9(a) (i.e. L_c increases simply with t_I regardless of s_M) since ${\overline{L}}_p$ is not much larger than L_b. However, if ε_M = ε _Ag2, for small s_M, L_c changes almost in the same way as L_b shown in Fig. 9(a) since ${\overline{L}}_p$ is much larger than L_b in this case. With regard to P _max, it decreases as s_M increases, and it decreases almost exponentially for large s_M. The exponential decrease of P _max for large s_M is explained from Eq. (3). If s_M is sufficiently large, ϕ ≈$\pi L_p/2/L_b$ << 1, and so P _max is approximately given by $\exp (-2){(\pi L_p/2/L_b)}^2$, which decreases almost exponentially. However, P _max increases with t_I since the propagation loss of the MISIM waveguide mode decreases so that ${\overline{L}}_p$ increases with t_I.

 

Fig. 10 (a) – (e) Relations of the coupling length L_c and the normalized, maximally-transferred power P _max at L_c to s_M for t_I = 20, 40, 60, and 80 nm. In the calculation, ε_M was set to ε _Au for (a), ε _Ag1 for (b), ε _Al for (c), ε _Cu for (d), and ε _Ag2 for (e). The correspondence between the line colors and the values of t_I is the same as in the above figures. (f) Relations of P _max in the case of s_M = 60 nm to t_I for the different values of ε_M. (g) Relations of the minimum values of s_M to t_I for the different values of ε_M. The correspondence between the symbol shapes and the values of ε_M is the same as in the above figures.

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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_M is 60 nm, and Fig. 10(f) shows P _max in the case of s_M = 60 nm as functions of t_I for the different values of ε_M. The best power transfer is obtained when ε_M = ε _Ag2. As t_I increases from 20 nm to 80 nm, P _max increases from 0.86 to 0.89, and L_c increases from 3.8 μm to 5.3 μm. The worst power transfer is obtained when ε_M = ε _Al. As t_I increases from 20 nm to 80 nm, P _max increases from 0.37 to 0.48, and L_c increases from 4.2 μm to 6.3 μm. If s_M is reduced below 60 nm, P _max increases a little, and L_c decreases. However, the reduction of s_M is limited by the following two factors. The one is that the distance between the two silicon lines in Fig. 8(a), which is s_M + 2t_I, should be larger than the minimum line width of the 193-nm optical lithography. (If t_I = 20 nm and the minimum line width is 100 nm, s_M = 60 nm.) The other is whether the narrow channel for the shared metal line is well filled with metal.

For the low crosstalk between the two MISIM waveguides, s_M should be sufficiently large. The minimum value of s_M, which is large enough for the low crosstalk, is determined as follows. First, the crosstalk is quantitatively defined as the ratio of P _max to the normalized power carried by the left MISIM waveguide at a distance of L_c, which is given by $\exp (-2\varphi \cot \varphi ){\cos }^2\varphi$. Then, the crosstalk is given by $\exp (2)P_{\max }$ for sufficiently large s_M. If the desired value of the crosstalk is 0.01, P _max becomes 0.00135 for the minimum value of s_M. This value is indicated as the dashed lines in Figs. 10(a) to 10(e), and the curves of P _max cross these dashed lines at the minimum values of s_M. Figure 10(g) shows the minimum values of s_M as functions of t_I for the different values of ε_M. On the one hand, the minimum value of s_M is smallest for ε_M = ε _Cu. It increases from 420 nm to 880 nm as t_I 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 SiO₂), 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_M is largest for ε_M = ε _Ag2. It increases from 660 nm to 1410 nm as t_I 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 SiO₂ shell, are placed on an Ag substrate whose dielectric constant is equal to ε _Ag2. Although the edge-to-edge distance between the SiO₂–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_M, and it may be necessary to polish them by using a numerical method.

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_I = 20 nm, and s_M = 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 ε_M = ε _Cu, t_I = 20 nm, the crosstalk between them becomes very low for s_M > 420 nm. The MISIM waveguides are expected to play an important role in connecting Si-based photonic and electronic circuits.