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Accumulating electrons in transparent conductive oxides such as indium tin oxide (ITO) can induce an ”epsilon-near-zero” (ENZ) in the spectral region near the important telecommunications wavelength of λ = 1.55μm. Here we theoretically demonstrate highly effective optical electro-absorptive modulation in a silicon waveguide overcoated with ITO. This modulator leverages the combination of a local electric field enhancement and increased absorption in the ITO when this material is locally brought into an ENZ state via electrical gating. This leads to large changes in modal absorption upon gating. We find that a 3 dB modulation depth can be achieved in a non-resonant structure with a length under 30 μm for the fundamental waveguide modes of either linear polarization, with absorption contrast values as high as 37. We also show a potential for 100 fJ/bit modulation, with a sacrifice in performance.
© 2013 Optical Society of America
1. Introduction
As the drive for ever-increasing computing power presses on, the performance bottlneck has shifted from individual transistors to the interconnection network between transistors. The time delay associated with charging the capacitances of traditional metallic interconnects to a signal voltage limits their operating bandwidth. Optical links, already the preferred solution at longer length scales, show promise to replace their electrical counterparts for on-chip interconnection provided their energy consumption can be reduced sufficiently [1].
This need to reduce the energy consumption of on-chip optical interconnects has led to much recent work on the development of CMOS-compatible modulators, specifically silicon waveguide-integrated devices [2]. Typical silicon waveguide modulators leverage electrically altering either the refractive or the absorptive properties of a material to modulate the transmission of light through a device. Due to silicon’s weak electro-optic effect, electro-refractive modulation requires either large device footprints and higher capacitances or high-Q resonant devices that suffer from thermal instability. Electro-absorption modulators, typically leveraging the Franz-Keldysh effect or the quantum confined Stark effect, require the incorporation of materials subject to complex processing steps; these include recent demonstrations of modulators integrating germanium into silicon devices [3].
Transparent conductive oxides (TCOs), specifically indium tin oxide (ITO), have been of recent interest as near-infrared optical materials due to their electrically-tunable permittivity [4, 5, 6, 7]. At infrared frequencies, the optical response of ITO is largely due to its free electrons, allowing a Drude model to accurately model the permittivity. The accumulation of electrons in ITO via electrical gating can swing its plasma frequency from the mid-infrared to the near-infrared. The resulting ability to dramatically change the optical properties of ITO from dielectric-like to metallic allows for the realization of new optical modulator designs.
In this paper, we introduce a concept for a silicon waveguide modulator where electrically inducing an epsilon-near-zero (ENZ) in an adjacent ITO film suppresses the transmission of a waveguide mode. This is accomplished by simultaneously inducing loss due to free carrier absorption in the ITO and by increasing the modal overlap with the lossy region in which the free carrier absorption occurs. We first provide a brief overview of the modulator, including its architecture and its operating principles. We go on to detail the optical properties of ITO and how electrical gating can induce an ENZ region. Next, we go into the physical implications of an ENZ region on a waveguide mode and our device’s operation. We then conclude with a discussion on the modulator’s performance, including its energy-performance tradeoff.
2. Modulator overview
The proposed modulator design is geared to operate at the 1.55 micron telecom wavelength and consists of a silicon strip waveguide (400×200 nm2) coated with a 5 nm thick HfO2 layer and a 10 nm thick ITO layer (Fig. 1(a)). The above structure sits atop a buried SiO2 layer, following standard silicon waveguide design principles [8]. This waveguide supports two modes: a transverse electric-like (TE) mode with its dominant electric field component along the horizontal direction (Fig. 1(b)) and a transverse magnetic-like (TM) mode with its dominant electric field component along the vertical direction (Fig. 1(c)) [9]. The ITO-HfO2-silicon stack forms a metal-oxide-semiconductor (MOS) capacitor with ITO effectively serving as a metallic gate electrode at GHz frequencies and HfO2 forming the gate insulator; this capacitor is formed on all three coated sides of the silicon waveguide. We choose an ITO doping of ND = 1019 cm−3 to best match its dielectric constant with that of the adjacent HfO2 layer at a 1.55 micron wavelength. Maximal optical transmission occurs in this waveguide structure with no applied electrical bias. We call this the ON state of the modulator (Fig. 1(a), upper inset). The optical transmission can be modulated by applying a negative gate bias between the ITO and the silicon waveguide to induce an electron accumulation layer at the ITO/HfO2 interface. We will demonstrate that this action results in a local change in the dielectric constant of ITO as an increasing number of electrons are accumulated, eventually reaching epsilon-near-zero at an applied gate voltage of −2.3 V. The boundary conditions imposed by the ENZ region alters the waveguided modes in such a fashion that overlap with the lossy electron accumulation layer is increased. This feature enables realization of a highly absorptive state of the modulator in which optical transmission is significantly reduced. This is distinct from the operation of conventional Si waveguide modulators where, due to the high dielectric constant of Si at 1.55 microns, accumulation of carriers only induces a small fractional change in this quantity and the shape of the optical mode remains unaltered [10, 11]. The absorptive state of the modulator will be termed the OFF state (Fig. 1(a), lower inset). We choose this simple structure to illustrate the physics of ENZ-based modulation; for a practical device, the silicon strip can be converted to a silicon rib where a thin silicon layer covers the substrate, allowing electrical contact without perturbation of the waveguide mode. In the rest of this letter we will elaborate on the physical underpinnings behind this modulator concept, discuss in detail the results of our calculations, and provide our thoughts on the future prospects for the practical implementation of such devices.
Fig. 1 The proposed modulator consists of a silicon-on-insulator (400 × 220nm2) waveguide coated with layers of HfO2 (5 nm) and ITO (10 nm), forming a MOS capacitor (a). Without an applied bias, the ITO absorbs little light leading to a highly tranmissive ON state (upper inset). With a negative bias between the ITO and the Si, an electron accumulation layer is induced at the ITO-HfO2 interface. This accumulation layer modifies ITO’s local optical permittivity, creating an epsilon-near-zero (ENZ) region that perturbs the waveguide mode into a highly absorptive OFF state (lower inset). This electro-absorption modulation occurs for both TE-like and TM-like modes supported by the waveguide structure (b), (c). The TE (TM) mode exhibits discontinuities in |Ex|(|Ey|) at interfaces due to dielectric constant mismatches.
3. Optical properties of ITO
The electrical and optical properties of transparent conductive oxides, ITO in particular, have been extensively studied. These wide band gap, amorphous semiconductors provide a unique combination of high transparency to visible light and high electrical conductivity, lending their application to a range of optoelectronic devices where optical access is required through an electrical contact. Example uses of ITO include the transparent electrodes in solar cells, photodetectors, and touchscreen displays.
In the near-infrared, ITO responds to light as a metal would, indicating that its free electrons dictate its optical response. In fact, many researchers have begun to explore ITO, and related TCOs, as plasmonic materials in the optical frequency range [12]. It has been shown that a Drude model accurately predicts the permittivity of ITO at wavelengths beyond 1 μm [13]. The permittivity of ITO, as defined by a Drude model, is given by
where ε∞ is the high-frequency dielectric constant, ω is the frequency of the illuminating light, γ is the electron scattering frequency, and ωp is the plasma frequency. Whereas ε∞ and γ depend on the deposition conditions, in our analysis we have taken ε∞ = 3.9 and γ = 1.8 × 1014 [13]. The plasma frequency term can be further expanded to reveal its square-root dependence on the electron concentration, where n is the electron concentration, m* is the electron effective mass, and e and ε0 are the fundamental charge and the permittivity of free space, respectively. This link between electron concentration and permittivity has been exploited to tune the near-infrared optical properties of ITO by electrical gating in a plasmonic waveguide [5]. This initial demonstration has since sparked the emergence of a new class of optoelectronic devices exploiting this strong electro-optic effect [6, 7, 4].In designing such devices, however, it is imperative to consider three operating constraints. First, the capacitively induced electron accumulation layer is only ∼ 1 nm thick, as determined by Thomas-Fermi screening theory [7]. Thus the volume of ITO that experiences a dramatic refractive index change is quite miniscule compared to the wavelength of visible and near-infrared light. Second, there is an upper bound on the concentration of electrons that can be capacitively accumulated that is determined by the electric field breakdown of the gate insulator. In order to maximize this upper bound, we may borrow an approach from CMOS electronics and use HfO2 as the gate insulator. HfO2’s high DC permittivity (εr = 25) allows an accumulated electron concentration of Δnmax = 1.4 × 1021 cm−3, assuming a breakdown field of 10 MV/cm and a 1 nm accumulation layer thickness [14]. This value compares quite favorably to that of a similar MOS stack with a SiO2 gate oxide instead, Δnmax = 2.1 × 1020 cm−3, under the same breakdown field and accumulation layer thickness assumptions. Last, the minimum bulk electron concentration attainable in ITO is 1 × 1019 cm−3. ITO is a Mott insulator below this carrier concentration and no longer exhibits free electron behavior [15]. To the best of our knowedge, no prior work on the electronic modulation of ITO’s optical properties has incorporated all three operating constraints.
Given the above constraints, we can map out the permittivities in an ITO electron accumulation region that are accessible via electrical gating (Fig. 2). Here we separate the complex permittivity into its real and imaginary components, ε = ε1 + iε2. We choose an ITO film with a bulk electron concentration of n0 = 1019 cm−3 (λp = 2πc/ωp = 6.25μm) to match its real permittivity, ε1, with that of its neighboring HfO2 layer (ε1,HfO2 = 3.92) at λ0 = 1.55μm. This choice to index-match the ITO and HfO2 balances two opposing needs: to minimize the ITO absorption by reducing its bulk carrier concentration and to minimize the ON-state electric field magnitude in ITO by reducing its refractive index. When additional electrons accumulate at the ITO/HfO2 interface via electrical gating (Δnacc), the electron concentration in that accumulation layer, n, is n = n0 + Δnacc. Such accumulation of electrons increases the plasma frequency of the accumulation region, causing its ε1 to decrease. Applying an electron-accumulating bias of up to −5 V to the MOS stack yields a Δnacc of up to 1.38×1021 cm−3, allowing ε1 to decrease from its bulk value of 3.92 down past −4. As per the Drude model, this electron accumulation also induces a corresponding increase in the imaginary permittivity, ε2, of the accumulation region. This increase in ε2 increases the optical loss in the accumulation region as the accumulated electron density increases.
Fig. 2 Accumulating electrons in ITO via electrical gating tunes its local permittivity for λ0 = 1.55μm light. Increasing the concentration of accumulated electrons by increasing the applied bias leads to a decreasing ε1 and an increasing ε2, as per the Drude model. At an applied voltage of −2.3 V (Δnacc = 6.33 × 1020cm−3), ε1 approaches zero; this indicates the formation of an ENZ region at the ITO-HfO2 interface.
4. ENZ physics and modulator operation
For an applied voltage of −2.3 V, corresponding to n = n0 + Δn = 6.33 × 1020 cm−3, we find that the real part of the accumulation layer’s permittivity approaches zero (Fig. 2). Physically this represents a transition between a material exhibiting a dielectric response and a metallic response to incident light; this permittivity range is referred to as the ENZ range. ENZ materials impose an unusual boundary condition upon electric fields along their surface normal as a consequence of the continuity of the normal component of the electric displacement field, Dn = εEn. This boundary condition leads to very large enhancements in the electric field magnitudes inside the ENZ material with respect to the adjacent materials. One finds that:
where εd and E0 are the permittivity of and the electric field inside, respectively, a material adjacent to an ENZ region with permittivity εENZ. The electric field enhancement, |EENZ/E0 in an ENZ material is proportional to 1/|εENZ|. The upper bound on this field enhancement is determined by how small |εENZ| can be, which itself is governed by how small the ENZ material’s γ can become.In order to effectively use an ENZ material in a high-index-constrast waveguide modulator, we must understand how the ENZ material’s presence affects the waveguide modes’ propagation. Using the formalism established to study gain in such waveguides [16], we can describe how a voltage-switchable ENZ material affects a waveguide’s modal loss.
We can express a waveguide’s modal loss, αm, as the product of two factors,
where αb is the bulk material loss and Γ is the confinement factor of the optical mode to the active, switching layer. αb represents the loss attributable to intrinsic, bulk material absorption and can be written as: where k0 is the free-space wavevector of light and ñ is the complex refractive index of the ENZ material (ñENZ ≡ [ε̃ENZ]1/2). Γ in Eq. 4 represents the degree to which the waveguide mode overlaps with the switching layer and can be expressed by: where ng is the group index (ng = c/vg) of the waveguide mode. We can plot αm and its constituents, αb and Γ, as a function of applied voltage between the ITO and the silicon waveguide to find the optimal operating bias of −2.3 V (Fig. 3). The data points in the figure are computed from finite-element simulations and the lines connecting them are cubic-spline interpolations between the data points. The modal loss was extracted from the computationally calculated complex mode index. We find that this operating voltage corresponds to an ENZ ITO permittivity (Fig. 3(c), inset). Maximal modal loss is observed where a local maxima in the confinement factor, Γ, occurs simultaneously with an appreciable bulk material loss, αb. At smaller voltages, Γ is greater than its local maxima at ITO’s ENZ point but αb is too small for significant modal loss, αm. Similarly at higher voltages, while αb is high the small Γ leads to a poorer overall αm. Note that we re-compute the mode profile for each applied voltage in order for Eq. 6 to remain valid.Fig. 3 As the gate bias is increased, the modal loss exhibits a local maximum at −2.3 V, precisely the voltage required to induce an ENZ region in ITO (inset), for both TE and TM modes (a). We can express the modal loss as the product between the bulk material loss (b) and the confinement factor (c). Maximal modal loss occurs when a combination of appreciable bulk material loss and modal confinement. At higher voltages where the bulk material loss is higher, but the mode is not as confined within the highly absorptive region does not lead to optimal modal loss. Lower voltages with lower bulk material loss but higher modal confinement are also not optimal.
We can further understand the basis for this electro-absorption by looking at the waveguide cross-sectional mode profiles in the ON and OFF states and their correspondence to the permittivity profile of the structure (Fig. 4). Here we focus our attention to just the TE mode for instructive purposes, as these operating principles apply similarly to the TM mode. We begin by looking at a horizontal cross-section of the magnitude of the x-component of the electric field, |Ex|, taken at the center of the silicon waveguide, under no applied bias (Fig. 4(a), left). Discontinuities exist in this mode profile at interfaces between materials with mismatched permittivities, but note the minimal discontinuity at the ITO/HfO2 interfaces (Fig. 4(b), left) due to our choice to match the bulk ITO permittivity with that of HfO2 via doping (Fig. 4(c), left). Under application of a −2.3 V gating bias, the mode profile develops a pair of sharp peaks at both ITO/HfO2 interfaces (Fig. 4(a), right). These peaks, or electric field enhancements, occur in the electron accumulation region (Fig. 4(b), right) where an ENZ region has been established (Fig. 4(c), right). It is this generation of high electric fields precisely in the region of highest absorption, the ENZ region, that allows high electro-absorption. These principles also hold for the TM mode with the exception that only one sharp electric field spike occurs due the single ITO/HfO2 interface along the vertical axis of the device.
Fig. 4 In the ON state, without applied bias, the highest electric field magnitude resides inside the silicon waveguide (a, left). In the OFF state, with a bias that induces an ENZ region, we find two large spikes in the electric field at each ITO-HfO2 interface due to the ENZ regions’ presence and their imposed continuity condition (a, right). A closer look at the region of interest (b) shows that the high electric fields are confined to the electron accumulation layer at the ITO-HfO2 interface, precisely where the permittivity drops near zero (c). This also coincides with the region that most effectively contributes to the modal absorption.
5. Modulator performance
Now that we have discussed the physical mechanisms behind the electro-absorption modulation and we have identified the operating bias yielding the largest modal loss, we are in a position to quantify the overall performance of the modulator.
We will evaluate the performance of this modulator using the absorption contrast as a figure of merit. The absorption contrast is defined as the ratio between the extinction ratio (Δα) and the insertion loss (α). The insertion loss is the modal loss in the ON state (α = αm,ON) and the extinction ratio is the difference between the modal losses in the OFF state and in the ON state (Δα = αm,OFF − αm,ON). Current state-of-the-art silicon-compatible electro-absorption modulators operate with absorption contrasts near 2 [17]. Here we find that our modulator exhibits a peak absorption contrast of 37.3 (5.6) for TE (TM) mode operation (Fig. 5(a)). The peak in absorption contrast occurs at an applied bias of 2.3 V, again, corresponding to ENZ conditions near the ITO/HfO2 interface. The TE mode exhibits superior absorption contrast than the TM mode since the TE mode has a lower insertion loss due to its better confinement within the silicon waveguide. The insertion loss for the TE (TM) mode is 7.05 × 10−4μm−1 (4.56 × 10−3μm−1).
Fig. 5 (a) The absorption contrast peaks for both TE and TM modes at an applied voltage of 2.3 V, corresponding to the generation of an ENZ region at the ITO/HfO2 interface. We find peak absorption contrasts of 37.3 (5.6) for the TM (TE) mode. (b) Making the waveguide modulator longer increases its modulation depth. We find that a device length of 27 μm yields a modulation depth of 0.5 (3 dB modulation) for both TM and TE operation.
We can use the extinction ratio (Δα) and the insertion loss (α) to calculate the modulation depth of the device as a function of its length. The modulation depth, M, is defined as
where l is the modulator length. We find that a device length of 26.4 (27.0) μm yields a modulation depth of 0.5 for TE (TM) mode operation, which corresponds to 3 dB switching (Fig. 5(b)). Even though there are two ENZ regions overlapping with the TE mode, compared with the one ENZ region overlapping with the TM mode, the TE and TM modulation depths are similar since the TM mode is less confined within the silicon waveguide, thus has a higher electric field magnitude overlapping spatially with the ENZ region.The contribution of the accumulated holes in the silicon region to the modulator loss is minimal, based on the carrier concentration-dependent absorption coeffecient in silion [10]. However, any additional loss induced in the silicon layer upon accumulation will only improve the modulator performance by further reducing the waveguide optical transmission.
6. Energy consumption
In addition to performance figures of merit, low modulator energy consumption is necessary for enabling optical interconnects. Our modulator as-designed consumes 1.3 pJ per average switching operation (pJ/bit) for 3 dB modulation. We calculate this value using the criterion [18], where V is the operating voltage and C is the device capacitance. We operate the modulator at −2.3 V, where the absorption contrast is maximized. The capacitance of our device is 1 pF, assuming a 27μm device length and all three coated sides of the waveguide contributing their areas to the capacitance. The chosen device length and operating voltage correspond to 3 dB modulation of the optical signal.
While this energy consumption is higher than what is required for optical interconnect integration, we can dramatically reduce the amount of energy consumed by increasing the bulk ITO doping. Increasing the ITO doping, thus increasing the electron concentration, increases ωp and moves ε1 closer to the ENZ point. This allows modulator operation at a lower voltage since fewer accumulated electrons are needed to reach the ENZ point. There is a trade-off, however, as increasing the bulk ITO doping also increases ε2, thus yielding a higher insertion loss.
We can plot the relation between the modulator’s absorption contrast and its energy consumption to illustrate the trade-off between the two (Fig. 6). We can achieve 100 fJ/bit operation in this device while maintaining a absorption contrast of 2.47 for the TE mode (1.78 for the TM mode). These values correspond to those reported on state-of-the-art devices in current literature [17]. The energy consumption can be reduced even further if a lower absorption contrast is acceptable.
Fig. 6 There is a trade-off between modulator performance and energy consumption. By increasing the bulk electron concentration in the ITO, we can reduce the voltage needed to reach the ENZ point. This comes at a cost of sacrificing absorption contrast since the ON state absorption would increase with increased bulk electron concentration. We do find that 100 fJ operation is achievable with an absorption contrast of 2.47 (1.78) for the TE (TM) mode. Even lower energy consumption is possible provided that a lower absorption contrast is acceptable.
7. Conclusion
To conclude, we have introduced a modulator concept where electrical gating of an ITO/HfO2− coated silicon waveguide induces an ENZ region. This ENZ region diminishes transmission of an optical mode through the waveguide by simultaneously altering the local refractive index and increasing the free carrier absorption. The ENZ boundary condition induces high electric field concentration precisely in the region of highest free carrier absorption, yielding promising optical modulation. After taking into account realistic device operating constraints, we calculate a peak absorption contrast of 37.3 (5.6) for the TE (TM) waveguide mode; this allows for 3 dB modulation in under 30 μm of length in a non-resonant structure. We also find that 100 fJ/bit operation is achievable with a sacrifice in performance.
8. Methods
To simulate the modulator, we used a commercially available finite-element method (FEM) software package. To accurately model our structure we chose a 0.25 nm mesh around the HfO2-ITO interface; the accuracy of such a mesh was verified using convergence tests. We modeled the silicon waveguide with a permittivity of 12.11 and the HfO2 layer with a permittivity of 3.92.
Acknowledgments
We acknowledge funding support from The Intel Corporation and from the Air Force Office of Scientific Research (G. Pomrenke; grant no. FA9550-10-1-0264).
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