Figure 1 shows the proposed modulator structure, which consists of three waveguides. Two of them are silicon waveguides, and one is a plasmonic waveguide, which is placed between the two silicon waveguides. The plasmonic waveguide comprises ITO and multilayer graphene separated by HfO2 dielectric and two silicon layers. The ITO and graphene layers are sandwiched between two layers of silicon.

The height and width of the two outer silicone waveguides HSi and WSi are assumed. The width and height of the middle waveguide are WPL and HPL, respectively. The height of the ITO layer is HITO, and the height of the HfO2 layer is HHfO2.

To investigate the epsilon near zero (ENZ) effect of two materials (ITO and graphene), we model the ITO and graphene using the Drude-Lorentz model and the Kubo formula, respectively.

At first, the refractive index of graphene will be investigated, which is related to its dynamic conductivity. The conductivity of graphene can be described using the Kobo equation, as follows (Shin and Kim, 2015, Gosciniak and Tan, 2013, Chen et al., 2016):

$$\sigma (\omega )=\frac{{ - i{e^2}}}{{\pi {\hbar ^2}(\omega +i2\Gamma )}}\left( {\frac{{{\mu _c}}}{{{k_B}T}}+2\left( {{e^{\frac{{ - {\mu _c}}}{{{k_B}T}}}}+1} \right)} \right)+\frac{{ - i{e^2}(\omega +i2\Gamma )}}{{\pi {\hbar ^2}}}\left[ {\int_{0}^{\infty } {\frac{{\partial {f_d}( - \varepsilon ) - \partial {f_d}(\varepsilon )}}{{{{(\omega +i2\Gamma )}^2} - 4{{\left( {{\varepsilon \mathord{\left/ {\vphantom {\varepsilon \hbar }} \right. \kern-0pt} \hbar }} \right)}^2}}}d\varepsilon } } \right]$$

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kB is the Boltzmann constant, T is the temperature, \({f_d}(\varepsilon )={\left( {{e^{{{\varepsilon - {\mu _c}} \mathord{\left/ {\vphantom {{\varepsilon - {\mu _c}} {{k_B}T}}} \right. \kern-0pt} {{k_B}T}}}}+1} \right)^{ - 1}}\) is the Fermi–Dirac distribution, \(\hbar\) is the reduced Planck’s constant, µc is the chemical potential, \(\Gamma ={1 \mathord{\left/ {\vphantom {1 \tau }} \right. \kern-0pt} \tau }\) is the charged particle scattering rate, and ω is the angular frequency.

The relationship between voltage and chemical potential is defined\({\mu _c}=\hbar {v_F}\sqrt {\pi \left| {{a_0}V} \right|}\), where vF is the Fermi velocity in graphene and \({a_0}=9 \times {10^{16}}{m^{ - 2}}{V^{ - 1}}\).

Surface permittivity is obtained using graphene conductivity (\(\sigma =\sigma ^{\prime}+i\sigma ^{\prime\prime}\)), at room temperature and λ = 1.55 µm:

$${\varepsilon _g}=1+\frac{{i\sigma (\omega ,{\mu _c})}}{{\omega {\varepsilon _0}\delta }}$$

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\(\delta =1nm\) is the effective thickness of graphene (Shin and Kim, 2015, Gosciniak and Tan, 2013). The real part of the permittivity changes from positive to negative values as the chemical potential increased (Shin and Kim, 2015).

An ITO layer has been used in the plasmonic waveguide to increase the interaction between the graphene and optical fields (Feigenbaum et al., 2010). The ITO permittivity is described using the Drude-Lorentz model (equation 1) (Chee et al., 2012, Lou et al., 2012).

$$\epsilon ={\epsilon }_{{\infty }}-\frac{{\omega }_{p}^{2}}{{\omega }^{2}+i\gamma \omega }$$

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$${\omega }_{p}^{2}=\frac{N {e}^{2}}{{\epsilon }_{{\infty }}{m}^{\text{*}}}$$

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\({\varepsilon _\infty }=3.9\)is the high-frequency permittivity of ITO.\({m^*}=0.35{m_0}\) is the effective mass of electrons, and, \({m_0}\) indicates the charge and mass of electrons.\({\omega _p}\) is plasma frequency, and \(\gamma =1.84 \times {10^{14}}{{rad} \mathord{\left/ {\vphantom {{rad} s}} \right. \kern-0pt} s}\)represents electron scattering rate. N is the carrier concentration of electrons in the ITOs accumulation layer, which its thickness is 1nm (Wa) according to the Tomas-Fermi screening theory (Krasavin and Zayats, 2012, Kim and Kim, 2016). N is described as a function of voltage (V) as below equation (Jiang et al., 2019a):

$$N={N_0}+\frac{{{\varepsilon _0}.{\varepsilon _{Hf{O_2}}}.V}}{{e.{H_{Hf{O_2}}}.{W_a}}}$$

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It was assumed that N0=1× 1019 cm−3, \({\varepsilon _{Hf{O_2}}}=25\), \({H_{Hf{O_2}}}\)=10nm, \({W_a}\)=1nm and ɛ0 is the vacuum permittivity. High charge concentration in accumulation layers is reachable because of the high DC permittivity of HfO2 (\({\varepsilon _{Hf{O_2}}}=25\)). As voltage increases, the accumulation layer's carrier concentration and permittivity's imaginary part increases. The real part of ITO's permittivity changes from positive to zero (ENZ state) and negative values, and as a result, the ITO changes from dielectric state to metallic state (Alù et al., 2008). In the ENZ state, the electrical displacement components must be continuous at the interface of the ITO and dielectric (\({\varepsilon _{Hf{O_2}}}{E_{Hf{O_2}}}={\varepsilon _{ITO}}{E_{ITO}}\)). It causes significant confinement of the electrical field in the ITO and dielectric interface, and it improves the interaction of light and matter.

Figure 2(a) shows the propagation loss, and figure 2(b) shows the effective index of the device with two mono-graphene layers. The propagation loss has two peaks (one in 0.65 V and the other in 2.62 V) related to the ENZ effect in graphene and the other to the ENZ effect in the ITO. The first peak is affected by the chemical potential of graphene, and the second peak is related to the ITO. Because the impact of ENZ on graphene has occurred at a lower voltage, if the two peaks are combined by increasing the number of graphene layers, a lower operating voltage is achieved. Modulators with this effect could have higher efficiency and lower energy consumption (Eslami et al., 2021).

As the number of graphene layers increases, the two peaks propagation loss due to the effect of ENZ on graphene and ITO merged, and the propagation loss at voltage 0.62 V increases (figure 2 (c)). Also, The use of a large number of graphene layers will cause the properties of graphite, and the effect of ENZ will be attenuation. Therefore, we consider 7 layers of graphene.

The intensity of the guided mode in On-state (0 V) and in the Off-state (0.62 V) is shown in figure 3(b) and 3(c), respectively. The intensity of the guided mode in 0 V is distributed in the active layers in the coupling region (Figure 3(b)). In the 0.62 V, it is concentrated in the ITO and graphene layer (Figure 3(c)), and high losses occur due to the ENZ effect.