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Abstract
The electrical bandwidth of an electro-optic modulator plays a vital role in determining the throughput of an optical communications link. We propose a broadband plasmonic electro-optic modulator operating at telecommunications wavelengths (λ0 ∼ 1550 nm), based on free carrier dispersion in indium tin oxide (ITO). The ITO is driven through its epsilon-near-zero point within the accumulation layers of metal-oxide-semiconductor (MOS) structures. The MOS structures are integrated into a pair of coupled metal-insulator-metal (MIM) waveguides aligned on a planarized silicon waveguide. The coupled MIM waveguides support symmetric and asymmetric plasmonic supermodes, excited adiabatically using mode transformation tapers, by the fundamental TM0 and TE0 modes of the underlying silicon waveguide, respectively, such that the modulator can operate in either mode as selected by the input polarisation to the silicon waveguide. The modulator has an active section 1.5 to 2 µm long, enabling the modulator to operate as a lumped element to bandwidths exceeding 200 GHz (3 dB electrical, RC-limited). The modulators produce an extinction ratio in the range of 3.5 to 6 dB, and an insertion loss in the range of 4 to 7.5 dB including input/output mode conversion losses. The AC drive voltage is ±1.75 V. The devices comprise only inorganic materials and are realisable using standard deposition, etching and nanolithography techniques.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The ongoing increase in demand for data requires modulators to be high speed and efficient [1]. Silicon modulators offer good performance while being amenable to well-established CMOS fabrication technologies [2–4]. However, in recent years, a variety of materials have shown improvement in different aspects of modulator performance, such as organic polymers [5,6], phase change materials [7], graphene [8], and indium tin oxide [9–12]. Plasmonic modulator structures allow light confinement beyond the diffraction limit, thus allowing extremely compact modulators [5,13,14]. They offer additional advantages such as small capacitances and resistances, and consequently, small associated RC time constants which enable lumped-element operation over extremely high electrical bandwidths.
Inorganic electro-optic materials offering strong index modulation and compatibility with silicon photonics represent ideal choices for plasmonic modulator design. One such material is indium tin oxide (ITO), which recently has shown potential in several photonic devices including modulators [9–12] and optical phased arrays [15–17]. In a metal-oxide-semiconductor (MOS) structure, the charge density within the ITO can be modulated by an applied bias, which in turn modulates the complex refractive index of ITO. Depending on the polarity and the magnitude of the applied voltage, the device can be driven into accumulation or depletion near the oxide-ITO interface. Given the low background permittivity of ITO, the epsilon-near-zero (ENZ) regime can be accessed under strong accumulation. As the carrier concentration crosses the ENZ point, ITO undergoes a transition from dielectric-like to metal-like behaviour resulting in a very large refractive index modulation at the telecommunication wavelengths. This effect can be combined with the strong light confinement produced by plasmonic waveguides, leading to deep optical modulation over a large electrical bandwidth in a compact footprint.
In most of the previously reported designs of ITO modulators based on plasmonic structures (cf. [8,10,11]), a hybrid plasmonic waveguide (HPW) configuration was used to manage the confinement-attenuation trade-off [18], implemented as a MOS structure. A design based on coupled hybrid plasmonic waveguides (CHPW) has also been proposed for modulator applications [19], as another approach to manage the confinement-attenuation trade-off. These designs use direct butt coupling from an input silicon waveguide, and do not include mode transformation sections to optimise coupling to the HPW mode. Due to mode mismatch, direct butt coupling from the input silicon waveguide produces insertion loss and may result in power coupling to other forward propagating modes such that the power output depends on the interference between the excited modes of the HPW modulator section. Also, this configuration gives rise to additional semiconductor resistance in the cathode contacts, which in turn limit the electrical bandwidth of the design.
We propose a design for an ITO based plasmonic modulator where the carrier density is modulated in metal-oxide-semiconductor (MOS) structures backed by a bottom metal film, which collectively forms a vertical plasmonic metal-insulator-metal (MIM) waveguide [18]. The MIM waveguide strongly confined fields in the thin HfO2-ITO regions, producing a strong overlap between the modal field and the ITO layer which is perturbed into the ENZ region, and thus requires a smaller length of the active region to achieve a given extinction ratio. A smaller active area enables the capacitances to be very small, leading to a very high electrical bandwidth. A pair of coupled MIM waveguides form the active section of the modulator, which operates with coupled plasmonic modes. Input and output taper sections are designed for efficient excitation of the relevant mode of the active section by the fundamental TE or TM mode of an underlying Si waveguide. The input and output tapered sections are kept electrically isolated from the active section by a gap in the anode metal film at the junction between them, such that they do not contribute to the device capacitance and do not limit the electrical bandwidth of the design.
We first give an overview of our design, then present charge transport results within the ITO layer to obtain the spatial charge density as a function of applied voltage. Subsequently, we transform the spatial charge density profiles into corresponding refractive index profiles and carry out modal analyses of the active section to determine the optical performance. We then discuss how the relevant modes of the active section of the device can be excited by an appropriately designed taper. We also calculate the parasitic capacitance and resistance, and estimate the electrical bandwidth of the design. Finally, we conclude with brief remarks.
2. Device concept
Our concept exploits an active section of length ${L_{\textrm{active}}}$, and input and output mode conversion tapers of length ${L_\textrm{T}}$, as shown in Fig. 1. The active section is responsible for modulating the MIM modes propagating through, whereas the input and output tapered sections transform adiabatically the modes of the underlying Si waveguide into corresponding modes of the active section, and vice versa. The modes supported by the active section are TM supermodes of the MIM waveguide pair, having symmetric or asymmetric lateral field distributions, evolving along the input taper from the fundamental TM or TE mode (respectively) of the Si waveguide. The same taper design can enable adiabatic transformation of the TE or TM mode of the Si waveguide, such that the modulator can operate with either, depending on which one is launched into the input Si waveguide. The same taper design is applied at the output of the active section. The computed mode profiles plotted on Fig. 1 show the transformations.
Fig. 1. Schematic of the modulator along with expanded cross-sectional views of the active section. The structure is covered by a 20 nm thick hafnia passivation layer (not shown) which also infills the slits (${s_w}$) created to isolate electrically the tapered sections from the active section.
The active section of the design is composed of a planarized silicon waveguide below two metal-oxide-semiconductor-metal stacks, separated from each other by a gap g. The top Au anode along with the hafnia ($\textrm{Hf}{\textrm{O}_2}$) and ITO films form a MOS structure. The Au cathode below the ITO serves three purposes: (i) forming MIM plasmonic waveguides producing strong confinement in the region between the two Au films to increase optical overlap with the accumulation region in the ITO; (ii) forming a low-loss Ohmic contact to the ITO; and (iii) enabling low-resistance electrical ground (cathode) contacts by extending this layer to the sides (rather than the ITO region). Anode contacts can be formed by extending the top Au/ $\textrm{Hf}{\textrm{O}_2}$ layers to their respective sides. The anode and cathode contacts on each side of the device can be thickened into signal (${V_s}$) and ground pads for contacting to a broadband electrical probe.
The applied voltage between the anode and cathode results in a vertical charge distribution profile within the ITO region. This leads to a corresponding change in the complex refractive index in the ITO, and the complex effective index of the propagating MIM modes. Depending upon the change in the real or imaginary part of the effective index, the propagating modes undergo a modulation of phase or absorption as a function of applied voltage. The modulation efficiency depends primarily on the interaction between the modal fields and the perturbed semiconductor region, which justifies our preference for MIM structures.
The active section is electrically isolated from the tapered sections by slits of width ${s_w}$ in the top metal layer at the junction of the active section and each tapered section. These slits are required to eliminate parasitic capacitance and resistance associated with the tapers, thereby enabling a broad electrical bandwidth. We assume a 20 nm thick hafnia passivation layer on top of the device (not shown in Fig. 1), which also infills the slits of width ${s_w}$.
Finally, an embedded planarized Si waveguide is selected for three reasons: (i) to provide a flat planar surface facilitating fabrication of the modulator stack; (ii) to minimise the parasitic series resistance associated with the anode and cathode contacts, and (iii) to enable effective mode conversion tapers to adiabatically transform the modes of the underlying Si waveguide into corresponding plasmonic modes of the active section.
Modelling this design requires a variety of simulation techniques. For the modal analysis, we use the vectorial finite difference eigenmode (FDE) technique. We use the eigenmode expansion (EME) method to model the adiabatic mode conversion tapers. To calculate the insertion loss associated with the gap region between the active and tapered sections, we use the finite difference time domain (FDTD) method as the absence of guided mode in the gap region makes it difficult to model using the EME method. For charge transport simulation, we find self-consistent solutions to Poisson’s and the drift-diffusion equations. All of these simulation techniques are available as different modules in Ansys Lumerical [20]. FDE and EME solvers are part of the MODE module whereas FDTD is a standalone module. Charge transport solver is available in the Multiphysics module.
Fabrication could proceed by applying nanofabrication techniques implemented using wafer processing tools [9,15,17,19]. For instance, embedded Si waveguides could be fabricated by etching a SOI wafer, depositing a SiO2 cladding, and planarizing using chemical-mechanical polishing. The MIM stack could be formed by depositing and lifting-off the bottom Au layer (cathode) and the thin ITO layer. The HfO2 layer could be formed by atomic layer deposition, and the top Au layer (anode) also by deposition and lift-off. The gap region of width g could be formed by focussed ion beam milling as well as the isolation slits (Sw).
3. Results and discussion
3.1. Electrostatic modeling
The detailed cross-sectional view of the active section of our proposed design is shown in Fig. 2. As the driving electric field is applied between the anode (top Au electrode) and cathode (bottom Au electrode), the carrier concentration within the ITO region starts to vary. Depending upon the applied voltage and its relation to the work function difference between the anode material and ITO (${\phi _{\textrm{Au}}} - {\phi _{\textrm{ITO}}}$) there may be accumulation or depletion of the majority carriers (electrons) within the ITO near the $\textrm{Hf}{\textrm{O}_2}$-ITO interface. For an applied voltage ${V_\textrm{s}} > {V_{\textrm{fb}}}$, where ${V_{\textrm{fb}}} = ({{\phi_{\textrm{Au}}} - {\phi_{\textrm{ITO}}}} )/e$ is the flat-band voltage, free electrons accumulate near the hafnia-ITO interface, whereas for ${V_\textrm{s}} < {V_{\textrm{fb}}}$, the ITO is depleted near the $\textrm{Hf}{\textrm{O}_2}$-ITO interface (bottom right inset of Fig. 2). Inversion is not typically observed in ITO given its large bandgap energy.
Fig. 2. Expanded cross-sectional view of the active section of the modulator. The bottom right inset illustrates two regimes of operation: accumulation ${V_\textrm{s}} > {V_{\textrm{fb}}}$ (left), and depletion ${V_\textrm{s}} < {V_{\textrm{fb}}}$ (right).
The 2D charge distribution over the device cross-section was computed by finding self-consistent solutions to the Poisson and drift-diffusion equations [20]. The length of the active section is expected to be larger (µm’s) than its width (100’s nm), so computationally expensive 3D charge simulations are not required.
ITO was modelled as an n-doped semiconductor of bandgap energy ${E_\textrm{g}} = 2.8$ eV, DC permittivity $\varepsilon _{\textrm{DC}}^{\textrm{ITO}} = 9.3$, electron affinity ${\chi _\textrm{s}} = 4.8$ eV, effective electron mass $m_\textrm{n}^\ast{=} 0.35{m_\textrm{e}}$, and effective hole mass $m_\textrm{p}^\ast{=} {m_\textrm{e}}$, where ${m_\textrm{e}}$ is the mass of a free electron [15]. In addition, we also assume an unperturbed carrier density of ${10^{19}}$ $\textrm{c}{\textrm{m}^{ - 3}}$. Corresponding to this value of carrier concentration, the Fermi level of ITO lies within the conduction band, so ITO behaves as a degenerate semiconductor. The work function of ITO (${\phi _{\textrm{ITO}}}$) with unperturbed carrier density of ${10^{19}}$ $\textrm{c}{\textrm{m}^{ - 3}}$ is obtained as 4.766 eV. The work function of gold is taken as ${\phi _{\textrm{Au}}} = 5.1$ eV. $\textrm{Hf}{\textrm{O}_2}$ was selected as the insulating layer because its DC relative permittivity is high, taken as 25 [21,22], such that the required charge perturbation in ITO can be achieved at a voltage much smaller than required for other dielectrics of the same thickness but lower permittivity, such as $\textrm{Si}{\textrm{O}_2}$ or $\textrm{A}{\textrm{l}_2}{\textrm{O}_3}$. The breakdown field in $\textrm{Hf}{\textrm{O}_2}$ strongly depends on the deposition process. For $\textrm{Hf}{\textrm{O}_2}$, we assume a breakdown field of 6.4 MV/cm [16], which corresponds to a breakdown voltage of ${V_\textrm{b}} = 3.2$ V for a 5 nm thick film. The underlying silicon waveguide has a negligible effect on the charge distribution in ITO, so it is not included in the charge simulations. The thicknesses of the $\textrm{Hf}{\textrm{O}_2}$ and ITO films (${h_{\textrm{Haf}}}$, ${h_{\textrm{ITO}}}$) are chosen as 5 and 20 nm, respectively. The cathode and anode metal films are assumed to be 50 nm thick. The width of the cathode (${w_{\textrm{cat}}}$) is chosen as 250 nm as below this value the MIM structure does not support the plasmonic supermodes of interest.
We simulate the structure to obtain the spatial carrier profile within the ITO as a function of applied voltage from - 4 V to + 4 V. We observe two different regimes: accumulation for ${V_s} > 0.334$ V, and depletion for ${V_s} < 0.334$ V, where $({{\phi_{\textrm{Au}}} - {\phi_{\textrm{ITO}}}} )/\textrm{e} = 0.334$ V is the flat-band voltage. In Fig. 3, we plot the 2D electron profiles within the ITO for −2 and 3 V, which are in the depletion and accumulation regimes, respectively. As the structure is symmetric along the horizontal ($y$), we show the charge distribution in the right half of the structure only.
Fig. 3. 2D electron concentration distributions in the ITO over the right half of the device for ${V_s} ={-} 2$ V (depletion, top) and 4 V (accumulation, bottom). The colorbar is in units of $\textrm{c}{\textrm{m}^{ - 3}}$.
As the width of the MOS stack is larger than the thickness of the ITO, the 1D carrier density profile in the z-direction does not vary much along the width of the ITO film except near the corners. For better visualization of the charge profile as a function of applied voltage, we plot the 1D carrier density along the thickness of ITO at the horizontal centre of one of the MOS stacks. Figure 4(a) shows the 1D carrier density distribution along the ITO thickness in the depletion regime ${V_\textrm{s}} < {V_{\textrm{fb}}}$, revealing that the depletion width extends deeper into the ITO as the applied voltage drops further below the flat-band voltage.
Fig. 4. 1D electron concentration distributions along the ITO thickness for different applied voltages in (a) the depletion and (b) accumulation regimes.
In Fig. 4(b), we plot the 1D charge profile in the accumulation regime ${V_\textrm{s}} > {V_{\textrm{fb}}}$, along with an expanded view of the distribution for better visualization. The charge density profile decays exponentially away from the $\textrm{Hf}{\textrm{O}_2}$-ITO interface, following classical drift-diffusion theory (the accuracy of the profile could be improved by using the more computationally intensive Schrödinger-Poisson theory [23]). The charge density is observed to increase near the $\textrm{Hf}{\textrm{O}_2}$-ITO interface as the voltage increases above the flat-band voltage. The horizontal dashed line in Fig. 4(b) represents the epsilon-near-zero (ENZ) carrier concentration at the free-space operating wavelength of $\lambda_0 = 1550$ nm (our design wavelength), where the real part of permittivity becomes zero (the ENZ point and its relation to the carrier concentration is discussed in greater detail in the next section). As the voltage increases, the ENZ point shifts away from the $\textrm{Hf}{\textrm{O}_2}$-ITO interface into the bulk ITO. On one side of the ENZ point the carrier concentration is greater than the ENZ concentration and ITO appears metallic, whereas on the other side it appears dielectric.
Next, we plot the potential distribution inside the $\textrm{Hf}{\textrm{O}_2}$ and ITO for different values of applied voltage, as shown in Fig. 5. The maximum voltage that can be applied is limited by the breakdown field of $\textrm{Hf}{\textrm{O}_2}$, and is determined by the potential drop across this layer. For example, at an applied voltage of 4 V, the voltage drop across $\textrm{Hf}{\textrm{O}_2}$ is ${\sim} 2.5$ V, which is lower than the breakdown voltage of 3.2 V for 5 nm thick $\textrm{Hf}{\textrm{O}_2}$. Thus, to evaluate the performance of our design, we keep the maximum applied voltage to +4 V to keep the voltage drop across $\textrm{Hf}{\textrm{O}_2}$ lower than the breakdown voltage. The values of potential plotted in Fig. 5 are referenced to the intrinsic Fermi level.
Fig. 5. Potential distribution within the ITO and $\textrm{Hf}{\textrm{O}_2}$ layers for different applied voltages.
In the next section, we use the 2D electron profiles obtained in this section at different values of applied voltage (cf. Figure 3) to obtain corresponding refractive index profiles which are then used in the optical mode calculations.
3.2. Modal analysis of the active section
To transform the charge distribution in the ITO film obtained via the electrostatic simulations to corresponding relative complex permittivity profiles, we use the Drude equation [16]:
We perform modal analyses of the full device cross-section (Fig. 2) based on a vectorial finite difference eigenmode (FDE) technique [20]. When g is large, the system of Fig. 2 presents three uncoupled waveguides (two MIM and one Si), supporting distinct MIM and Si waveguide modes. As g decreases the system becomes coupled, the modes hybridise and evolve into coupled modes. We investigate the modes for the case where g is small, corresponding to the active modes of the modulator. We present results for two cases of Si core dimensions $w \times h$: 500 nm ${\times} $ 300 nm (Design I) and 350 nm ${\times} $ 500 nm (Design II).
The fundamental modes of the active section are the TM symmetric ($\textrm{T}{\textrm{M}_\textrm{s}}$) and TM antisymmetric ($\textrm{T}{\textrm{M}_\textrm{a}}$) supermodes, both strongly confined within the MIM structures, but distributed symmetrically and asymmetrically laterally along y, over the gap $g = 200$ nm separating the left and right MIMs, as plotted in Fig. 6(a) for Design I. In Fig. 6(a), we show the dominant transverse electric field component (${E_z}$) of these modes for an applied voltage of 0.25 V, which is near the flat-band voltage. The field is strongly confined within the MIM structures, overlapping essentially perfectly with the $\textrm{Hf}{\textrm{O}_2}$ and ITO regions, thus allowing strong interaction with the perturbed ITO layer, ultimately resulting in strong modulation.
Fig. 6. (a) Modal profiles of the $\textrm{T}{\textrm{M}_\textrm{s}}$ (top) and $\textrm{T}{\textrm{M}_\textrm{a}}$ (bottom) modes of the active section for ${V_\textrm{s}} = 0.25$ V, and (b) modal profile of the $\textrm{T}{\textrm{M}_\textrm{s}}$ mode for ${V_\textrm{s}} = 2.5$, $3.0$, $3.5$, and $4.0$ V in the accumulation regime, plotted in expanded view to show the shift in the position of $\textrm{max}|{{E_z}} |$.
In Fig. 6(b), we show the modal profile of the $\textrm{T}{\textrm{M}_\textrm{s}}$ mode near the hafnia-ITO interface, for applied voltages of 2.5, 3.0, 3.5, and 4.0 V in the accumulation regime. Highly localised enhancement of the ${E_z}$ component is observed at the ENZ location compared to the flat-band case of Fig. 6(a). Since the ENZ point shifts away from the $\textrm{Hf}{\textrm{O}_2}$-ITO interface into the bulk ITO with an increase in ${V_\textrm{s}}$ (cf. Fig. 4(b)), the peak of ${E_z}$ follows. (There is no ENZ point for ${V_s} = 3.0$ V, but the real part of permittivity is small and close zero, thereby still producing significant field localisation and enhancement).
Next, we plot the variation in the complex effective indices of these modes ($\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$) as a function of applied voltage for Designs I and II, in Figs. 7(a) and 7(b), respectively. The variation in the imaginary part of the complex effective index is smaller for values of ${V_\textrm{s}}$ in the depletion regime (${V_s} < 0.334$ V). As the applied voltage drops below 0.334 V, the imaginary part becomes almost constant, as the depletion region in ITO reaches its maximum depth. In the accumulation regime (${V_s} > 0.334$ V), the imaginary part increases by more than 4 ${\times} $ over the voltage range investigated for both modes in both designs. In the accumulation regime, the most significant change in the imaginary part occurs between 2 and 3 V due to the emergence of the ENZ point in this voltage range.
Fig. 7. Complex effective indices of the $\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$ modes of the active section as a function of applied voltage for (a) Design I, and (b) Design II.
For Design I, the complex effective indices of the $\textrm{T}{\textrm{M}_\textrm{s}}$ mode are $2.9142 - j0.02646$ and $2.8981 - j0.1140$ for applied voltages of 0.5 and 4.0 V, resulting in an insertion loss $\textrm{I}{\textrm{L}_{\textrm{Active}}} = 0.9$ dB/µm at 0.5 V, and an extinction ratio of $ER = 3.1$ dB/µm. The corresponding complex effective indices of the $\textrm{T}{\textrm{M}_\textrm{s}}$ mode of Design II are $2.9168 - j0.01958$ and $2.9095 - j0.08536$ resulting in $\textrm{I}{\textrm{L}_{\textrm{Active}}} = 0.7$ dB/µm and $\textrm{ER} = 2.3$ dB/µm. Design I produces a higher extinction ratio and insertion loss due to the slightly stronger confinement of the modal field in the MIM structures. The AC drive voltage required to achieve this performance in both designs is ${V_s} ={\pm} 1.75$ V, about a DC bias voltage of 2.25 V.
3.3. Modal evolution and hybridization
In plasmonic ITO modulators, the efficient and selective excitation of the plasmonic mode of the active section is an important design consideration. Improper excitation results in issues such as a high insertion loss due to loss of power into other modes including radiative modes, low extinction ratio due to forward propagating unmodulated modes, or interference between multiple modes excited at the input of the active section.
To achieve efficient and selective excitation of the modes in the active section, we design an adiabatic taper which transforms the $\textrm{T}{\textrm{M}_0}$ and $\textrm{T}{\textrm{E}_0}$ modes of the input silicon waveguide into the $\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$ plasmonic supermodes of the active section, respectively. To show how modal transformation occurs, we plot the real part of the effective index of the modes as a function of g for Designs I and II in Fig. 8. As we decrease the gap from $g = 1200$ nm, the active region couples to the Si waveguide as the two MIM stacks approach, and the modes of the silicon waveguide transform into the corresponding plasmonic supermodes of the active section. As observed from Fig. 8, the largest change in $\textrm{Re}\{{{n_{\textrm{eff}}}} \}$ occurs for $g < w$ ($w = 500$ nm for Design I, $w = 350$ nm for Design II), indicating that most of the transformation occurs for $g < w$. For $g = 1200$ nm ${\gg} w$, the effective indices of the modes are those of the corresponding $\textrm{T}{\textrm{M}_0}$/$\textrm{T}{\textrm{E}_0}$ modes of the input silicon waveguide. On the left side of the plots, we show the modal profiles of the $\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$ modes at $g = 200$ nm, and on the right, we show the modal profiles of the corresponding $\textrm{T}{\textrm{M}_0}$ and $\textrm{T}{\textrm{E}_0}$ modes at $g = 1200$ nm.
Fig. 8. Real part of mode effective index as a function of g for (a) Design I and (b) Design II. The corresponding field distributions are shown on the left for $g = 200$ nm and on the right for $g = 1200$ nm (the ${E_z}$ field component is also plotted on the right for the $\textrm{T}{\textrm{E}_0}$ mode).
It is intuitively clear that the $\textrm{T}{\textrm{M}_0}$ mode of the Si waveguide should transform into the $\textrm{T}{\textrm{M}_\textrm{s}}$ supermode of the MIM pair, because the ${E_z}$ field is the dominant transverse field component of both modes and it is distributed symmetrically along the lateral dimension. However, that the $\textrm{T}{\textrm{E}_0}$ mode of the Si waveguide transforms into the $\textrm{T}{\textrm{M}_\textrm{a}}$ mode of the MIM pair is surprising given that the dominant transverse field component of the former is ${E_y}$ and orthogonal to that of the later which is ${E_z}$. We recall that modes in rectangular dielectric waveguides are not strictly TE or TM as in a dielectric slab - all field components are present in all modes, including the ${E_z}$ component of the $\textrm{T}{\textrm{E}_0}$ mode. Indeed, this field component is asymmetrically distributed along the lateral dimension, as shown in Fig. 8, and although weak, enables transformation to the $\textrm{T}{\textrm{M}_\textrm{a}}$ mode of the MIM pair. Thus, the mode pairs ($\textrm{T}{\textrm{M}_0}$, $\textrm{T}{\textrm{M}_\textrm{s}}$) and ($\textrm{T}{\textrm{E}_0}$, $\textrm{T}{\textrm{M}_\textrm{a}}$) evolve smoothly with g, implying that a taper can be designed to adiabatically transform them.
The taper is created by decreasing the gap g between the two MIM stacks, linearly from 1000 nm at the input of the taper, to 200 nm in the active section, over a length ${L_T}$ which should be chosen to ensure adiabatic mode transformation. To evaluate the taper performance and optimise its design, we use the eigenmode expansion (EME) technique [20].
In Fig. 9 we plot the power in the $\textrm{T}{\textrm{M}_\textrm{s}}$/$\textrm{T}{\textrm{M}_\textrm{a}}$ plasmonic supermodes at the output of the tapered section due to excitation at the input by the $\textrm{T}{\textrm{M}_0}$/$\textrm{T}{\textrm{E}_0}$ modes of the silicon waveguide, as a function of the taper length ${L_\textrm{T}}$. The curves show a maximum at a specific value of ${L_\textrm{T}}$. At short lengths, loss is due to power coupling into other modes (including radiative) due to non-adiabatic modal evolution, and at long lengths propagation loss dominates. There is always a trade-off between them, and thus an optimum length exists minimising the overall loss. The powers were normalised such that the output power plotted in Fig. 9 can be interpreted as the transmittance of the tapers for the corresponding mode pairs ($\textrm{T}{\textrm{M}_0}$, $\textrm{T}{\textrm{M}_\textrm{s}}$) and ($\textrm{T}{\textrm{E}_0}$, $\textrm{T}{\textrm{M}_\textrm{a}}$). For Design I, the taper transmittance maximizes at about 0.6 for ${L_T} = 6$ µm for both mode pairs. For Design II, the taper transmittance maximizes slightly above 0.7 for ${L_T} = 11$ µm for mode pair ($\textrm{T}{\textrm{E}_0}$, $\textrm{T}{\textrm{M}_\textrm{a}}$), and at 0.73 for ${L_T} = 15$ µm for mode pair ($\textrm{T}{\textrm{M}_0}$, $\textrm{T}{\textrm{M}_\textrm{s}}$). In the case of Design I, the same taper could be used to operate optimally with both mode pairs, whereas for Design II, a good taper length would be ${L_\textrm{T}} = 11$ µm, yielding an equal transmittance of about 0.7 for both mode pairs.
Fig. 9. Normalised output power (transmittance) in the $\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$ plasmonic supermodes at $g = 200$ nm vs. the length of the tapered section ${L_T}$, due to excitation at the input ($g = 1000$ nm) by the $\textrm{T}{\textrm{M}_0}$ and $\textrm{T}{\textrm{E}_0}$ modes of the silicon waveguide. (a) Design I, and (b) Design II.
In Fig. 10(a), we show evolution of the ($\textrm{T}{\textrm{M}_0}$, $\textrm{T}{\textrm{M}_\textrm{s}}$) mode pair through a tapered section of length ${L_\textrm{T}} = 15$ µm, and in Fig. 10(b) of the ($\textrm{T}{\textrm{E}_0}$, $\textrm{T}{\textrm{M}_\textrm{a}})$ mode pair for ${L_\textrm{T}} = 11$ µm (both cases for Design II). We plot the field profiles of the dominant components on two planes: one passing through the vertical centre of the silicon core (left panels), and one passing through the vertical centre of the ITO (right panels). From these plots we confirm that the $\textrm{T}{\textrm{M}_0}$ mode input at $g = 1000$ nm gets transformed into the $\textrm{T}{\textrm{M}_\textrm{s}}$ mode at $g = 200$ nm, whereas the $\textrm{T}{\textrm{E}_0}$ mode input at $g = 1000$ nm gets transformed into the $\textrm{T}{\textrm{M}_\textrm{a}}$ mode at $g = 200$ nm. We also notice the expected enhancement in the field (the colour bars can be directly compared), as the $\textrm{T}{\textrm{M}_{\textrm{s},\textrm{a}}}$ modes are strongly confined within the $\textrm{Hf}{\textrm{O}_2}$ and ITO regions of the MIM structures.
Fig. 10. Electric field profiles of the dominant component on two planes, one passing through the vertical centre of the silicon core (left panels), and one passing through the vertical centre of the ITO (right panels), showing propagation through tapered sections for Design II, where g changes from 1000 nm to 200 nm. (a) Input $\textrm{T}{\textrm{M}_0}$ mode transforms adiabatically into the output $\textrm{T}{\textrm{M}_\textrm{s}}$ mode for ${L_\textrm{T}} = 15$ µm. (b) Input $\textrm{T}{\textrm{E}_0}$ mode transforms adiabatically into $\textrm{T}{\textrm{M}_\textrm{a}}$ mode for ${L_T} = 11$ µm.
3.4. Electrical isolation of the active section
The tapered sections at the input and output of the active section are strictly passive and must be isolated electrically to maximise the modulator bandwidth. Specifically, it is undesirable for the tapered sections to contribute capacitance or series resistance to the modulator.
To electrically isolate the active section, we create slits of width ${s_w}$ in the top metal film comprising the MIM sections, perpendicular to the direction of propagation, at the junction of the taper and the active section, as shown in Fig. 1. These slits electrically isolate the active section, as they ensure that the drive voltage (${V_s}$) is applied thereon only. However, there is an insertion loss ($\textrm{I}{\textrm{L}_{\textrm{slit}}}$) associated with this discontinuity in the propagation direction. In Figs. 11(a) and 11(b), we plot $\textrm{I}{\textrm{L}_{\textrm{slit}}}$ computed using the FDTD technique as a function of slit width (${s_w}$) for Designs I and II. As ${s_w}$ is reduced, $\textrm{I}{\textrm{L}_{\textrm{slit}}}$ is also reduced. For ${s_w} = 10$ nm in Design I, $\textrm{I}{\textrm{L}_{\textrm{slit}}} = 0.37$ dB for both $\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$ supermodes. For Design II, $\textrm{I}{\textrm{L}_{\textrm{slit}}} = 0.24$ dB for the $\textrm{T}{\textrm{M}_\textrm{s}}$ mode and 0.33 dB for the $\textrm{T}{\textrm{M}_\textrm{a}}$ mode. $\textrm{I}{\textrm{L}_{\textrm{slit}}}$ accounts for a pair of slits through both MIM structures along the same plane y-z plane. Slits are needed at input and output planes of the active section, so $\textrm{I}{\textrm{L}_{\textrm{slit}}}$ must be doubled when computing the full insertion loss of the device.
Fig. 11. Insertion loss of isolation slits, $\textrm{I}{\textrm{L}_{\textrm{slit}}}$, vs. ${s_w}$ for (a) Design I and (b) Design II.
3.5. Electrical bandwidth
As the active device length is very small, the active section of the modulator can be modelled as the lumped RC network shown as the inset to Fig. 12. The 3-dB electrical bandwidth associated with this model is given by [27]:
Fig. 12. Total charge per unit length of the modulator active section vs. applied voltage. The inset gives a sketch of the lumped-element equivalent circuit of the modulator.
We performed 2D electrostatic modelling (cf. Sub-section 3.1) to obtain the total charge per unit length in the simulation domain comprising both MIM structures, as a function of applied voltage, which we plot in Fig. 12. We then used these results to calculate the capacitance per unit length of the modulator, as the derivative of the total charge with respect to applied voltage. The capacitance per unit length in the strong accumulation regime is 18 fF/µm, which matches well with the oxide capacitance calculated using the parallel plate capacitor model of Eq. (3). The capacitance ($C$) of the modulator is then simply obtained by multiplying this capacitance per unit length by the active length (${L_{\textrm{active}}}$).
To calculate the parasitic series resistance, we set the thickness of the oxide layer to zero and calculate via 2D electrostatic modelling the current density flowing through the modulator incorporating both MIM structures for a specific applied voltage. Using this data, we obtain the series resistance per unit length of the modulator as 5.4 Ω·µm. The series resistance is dominated by the resistance through the ITO layer, which decreases with modulator length as it is inversely proportional to the modulator area A. The series resistance of the modulator R is then obtained by dividing the resistance per unit length by the active length (${L_{\textrm{active}}}$).
To estimate the 3-dB electrical bandwidth, we assume a source of impedance ${R_\textrm{S}} = 50$ Ω and a load resistance ${R_\textrm{L}}$ of 50 Ω, resulting in a simplified expression for the bandwidth (Eq. (2)): $\textrm{BW} = {[{2\pi ({R + 25} )C} ]^{ - 1}}$. Using the calculated values of capacitance and resistance, we obtain the corresponding bandwidths tabulated in Table 1 for various design options.
Table 1. Summary of modulator results.
3.6. Summary of designs
We summarize in Table 1 several important modulator characteristics, IL, ER, and BW, for Designs I and II, and for both modes pairs of operation. The total insertion loss (dB) of the design is: $\textrm{I}{\textrm{L}_{\textrm{total}}} = 2 \times \textrm{I}{\textrm{L}_{\textrm{taper}}} + \textrm{I}{\textrm{L}_{\textrm{active}}} + 2 \times \textrm{I}{\textrm{L}_{\textrm{slit}}}\; $ where $\textrm{I}{\textrm{L}_{\textrm{taper}}}$ (dB) is the insertion loss of each tapered section at the input and output, $\textrm{I}{\textrm{L}_{\textrm{active}}}$ (dB) is the insertion loss of the active section, and $\textrm{I}{\textrm{L}_{\textrm{slit}}}$ (dB) is the insertion loss due to one slit pair at the junction of a tapered section and the active section.
The total insertion loss of Design I ($\textrm{I}{\textrm{L}_{\textrm{total}}}\sim 7$ dB) is larger than for Design II ($\textrm{I}{\textrm{L}_{\textrm{total}}}\sim 4.5$ dB), mostly because the insertion loss of the tapered sections ($\textrm{I}{\textrm{L}_{\textrm{taper}}}$) is larger in the former. The insertion loss of the modulator is dominated by the taper losses, especially as two tapers are needed per modulator, one at the input of the active section and another at the output. However, Design I provides a larger ER compared to Design II.
Two cases of length for the active section are considered to illustrate the trade-off between ER and BW. For ${L_{\textrm{active}}} = 1.5$ µm, the capacitance and series resistance of the modulator are $C = 27$ fF and $R = 3.6$ Ω yielding an electrical BW of 206 GHz - in this case, Design I yields $\textrm{ER}\sim 4.5$ dB for operation with both mode pairs. For ${L_{\textrm{active}}} = 2$ µm, the capacitance and series resistance of the modulator are $C = $ 36 fF and $R = $ 2.7 Ω yielding an electrical BW of 160 GHz - in this case, Design I yields ER ∼ 6 dB for operation with both mode pairs. The overall length of the designs, including the active length and both tapers, range from 13 to 24 µm.
We note from Table 1 that the performance of mode pair $({\textrm{T}{\textrm{M}_0},\textrm{T}{\textrm{M}_\textrm{s}}} )$ is very similar to the performance of mode pair $({\textrm{T}{\textrm{E}_0},\textrm{T}{\textrm{M}_\textrm{a}}} )$ within the same design. This is due to the facts that the plasmonic supermodes ($\textrm{T}{\textrm{M}_\textrm{s}}$, $\textrm{T}{\textrm{M}_\textrm{a}}$) behave similarly in the active section in terms of $\textrm{Im}\{{{n_{eff}}} \}$ (cf. Figure 7), and the modes of the silicon waveguide ($\textrm{T}{\textrm{M}_0}$ and $\textrm{T}{\textrm{E}_0}$) transform similarly into the corresponding plasmonic supermodes (cf. Figures 8 - 10). Thus, relative to the mode of the input silicon waveguide ($\textrm{T}{\textrm{M}_0}$ or $\textrm{T}{\textrm{E}_0}$) the modulator performance is polarization independent.
In Table 2, we compare our design with other plasmonic modulators including different designs of plasmonic ITO modulators. We also include GeSi [28], Ge [29] and graphene [30] based electro-absorption modulators. For comparison, we use specifications of importance to assess the performance of modulators, such as $\textrm{I}{\textrm{L}_{\textrm{Active}}}$, ${L_{\textrm{Active}}}$, ER, electrical bandwidth, drive voltage, and supported polarizations.
Table 2. Comparison with other modulatorsa
4. Conclusions
We propose a broadband plasmonic electro-optic modulator operating at ${\lambda _0} = 1550$ nm, exploiting the free carrier dispersion effect in ITO to access its ENZ point within the accumulation layers of MOS structures integrated into a pair of coupled MIM waveguides aligned on a planarized Si waveguide. The coupled MIM waveguides support symmetric and asymmetric plasmonic supermodes, excited adiabatically by the TM0 and TE0 modes of the underlying Si waveguide, respectively.
Through a combination of strong index modulation of ITO near its ENZ point, and a strong interaction of the propagating plasmonic supermodes with the perturbed ITO, the active section of the modulator can be extremely compact, of length between 1.5 and 2.0 µm, such that it may be driven as a simple capacitive lumped element, with a broad electrical bandwidth ranging between 160 and 200 GHz, and an extinction ratio in the range of 3.5 to 6 dB. The modulator section is isolated electrically from passive mode conversion tapers at the input and output which adiabatically transform the input TM0 and TE0 modes of the Si waveguide to the symmetric and asymmetric plasmonic supermodes, $\textrm{T}{\textrm{M}_\textrm{s}}$ and $\textrm{T}{\textrm{M}_\textrm{a}}$, of the modulator. The total insertion loss of the modulators, including the mode conversion sections, ranges from 4 to 7.5 dB. The AC drive voltage required is ${\pm} 1.75$ V about a DC bias of 2.25 V. The modulators make use of inorganic materials only and are essentially polarization independent.
Funding
Huawei Technologies Canada.
Acknowledgment
Fruitful discussions with Dominic Goodwill, formerly of Huawei Technologies Canada, and with Eric Bernier of Huawei Technologies Canada are gratefully acknowledged.
Disclosures
The authors declare no conflicts of interest.
Data availability
All data underlying the results or needed to evaluate the conclusions of the paper are present in the paper.
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