1. Introduction

A potential approach for future minimization of optical devices beyond the diffraction limit is to utilize surface plasmon polaritons (SPPs), which are electromagnetic surface waves coherently coupled to collective electron oscillations in metal-dielectric interfaces [1]. Plasmonic waveguides, which comprise one or more metal-dielectric interfaces, are a basic element to accommodate various passive and active plasmonic devices. In the past decades, many waveguding structures have been explored such as channel plasmon polariton (CPP) [2], metal-insulator-metal (MIM) [3], dielectric-loaded SPP (DLSPP) [4], and hybrid plasmonic waveguides [5], etc. A typical challenge in all these plasmonic waveguides is the tradeoff between strong optical field confinement and long propagation distance. Moreover, the plasmonic waveguide should be flexible enough to allow active functions such as modulation and detection to be implemented. For monolithic integration in existing Si electronic-photonic integrated circuits (EPICs), the fabrication of these plasmonic waveguides as well as waveguide-based devices should be compatible with the mature complementary metal-oxide-semiconductor (CMOS) technology. The hybrid plasmonic waveguide, which consist of a high-index dielectric core separated from a metal surface by a nanoscale low-index dielectric gap, is regarded as an attractive candidate to meet the above requirements. Theoretical analyses show that the nanoscale gap can support a relatively low-loss compact plasmonic-like mode whereas the high-index dielectric core, which is typically patterned to a rectangle or a cylinder, supports a photonic-like mode [5–7]. For seamless integration in Si EPICs, the high-index dielectric core is naturally Si and the low-index dielectric gap is naturally SiO₂ or Si₃N₄. Such a vertical metal-SiO₂-Si hybrid waveguide, as well as various waveguide-based devices such as couplers, power splitters, and filters, etc., has been extensively studied theoretically [7–9]. Experimentally, conductor-gap-silicon (CGS) plasmonic waveguides have been demonstrated using patterned Au as the metal [10]. The nanoscale Au patterning requires expensive electron beam lithography (EBL), which is not an industry-standard technology. A self-aligned approach is proposed to avoid EBL [11], but it is only viable for Si waveguides formed by the local oxidation process which suffer from blunt sidewalls. However, theoretical studies have revealed that the metal layer in hybrid plasmonic waveguides can be infinitely wide because the lateral field confinement of the plasmonic-like mode can be solely achieved by the Si core, similarly to the photonic mode in standard Si waveguides [8, 9]. Moreover, although Au is a good metal for plasmonics, it is not a CMOS compatible material. In terms of CMOS compatibility, Al or Cu should be used as the metal. At the telecom wavelengths of 1550 nm, Cu is found to be much better than Al as it provides much lower propagation loss in the same waveguide configuration [12]. Considering the above two points, a vertical Cu-SiO₂-Si hybrid plasmonic waveguide is proposed and has been experimentally demonstrated on silicon-on-insulator (SOI) substrates using standard CMOS technology [13]. The Cu layer above the Si core, which is defined by the industry-standard ultraviolet (UV) lithography, is much wider than the underlying Si core. The fabricated Cu-SiO₂-Si plasmonic waveguide exhibits a relatively low propagation loss of ~0.12 dB/μm at telecom wavelengths and a high coupling efficiency of ~86% with the conventional Si strip waveguide [13]. It indicates that this hybrid plasmonic waveguide is a more feasible candidate for seamless integration in existing Si EPICs than the CGS waveguide.

Waveguide-ring resonator (WRR) is a key element to accommodate many important building blocks such as filters, switchers, and modulators, etc. in integrated photonics [14]. Many kinds of plasmonic WRRs have been demonstrated. Their characteristics are mainly depended on the plasmonic waveguides which they are based. An early sample is CPP based WRR, which has radius of ~5 μm and exhibits a small quality factor [2]. The DLSPP-based WRRs, which are the mostly studied plasmonic WRR type to date, have minimum R of several micrometers and the demonstrated Q factor is in the range of 100–200 [15–17]. Although a very high Q factor of ~1376 has been demonstrated on a plasmonic microcavity formed by a metal disc, it works on a whispering gallery mode, thus making it inappropriate for applications that require a single resonance operation [18]. Plasmonic WRRs based on horizontal MIM plasmonic waveguides can reach a submicron radius, but they require a small aperture for coupling between the ring and bus waveguides because of the very small field penetration depth in the metal [19]. The aperture is difficult to be precisely controlled using standard CMOS technology. The vertical Cu-SiO₂-Si waveguide supports evanescently codirectional coupling similarly to the conventional Si waveguide, which can dramatically simplify the design and optimization of the plasmonic WRRs as compared with the horizontal MIM counterpart. Moreover, the Cu-cap can be directly used as a thermal heater (if it is thin enough for sufficiently large resistance) or an additional thermal heater, e.g., TiN, can be placed just above the Cu cap because the Cu cap can completely isolate the optical field from the thermal heater. Using the approach, also combined with the good thermal conductivity of Cu and the ultracompact size of vertical Cu-SiO₂-Si based WRRs, various effective thermo-optical (TO) devices could be realized. To achieve this goal, the optical characteristics of various Cu-SiO₂-Si plasmonic WRRs including their TO effect need to be evaluated. In this paper, vertical Cu-SiO₂-Si based WRRs with different structural parameters are fabricated and their characteristics are systematically studied.

2. Design, fabrication, and measurement

The schematic layout and cross section of the Cu-SiO₂-Si plasmonic WRRs studied in this paper are depicted in Fig. 1(a) and 1(b), respectively. The Cu-capped rectangular window is defined as the plasmonic area. The straight bus plasmonic waveguide has length of 7 μm and width (W_P) of 0.18 μm, which is linked with input/output 0.5-μm-wide Si strip waveguides through 1-μm-long tapered couplers. A microring adjacent to the bus waveguide has the same width of 0.18 μm and the central radius (R) of 1.09, 1.59, 2.09, or 2.59 μm, respectively. The separation gap between the ring and bus waveguide is 0.2 or 0.25 μm, respectively.

 

Fig. 1 Vertical Cu-SiO₂-Si hybrid plasmonic waveguide-based WRRs fabricated in this paper. (a) Schematic layout, (b) Cross section, and (c) SEM image of the patterned Si core of a typical WRR with R of 2.59 μm. The parameters in the layout are W_P = 0.18 μm, R = 1.09, 1.59, 2.09, or 2.59 μm, and gap = 0.2 or 0.25 μm. The parameters which are determined by fabrication are h_Si = ~340 nm and t_SiO2 = ~17 nm.

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The devices are fabricated on SOI wafers with 340-nm top-Si and 2-µm buried SiO₂. During patenting the Si core of the plasmonic WRRs and tapered couplers (as well as the input/output Si strip waveguides simultaneously) using UV lithography, the expose condition is intentionally varied to result in different critical dimensions of the Si core. A photoresist trimming process is carried out to further reduce the critical dimension. Using this method, WRRs with different Si core widths are fabricated using the same mask. Figure 1(c) shows a scanning electron microscope (SEM) image of a patterned Si core of a typical plasmonic WRR with 2.59-μm R, for example. Then, SiO₂/Si₃N₄ deposition, SiO₂ chemical mechanical polishing (CMP) (using Si₃N₄ as the CMP stopping layer), SiO₂/Si₃N₄ deposition again, SiO₂ window opening (using Si₃N₄ as the SiO₂ dry etching stopping layer), wet etching of remaining Si₃N₄ in the windows, thermal oxidization to grow a thin SiO₂ layer, Cu deposition, and Cu-CMP (to remove Cu outside the windows) are carried out sequentially. The details of the fabrication have been described elsewhere [13]. The thickness (t_SiO2) of the thin SiO₂ layer between the Si core and the metal is a critical parameter to determine the property of the hybrid plasmonic waveguide [5], thus it needs to be precisely controlled. In this work, for simplicity, this layer is a ~17-nm-thick thermal oxide, as shown in Fig. 2 . It is worthy to be noted that if this thin SiO₂ layer is replaced by a functional dielectric, e.g., a dielectric with a large electro-optic (EO) or TO coefficient, effective active plasmonic devices may be realized because of the strong mode confinement in this layer. Figure 2 shows cross sectional transmission electron microscope (XTEM) images of four fabricated plasmonic WRRs. Their Si core widths at the middle of the height (W_P) are ~163 nm, ~156 nm, ~133 nm, and ~126 nm, respectively. The Si cores are not a perfect rectangle and contain round shoulders at upper two corners due to the imperfection of the Si dry etching process. Because the plasmonic mode is strongly confined near the upper surface of the Si core, the round shoulder makes the effective Si core width smaller than the above W_P value measured on the middle height of the Si core, especially for the devices with narrower W_P. For the S4 device shown in Fig. 2(d), the flat region of the Si core top is only ~27 nm. WRRs with W_P narrower than S4 device are also fabricated, but they exhibit poor resonant characteristics, which can be explained by the fact that their flat region of the Si core top has approached to zero so that no plasmonic mode can be excited by the transverse-magnetic (TM)-polarized light (the electric field is perpendicular to the chip surface plane). Hereafter, WRRs with Si cores shown in Figs. 2(a), 2(b), 2(c), and 2(d) are referred as S1, S2, S3, and S4 respectively for analysis in detail. Because the Si core shrinks from the nominal width of 180 nm in the mask to a certain width as shown in Fig. 2, the gap between the bus and ring waveguides is widened accordingly. Assuming that the Si core shrinks equally at both sides during fabrication, the gaps of these four WRRs measured at the middle height of the Si core are widened by ~17 nm, ~24 nm, ~47 nm, and ~54 nm, respectively. Be noted that due to the round shoulder of the Si core, the effective gap is larger and the effective Si core width is smaller than the abovementioned values.

 

Fig. 2 XTEM images of vertical Cu-SiO₂-Si hybrid plasmonic waveguides with different Si core widths, which are caused by the different expose conditions during UV lithography. A thin Si₃N₄ layer between Si and SiO₂ in the bottom Si core region is for the fabrication process control, not for the function. The Si core height is ~300 nm and the thermal SiO₂ layer between the Si core and the Cu cap is ~17 nm. These four kinds of plasmonic WRRs are referred as S1, S2, S3, and S4, respectively.

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The diced chips are characterized using standard fiber-chip-fiber measurement setup [20]. Quasi-TM-polarized light from a broadband EXPO laser source (whose the spectral range is ~1520–1620 nm) is coupled into the input Si waveguide through a lensed polarization-maintaining (PM) sing-mode fiber. Light transmitted from the output Si waveguide is coupled to another single-mode fiber and to be measured by a power meter and an AQ6317B optical spectrum analyzer (OSA). A semi-auto micrometer piezo-stage is used to adjust the fibers to search the maximum output power. A thermal chip holder, which can be heated by an external current and its temperature can be measured by a thermocouple, is used to heat the chip under test from room temperature up to 100°C. The temperature stability is ~ ± 1°C. Besides plasmonic WRRs, straight plasmonic waveguides with length ranging from 0 to 200 μm are also included in the same chip, from which the propagation losses are extracted using the conventional cut-back method, as shown in Fig. 3 . The propagation loss is measured to be ~0.10–0.12 dB/μm, close to that predicted from the full-vectorial finite-difference method [13]. The simulation results (see Section 3.4) predict that the propagation loss decreases slightly with W_P decreasing from 160 nm to 80 nm. The observed non-monotonic behavior of propagation loss versus W_P in Fig. 3 may be simply attributed to the experimental error [20]. Nevertheless, the dependence of propagation loss on W_P is very weak, at least at the range of W_P studied in this work. On the other hand, the propagation loss increases substantially with temperature increasing, which can be mainly attributed to the increase of Cu imaginary permittivity with temperature [21], as will be discussed in Section 3.5.

 

Fig. 3 Transmitted powers measured on straight Cu-SiO₂-Si hybrid plasmonic waveguides with different Si core widths, normalized by that measured on the corresponding reference waveguide (i.e., the Si strip waveguide without the plasmonic area). Each data point is averaged from three identical waveguides and the standard deviation is indicated as the error bar. From the linearly fitting, the propagation losses are extracted, as indicated.

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3. Results and discussions

Because the plasmonic WRRs have 7-μm-long bus waveguide, the transmission spectra measured on the plasmonic WRRs are normalized by that measured on the corresponding 7-μm-long straight plasmonic waveguide. The normalized transmission spectrum can in general be expressed by [22]:

3.1. Plasmonic WRRs with different radii and silicon core widths

Figure 4 plots the normalized transmission spectra of plasmonic WRRs with the same nominal gap of 200 nm. Due to the abovementioned Si core shrinkage, the real gaps (measured at the middle height of the Si core) become ~217, ~224, ~247, and ~254 nm for the S1, S2, S3, and S4 WRRs, respectively. From Fig. 4, we can see that the small-ring WRRs exhibit only one resonance while the large-R WRRs exhibit two resonances in the spectral range of 1520–1620 nm. The free spectral range (FSR) between these two resonances is solely determined by n_eff:

 

Fig. 4 Power transmission spectra for plasmonic WRRs with the same nominal gap of 200 nm but with different Si core widths (S1, S2, S3, and S4) and different ring radii (R = 1.09, 1.59, 2.09, and 2.59 μm). The experimental curves (in black) are normalized by the transmission spectrum of the corresponding 7-μm-long straight plasmonic waveguide. The fitting curves (in red) are based on Eq. (1). The fitting parameters of n_eff, ${\alpha }_b$, |t|, and θ are indicated for each plasmonic WRR.

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The fitting curves as well as the fitting parameters of n_eff, ${\alpha }_b$, |t|, and θ for each WRR are shown in Fig. 4 accordingly. Because the value of ${\alpha }_b$ increases and the value of ${\alpha }_{prop}$ decreases with R increasing, ${\alpha }^2={\alpha }_b\times {\alpha }_{prop}$ depends on R more weakly than |t|, thus the different spectral characteristics for different-R WRRs is mainly attributed to the variation of the |t| value with R and W_P. The different spectral characteristics for various WRRs observed in Fig. 4 are explained as below:

3.2. Plasmonic WRRs with a large gap

Because the |t| value increases quickly with the gap between the bus and ring waveguides increasing, according to the above analysis, it is predicted that the S1 WRRs will have a larger ER, while S4 WRRs will have a smaller ER if the gap of WRRs increases. The plasmonic WRRs with the nominal gap of 250 nm (whereas the other structural parameters keep the same as before) are fabricated and characterized. Again, due to the abovementioned Si core shrinkage, the real gaps (measured at the middle height of Si core) become ~267, ~274, ~297, and ~304 nm for the S1, S2, S3, and S4 WRRs, respectively. The transmission spectra measured on these WRRs are plotted in Fig. 5 , verifying the above prediction. The same fitting method is used to fit the measured spectra. The fitting curves as well as the fitting parameters for each WRR are shown in Fig. 5 accordingly. We can see that the ER value increases with R increasing for all four kinds of WRRs, indicating that all of these four kinds of WRRs are undercoupling. In particular, the S1 WRR with 2.59-μm R shown in Fig. 5(d) exhibits ER = ~11.5 dB, IL = ~0.1 dB, FSR = ~44 nm, FWHM = ~6.0 nm, and Q = ~260, respectively, and the S2 WRR with 2.59-μm R shown in Fig. 5(h) exhibits ER = ~12.8 dB, IL = ~0.1 dB, FSR = ~47 nm, FWHM = ~5.7 nm, and Q = ~275, respectively. Because the |t| value increases with W_P decreasing, the S3 and S4 WRRs exhibit a small ER value, especially for the small-R WRRs whose |t| value is much larger than the α value. Compared with the WRRs shown in Fig. 4, we can see that the insertion loss is significantly reduced and the Q-value is significantly improved with the gap increasing. The property can be mainly attributed to the large |t| value for these WRRs. The large |t| value also results in the moderate ER value for these WRRs. It is expected that the ER value could be significantly improved by adjusting the gap to match the critical coupling condition of t ≈α.

 

Fig. 5 Power transmission spectra for plasmonic WRRs with the same nominal gap of 250 nm but with different Si core widths (S1, S2, S3, and S4) and different ring radii (R = 1.09, 1.59, 2.09, and 2.59 μm). The experimental curves (in black) are normalized by the transmission spectrum of the corresponding 7-μm-long straight plasmonic waveguide. The fitting curves (in red) are based on Eq. (1). The fitting parameters of n_eff, ${\alpha }_b$, |t|, and θ are indicated for each plasmonic WRR.

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3.3. Plasmonic WRRs with a dual-ring configuration

Another way to adjust the |t| value is by using a symmetric dual-ring configuration, as shown schematically in Fig. 6(a) , where two ideal rings are symmetrically placed at both sides of the bus waveguide with the separation gap of 200 nm. The plasmonic WRRs with such a dual-ring configuration are also fabricated and characterized. The other structural parameters keep the same as before. Figure 6(b) is a SEM image of the Si core pattern for such a dual-ring WRR, for example. Again, due to the abovementioned Si core shrinkage, the real gaps (measured at the middle height of the Si core) become ~217, ~224, ~247, and ~254 nm for the S1, S2, S3, and S4 WRRs, respectively. The S1 and S2 WRRs exhibit good resonant property, as shown in Fig. 7 , whereas the S3 and S4 WRRs exhibit very weak resonance, thus they are not shown here. For simplicity, the experimental spectra of S1 and S2 WRRs are also fitted by Eq. (1) using the same fitting method as before. The fitting curve and the fitting parameters for each WRR are shown in Fig. 7 accordingly. Compared with the single-ring WRRs with the same nominal gap of 200 nm shown in Fig. 4, the |t| value becomes large and its dependence on R becomes weakly, leading to a much large ER value for the WRRs with R of 1.59–2.59 μm. In particular, the S1 WRR with 2.59-μm R exhibits ER = ~21.8 dB, IL = ~0.5 dB, FSR = ~43 nm, FWHM = ~7.2 nm, and Q = ~218, respectively. The relatively weak R dependence makes these dual-ring WRRs have large tolerance in design and fabrication. Moreover, the n_eff value extracted from these dual-ring WRRs is substantially larger than that extracted from the single-ring WRRs, which may be attributed to the dispersion of the Cu-SiO₂-Si hybrid plasmonic waveguide, as will be explained in the next section.

 

Fig. 6 Plasmonic WRRs with a dual-ring configuration. (a) Schematic layout and (b) SEM image of the patterned Si core of a plasmonic WRR with 2.59-μm R. The structural parameters are as before.

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Fig. 7 Power transmission spectra for dual-ring plasmonic WRRs with the same nominal gap of 0.20 μm but with different Si core widths (S1 and S2) and different ring radii (R = 1.09, 1.59, 2.09, and 2.59 μm). The experimental curves (in black) are normalized by the transmission spectrum of the corresponding 7-μm-long straight plasmonic waveguide. The fitting curves (in red) are based on Eq. (1). The fitting parameters of n_eff, ${\alpha }_b$, t, and θ are indicated for each plasmonic WRR.

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3.4. Wavelength dependence of the real effective index

The real effective index of the straight plasmonic waveguide calculated using the full-vectorial finite-difference method is ~2.27 at 1550 nm, as shown in Fig. 8(a) . This value is substantially smaller than those extracted from the above WRRs. For comparison, the horizontal Cu-SiO₂-Si-SiO₂-Cu plasmonic waveguide has the similar n_eff value for that obtained from numerical simulation of the straight waveguide theoretically and that extracted from the fabricated WRRs experimentally [19]. It is well known that the effective index n_eff in Eqs. (1) and (2) should be replaced by the group index n_g if the waveguide is dispersive [14], which is defined by

 

Fig. 8 (a) Electrical field |E_y| distribution of the fundamental quasi-TM mode in the Cu-SiO₂-Si hybrid plasmonic waveguide; (b) Real effective index of straight waveguides with different Si core widths versus wavelength, obtained by the 3-D FDTD numerical simulation.

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It is difficult to experimentally extract n_eff value at each wavelength. Instead, we calculate n_eff in the wavelength range of 1520–1620 nm with a step of 10 nm using the three-dimension (3-D) finite-difference time-domain (FDTD) method [23]. For simplicity, the Si core of the hybrid plasmonic waveguide is assumed to be a rectangle with height h_Si = 340 nm and width W_P = 160, 120, and 80 nm, respectively, and the SiO₂ thickness t_SiO2 = 17 nm, as shown in Fig. 8(a). The electrical field |E_y| distribution calculated by a mode solver is also shown in Fig. 8(a). The wavelength-dependent complex permittivity of Cu is cited from Ref [21], which matches the experimental results well in the case of horizontal Cu-SiO₂-Si-SiO₂-Cu plasmonic waveguide [24]. Figure 8(b) plots the calculated n_eff value as a function of wavelength. As expected, n_eff is wavelength dependent and the $d{n}_{eff}/d\lambda$ value is negative, leading to a significantly larger n_g value than the n_eff value. However, the dependence of n_eff on λ is not clear in the wavelength range of 1520–1620 nm, thus the n_g value is also wavelength dependent. Nevertheless, by linearly fitting the data points in Fig. 8(b), an approximate $d{n}_{eff}/d\lambda$ value of ~0.0008 nm⁻¹ is estimated in the wavelength range of 1520–1620 nm, thus the n_g values for the plasmonic waveguides with W_P of 160, 120, and 80 nm are estimated to be ~3.49, ~3.21, and ~3.03, respectively, which agree well with those extracted from the above plasmonic WRRs experimentally. We suspect that the even larger n_g value extracted from the dual-ring WRRs may also be attributed to the waveguide dispersion effect as the dual-ring configuration may result in a larger dispersion effect apparently. The imaginary part of the modal index calculated from simulation is also plotted in Fig. 8(b). It increases monotonically with wavelength increasing. Moreover, we can see that the waveguide with a larger W_P has a larger imaginary part of the modal index, corresponding to a larger propagation loss. The calculated propagation loss at λ = 1550 nm decreases from 0.09 dB/μm to 0.08 dB/μm for W_P decreasing from 160 nm to 80 nm.

3.5. Thermo-optical response of plasmonic WRRs

The plasmonic WRRs are measured again when the chip holder is heated to a certain temperature (keeping for a sufficiently long time before measurement to make sure that the holder and the chip have the same temperature). Figure 9 plots the transmission spectra of the S1 single-ring WRR with 2.59-μm R and 250-nm nominal gap measured at 27°C, 50°C, 82°C, and 101°C, respectively. At 27°C, the central resonance (λ_r) is peaked at 1536.2 nm, whereas at 82°C, λ_r is redshifted by ~4.6 nm. This redshift corresponds to an increase of the transmitted power level by ~7.5 dB. λ_r shifts with temperature almost linearly, as shown in Fig. 10(a) . From linearly fitting the experimental data read from Fig. 9, the TO coefficient (dλ_r/dT) is estimated to ~80 pm/°C. It is worthy to be noted that this TO coefficient is smaller than that for a conventional strip-Si WRR (~107 pm/°C) but is larger than that for the DLSPP WRR reported in Ref [17]. (~60 pm/°C). In Eq. (1), we can see that λ_r is determined by n_g and θ. Assuming that θ is temperature-independent, the shift of λ_r is solely contributed by the temperature-induced n_g variation. To extract the dn_g/dT value, the spectra in Fig. 9 are fitted using Eq. (1) by varying the n_g value only whereas keeping other fitting parameters of ${\alpha }_b$, |t|, and θ the same. Figure 10(b) shows that n_g depends on temperature almost linearly with the TO coefficient (dn_g/dT) of ~1.69 × 10⁻⁴/°C.

 

Fig. 9 Transmission spectra of the S1 single-ring WRR with 2.59-μm R and 250-nm nominal gap measured at temperature of 27, 50, 82, and 101°C, respectively.

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Fig. 10 (a) Two resonant wavelengths,λ_r, read from Fig. 9 versus temperature, the thermo-optical coefficient dλ_r/dT is extracted from linearly fitting; (b) Group index n_g of the plasmonic WRR versus temperature, n_g is obtained by fitting the spectra in Fig. 9 using Eq. (1), the thermo-optical coefficient dn_g/dT is then extracted from linearly fitting.

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Other plasmonic WRRs are also measured at different temperatures and their transmission spectra are analyzed using the same method. The extracted TO coefficient dn_g/dT values for these plasmonic WRRs are plotted in Fig. 11 . The relatively large discrepancy of the dn_g/dT data may be attributed to the temperature variation (~ ± 1°C) during measurement and the uncertainty of λ_r read from the spectra, especially for those with large FWHM. Nevertheless, the average dn_g/dT value of ~1.6 × 10⁻⁴ /°C can be estimated from Fig. 11 for all plasmonic WRRs studied in this work. Moreover, the dn_g/dT values extracted from the dual-ring WRRs is slightly larger than those extracted from the single-ring counterparts. This observation is consistent with the above observation that the n_g values extracted from the dual-ring WRRs are larger than those extracted from the single-ring counterparts.

 

Fig. 11 Thermo-optical coefficients (dn_g/dT) extracted from different plasmonic WRRs studied in this work, which are extracted from linearly fitting the data points of n_g versus temperature, as shown in Fig. 10(b). The standard deviation of the linear fitting is indicated as the error bar.

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The dn_g/dT value is related with the dn_eff/dT value through Eq. (3), whereas n_eff at different temperatures can be calculated using the abovementioned FDTD method if the temperature-dependence of optical properties of Cu, SiO₂, and Si are known. For Si, n_Si = ~3.52 at room temperature and dn_Si/dT = ~2 × 10⁻⁴ /°C. For SiO₂, n_SiO2 = ~1.44 at room temperature and dn_SiO2/dT = ~1 × 10⁻⁵ /°C. However, for Cu, the permittivity reported in literature is discrepancy and its temperature dependence is lack. Ref [21]. reports that the Cu permittivity at 1500 nm is −114.0 + i2.56 at 300°K and becomes −11.4 + i7.51 at 500°K. The significant increase of the imaginary part of Cu permittivity with temperature explains the observation in Fig. 3 that the propagation loss of the Cu-SiO₂-Si plasmonic waveguides increases substantively with temperature increasing. For the Cu-SiO₂-Si plasmonic waveguide with 160-nm × 340-nm Si core and 17-nm SiO₂, n_eff at 1500 nm is calculated to ~2.275 at 300°K and ~2.299 at 500°K using the above parameters, which corresponds to the dn_eff/dT value of ~1.21 × 10⁻⁴ /°C, which is close to the above experimental dn_g/dT value after taking the dispersive effect into account.

Because both Si and Cu have large TO effect, it is expected that dn_eff/dT is contributed by both the Si core and the Cu cap. To distinguish the Cu-cap contribution to dn_eff/dT, the above plasmonic waveguide is simulated again by setting the Si and SiO₂ indices at 500°K while the Cu permittivity at 300°K. The calculated n_eff is ~2.297, which corresponds to the dn_eff/dT value of ~1.10 × 10⁻⁴ /°C. It indicates that the Cu cap provides ~10% contribution in the overall dn_eff/dT.

As shown in Fig. 8(a), the plasmonic mode is mainly confined in the thin SiO₂ layer between the Si core and the Cu cap. However, SiO₂ has a very small TO coefficient. It is expected the overall dn_eff/dT can be significantly improved if this layer is replaced by a dielectric with higher TO coefficient, such as silicon-rich SiO₂ [25].

To design real thermo-optical devices, a heater should be implemented to control temperature through an external current or voltage. For the Cu-SiO₂-Si plasmonic waveguide studied in this work, the Cu layer may be directly used as the heater, but due to the very low resistivity of Cu, the Cu layer should be thin enough to offer a sufficiently large resistance. The other approach is to add an additional layer such as TiN for heating, as the conventional Si-waveguide-based TO devices. An advantage offered by the hybrid plasmonic waveguide is that the TiN layer can be placed just above the Cu cap because the Cu cap can isolate the optical field completely. Combined with the high thermal conductivity of Cu and the ultracompact size of the plasmonic WRRs, TO devices based on the vertical Cu-SiO₂-Si plasmonic WRRs should be more effective and fast than those based on the conventional Si-waveguide WRRs.

4. Conclusions

We have experimentally investigated the optical properties and thermo-optical effect of ultracompact plasmonic WRRs based on the Cu-SiO₂-Si hybrid plasmonic waveguide. The effect of various structural parameters such as the ring radius, the separation gap, the Si core width, and the ring configuration (i.e., single-ring or dual-ring) on the spectral characteristics of WRRs is evaluated. Some of the demonstrated plasmonic WRRs exhibit very good performance. For example, a single-ring WRR with 2.59-μm R, 156-nm W_P, and 250-nm nominal gap exhibits ER of ~12.8 dB, IL of ~0.1 dB, FSR of ~47 nm, FWHM of ~5.7 nm, and Q of ~275, respectively. ER larger than 20 dB has been achieved using the dual-ring configuration. The plasmonic WRRs exhibit a redshift of the resonance of ~4.6 nm for a temperature increase of ~55°C, which leads to a ~7.5-dB variation of the transmitted power. The overall TO coefficient dn_g/dT is estimated to ~1.6 × 10⁻⁴ /°C, which is contributed by the thermo-optical effect of both the Si core and the Cu cap. It is expected that the dn_g/dT value could be significantly improved if a dielectric with large TO coefficient replaces the thin SiO₂ layer between the Si core and the Cu cap. Combined with the Si-CMOS compatibility and the ultracompact size, the Cu-capped hybrid plasmonic WRRs are a very promising element for various effective TO devices for seamless integration in existing Si EPICs.