## Abstract

An ultracompact silicon electro-optic modulator operating at 1550-nm telecom wavelengths is proposed and analyzed theoretically, which consists of a Cu-TiO_{2}-Si hybrid plasmonic donut resonator evanescently coupled with a conventional Si channel waveguide. Owing to a negative thermo-optic coefficient of TiO_{2} (~-1.8 × 10^{−4} K^{−1}), the real part of effective modal index of the curved Cu-TiO_{2}-Si hybrid waveguide can be temperature-independent (i.e., athermal) if the TiO_{2} interlayer and the beneath Si core have a certain thickness ratio. A voltage applied between the ring-shaped Cu cap and a cylinder metal electrode positioned at the center of the donut, − which makes Ohmic contact to Si, induces a ~1-nm-thick free-electron accumulation layer at the TiO_{2}/Si interface. The optical field intensity in this thin accumulation layer is significantly enhanced if the accumulation concentration is sufficiently large (i.e., > ~6 × 10^{20} cm^{−3}), which in turn modulates both the resonance wavelengths and the extinction ratio of the donut resonator simultaneously. For a modulator with the total footprint inclusive electrodes of ~8.6 μm^{2}, 50-nm-thick TiO_{2}, and 160-nm-thick Si core, FDTD simulation predicts that it has an insertion loss of ~2 dB, a modulation depth of ~8 dB at a voltage swing of ~6 V, a speed-of-response of ~35 GHz, and a switching energy of ~0.45 pJ/bit, and it is athermal around room temperature. The modulator’s performances can be further improved by optimization of the coupling strength between the bus waveguide and the donut resonator.

© 2013 OSA

## 1. Introduction

A CMOS compatible integrated Si electro-optical (EO) modulator is a key component in Si electronic photonic integrated circuits (EPICs) [1]. Many kinds of Si EO modulators have been reported recently [2,3], mostly relying on the free-carrier dispersion effect of Si to modulate the Si refractive index based on a MOS capacitor [4, 5], a PIN diode [6], or a PN junction [7]. Either a Mach-Zehnder interferometer (MZI) [4,7] or a waveguide-ring resonator (WRR) [5,6] is utilized to convert the phase variation into the intensity modulation. The WRR modulators offer smaller footprints than the MZI modulators but at the price of narrower optical bandwidth, higher temperature sensitivity due to the relatively large thermo-optic (TO) coefficient of Si (~1.8 × 10^{−4} K^{−1}), and limited modulation speed due to the long photon lifetime in the resonator if the resonator has a very high quality factor (Q value).

One approach to suppress the temperature sensitivity of Si resonators is by overlaying a polymer coating with a negative TO coefficient [8], but polymers are currently not compatible with CMOS process. Another approach is by overcoupling the ring resonator to a balanced MZI [9], but it requires complex design and sacrifices the footprint. Moreover, due to the fundamental diffraction limit of light propagation along Si waveguides, the WRR modulators are still quite large as compared with the nanoscale electronic devices. The minimum bending radius is ~1.5 μm for Si single-mode channel waveguides [10] and is usually larger than ~5 μm for Si rib waveguides in which the EO modulators are implemented. The total footprint of Si WRR modulators inclusive of the electrodes is usually larger than ~200 μm^{2}.

A technology emerging recently which can scale down the dimension of optical devices far beyond the diffraction limit is plasmonics, which deals with surface plasmon polariton (SPP) signal propagating along the metal-dielectric interfaces [11]. Several ultracompact plasmonic EO modulators have been proposed and/or demonstrated [12,13]. However, they mostly rely on active materials other than Si and/or require non-standard CMOS techniques for fabrication. For ease of implementation into the exiting Si EPICs, it is preferred to use Si as the active material and the modulator is waveguide-based. A horizontal Cu-insulator-Si-insulator-Cu nanoplasmonic waveguide is a plasmonic waveguide enabling to realize plasmonic modulator [14], which has a MOS capacitor structure and the Si core can be used as the active material. Electro-absorption (EA) and phase modulations have been experimentally demonstrated based on this plasmonic waveguide [15,16], but a relatively large driving voltage is required to reach 3-dB modulation in the EA modulators or π-phase shift in the MZI modulators. Another feasible plasmonic waveguide is a vertical Cu-insulator-Si hybrid plasmonic waveguide (HPW) [17,18]. WRRs with radius of ~1-2 μm and Q-value of ~200-300 have been experimentally demonstrated based on the Cu-SiO_{2}-Si HPW [19]. Theoretically, the radius of such plasmonic WRRs can be reduced to submicron (e.g., ~0.8 μm) [20]. Moreover, if TiO_{2}, – which has a negative TO coefficient of ~-1.8 × 10^{−4} K^{−1} and is transparent at near-infrared wavelengths [21], is used as the insulator between the Cu-cap and the Si core, the plasmonic WRRs can be athermal. TiO_{2} is also used as a gate dielectric in MOS electronics, whose dielectric constant ranges from 4 to 86 depending on the detailed fabrication processes [22]. The Cu-TiO_{2}-Si HPW is also a MOS capacitor, thus enabling a voltage to be applied between the Cu cap and the Si core to modulate its propagation property. It is expected that the WRR modulators based on this hybrid plasmonic WRR may overcome the abovementioned two critical issues of the conventional WRR modulators, i.e., footprint miniaturization and temperature-sensitivity suppression. This paper presents a systematical investigation on ultracompact WRR modulators based on the Cu-TiO_{2}-Si HPW.

## 2. Device structure

The structure of the proposed modular is shown in Fig. 1 schematically. It consists of a Cu-TiO_{2}-Si hybrid plasmonic donut resonator and a bus waveguide. The modulator can be seamlessly inserted in a dense Si channel waveguide-based photonic circuit. To reduce the insertion loss, the bus waveguide is a conventional Si channel waveguide. To reduce the overall footprint, a donut rather than a ring is used, thus the electrode for Si Ohmic contact can be positioned at the center of the donut and the donut forms a standard MOS capacitor (the other electrode is the Cu cap).

The cylindrical electrode for Si Ohmic contact has radius of *r _{0}*. The outer radius of the Si donut is

*R*and the inner radius is

*(R – W*, where

_{P})*W*is the width of Si core of the curved hybrid plasmonic waveguide. The separation between the resonator and the bus waveguide is “

_{P}*gap*”. The Si height is

*H*, the slab thickness in the Si donut is

*t*, and the TiO

_{slab}_{2}thickness is

*t*. The Cu-cap thickness is set to be much larger than the penetration depth of the SPP mode in the metal (~26 nm). Because the plasmonic mode can only be excited by the electric field of optical mode perpendicular to the metal/dielectric interface, the proposed modulator is valid only for the transverse magnetic (TM)-polarized light.

_{ox}The modulators are fabricated on silicon-on-insulator wafers. The Si pattern of the bus and donut waveguides are defined by partially dry etching of Si down to *t _{slab}* using a thin SiO

_{2}layer as the etching mask, following by dry etching of the remaining Si down to the buried SiO

_{2}using both SiO

_{2}and photo-resistor as the etching mask. Using this etching method, there is no misalignment issue between the inner and outer rings of the donut. After Si patterning, a thick SiO

_{2}is deposited and a ring-shaped window is opened to expose the surface of the Si core. There exists possible misalignment between the SiO

_{2}window and the beneath Si core due to fabrication imperfection. Here, the SiO

_{2}window (hence the width of the TiO

_{2}/Cu cap,

*W*) is intentionally designed to be larger than the beneath Si core by

_{ox}*ΔW*in each side, thus

_{P}*W*. TiO

_{ox}= W_{P}+ 2ΔW_{P}_{2}is then deposited on the Si core through the windows, followed by Cu deposition and Cu chemical mechanical polishing (CMP) to remove TiO

_{2}and Cu outside the windows. The structural parameters are initially chosen based on our experience [23], as listed in Table 1. Then, one of these parameters is varied while the others keep the same to investigate its effect on the whole performance.

The refractive indices of Si, SiO_{2}, TiO_{2}, and Cu depend both on wavelength and temperature. For simplification, the indices as well as the TO coefficients at 1550-nm wavelength and room temperature (RT) are used here, as listed in Table 2. The validity of these optical parameters has been verified as the calculated propagation losses of plasmonic waveguides agree well with the experimental results measured at 1550-nm wavelength and RT [23]. Be noted that the quantitative results in this paper are accurate only near 1550-nm wavelengths and room temperature.

## 3. Thermo-optic simulations

The effective modal indices of the curved Cu-TiO_{2}-Si HPWs are calculated using the eigenmode expansive (EME) method [25]. The Si core has an asymmetric rib structure to mimic the donut resonator shown in Fig. 1(b). The electrical field intensity distribution of 1550-nm fundamental TM mode in the curved HPW with parameters as listed in Table 1 is depicted in Fig. 2(a), showing that the electric field is enhanced in the TiO_{2} layer as well as the Si core just beneath the TiO_{2} layer. The lateral confinement is well provided by the Si core as the extended TiO_{2} region (i.e., over the Si core region laterally) contains weak electric field. The effective modal index is calculated to be 2.275 + *i*0.00577, corresponding to a propagation loss of 0.203 dB/μm, which is close to the experimental result [23]. The ratios of optical intensity in the TiO_{2} layer, the Si rib, the Si slab, and the surrounding SiO_{2} cladding layer are 41.3%, 42.0%, 0.04%, and 16.7%, respectively. As expected, the thin Si slab has negligible effect on the optical mode.

The real (n_{eff}) and imaginary part (k_{eff}) of effective modal indices are plotted in Fig. 2(b) as a function of temperature in the range of ± 20°C deviated from RT for two curved HPWs which have TiO_{2} thicknesses *t _{ox}* of 10 nm and 50 nm, respectively. Both n

_{eff}and k

_{eff}depend on temperature almost linearly. Thus, TO coefficients of the effective modal index (i.e.,

*d*n

_{eff}/

*d*T and

*d*k

_{eff}/

*d*T) can be deduced from only two temperature points. n

_{eff}determines the resonant wavelengths (λ

_{r}) and k

_{eff}determines the extinction ratio (ER) of WRRs [26]. Since

*d*k

_{eff}/

*d*T is relatively small for our HPWs, e.g., ~1.5 × 10

^{−5}K

^{−1}for the 10-nm-TiO

_{2}HPW and ~8.0 × 10

^{−6}K

^{−1}for the 50-nm-TiO

_{2}HPW, as read from Fig. 2(b), a WRR can be claimed to be athermal if its

*d*n

_{eff}/

*d*T is zero even its

*d*k

_{eff}/

*d*T is not zero.

Figure 3 plots *d*n_{eff}/*d*T versus *t _{ox}* for curved Cu-TiO

_{2}-Si HPWs. As expected,

*d*n

_{eff}/

*d*T decreases monotonically with t

_{ox}increasing because the ratio of optical intensity in the TiO

_{2}layer increases. However, the slope of the

*d*n

_{eff}/

*d*T~

*t*curves decreases with

_{ox}*t*increasing. This observation can be explained by the fact that the hybrid mode shown in Fig. 2(a) is a superposition of a pure SPP mode located at the Cu/TiO

_{ox}_{2}interface (i.e., waveguide without the Si core) and a pure optical mode located at the Si core (i.e., waveguide without the metal). The hybrid mode becomes more optical-like when t

_{ox}increasing [27]. In the extreme case when t

_{ox}is sufficiently larger (e.g., > ~200 nm), it behaves as a pure optical mode as the conventional Si waveguide with TiO

_{2}behaving as a cladding layer, thus

*d*n

_{eff}/

*d*T will be independent on

*t*. One sees from Fig. 3 that the

_{ox}*d*n

_{eff}/

*d*T~

*t*curve depends on the Si core width

_{ox}*W*weakly, while depends on the Si height

_{P}*H*strongly. The critical

*t*at which

_{ox}*d*n

_{eff}/

*d*T = 0 (i.e., athermal point) depends on

*W*weakly, while increases with

_{P}*H*increasing. Electrically, a thin gate dielectric is preferred to reduce the driving voltage. Optically, the height of the Si waveguide should be thick enough for vertical optical confinement. To balance the electric and optical requirements, our modulator is set to be

*H*= 160 nm and

*t*= 50 nm. It is athermal as read from Fig. 3(b).

_{ox}## 4. Electrical simulations

A semiconductor device simulation software MEDICI is used to obtain the two-dimensional (2D) dynamic free carrier distribution in the MOS capacitor at different biases, as in the case of conventional Si MOS modulators [28]. The dielectric constant of TiO_{2} is set to 80, which is reachable for a high-quality TiO_{2} film [22]. The Si core of the resonator is n-type doped with concentration (*N _{D}*) of 5 × 10

^{18}cm

^{−3}in the rib and slab region and 2 × 10

^{20}cm

^{−3}in the contact region for good Ohmic contact. Auger recombination, Shockley-Hall-Read recombination, surface recombination, Fermi-Dirac statics, and the Modified Local Density Approximation (MLDA) method in the MEDICI are included to account for the heavy doping and the quantum confinement effect on the carrier concentration near the TiO

_{2}/Si interface [29]. The 2D free carrier distribution in the MOS capacitor under 5-V bias is shown in Fig. 4(a). The accumulated electrons are located near the TiO

_{2}/Si interface. To see the 2D distribution more clearly, the figure near the interface is enlarged, as shown in Fig. 4(b). We can see that the electron concentration contours are almost in parallel with the TiO

_{2}/Si interface. Therefore, the 2D distribution of free electron distribution

*N(x,y)*can be simplified by a one-dimensional (1D) distribution

*N(y)*. Figure 4(c) plots 1D electron distributions along y-axis for the Cu-TiO

_{2}-Si MOS capacitor under different biases ranging from −1 V to 8 V. At the −1 V bias, the electrons are depleted from the interface. The depletion width

*W*($\propto {N}_{D}{}^{-0.5}$) is ~16.3 nm when

_{dep}*N*= 5 × 10

_{D}^{18}cm

^{−3}. With the gate voltage increasing, the electrons are accumulated at the interface, maximizing at a short distance (~0.3-0.5 nm) away from the interface due to the quantum mechanical effect and then decreasing to N

_{D}quickly with the distance from the interface increasing. The free-electron distribution approaches to the interface more closely when V increases. These results agree well with the experimental observation [30]. As a first approximation, the electron distribution is approximated by a step function to define an accumulation layer (AcL) which has width of

*t*and average concentration of

_{AcL}*N*as:

_{AcL}*ε*is the vacuum permittivity,

_{0}*ε*is the dielectric constant of the gate dielectric,

_{d}*e*is electronic charge,

*V*is the flat-band voltage, and

_{FB}*E*is the electric field in the gate dielectric. Here, we simply assume

_{d}*t*= 1nm. At large V,

_{AcL}*t*will be smaller than 1 nm and

_{AcL}*N*will be larger than that predicted from Eq. (1). The achievable

_{AcL}*N*depends on the breakdown field of the gate dielectric, and it can be larger than 10

_{AcL}^{20}cm

^{−3}for modern CMOS devices.

For transient state simulations, the gate voltage V is increased from 0 to 8 V with the ramp time of the gate voltage of 10 fs. The free electron concentration in the 1-nm AcL is plotted in Fig. 5 as a function of time. Rise time (*t _{r}*) of the electron concentration is defined as the time for the electron concentration to increase from 10% to 90%, and fall time (

*t*) is defined as the time for the electron concentration to drop from 90% to 10%. In the case of Si n

_{f}^{+}-contact just below the electrode, as shown in Fig. 4(a), the sum of these times

*(τ*) is ~29 ps. The modulation speed estimated from the inverse of

_{s}= t_{r}+ t_{f}*τ*is ~34 GHz. The speed can be improved by shortening the distance between the accumulation layer and the n

_{s}^{+}contact. In the case that the n

^{+}contact in the Si slab is extended to the Si rib, both

*t*and

_{r}*t*decrease, as shown by the dash curve in Fig. 5.

_{f}*τ*is read to ~9.3 ps for this MOS capacitor, which corresponds to a ~107-GHz modulation speed. However, the propagation loss of the curved HPW will increase from 0.27 dB/μm to 0.31 dB/μm when the doping level in the Si slab is increased from 5 × 10

_{s}^{18}cm

^{−3}to 2 × 10

^{20}cm

^{−3}(while keeping the doping level in the Si rib to be 5 × 10

^{18}cm

^{−3}). To balance the propagation loss and the speed, the n

^{+}doping may be extended to a certain location between the Si rib and the electrode.

The proposed EO modulator is a MOS capacitor working between the depletion and accumulation states. The switching energy *E _{s}* per bit of the MOS modulator can be roughly estimated as:

*C*and

_{dep}*C*are capacitances under the depletion and accumulation states respectively. Because

_{accu}*C*is smaller than

_{dep}*C*and

_{accu}*V*is smaller than

_{dep}*V*,

_{accu}*E*is mainly determined by the second term of Eq. (2), namely the accumulation state.

_{s}*C*can be approximated to the gate oxide capacitance as ${C}_{accu}\approx A\frac{{t}_{ox}}{{\epsilon}_{0}\cdot {\epsilon}_{d}}$, where

_{accu}*A*is the active area. For the modulator with structural parameters as listed in Table 1,

*A*is 1.76 μm

^{2}and

*C*is ~25 fF. If the driving voltage is 6 V,

_{accu}*E*is estimated to be ~0.45 pJ.

_{s}## 5. Electro-optic simulations

The modification of Si reflective index (n_{Si} + *i*k_{Si}) at 1550 nm depends on the free carrier concentration (*ΔN _{e}* for electrons and

*ΔN*for holes) almost linearly as:

_{h}However, when the free carrier concentration becomes very large, e.g., > ~10^{20} cm^{−3}, Δn and Δα will deviate significantly from the above linear dependence. Instead, the well-known Drude model can be used to estimate Si complex permittivity (*ε*) and index at this region [31]:

*ε*( = 11.7) is the Si static permittivity,

_{∞}*m**( = 0.272

*m*) is the electron effective mass, and

_{0}*τ*is the electron relaxation time. The calculated n

_{Si}and k

_{Si}are plotted in Fig. 6(a) as a function of

*ΔN*at λ = 1550 nm (

_{D}*ω*= 1.2 × 10

^{15}s

^{−1}).

The effective modal index is calculated as a function of *N _{AcL}* in the 1-nm AcL for curved Cu-TiO

_{2}-Si HPWs with W

_{P}of 100, 200, and 300 nm, respectively (the Si height

*H*is 160 nm and the other parameters are as listed in Table 1). The results are shown in Fig. 6(b). The index of the 5 × 10

^{18}cm

^{−3}-doped Si is set to 3.4506 +

*i*0.00123 based on Eq. (3) and the index of the depleted Si is 3.455. We can see that the real part of effective modal index decreases and the imaginary part increases almost exponentially with

*N*increasing, which can be explained by the increase of optical modal confinement in the 1-nm AcL. For example, Fig. 7(a) shows electric field (|E

_{AcL}_{y}|) 2D distribution of fundamental TM mode in the curved Cu-TiO

_{2}-Si HPW with

*N*= 1 × 10

_{AcL}^{20}cm

^{−3}. To see the field distribution in the 1-nm-thick AcL more clearly, normalized |E

_{y}| 1D distributions along the y-axis at x = 0, shown as the dash line in Fig. 7(a), are plotted in Fig. 7(b) for HPWs with

*N*= 1 × 10

_{AcL}^{20}cm

^{−3}and 8 × 10

^{20}cm

^{−3}respectively. In the case of

*N*= 8 × 10

_{AcL}^{20}cm

^{−3}, the electric field in the 1-nm AcL is dramatically enhanced. Because the continuity of electric displacement normal to the interfaces makes

*E*is inversely proportional to the permittivity roughly, the Si permittivity becomes smaller than the permittivity of TiO

_{y}_{2}when

*N*is larger than ~6 × 10

_{AcL}^{20}cm

^{−3}as read from 6(a), which results in dramatic enhancement of optical field in this thin layer. To see this point more clearly, Fig. 7(c) plots the optical intensity ratio in the 1-nm AcL, which is defined as the optical power contained in this region over the total optical power, as a function of

*N*. From Figs. 6(c) and 7(c), one sees that the modulation efficiency is

_{AcL}*W*dependent. The HPW with 100-nm

_{P}*W*has smaller modulation efficiency than the HPWs with larger

_{P}*W*s while the HPW with 200-nm

_{P}*W*provides similar modulation efficiency as HPW with 300-nm

_{P}*W*. This is because the lateral confinement is weak when

_{P}*W*is too small (e.g., < ~100 nm) while it becomes saturated when

_{P}*W*is large enough (e.g., > ~200 nm). On the other hand, the modulator with smaller

_{P}*W*has a smaller active area, which means a faster modulation speed. To balance the optical and electric performance,

_{P}*W*is set to be 200 nm for our modulator.

_{P}It has been experimentally demonstrated that the transmission spectrum of HPW-based WRRs can in general be expressed as [18,19]:

^{2}is the power loss factor per roundtrip around the ring, $t=\left|t\right|\mathrm{exp}\left(i\vartheta \right)$ is the self-coupling coefficient, and n

_{g}is the group index.

*α*at different states can be calculated from k

^{2}_{eff}read from Fig. 6(b). n

_{g}in the depletion state is calculated to ~3.782, close to the experimental result [19]. Because it is difficult to calculate the n

_{g}value accurately, we simply assume that the free-carrier effect induced n

_{g}modification is the same as the n

_{eff}modification, namely

*Δn*, thus n

_{g}= Δn_{eff}_{g}at different states and different

*N*s can also be read from Fig. 6(b). Since n

_{AcL}_{g}is larger than n

_{eff}, this assumption may underestimate the modification efficiency of our modulator. The self-coupling coefficient

*t*is related with the cross-coupling coefficient

*k*as ${t}^{2}+{k}^{2}=1$

**.**The coupling between the bus waveguide and the resonator depends on the separation between them (

*gap*) and the effective modal index difference between the bus waveguide and the curved plasmonic waveguide [26]. The

*|t|*value can be varied in a large range from over-coupling

*(|t| < α)*to under-coupling

*(|t| > α*), depending on the detailed structural parameters of WRRs [19]. The relationship between the coupling strength and the WRR’s structural parameters has been well studied [26], and will not be studied in detail in this paper. Instead, we simply assume

*|t|*to a certain value and assume $\vartheta =0$ (i.e., no phase shift due to coupling). Figure 8(a) plots the calculated spectra of our modulator in the case of

*|t|*= 0.8. Because

*α*is ~0.75 in the depletion state, the resonator is slightly under-coupling with ER of ~18 dB. With

*N*increasing, n

_{AcL}_{g}decreases and

*α*increases simultaneously, which leads to

*λ*blue-shift and ER reduction as the resonator becomes more under-coupled. In contrast, for conventional Si WRR modulators, only the shift of

_{r}*λ*occurs and ER is not changed because

_{r}*α*is almost independent on the applied voltage. One can see from Fig. 8(a) that the EO induced modulation (modulation depth) is enhanced in the region of wavelength larger than

*λ*due to this additional

_{r}*α*modification, while in the region of wavelength smaller than

*λ*, the modulation depth is weakened. Figure 8(b) plots the calculated spectra in the case of over-coupling by assuming

_{r}*|t|*= 0.5 which has ER of ~8 dB in the depletion state. With

*N*increasing, the resonance wavelength is blue-shifted and the ER increases. ER becomes very large when

_{AcL}*α*approaches to |

*t|*. The modulation depth is enhanced in the region of wavelength smaller than

*λ*and is weakened in the region of wavelength larger than

_{r}*λ*due to this additional

_{r}*α*modification. It indicates that the performance of our plasmonic modulators depends on

*|t|*more sensitively than the conventional Si WRR modulators. Moreover, because of the relatively large propagation loss of the plasmonic waveguide (hence the small

*α*value) as compared with the conventional Si waveguide, our plasmonic modulators require strong coupling (hence the small

*|t|*value) between the bus waveguide and the resonator to meet the critical coupling condition.

The simple calculation based on Eq. (5) represents an ideal condition of WRRs in which many effects are ignored. To verify the results observed in Fig. 8, three-dimensional (3D) full-difference time-domain (FDTD) simulation is performed. To enhance the coupling between the bus waveguide and the resonator, a race-track shaped resonator is used. The gap between the bus waveguide and the resonator is set to 10 nm and the directional coupling length is set to be 500 nm. The total footprint of the modulator inclusive electrodes is ~8.6 μm^{2}, and the active area *A* is ~1.96 μm^{2}. To minimize the simulation error during simulation in different states, only the Si complex index in the 1-nm-thick AcL is changed as read from Fig. 6(b), while all other settings including the grid size keep the same. Figure 9(a) plots the transmission spectra for the modulator under accumulation states with *N _{AcL}* = 1 × 10

^{20}cm

^{−3}or 6 × 10

^{20}cm

^{−3}, which corresponds to a bias of 1.1 V or 6.8 V, respectively, according to Eq. (1). In the case of

*N*= 1 × 10

_{AcL}^{20}cm

^{−3}, ER is ~9 dB near 1550 nm, which corresponds to

*α*= ~0.74 and

*|t|*= ~0.85. When

*N*increases to 6 × 10

_{AcL}^{20}cm

^{−3}, the resonant waveguides are blue-shifted by ~13.5 nm and ER is reduced to be ~2.5 dB because of the reduction of

*α*to ~0.67. At 1546 nm wavelength, the output power is modified from ~-10 dB to ~-2 dB by increasing

*N*from 1 × 10

_{AcL}^{20}to 6 × 10

^{20}cm

^{−3}. The optical power distributions in the modulator with

*N*of 1 × 10

_{AcL}^{20}and 6 × 10

^{20}cm

^{−3}are shown in Figs. 9(b) and 9(c) respectively, and dynamic light propagation through the modulator is shown by the attached movie (Media 1). One sees that in the case of

*N*= 1 × 10

_{AcL}^{20}cm

^{−3}, the modulator is almost in on-resonance as

*α*being close to

*|t|*, which results in optical trap in the resonator and small output power of ~-10 dB. In the case of

*N*= 6 × 10

_{AcL}^{20}cm

^{−3}, the modulator is in off-resonator as

*α*being larger than

*|t|*, which results in large output power of ~-2 dB. Thus this modulator operating at 1546 nm has modulation depth of ~8.0 dB and insertion loss of ~2 dB. The relatively low insertion loss (as compared with other plasmonic modulators [15,16]) is benefited from the conventional Si channel bus waveguide. As the abovementioned, both the modulation depth and the insertion loss can be further improved by adjusting the coupling between the bus waveguide and the resonator.

## 6. Conclusions

In summary, an ultracompact Si WRR modulators based on the recently developed Cu-insulator-Si HPW is proposed. The modulator is atheraml when TiO_{2} with a certain thickness is used as the insulator. The EO modulation is achieved by free-electron accumulation near the TiO_{2}/Si interface. Significant modification of both the real and imaginary parts of effective modal index of the curved Cu-TiO_{2}-Si HPW can be obtained if *N _{AcL}* is large enough (e.g., > ~6 × 10

^{20}cm

^{−3}), which leads to the resonance wavelength blue-shift and the extinction ratio variation simultaneously. The modulator provides a large modulation depth and a small insertion loss after optimization of the coupling between the bus waveguide and the resonator. The modulator offers high speed, up to ~100 GHz dependent on the doping scheme, and low power consumption of ~0.45 pJ/bit owing to its ultracompact footprint. Moreover, the resonator can be further scaled down to submicron radius and offers a relatively large fabrication tolerance. These promising performances combined with the fully CMOS compatibility make the proposed modulator very attractive for dense Si EPICs.

## Appendix

In this appendix, we discuss the further miniaturization of the ring radius and the fabrication tolerance.

In the above analysis, the radius of the plasmonic resonator is set to *R* = 1.5 μm. The radius (hence the footprint of modulator) can be further reduced. Figure 10 plots the effective modal index and the TO coefficient as a function of the bending radius for curved Cu-TiO_{2}-Si HPWs with different *W _{P}*s. The real part of effective modal index decreases and the imaginary part increases with

*R*decreasing due to outward shift of the optical mode [20], especially for the HPW with wider

*W*. A noticeable variation of the effective modal index occurs when

_{P}*R*is less than ~1.0 μm for HPW with W

_{P}of ≤ ~200 nm. It indicates that the bending radius of our modulator can be reduced to ~1.0 μm without noticeable performance degradation. Moreover, the thermo-optic property keeps almost unchanged for bending radius as small as 0.5 μm, as shown in Fig. 10(b).

The TiO_{2}/Cu cap of the Cu-TiO_{2}-Si waveguide is designed to be wider than the beneath Si core by *ΔW _{P}*, as shown in the inset of Fig. 11(a). In literature, the metal/insulator width in the metal-insulator-Si hybrid plasmonic waveguides is either infinite [18,19,27] or is the same as the Si core [32]. Here, the effect of the width of the TiO

_{2}/Cu cap is studied. Figure 11 plots the real and imaginary parts of effective modal index (compared to that with the infinite wide TiO

_{2}/Cu cap, namely

*ΔW*= ∞) as a function of

_{P}*ΔW*for curved Cu-TiO

_{P}_{2}-Si HPWs with

*W*of 100, 200, and 300 nm, respectively. With

_{P}*ΔW*increasing, n

_{P}_{eff}increases first and then decreases slowing while the k

_{eff}increases continuously. Nevertheless, the dependence of the effective modal index on

*ΔW*is weak, especially for HPWs with wider

_{P}*W*, in consistent with the fact that a good lateral confinement can be provided by the beneath Si core in the Cu-TiO

_{P}_{2}-Si HPWs when

*W*is wide enough (> ~200 nm).

_{P}In fabrication, the central line of the TiO_{2}/Cu cap may misalign from the central line of the beneath Si core, as shown schematically in the inset of Fig. 12(a). In the case that the TiO_{2}/Cu cap is intentionally designed to be wider than the beneath Si core by 50 nm in each side, the over-width in one side will be (50nm - *ΔW _{P}’*) and that in the other side will be (50nm +

*ΔW*) if the misalignment is

_{P}’*ΔW*. Figure 12 plots the real and imaginary parts of effective modal index (compared to that without misalignment, namely

_{P}’*ΔW*= 0) as a function of

_{P}’*ΔW*for curved Cu-TiO

_{P}’_{2}-Si HPWs with

*W*of 100, 200, and 300 nm, respectively. Both the real and imaginary parts of the modal index decrease with

_{P}*|ΔW*increasing. While in the

_{P}’|*ΔW*range from −10 nm to 20 nm, the variation of the effective modal index with

_{P}’*ΔW*is very small, especially for HPWs with wider

_{P}’*W*. It indicates that our EA modulator has a relatively large misalignment tolerance of ~15 nm for the TiO

_{P}_{2}/Cu cap fabrication. It should be noted that fabrication of the coupling region will have small tolerance, as the conventional Si WRR modulators.

## Acknowledgment

This work was supported by the Science and Engineering Research Council of A*STAR (Agency for Science, Technology and Research), Singapore Grant 092-154-0098.

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