1. Introduction

Plasmonic metasurfaces exploit metal nanostructures that confine light beyond the diffraction limit while providing strong field enhancement [1,2]. However, most plasmonic metasurfaces fabricated to date are passive, and their optical properties are fixed and cannot be tuned after fabrication [3–9]. Yet applications abound for metasurfaces if they could dynamically tune the properties of light, such as phase, amplitude, and polarization, over a wide range of frequencies.

Several approaches have been utilized to actively and dynamically tune the optical response of metasurfaces in the range of wavelengths from mid-infrared to visible [10–19]. Based on their main mechanisms, these approaches can be divided into categories, for example, thermo-optical [10,18], electro-absorption [16,17] and electro-optical [11,13–15,19]. In the first approach, the optical properties of the metasurface are actively modulated by temperature. However, a major drawback is the limited speed of operation due to heat dissipation [18], rendering such devices too slow for applications requiring high speed, such as LIDAR for self-driving cars and autonomous machines. In the electro-absorption approach, an electric field is applied to a multiple quantum well structure to alter the bandgap of the semiconductor, thereby providing an opportunity to actively tune the imaginary part of the refractive index [16,17]. As an example, Chen et al. predicted that due to this effect, the modulating speed of germanium quantum wells with silicon-germanium barriers grown on a silicon substrate can be increased up to $100\;\textrm{GHz}$ when it is exposed to a sufficiently high intensity electric field [16]. In the electro-optical method, the carrier density and refractive index of a doped semiconductor are tuned by applying a voltage to a suitable device structure, such as a metal-oxide-semiconductor (MOS) capacitor. By changing the applied voltage, the structure can be driven from depletion to strong accumulation, which dynamically tunes the optical response of the structure due to perturbation of the carrier density, as reported in a plasmonic metasurface producing high-speed reflection modulation based on the carrier refraction in silicon [14]. The modulation relies on voltage-tuning of the complex refractive index of the semiconductor, which modulates the coupling efficiency between the incident light beam and the surface plasmon polaritons (SPPs) guided at the interface of the metal and dielectric in the MOS structure.

In recent years, there has been growing interest in transparent conductive oxides (TCOs), e.g., indium tin oxide (ITO), as a tunable material in plasmonic waveguides and metasurfaces. Interestingly, the behavior of these highly doped semiconductors can change from dielectric to metallic in the near infrared by increasing the electron density. This can be achieved, e.g., by applying a gate voltage to a MOS structure incorporating a TCO [11,20–23], where the real part of permittivity switches from positive to negative in the accumulation layer, inducing the dielectric-metal transition. When the real part of permittivity of a TCO is near zero (epsilon-near-zero, ENZ), the electric field becomes highly localized and enhanced [22,23]. The large index change of TCOs at ENZ motivates their use in plasmonic metasurfaces to dynamically and actively control the phase, amplitude and polarization of light by applying a voltage [11,20,21]. However, ITO has some drawbacks, the main one being that it has poor mechanical durability due to its brittle nature, so it should be deposited on a rigid substrate [24].

In one study, ITO was integrated into a metasurface antenna element producing a phase shift of $180^\circ $ and tuning the reflectance by ${\sim} 30\%$ through perturbation of the carrier density of the ITO by applying a gate voltage of $2.5\; \textrm{V}$ [20]. By incorporating ITO in a dual-gated reflectarray metasurface architecture, they were able to increase the phase shift and reflectance modulation to $303^\circ $ and $89\%$, respectively, under a gate voltage of 6.5 V [21]. Recently, a gate-tunable plasmonic (Au) nanoantenna array covered by hafnia (HfO₂) and ITO was proposed as a beam steering reflectarray [11], where varying the gate voltage from −3.2 to 3.2 V produced (theoretically) a large phase shift of ${\sim} 330^\circ $. Such devices are typically fabricated on a suitable substrate, e.g., Si or SiO₂. The HfO₂ film can be formed using atomic layer deposition (D) [14,21], the ITO film can be deposited via RF (radio frequency) sputtering [21], and plasmonic metallic features can be fabricated via lithography and lift-off of evaporated Au [14].

To date, various theoretical models have been developed and introduced to simulate the carrier density profile in biased semiconductor devices [11,25–29]. Conventional drift-diffusion (CDD) theory, which is defined by the Poisson and continuity equations [23,25], is a classical approach commonly used to simulate the physics of semiconductor devices [23]. To reduce computational cost in optical simulations, a step carrier density (SCD) model has been widely used, based on homogenizing the carrier density distribution obtained from CDD relative to the bulk [11,30]. As the dimensions of semiconductor devices decrease and become comparable to the electron wavelength, quantum effects play important roles and must be taken into account when computing the carrier density profile [25,31–33]. The self-consistent coupled Schrödinger-Poisson (SP) quantum model provides a robust theoretical framework for computing the density of carriers in small devices [26,27]. This approach solves the Poisson and Schrödinger equations iteratively to achieve self-consistency. However, the SP model becomes computationally prohibitive as dimensionality increases (i.e., 1D to 3D), motivating alternative models accounting for quantum effects. The density-gradient (DG) approach is computationally efficient and provides a macroscopic model of first- and second-order quantum effects in small devices [25,32] – specifically, the approach modifies CDD theory by adding terms that model quantum confinement and tunneling.

Extensive research has been carried out on optoelectronic MOS structures using carrier density profiles computed via CDD or SCD [11,23,30]. In this paper we show that neglecting quantum effects in carrier density distributions can lead to significant deviations in predicted device performance, especially in optoelectronic MOS structures involving conductive oxides, ENZ phenomena, and the propagation or resonance of SPPs. After summarizing the four methods of interest (CDD, SCD, DG, SP) in Section 2, we present in Section 3 modelling results for single-gated MOS, and single- and double-gated MIM (metal-insulator-metal) structures, and compare the predictions made by these models on the propagation characteristics of SPPs. Section 4 gives concluding remarks.

2. Carrier density models

Modeling the carrier density distribution in MOS structures driven by a gate voltage is essential to the engineering of nanoscale optoelectronic MOS devices. Applying a voltage alters the carrier distribution therein, inducing changes in the electrical and optical properties of the semiconductor [11,33]. In this section, we summarize four approaches for modelling the carrier density profile in a MOS structure: CDD theory, SCD, DG theory, and self-consistent SP theory. The CDD, DG and SP models are available in the semiconductor module of a commercial modelling tool [32].

We consider the MOS structures sketched in cartesian coordinates in Fig. 1, where ITO is assumed as the semiconductor, Au as the metal, and HfO₂ as the insulator. Figure 1(a) shows a single-gated MOS structure on the left panel and its band diagram at gate voltage of 0 V on the right panel, Fig. 1(b) shows a single-gated MOS structure integrated into an MIM waveguide, and Fig. 1(c) shows double-gated back-to-back MOS structures integrated into an MIM waveguide. Applying the gate voltage to the structures creates a perturbed region within ITO at the HfO₂-ITO interfaces, shown as the rectangular regions enclosed by the red dashed boundaries, where the carrier density changes occur.

 

Fig. 1. (a) Sketch of a MOS structure on the left panel and its band diagram at a gate voltage (V_g) of 0 V on the right panel; ${W_m}({{W_s}} )$ is the work function of the metal (semiconductor), ${E_{fm}}({{E_{fs}}} )$ is the Fermi energy level of the metal (semiconductor), ${\chi _s}({{\chi_{ox}}} )$ is the electron affinity of the semiconductor (oxide), ${E_c}({{E_v}} )$ is the conduction (valence) band of semiconductor, ${E_g}$ is the bandgap energy of the semiconductor and ${E_0}$ is the vacuum energy level. ${\phi _s}$, ${\phi _{fb}}$ and ${\phi _{ox}}$ are the surface potential, the flat band voltage and the voltage drop across oxide, respectively. Sketch of (b) single-gated MOS-MIM and (c) double-gated MOS-MIM structures. Applying a gate voltage to the structures perturbs the carrier density in ITO near the HfO₂ within the perturbation regions outlined as the dashed red rectangles.

Download Full Size | PDF

2.1. Electron effective mass in ITO

The electron effective mass is required in all of the carrier density models adopted. By changing the carrier density, the effective mass of electrons (${m_{eff}}$) varies due to the degeneracy and nonparabolicity of the ITO conduction band [34]. Several methods have been used to define the electron effective mass in ITO [34–36]. Inspired by Fujiwara and Kondo [34], we took into account the variation of ${m_{eff}}$ with carrier concentration in ITO via

Since ITO is a heavily-doped degenerate semiconductor, the conduction band effective density of states depends on the degeneracy factor g, and on the carrier density through ${m_{eff}}$

The electron mobility in ITO also varies with carrier density through m_eff,

2.2. Conventional drift-diffusion (CDD)

The conventional drift-diffusion theory is macroscopic in character and follows the electrostatics and current conservation laws commonly used for simulating the physics of semiconductor devices [23]. The steady-state equations of this classical approach incorporate a set of current density, current continuity, and Poisson’s equations [23,25],

Only one quantum effect – quantum compressibility – is included in this model. This quantum effect is induced via the Pauli exclusion principle and incorporated through the Fermi-Dirac distribution [25].

To obtain the electronic response of a MOS structure, we used the semiconductor physics interface and the equilibrium study in COMSOL 5.6 (semiconductor module) [32]. The physical and study settings used to implement our CDD model are provided in Supplement 1, Sub-Section S.1.1.1.

2.3. Step carrier density (SCD)

In this approach, we divide the ITO into a perturbed region with homogenized carrier density (dashed red bounding box in, e.g., Fig. 1(a), left panel) and an unperturbed (bulk) region of uniform doping ${N_b}$. This is done by defining a constant change in carrier density [11]

2.4. Density-gradient (DG)

Density-gradient theory is a macroscopic-continuum classical theory of electron transport that efficiently accounts for quantum effects [25,32]. DG theory builds on CDD and (similarly) respects the conservation of charge and mass, the balance of angular and linear momenta, and satisfies electrostatic and current conservation laws [25]. In contrast to CDD theory, in the DG description, the electron and hole concentrations not only depend on the quasi-Fermi levels but also on density gradient potentials. The density gradient potentials for electrons and holes are dependent on the gradient of their concentration and are defined by [25]

2.5. Schrödinger-Poisson (SP)

At length scales comparable to the electron wavelength, quantum effects (e.g., quantum confinement and tunneling) strongly affect the characteristics of a device, such as the carrier density distribution, the capacitance-voltage response, and the energy levels within the device [25,31–33]. Such effects appear in, e.g., tunnel junctions or in very small devices, but also in MOS devices where the carrier density profile spans a short length. The self-consistent SP model can be used to model quantum effects in such devices [26,27].

The SP model is a self-consistent iterative approach that provides a robust theoretical framework for modeling quantum electronic properties of semiconductor devices [26–29]. The approach includes quantum compressibility, quantum confinement, and quantum tunneling effects of an electron gas in quantum-sized semiconductors. The SP model also naturally includes electron diffraction/interference effects, which are not incorporated in the CDD and DG approaches [25–29]. To achieve self-consistency, the Poisson and Schrödinger equations are solved iteratively. The electrostatic potential $\phi $ satisfying the nonlinear Poisson equation couples to the Schrödinger equation via the potential energy term

Solving the system of coupled nonlinear second-order differential equations described by Eqs. (4.e) and (12) is nontrivial, so solutions are obtained iteratively. The iterations start by using a solution for the electric potential distribution $\phi $ satisfying the DG model (Eqs. (8)–10). The potential distribution is then used to update the potential energy term (Eq. (11)). After solving Schrödinger’s equation (Eq. (12)) with the updated potential energy, the calculated eigenenergies, ${E_j}$, and the noized wavefunctions, ${\Psi _j}$, are used to compute the carrier density profile, N, using a statistically weighted sum of the probability densities [26,39] via

3. Results and discussions

3.1. Electrostatic modelling

With reference to the MOS structures of Fig. 1, the thickness and static relative permittivity of HfO₂ were set to 5 nm and, respectively. The work function of Au (${W_m}$) was set to 5.1 eV [11]. The bulk doping concentration (${N_b}$), the bandgap energy (${E_g}$), the electron affinity $({{\chi_s}} )$ and the static relative permittivity (${\varepsilon _s}$) of ITO were taken as $2.65 \times {10^{20}}\textrm{c}{\textrm{m}^{ - 3}}$, $2.8$ eV, $4.8$ eV and 9.1, respectively [11]. The high doping concentration of ITO leads to its Fermi level residing within its conduction band, at a level higher than the Fermi level of Au, so short-circuiting the terminals allows electrons to flow from ITO to Au such that the Fermi levels are aligned, creating a depletion region in ITO and a non-zero electric field in HfO₂ at zero gate voltage. The maximum gate voltage that the structure can withstand is limited by the breakdown field (${E_{bk}})$ of HfO₂, taken as ${E_{bk}} = 6.4$ MV/cm [14]. We limit the gate voltage to the range −3 to 4 V, over which the electric field in HfO₂ remains below ${E_{bk}}$. We discretized the structure with mesh sizes of 0.5, 0.05 and 0.1 nm in the Au, HfO₂ and ITO regions, respectively.

We applied gate voltages to the 1D MOS structure shown in Fig. 1(a) and calculated the carrier density distribution (profile) in the ITO via the CDD, DG and SP models. Figure 2 shows the electron density profile over distance from the HfO₂-ITO interface at different gate voltages (${V_g}$) computed with the CDD (Fig. 2(a)), DG (Fig. 2(b)) and SP (Fig. 2(c)) models. As noted from Fig. 2, the carer density in ITO is perturbed over a region localized to ${\sim} \; 4$ nm from the HfO₂-ITO interface. The perturbed region undergoes depletion for ${V_g} < {V_{fb}}$ and accumulation for ${V_g} > {V_{fb}}$, where ${V_{fb}} = $. 0.7 V is the CDD flat-band voltage. The models agree well with each other under depletion but deviate significantly under accumulation.

 

Fig. 2. 1D carrier density vs. distance from the HfO₂-ITO interface into ITO, for the MOS structure of Fig. 1(a), at different gate voltages (V_g), obtained using: (a) CDD, (b) DG, and (c) SP models. The carrier density required to produce NZ in ITO is highlighted as the horizontal black dash-dot line.

Download Full Size | PDF

The horizontal black dash-dot line shows the electron concentration required to produce ENZ in ITO (∼ 6.4 × 10²⁰ cm⁻³). This carrier density is achievable under accumulation. From Fig. 2(a) we note that the CDD model predicts a carrier density under accumulation that is maximum at the HfO₂-ITO interface, decaying exponentially from the HfO₂ boundary in qualitative distinction from the predictions of the DG (Fig. 2(b)) and SP (Fig. 2(c)) models that predict a low density on the interface followed by a maximum into the ITO. The CDD model treats carriers as a classical free electron gas, ignoring quantum effects such as electron quantization and confinement, and electron diffraction/interference, which are accounted for by the SP model. The SP model (Fig. 2(c)) predicts a low carrier density at the HfO₂-ITO interface due to quantization of electron states thereon induced by quantum confinement. It is interesting to note that the carrier density profiles predicted by the SP model show an oscillatory character (Friedel-like oscillations [28]) at higher gate voltages due to electron diffraction and interference effects that are included in this model (Fig. 2(c)). Comparing Figs. 2(b) and 2(c) reveals that the peak carrier density predicted by the DG model is closer to the HfO₂-ITO interface relative to the SP model. However, the general shape of the carrier density distributions obtained using the DG model (Fig. 2(b)) agree well with those obtained with the SP model (Fig. 2(c)), validating the former as a good approximation to the latter.

3.2. Relative permittivity

To study the optical response of the structures, we use the spatially-dependent complex relative permittivity of ITO as computed using the Drude equation [41]

Figures 3(a)–3(c) and Figs. 3(d)–3(f) plot the real and imaginary parts of $\varepsilon $ of ITO, respectively, computed using the electron density profiles given in Fig. 2, at the free-space operating wavelength of $= 1550$ nm. We define ${V_{ENZ}}$ to be the minimum gate voltage required for the real part of the permittivity of ITO to approach ENZ ($\textrm{Re}(\varepsilon )= 0$). The profiles of $\textrm{Re}(\varepsilon )$ reveal that ${V_{ENZ}}$ is ${\sim} 1.9$ V for CDD (Fig. 3(a)) and ${\sim} 2.2\; $V for the DG and SP models (Figs. 3(b) and 3(c)). By applying negative voltages (depletion), the real and imaginary parts of $\varepsilon $ reach 4 and 0. Figure S1 (Supplement 1, section S.2) plots the corresponding complex refractive index.

 

Fig. 3. Relative permittivity in the perturbed region of ITO computed with the Drude model using the carrier density profiles plotted in Fig. 2 computed using (a) and (d) the CDD, (b) and (e) the DG, and (c) and (f) the SP models, for different gate voltages (V_g) at the operating free-space wavelength of $\lambda = 1550$ nm. The blue rectangle represents the metallic layer created within ITO at ${V_g} = 4$ V.

Download Full Size | PDF

From Fig. 3(a), we note that for ${V_g} < 2$ V, $\textrm{Re}(\varepsilon )$ throughout the perturbed ITO region is positive, indicating a dielectric. However, for ${V_g} \ge 2$ V, $\textrm{Re}(\varepsilon )$ crosses the ENZ line, such that a thin metallic layer develops directly adjacent to HfO₂, illustrated by the blue rectangle for ${V_g} = 4$ V in Fig. 3(a). The metallic character and thickness of this layer increase with increasing ${V_g}$, reaching ${\sim} 0.5$ nm at ${V_g} = 4$ V. The metallic character of this layer is strongest at the HfO₂-ITO interface, decreasing with distance and becoming dielectric after crossing the ENZ line.

Figures 3(b) and 3(c) for the DG and SP models show that $\textrm{Re}(\varepsilon )$ within ITO crosses the ENZ line at two points for ${V_g} > 2.2$ V, leading to the creation of a thin metallic layer a slight distance away from the HfO₂-ITO interface. This thin metallic layer is sandwiched between two thin dielectric gions within the ITO, as illustrated for ${V_g} = 4$V by blue rectangles in Figs. 3(b) and 3(c). The existence of two ENZ crossings is a significant qualitative difference in the predictions made by the DG and SP models relative to the CDD model, which has important implications on the plasmonic behavior of devices (as discussed below). This difference in behavior is due to quantum confinement, which is ignored in the CDD model (cf. Fig. S2, section S.3, Supplement 1). The DG and SP models also predict an increase in the metallic character and thickness of the metallic layer with increasing ${V_g}$ due to increasing electron density, with the thickness reaching ∼ 0.7 nm at V_g = 4 V (vs. ∼ 0.5 nm for the CDD model).

Although we find generally that the results of the DG model agree well with those of the SP model, there are small discrepancies between them. A close look at Figs. 3(b) and 3(c) reveals that the metallic layer is created closer to the interface in the DG model compared to the SP model. Interestingly, the SP model produces a small oscillatory behavior on the right side of the metallic region which are absent in the DG results, due to electron diffraction and interference effects that are ignored in the DG model (as mentioned previously).

3.3. Optical (plasmonic) mode analysis

In this subsection, we study the plasmonic modes supported by the single-gated MOS, single-gated MOS-MIM, and double-gated MOS-MIM waveguides sketched in Fig. 1. The structures vary along the x-axis but are invariant along the y-axis and along the propagation direction taken as the z-axis. We analyzed the plasmonic modes of the above structures for different gate voltages at the free-space optical wavelength of $= 1550\; \textrm{nm}$. At this wavelength, the relative permittivity of Au is ${\varepsilon _{Au}} ={-} 114.2 - 10.9i$ [42] (${e^{ + i\omega t}}$ time-harmonic form implied) and the relative permittivity of HfO₂ is ${\varepsilon _{ox}} = 4$ [11]. The relative permittivity in the perturbed region of ITO was taken as calculated from the carrier density profiles obtained using the CDD (Figs. 3(a) and 3(d)) and SP (Figs. 3(c) and 3(f)) models to facilitate comparisons and investigate the consequences of neglecting quantum effects. In addition, we consider the step index perturbation (SIP) model that follows the SCD approximation to the CDD profile (Sub-section 2.2, Eq. (7) in Eq. (16)), given its prevalence in the literature. Since the permittivity distribution obtained via the DG model approximates well that obtained with the SP model, we carry out the optical computations for the latter only. We solved the frequency-domain vector wave equations (derived from Maxwell's equations) subject to boundary conditions to find the plasmon modes supported by these structures [43],

The structures of interest are invariant along y such that they may be considered 1D (planar) waveguides (Fig. 1), so the plasmonic modes supported are purely TM (transverse-magnetic) with ${H_z} = {H_x} = {E_y} = 0.$ The structure may be modelled as an infinitesimal slice, bounded laterally by perfect magnetic conductor (PMC) boundary conditions

To obtain the modes of interest we used the electromagnetic waves (frequency domain) physics interface along with the mode analysis study in COMSOL 5.6 [32]. The physical and stuy settings used to carry out our computations are given in Supplement 1, Sub-Section S.1.2. The voltage-dependent relative permittivity distributions in the perturbed region of ITO (Figs. 3(a), 3(d), and 3(c), 3(f)) were incorporated in the optical simulations. Mesh sizes were set to 0.02 nm in perturbed ITO regions, 0.1 nm in the unperturbed ITO and HfO₂ regions, with minimum and maximum element sizes of 0.1 nm and 5 nm along the boundaries and in the Au regions.

3.3.1. Single-gated MOS waveguide

The structure of interest (Fig. 1(a)) consists of an optically semi-infinite Au layer, a 5 nm thick HfO₂ layer and a semi-infinite ITO layer. The mode field distributions were normalized to carry 1 W/m of power, enabling direct comparison of fields among cases.

We plot in Fig. 4 the distribution of the electric field magnitude of the plasmonic mode supported by the MOS structure at $= 1550$ nm, for gate voltages ranging from −3 V (depletion) to 4 V (accumulation). The fields are bounded, decaying into the Au and ITO region. The field decay length in Au is ∼ 5 nm, much shorter that the decay length in ITO, as can be appreciated from the insets to Fig. 4.

 

Fig. 4. Distribution of the electric field magnitude of the plasmonic mode supported by the MOS structure, for different gate voltages, at the free-space operating wavelength of $\lambda = 1550$ nm, computed using (a) the SIP, (b) CDD and (c) SP models. The insets show field distributions over the entire thickness of the MOS structure.

Download Full Size | PDF

As shown in Fig. 4(a), the SIP model produces a constant electric field throughout the perturbed ITO region due to the constant carrier density associated with this model (Eq. (7)). Recall that ${\tilde{t}_{pert}}$ was set to 0.5 nm under accumulation based on the CDD carrier distribution (Fig. 2(a)), and that ${\tilde{t}_{pert}}$ follows the depletion width under depletion. With increasing gate voltage (accumulation), the electric field magnitude initially increases, reaching a maximum value at ${V_g} = 1.7$ V, then decreases. This behavior can be understood by considering the continuity of the normal component of the electric displacement vector (D): By increasing the gate voltage from $- 3$ to $1.7\; \textrm{V}$, $\textrm{Re}(\varepsilon )$ in the perturbed region decreases, approaching ENZ at ${V_{ENZ}} = 1.7\; \textrm{V}$ (SIP), resulting in field enhancement therein. By further increasing the voltage (${V_g} > {V_{ENZ}})$, the electron density in the perturbed region increases, changing its character from dielectric to metallic, increasingly suppressing and expelling the electric field from this region.

In contrast, Fig. 4(b) computed via the CDD model predicts a single field extremum within the perturbed region of ITO for ${V_g} \ge 2V$ (${V_{ENZ}}$ ${\sim} $1.9 V for CDD). Over this voltage range, the electric field is strongly enhanced (${\sim} 5 \times {10^5}\; \textrm{V}/\textrm{m})$ and localized to the ENZ point. As shown in Fig. 2(a), increasing the gate voltage from $2$ to $4$ V increases the thickness of the metallic layer and shifts the ENZ point into the ITO, consequently shifting the peak electric field away from the HfO₂-ITO interface, as observed in Fig. 4(b).

The plasmonic mode fields computed via the SP model, shown in Fig. 4(c), differ significantly from those computed via the CDD model. The differences are due to the different carrier density profiles within the ITO (cf. Figures 2(a) and 2(c)). As noted relative to Fig. 3(c), ENZ occurs for ${V_g} = {V_{ENZ}}\; \sim \; 2.2\; \textrm{V}$ at the same position ($x)$ as the corresponding maximum in the carrier distribution of Fig. 2(c). However, further increasing the gate voltage (${V_g} > 2.2$ V), leads to ENZ at two positions, with $\textrm{Re}(\varepsilon )$ becoming negative in between, thus creating a thin metallic layer within the perturbed region (blue rectangle in Fig. 3(c) at ${V_g} = 4\; \textrm{V}$). As this metallic layer forms (${V_g} > 2.2\; \textrm{V}$), the electric field distribution changes in character, exhibiting two field extrema located at the two ENZ points (Fig. 4(c)). As the voltage increases further, the two field extrema are pulled apart due to the increasing thickness of the metallic layer.

We plot the spatial distribution of Re(${S_z})$ (Eq. (18)) for different gate voltages in Fig. 5. As shown in Fig. 5, significant wave power is localized to the low-permittivity region of ITO, consistent with the electric field behavior observed in Fig. 4. For ${V_g} < {V_{ENZ}}$, the perturbed region of the ITO is dielectric, and all models predict that the mode carries power along the positive z-axis, consistent with the direction of propagation. However, when part of the perturbed region transitions to metallic $({V_g} > {V_{ENZ}})$, the direction of power flow reverses therein bounded by ENZ points, because $\textrm{Re}(\varepsilon )$ changes sign. From the continuity requirement on normal D (i.e., D_x must be continuous), it can be understood that changing the sign of the permittivity reverses the electric field direction, consequently reversing the direction of power flow. However, integrating Re(${S_z}$) over its entire distribution, yields a net power flow along the positive z-axis for all cases, consistent with the direction of mode propagation.

 

Fig. 5. Distribution of the real part of the z-component of Poynting vector, Re(${S_z}$), of the plasmonic mode supported by the MOS structure, for different gate voltages, at the free-space operating wavelength of $\lambda = 1550$ nm, computed using (a) the SIP, (b) CDD and (c) SP models. The insets show Re(${S_z}$) over the entire thickness of the MOS structure.

Download Full Size | PDF

Contrary to the SIP model (Fig. 5(a)), Re(${S_z}$) computed for the CDD model (Fig. 5(b)) varies significantly within the perturbed ITO region due to the nonuniform permittivity therein. The CDD model predicts that for ${V_g} \ge {V_{ENZ}} = 1.9$ V, the mode power flows in the negative z-direction from the HfO₂-ITO interface to the ENZ point, then abruptly flips to flow in the positive z-direction into the ITO. This change in the sign of $\textrm{Re}({{S_z}} )$ is consistent with the change isign of $\textrm{Re}(\varepsilon )$ (Fig. 3(a)). Interestingly, $|{\textrm{Re}({{S_z}} )} |$ is maximum at the ENZ point.

Re(${S_z}$) computed for the SP model (Fig. 5(c)) also varies significantly within the perturbed ITO region. However, this model predicts a bump in the profile of $\textrm{Re}({S_z})$ for ${V_g} \le {V_{ENZ}}\; \sim 2.2$ V that increases with increasing gate voltage. This bump is consistent with the dip in $\textrm{Re}(\varepsilon )$ (Fig. 3(c)) and the bump in the carrier density (Fig. 2(c)) due to quantum confinement effects. For ${V_g} > {V_{ENZ}}\; \sim 2.2$ V, we note significant power localized at the ENZ points, with $\textrm{Re}({{S_z}} )$ becoming negative in between such that mode power flows in the negative z-direction therein, opposite to the direction of the power flow in the surrounding dielectric-like ITO.

We now investigate the effects of the gate voltage on the effective index of the plasmonic mod. Figure 6 plots ${n_{eff}}$ vs. gate voltage for the SIP, CDD and SP models. The predictions made by the SIP model depend on the thickness adopted for the step (cf. Fig. S3, section S.4, Supplement 1) - recall we chose ${\tilde{t}_{pert}} = $ 0.5 nm under accumulation and ${\tilde{t}_{pert}}$ equal to the depletion width under depletion. For all models $\textrm{Re}({{n_{eff}}} )$ is larger than the bulk refractive index of ITO (1.516), so the mode is bound (non-radiative). The three models agree very well over the voltage range of $- 3$ to $1$ V, in the depletion and flat-band regions of the MOS.

 

Fig. 6. (a) Real and (b) imaginary parts of the effective refractive index of the plasmonic mode supported by a single-gated MOS waveguide vs. gate voltage, calculated using the SIP, CDD, and SP carrier distribution models. ${V_{ENZ}}$ ${\sim} 1.7$, $1.9,$ and $2.2$ V for the CDD and SP models (gray, red and blue dotted lines), respectively. The perturbations in Re(${n_{eff}}$) and Im(${n_{eff}}$) relative to their values under depletion (${V_g} ={-} 3$ V) are shown on the right axes.

Download Full Size | PDF

However, the models differ significantly under accumulation, especially as the gate voltage approaches and exceeds ${V_{ENZ}}$ where features in ${n_{eff}}$ develop (${V_{ENZ}}$ ${\sim} 1.7$, $1.9$ and $2.2$ V for the SIP, CDD and SP models). All models show that $\textrm{Re}({{n_{eff}}} )$ decreases with increasing voltage, reaching a minimum at their ${V_{ENZ}}$ (Fig. 6(a)), due to increasing field localition (overlap) near the ENZ point (Fig. 4). For ${V_g} > {V_{ENZ}}$, $\textrm{Re}({{n_{eff}}} )$ of the CDD and SP models initially rise, then saturate at higher voltages. This behavior is due to creation of the metallic layer in the perturbed region of ITO that expels the electric field into the surrounding dielectric regions of higher refractive index. From the perturbations plotted on the right axes of Fig. 6, we note ΔRe(${n_{eff}}$) ∼ −7 × 10⁻³ and ΔIm(${n_{eff}}$) ∼ 7 × 10⁻³ predicted by the SP model, with the CDD model predicting (at most) half of this perturbation.

The SIP model fails to reproduce even the qualitative behavior of the CDD model that it approximates (Fig. 6), and due to its nature, it cannot reproduce any detail in the plasmonic field structure near ENZ (Fig. 4). The CDD model leads to underestimates in Δ${n_{eff}}$ (Fig. 6) and incorrect details in plasmonic field distributions near ENZ (Figs. 4 and 5) relative to the SP model, due to the prediction of a single ENZ point vs. two for the SP model. These results emphasize the need for a quantum carrier model to accurately simulate the accumulation regime of plasmonic MOS structures, particularly when the semiconductor is a TCO driven to ENZ.

3.3.2. Single-gated MOS-MIM waveguide

We now study the plasmonic mode supported by the single-gated MOS-MIM waveguide of Fig. 1(b). An MIM waveguide supports the symmetric plasmonic mode down to zero core thickness [44] (separation between the Au layers bounding the HfO₂ and ITO layers), thereby producing very strong optical confinement and increased overlap with the perturbed region in ITO. The thickness of the HfO₂ layer was maintained to $5$ nm but the ITO layer was set to a thickness of $15$ nm, for a total core thickness of 20 nm between two (optically) semi-infinite Au layers. Here, the perturbation in the carrier density of ITO is induced by single gating, by connecting the bottom Au layer to ${V_g}$ and grounding the top one (Fig. 1(b)). Since the thickness of the perturbed layer is ${\sim} 4$ nm (<15 nm), the single-gated voltage-dependent relative permittivity distributions given in Fig. 3 can be used.

Figure 7 gives the electric field and Re(S_z) distributions of the plasmonic mode in this structure for different gate voltages at $= 1550$ nm for the CDD and SP models. The field distributions (Figs. 7(a) and 7(b)) in the perturbed region exhibit similar trends as in the MOS structure (Fig. 4). However, the magnitude of the electric field in the MOS-MIM structure is significantly higher (recall that all modes are normalized to carry the same power), due to the modelling confined to the very small core region (20 nm). Correspondingly, Figs. 7(c) and 7(d) reveal that Re(S_z) in the MOS-MIM structure is almost 10× larger than in the MOS structure (Fig. 5).

 

Fig. 7. Distribution of the electric field magnitude and Re(${S_z}$) of the plasmonic mode supported by the single-gated MOS-MIM structure, for different gate voltages, at the free-space operating wavelength of $\lambda = 1550$ nm, computed using (a) and (c) the CDD, and (b) and (d) the SP models. The insets show field and Re(${S_z}$) distributions over the core thickness and in the Au layers of the single-gated MOS-MIM structure.

Download Full Size | PDF

Figure 8 plots ${n_{eff}}$ of the plasmonic mode in the single-gated MOS-MIM structure vs. gate voltage, revealing a qualitatively similar voltage response as the single-gated MOS structure (Fig. 6). However, the significantly higher confinement provided by the MOS-MIM structure is evident from the larger ${n_{eff}}$, and from the much larger perturbation in ${n_{eff}}$ (>50×) due to the stronger mode overlap with the perturbed region in ITO. We note perturbations of ΔRe(${n_{eff}}$) ∼ −0.4 and ΔIm(${n_{eff}}$) ∼ 0.4 predicted by the SP model, with the CDD model predicting (at most) half of this perturbation. The strong confinement of the MOS-MIM structure amplifies differences between the carrier density models under accumulation (especially near ${V_{ENZ}}$). The need for quantum modelling of the carrier density in such structures is manifest – the discrepancy between the results obtained using the CDD model and the SP model near ${V_{ENZ}}$ is large enough to invalidate use of the former in this regime.

 

Fig. 8. (a) Real and (b) imaginary parts of the effective refractive index of the plasmonic mode supported by a single-gated MOS-MIM waveguide vs. gate voltage, calculated using the CDD and SP carrier distribution models. ${V_{ENZ}}$ ${\sim} 1.9,$ and $2.2$ V for the CDD and SP models (red ande dotted lines), respectively. The perturbations in Re(${n_{eff}}$) and Im(${n_{eff}}$) relative to their values under depletion (${V_g} ={-} 3$ V) are shown on the right axes.

Download Full Size | PDF

3.3.3. Double-gated MOS-MIM waveguide

We now study the double-gated MOS-MIM structure sketched in Fig. 1(c) at $= 1550$ nm. The structure is similar to the single-gated structure of Fig. 1(b), except for an additional HfO₂ layer of thickness 5 nm and an ITO thickness reduced to 10 nm such that the core thickness (separation between the Au layers) is maintained to 20 nm. Both Au layers are assumed connected to the same gate voltage ${V_g}$ (for simplicity) and the ITO layer is grounded. The gate voltage induces mirror-symmetric perturbed regions within the ITO, along the top and bottom interfaces with HfO₂ (red dashed boxes in Fig. 1(c)). Since the thickness of the ITO layer (10 nm) is larger than the total thickness of both perturbed regions (∼8 nm), the single-gated voltage-dependent relative permittivity distributions given in Fig. 3 can be used for the mirror-symmetric perturbed regions.

Figure 9 gives the distributions of the electric field and Re(${S_z}$) of the plasmonic mode in this structure for different gate voltages at $= 1550$ nm for the CDD and SP models. In this symmetric structure, the modes have a symmetric field distribution about the midpoint of the y-axis, as shown in the insets to Fig. 9. The trends within each perturbed region are similar to those seen in the single-gated MOS-MIM structure. However, the magnitude of the electric field in the double-gated MOS-MIM is smaller than in the single-gated MOS-MIM, because there are two perturbed regions in the former, each producing ENZ regions and field enhancement (recall that all modes are normalized to carry the same power). We also observe for ${V_g} > {V_{ENZ}}$ that the fields localized near the ENZ points begin to couple (cf. insets to Fig. 9).

 

Fig. 9. Distributions of the electric field magnitude and Re(${S_z}$) of the plasmonic mode supported by the double-gated MOS-MIM structure, for different gate voltages, at the free-space operating wavelength of $\lambda = 1550$ nm, computed using (a) and (c) the CDD, and (b) and (d) the SP models. The insets show field and Re(${S_z}$) distributions over the ore thickness and in the Au layers of the double-gated MOS-MIM structure.

Download Full Size | PDF

Figure 10 plots ${n_{eff}}$ of the plasmonic mode in the double-gated MOS-MIM structure vs. gate voltage, revealing a similar voltage response as the single-gated MOS-MIM structure (Fig. 8), but with double the perturbation in ${n_{eff}}$ due to the two perturbation regions. We note perturbations of ΔRe(${n_{eff}}$) ∼ −0.9 and ΔIm(${n_{eff}}$) ∼ 0.8 predicted by the SP model, with the CDD model predicting (at most) half of this perturbation. Compared to the single-gated MOS structure (Fig. 6) the doubled-gated MOS-MIM produces a perturbation ∼100× larger.

 

Fig. 10. (a) Real and (b) imaginary parts of the effective refractive index of the plasmonic mode supported by a double-gated MOS-MIM waveguide vs. gate voltage, calculated using the CDD and SP carrier distribution models. ${V_{ENZ}}$ ${\sim} 1.9,$ and $2.2$. V for the CDD and SP models (red and blue dotted lines), respectively. The perturbations in Re(${n_{eff}}$) and Im(${n_{eff}}$) relative to their values under depletion (${V_g} ={-} 3$ V) are shown on the right axes.

Download Full Size | PDF

4. Conclusions

We modelled the gate-induced 1D carrier density profile in a Au-HfO₂-ITO MOS capacitor using two classical models - CDD and DG - and one quantum model - SP. Under depletion, all models agree well, such that quantum modelling in this regime is unnecessary. However, under accumulation the CDD and SP carrier profiles differ considerably. The difference gains significance in semiconductors with a low background refractive index such as TCOs, e.g., ITO, because ENZ can be reached and exceeded therein under a sufficiently large gate voltage. The CDD model yields one ENZ point in the permittivity profile, whereas the SP model yields two such points - this difference impacts the predicted performance of optoelectronic devices.

The surface plasmon mode in three waveguides incorporating Au-HfO₂-ITO structures (single-gated MOS, single- and double-gated MOS-MIM) was investigated as a function of gate voltage using the CDD and SP carrier profiles. The mode field (|E|) and power (Re(${S_z}$)) distributions in the structures reveal high localization and strong enhancement at ENZ points, with significant differences evident in field details associated with the CDD and SP models due to the different number of ENZ points. The gate voltage response of ${n_{eff}}$ of the plasmon modes is similar for both models under depletion, but differs significantly under accumulation, where the SP model predicts a much larger perturbation in ${n_{eff}}$ than the CDD model (by ∼4×). The discrepancy is large enough to invalidate the CDD model in MOS structures on ENZ materials under accumulation, strongly motivating a quantum carrier model in this regime.

The SIP model failed under accumulation to reproduce the qualitative behavior of the CDD model that it approximates, and due to its nature, cannot reproduce details in the plasmon field structure at ENZ points. This simple model, widespread in MOS structures on Si, is of limited use in MOS structures on TCOs involving ENZ effects. The DG model approximates well the SP model and can be used as an alternative (once calibrated) as it is easier to implement and more efficient numerically.

The results of this study emphasize the necessity of including quantum effects, as captured, e.g., by the SP model, when computing carrier density profiles under accumulation in gate-tunable MOS structures on TCOs. Incorporating the MOS structure in a high confinement surface plasmon waveguide amplifies the need for quantum carrier modelling. Although we limited our investigation to waveguides, we expect our conclusions to hold for plasmonic and dielectric resonators incorporating similar MOS structures.