Enhancing modulation performance by design of hybrid plasmonic optical modulator integrating multi-layer graphene and TiO2 on silicon waveguides

A novel way to enhance modulation performance is through the design of a hybrid plasmonic optical modulator that integrates multi-layer graphene and TiO2 on silicon waveguides. In this article, a design is presented of a proposed modulator based on the use of the two-dimensional finite difference eigenmode solver, the three-dimensional eigenmode expansion solver, and the CHARGE solver. Leveraging inherent graphene properties and utilizing the subwavelength confinement capabilities of hybrid plasmonic waveguides (HPWs), we achieved a modulator design that is both compact and highly efficient. The electrical bandwidth f 3dB is at 460.42 GHz and it reduces energy consumption to 12.17 fJ/bit with a modulator that functions at a wavelength of 1.55 μm. According to our simulation results, our innovation was the optimization of the third dielectric layer’s thickness, setting the stage to achieve greater modulation depths. This synergy between graphene and HPWs not only augments subwavelength confinement, but also optimizes light–graphene interaction, culminating in a markedly enhanced modulation efficiency. As a result, our modulator presents a high extinction ratio and minimized insertion loss. Furthermore, it exhibits polarization insensitivity and a greater bandwidth. Our work sets a new benchmark in optical communication systems, emphasizing the potential for the next generation of chip-scale with high-efficiency optical modulators that significantly outpace conventional graphene-based designs.

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Introduction
Significant advancements in communication bandwidth have continuously been made, particularly in alignment with the evolving trends of communication technology, supported by substantial research efforts [1][2][3][4].This enables development of high-speed and compact devices for optical communication systems [1].Modulators in silicon (Si) photonics have become a highly popular solution because they can transmit signals and transfer data at high rates [1][2][3][4].In the photonics domain, the optical modulator stands out as a preeminent component, playing a pivotal role in converting electrical signals into optical information.High-speed modulators have broadband capabilities and high modulation depths which are essential for integrated optoelectronic devices [1,2,[5][6][7].However, such devices do have some limitations that include low switching rate, low bandwidth, or large dimensions.Overcoming these constraints is crucial for advanced developments [1,2].Hybrid plasmonic waveguide (HPW) modulators, integrated with graphene electro-absorption on a silicon-on-insulator (SOI) platform represent an approach for augmenting the operational capabilities of modulators [1,8].Owing to graphene's two-dimensional nature and its exceptional optical and electrical properties, it exhibits distinctive characteristics, particularly in terms of its remarkably high electron mobility, exceeding 200 000 cm 2 V −1 s −1 [1,2].Moreover, graphene serves as a highly suitable material for electro-optical devices owing to its nearly 2.3% optical absorption over the ultraviolet to infrared frequency range, high carrier mobility, transparency, mechanical flexibility, and low contact resistance [9,10].The absorption properties of graphene can be significantly augmented through its integration into a waveguide structure.In this context, the most widely adopted method for graphene growth is chemical vapor deposition [11].The tunable electro-absorption and electro-refraction properties of graphene make it highly suitable for photonic applications, such as integrating graphene into optical modulators on Si photonic platforms [12].Numerous research studies have endeavored to design optical devices with enhanced performance characteristics by incorporating graphene, exemplified by optical switches and plasmonic absorbers [13,14].This phenomenon results in gate-controlled optical absorption in graphene within a hybrid structure, demonstrating an essential part of on-chip integrated photonic devices [15].However, integrating modulator design onto Si waveguides poses challenges and remains a focus of ongoing research.
Numerous research endeavors have been dedicated to advancing the development of hybrid graphene plasmonic waveguide modulators [8][9][10].For the first time and with an innovative design, a modulator incorporates a mono-layer graphene sheet and an aluminium oxide (Al 2 O 3 ) buffer layer on top of a Si waveguide, with a silicon dioxide (SiO 2 ) substrate.This device has a modulation depth (MD) of 0.1 dB μm −1 and a modulation speed of 3 dB with an electrical bandwidth ( f 3dB ) of 1.2 GHz [1].A double-layer graphene modulator has been proposed, which had an MD of ∼0.16 dB μm −1 , and a modulation speed f 3dB of 1 GHz [2].Nonetheless, responses of both modulators are notably limited, primarily attributed to their absorption characteristics that are constrained by the dispersion of propagated light within the field [1,2].Additionally, there have been reports on the proposed optical modulator utilizing double-layer graphene based on a Si waveguide.For instance, the modulator incorporates a double-layer graphene-Al 2 O 3 -graphene stack within an inverted-rib-type Si waveguide.It achieves a remarkable depth of modulation equal to 0.41 dB μm −1 , surpassing previous Si waveguide capabilities.Additionally, it maintains a low insertion loss (IL), 0.42 dB μm −1 , and offers an electrical bandwidth ( f 3dB ) of up to 46.4 GHz.Furthermore, its energy consumption per bit (E bit ) is less than 630 fJ [16].An optical modulator developed on the principles of strong coupling within a waveguide featuring a doublelayer graphene-hBN-MoS 2 -graphene buffer layer exhibits excellent performance characteristics.These include a high MD, 0.58 dB μm −1 , IL that is also 0.58 dB μm −1 , f 3dB of 22.1 GHz, and E bit of 52.7 fJ [17].These exceptional attributes underscore the potential significance of this modulator in the context of both off-chip and on-chip optical interconnection applications.
A novel modulator design has recently been presented that involves the use of multi-layer graphene as an absorber layer within a high-performance HPW structure [18].The plasmonic-enhanced modulator utilizes a waveguide structure with a stack of graphene/hBN layers, where hBN serves as a flat substrate for graphene [19,20].This can increase interaction between graphene layers with propagating light.Then, it yields a modulation extinction ratio (ER) as high as 36 dB/ μm with an E bit value of about 14 fJ/bit [18].This design approach has the potential to significantly enhance the modulator's speed, achieving an f 3dB of up to 262.9 GHz, while simultaneously reducing the IL to 4 × 10 −3 dB μm −1 in the transverse electric (TM) mode for a 40 μm long device [21].
However, there is an ongoing necessity for additional research and analysis on how the quantity of multi-layer graphene influences the behavior of the modulator.Achieving efficient control over the intensity or phase of incoming light is critical.It has been observed that graphene modulator performance is significantly influenced by polarization of incident light.Furthermore, optimizing the structural parameters can further enhance the modulation speed of the modulator.Therefore, a new design is proposed and a simulation presented of multi-layer graphene on a titanium dioxide (TiO 2 ) substrate in an HPW structure to enhance the modulation depth.It consists of a stack of graphene-TiO 2 -graphene as an absorption layer.The graphene layers are separated by TiO 2 spacers for an electro-absorption optical modulator that uses Si waveguides.We studied the number of graphene layers and their effect on the ER and the bandwidth.Also, the proposed modulator employs the two-dimensional finite difference eigenmode (2D FDE) solver, the three-dimensional eigenmode expansion (3D EME) solver, and the CHARGE solver.The wavelength used is 1550 nm under ambient conditions.Multilayered graphene acts as a transparent gate electrode modulating absorption in the bottom graphene layer.By electrically tuning graphene permittivity, the modulator switches between dielectric and graphene.This changes the waveguide's effective mode index (EMI) and loss.This design approach significantly improves modulator speed, which is an advantage of electrically switching ON/OFF graphene interband absorption.It also further increases the optical absorption and MD while reducing the IL.This analysis points to a very important role of the modulator for the structure of multi-layer graphene.It offers a promising pathway for realization of next-generation highspeed, low-power optical communication systems.

Device structure and operating principle
The selected material for the modulator structure under investigation is presented in figure 1.This structure comprises an SiO 2 layer having a thickness H SiO2 of 800 nm as the substrate layer.It includes an Si layer as a waveguide with a width W Si of 600 nm and thickness H Si of 250 nm [1].The width of the graphene ensures a dependable electrical connection between the platinum/gold (Pt/Au) electrode and the waveguide.They are positioned at 500 nm on the left slab W slab-L and on the right slab W slab-R , to avoid impacting the propagation mode.A dielectric layer with width W D of 20 nm, serves as an insulator between the graphene and the waveguide.In this research, we chose TiO 2 as the dielectric layer due to its high transparency within the telecommunications band [22], strong nonlinear optical properties, and its capability to reduce waveguide propagation losses caused by light scattering [23,24].On the top surface, the Si waveguide is covered with a graphene monolayer, as can be seen in figure 1(b).For structural configuration, multiple graphene monolayers are separated by layers of TiO 2 .Three TiO 2 layers of 10 nm fixed thicknesses are defined in figure 1 varies in the 10-70 nm range.The electrode to Si waveguide distance is set at 500 nm.This is to avoid the impacts of metal electrodes on the light mode.To ensure a reliable electrical connection between the metal and graphene, we utilized Pt/Au metal contacts on a TiO 2 substrate, following the methodology outlined in [1].The resultant device has a compact length L of only 40 μm, along with a scattering rate of 15 meV, at a temperature T of 300 K, as corroborated by references [25,26].The charge and core electron-hole density are applied in relation to a biasing voltage range of −5.5 to 4.5 V.In this investigation, we utilized a set of simulation tools available in ANSYS Lumerical software [25].We determined the refractive indices for a single graphene layer and Si.The refractive indices of Si, SiO 2 , and TiO 2 are documented as 3.47, 1.55, and 2.50, respectively [27].At a specific wavelength λ 0 of 1550 nm, equivalent to a frequency range of 193 to 200 THz, the relative permittivity coefficients of Si, SiO 2 , and TiO 2 , are 11.7, 3.9, and 80, respectively.TiO 2 displays nonlinear optical activity and the presence of a sharp waveguide edge serves to mitigate waveguide propagation losses resulting from light scattering [22,24].Consequently, TiO 2 is a viable candidate for use as a graphene substrate, given that its lattice constant is similar to graphene [28,29].Oxides restrict the mobility of carriers in graphene, however TiO 2 functions as an insulator, providing a solution to bypass this limitation [30].
Graphene has distinctive optical properties that are altered with applied voltage.The modulation rate is adjusted with a changing absorption rate that is in proportion to an applied voltage.This can be estimated using the Kubo formula, which describes the complex electrical surface conductivity σ of graphene.When applied to the SOI waveguide model, the following formula accounts for intraband and interband absorption [25,31,32]: , where ω is the light angular frequency, Γ is a phenomenological scattering rate (or relaxation time τ = 1/2Γ) that is independent of energy E. For the simulation in this work, the relaxation time (τ) of graphene in contact with TiO 2 was set at τ = 1 ps at T = 300 K [33].However, the relaxation time may be influenced by a scattering mechanism that dominates the material behaviour [34].E F is the Fermi level, which depends on graphene charge carrier concentration.
The chemical potential μ c is controlled using electrical gating.Basically, when μ c < ÿω2 (ÿ is the reduced Planck's constant), μ c is chemical potential.This is determined by the carrier concentration n 0 (n 0 = (μ c /ħv) 2 /π), and v is the constant velocity of the graphene sheet (v = 10 8 cm s −1 ) [23,31,35], j is an imaginary number, e is the charge of an electron, ħ is the reduced Planck constant (ħ = h/2π), and k B is Boltzmann's constant.
In equation (1), the first and second terms correspond to the intraband and interband contributions, respectively.Earlier studies [35,36] demonstrated that in the infrared spectral range, the intraband Drude-Boltzmann conductivity dominates, particularly at lower temperatures and higher carrier densities.Additionally, higher frequencies are influenced by interband absorption.Previous investigations assert that, under conditions of low temperatures and elevated carrier densities, the reflectance of multilayers experiences a precipitous decline followed by a subsequent plateau [35,36].These distinctive features arise from excitations of weakly damped waves in the context of direct interband electron transitions.At longer wavelengths, the influence of intraband transitions becomes stronger and the characteristic of these transitions in the transient response of photoexcited graphene in the mid-infrared and terahertz spectral ranges [37].Determination of the intraband and interband terms for the graphene layer employs the Boltzmann-Drude expression in conjunction with Fermi-Dirac statistics.This calculation is carried out under the specific condition where the chemical potential μ c significantly exceeds the temperature T and applied voltage [35].The surface conductivity of a single graphene layer in equation (1) can be given by the Kubo formula for considering the intraband transition terms in equations (2) and (3), where σ intra is intraband conductivity and σ inter is interband conductivity [35,36,38] The effective permittivity model provides a more accurate representation of the material's behavior and is well-suited for longer wavelengths.However, in the telecom range, shorter wavelengths are employed, as reported by Chatzidimitriou et al [39].The use of the surface tensor conductivity model can introduce complexities in the simulations, which often require a more rigorous treatment of the graphene material properties.However, since this material type requires explicit expressions for the entries of the permittivity tensor, it is necessary to use an approximation of the surface conductivity.Monolayer graphene's electric permittivity can be represented using a uniaxial anisotropic permittivity model for the linear surface conductivity X 1 ¯( ) [34], which is adequate for accurate predictions [23,29,30], given by: || where d g denotes the thickness of monolayer graphene.Generally, the thickness of monolayer graphene is in the range of 0.4-1.7 nm [2,37,40].For our model, we utilized a thickness of 0.75 nm.ε 0 represents the permittivity of vacuum, and ε r denotes the background relative permittivity.The dispersion of an SOI waveguide is influenced by the density of free carriers in the semiconductor, leading to variations in the refractive index.In the context of a modulator, Soref and Bennett [41] detailed the refractive index of Si.Their description connects refractive index changes to variations in the absorption coefficient at a specific pump wavelength, λ 0 of 1550 nm, as follows: The electro-optical modulator performance is determined by optical and electrical factors.Its structure was simulated to set voltage or surface recombination boundary conditions.Waveguide propagation losses are influenced by material attributes, geometry, and external voltage.The chemical potential is controlled by applying an external voltage to the graphene/TiO 2 modulator and using a dielectric in the capacitive stack [26].For modulation, a biased gate voltage changes the Fermi level of graphene, adjusting optical absorption and the waveguide's optical response.For electronic simulation, the waveguide core possesses electron-hole density traits based on the bias voltage, while the charge mode utilizes semiconductor 3D density for modelling semiconductor materials and waveguide shapes.The 3D electron density n 3D for graphene is [1,43,44] where * m is the effective mass of an electron or hole, E c signifies the conduction band base energy, and E F represents the Fermi level.Electron density in a graphene layer addresses discrepancies between n 3D and the actual electron density determined by the 2D density of states, notably if the Fermi level is either below (E F 0.05 eV) or above the Dirac point (E F > 0.05 eV).The fitting parameters incorporate a straightforward scaling factor, with electron and hole effective masses of 1.768 and 0.4614, respectively [27,44].
The n 3D model aligns well with the actual electron density.Charge variations in the Si waveguide, influenced by biasing voltage, impact the μ c of monolayer graphene.The optical mode maintains a waveguide geometry consistent with the CHARGE solver's specifications.The acceptor concentration in the Si waveguide is 10 18 cm −3 [26,44].Electron-hole density is loaded based on bias voltage and matches the CHARGE solver's output, derived from the designed waveguide and chemical potential of graphene sheets on the device's top and side.Cross-sectional field profiles and long waveguide propagation simulations were done for the modulator.Such techniques support the development of an electro-optical modulator, characterizing propagation losses for each biasing voltage [1,42].

Simulation results and discussion
According to previous reports, single-layer [1] and double-layer graphene [2] designs have been made with a graphene layer associated with the waveguide, as light appears on the graphene surface.However, in this work, we designed a modular multilayer graphene waveguide, as shown in figure 1.Although this waveguide can support both the TE [19] and TM modes [19,45], published research that is similar to our work shows that the absorption of the TM mode is greater than the TE mode because of its better overlap with graphene [45].Moreover, graphene and dielectric layers can increase absorption [19].Therefore, we focus on the waveguide for the TM mode because this mode maintains a lower IL and a higher MD than the TE mode, while optical losses are predominantly directed to the TE mode.The TM mode in our proposed modulator allows for a high polarization loss ratio, low IL and modulation independent of polarization.The simulation results show that a W Si of 600 nm and H Si of a 250 nm waveguide can support propagation of the optical mode.For the propagation field in the TM mode, most optical light can be concentrated on the graphene surface of the waveguide.
A previous design report gives important insights [1,2].Its optical modulator design consists of a single sheet or two graphene layers.A phenomenon occurs where a large amount of light is incident on the top surface layer of the dielectric.Therefore, to absorb a large amount of light, adding another graphene layer was necessary.However, an additional graphene layer alone is insufficient to control the electric charges generated in that layer.Therefore, dielectric layer thickness was increased.This is to elucidate the reactions that occur and how they affect light limitations.Therefore, the area under study is limited by only increasing the dielectric layer's thickness in a third layer.
We adjusted the thickness of various dielectrics located in the h (3)TiO2 , as shown in the structure of figure 1.For each site, we can calculate the profile mode, as shown in figure 2. To optimize light penetration into the layer or enhance deep absorption, we can manipulate the voltage drift to redirect the light toward the top graphene surface as effectively as possible.Also, the results of the simulation show that the intensity of light that appears inside the waveguide is where the light intensity is focused on the top of the graphene layer.In the inset of figure 2, the profile mode confirms that our designed waveguide can support light and that light can propagate in the waveguide.Moreover, it was found that there is apparently almost no light loss at a h (3)TiO2 value of 10 nm thickness.A driving voltage V D is applied with varying h (3)TiO2 values for thicknesses of 10 nm, 30 nm, 50 nm, and 70 nm.There is a loss of light, which shows that the V D value affects the absorbance, as can be seen in figure 2. It can be concluded that at a 30 nm thickness, the maximum of the imaginary part (Im(n eff )) increase is ∼0.013 and the real part (Re(n eff )) of the effective index is ∼1.91, which characterizes Table 1.Soref and Bennett fitting coefficient parameters [32,34,35].When examining the mode profiles with the insert images in figures 2(c) and (d), it is clear that charges accumulate near the dielectric area in h (3)TiO2 or on the surface of h (2)TiO2 .This accumulation occurs when the thickness of the layer is either 50 nm or 70 nm.The applied voltage is not strong enough to cause charge conduction or shift the Fermi energy level of graphene.As a result, these charges do not migrate to the uppermost graphene layer.Instead, residual charges or electric filtration remains in the third layer, leading to a profile mode on the h (2)TiO2 graphene layer.

Parameters
In figure 3, the thickness of h (3)TiO2 varies with values of 10, 30, 50, and 70 nm.Specifically, with a h (3)TiO2 of 30 nm, there is a modulation depth reaching 0.42 dB μm −1 in the TM mode, signifying the EMI of the structure.This EMI is influenced by the chemical potential, depending on the number of graphene layers present.At a wavelength of 1550 nm, incorporating additional graphene layers enhances the structure's EMI performance, which is driven by changes in its chemical potential.This discovery has been harnessed to bolster modulator performance, especially in structures with two graphene layers.Factors such as refractive index in both the Si and TiO 2 layers, THz frequencies along with their associated losses, have a profound effect on EMI.According to the relationship between the absorption depths and the modulator drive voltages, chemical potential experiences the least loss, and the mode profile is set to the 'OFF' state, ranging between −4 and −1.5 V. Conversely, at the highest chemical potential loss, the mode profile switches to the 'ON' state, fluctuating between −1 and 2.5 V.This data suggests that graphene's metallic properties are amplified, and the optical profile is concentrated within the topmost graphene  layer in the 'ON' state mode, which has particular implications from an optical standpoint, as depicted in figure 3.
These results correspond to the Im(n eff ) in figures 2(a) and (d) for the designed waveguides related to the application of V D on the graphene layer and TiO 2 with high permittivity [38,46].The optical absorption of graphene is determined by the position of the Fermi level, which is tuned by applying the drive voltage for the graphene/TiO 2 waveguides, resulting in modulation depths of 0.02, 0.42, 0.34, and 0.27 dB μm −1 for h TiO2 values of 10, 30, 50, and 70 nm, respectively, as shown in figure 3.For the best modulation depth result, the drive voltages at an h (3)TiO 2 value of 30 nm cause a change in its chemical potential, equivalent to shifting the Fermi level, resulting in an optimal modulation depth.The physical process behind the modulation depth relates to the transmission of 1550 nm photons through the wavelength at different applied , and electrons are excited by the incoming photons (hν 0 ).At large negative voltages (V D < −1.75 V), the Fermi level is lowered below the transition threshold (E F (V D ) = hν 0 /2) due to positive charge accumulation.The result is that the graphene is transparent because no electrons are available for interband transitions.Alternatively, at large positive voltages (V D > 3 V), all states of the electrons are filled, and no interband transitions are allowed [1].
The relationship between the Fermi energy of graphene and the V D value indicates an important role in the absorption depth and full width at the half maximum (FWHM) at higher voltages following where ν F denotes the Fermi velocity (ν F is 10 6 m s −1 ), ε TiO 2 represents a spacer's relative dielectric constant, V 0 depicts the voltage offset caused by natural doping (zero V 0 for pure graphene), and e denotes the elementary charge [47][48][49].
This demonstrates that graphene's Fermi energy is adjustable by increasing V D .Additionally, this energy level decreases with an increased dielectric thickness.Such variations have a direct impact on the waveguide's effective refractive index, which correlates with the dielectric constant e of graphene.Specifically, by altering the dielectric thickness in the h (3)TiO2 layer of the waveguide, one can control both the absorption and bandwidth of the modulator.Interestingly, when the thickness is 10 nm, there is a notable decrease in absorption tendency.This likely arises from an absence of absorbance loss.The absorption remains unaffected by the V D , as depicted in figure 3.
In figure 4, the electro-optical modulator devices are simulated by driving different voltages over the graphene layer with the FDE.The modulation depth (α) can be described by the ratio of the maximum to the minimum absorption or the modulation efficiency in decibels per micro unit length (dB/μm) which is α 100% − α 5% [38,50].At an h (3)TiO2 of 30 nm, thin TM mode absorption of the devices as a function of voltages ranging between −5 and 4.5 V is shown in figure 4 for different waveguide widths.These absorption depths for waveguide structures increase with the waveguide width, but do not affect absorption at higher voltages.The modulator efficiency is 1 dB V −1 , its loss ratio is 5 dB, and its insertion loss is 0.025 dB, where L is 40 μm.These optical modulators can operate at 1550 nm in a manner similar to [37,42].Moreover, changing the operational wavelength of the modulator device can influence its performance.This occurs since the operational wavelength plays a crucial role in dictating the device's interaction with light, encompassing absorption and extinction.In the context of this focus, where minimizing optical losses is paramount to preserving signal integrity in optical modulators, alterations in the operational wavelength can impact both modulation depth and the requisite optical power for effective modulation [51,52].
The modulation depth physical effect is linked to the Fermi energy level of graphene and the waveguide structures, as seen in the Im(n eff ) for the waveguides depicted in figure 2(b).The modulator efficiency, defined as (α 100% − α 5% ) × L/(V D(α5%) -V D(α100%) ), can be determined in a manner similar to that previously reported in [38].Figure 4(a) displays the modulator's performance.The design with multi-layer graphene significantly impacts the waveguide functionality.Efficiency peaks at a 30 nm thickness and diminishes with increased thickness of the dielectric layer h (3)TiO2 .At a 10 nm thickness, there is a deviation from figure 3 due to low absorbance.This leads to reduced modulator efficiency.Additionally, the ER and IL are calculated as (α 100% /α 5% ) × L and (α 5% −α 100% ) × L, respectively [38,50].These can be seen in figures 4(c) and (d).While ER values are in line with the calculated efficiency, IL values display an inverse trend.A thickness of 70 nm is most prone to loss and the loss decreases as the dielectric layer in the h (3)TiO2 layer becomes thinner.
In the modulator structure with four multi-layers of graphene, RC bandwidth is constrained due to graphene sheet resistance, contact resistance, and capacitance of the parallelplate capacitor.These frequency responses are depicted with an equivalent electrical circuit, illustrated by figure 5(a), which is similar to that of [53].
The f 3dB and power consumption (P) are calculated using equations (9) and (10), respectively [38] where V pp represents the difference in applied voltage between the 'ON' and 'OFF' switching stages of ∼3 V, calculated using equation (8).R total is the modulator structure's total resistance, evaluated using This equation can be summarized as follows: where the variable R 1 can be calculated from: where R c denotes the contact resistance between metal and graphene, 0.02 Ωm [54].R g represents the sheet resistance of graphene, 300 Ω/sq [55], where the sheet resistances, R g 1 to R g 6 can be calculated as: / from equation (11).R sub is substrate resistance of 400 Ω [56], R z is load resistance of 50 Ω [56], R Si is Si resistance which is equal to 640 Ωm [57], and C total represents the modulator's active area total capacitance, calculated using the graphene-TiO 2 -graphene area: Therefore, for capacitors C 1 and C 2 = 3ɛ 0 ɛ r S/h, ɛ 0 is the vacuum dielectric permittivity, ɛ r is the dielectric constant of TiO 2 , 6.14 [58].S is the overlapping area of graphene in the graphene-TiO 2 -graphene structure, which is the active area of capacitance, and can be calculated by S = L * (W Si + H Si + h).The h value is the spacer thickness in each layer ( h (1)TiO2 , h (2)TiO2 and h (4)TiO2 have an h value of 10 nm), which is used to calculate the C 1 value.The h (3)TiO2 is h at the varied dielectric lengths of 10-70 nm and is used to calculate the C 2 value.C ox is the bandwidth of Si oxide capacitance of 120 fF [56], and C air is the electrode-air capacitance, 12 fF [56].With these values, we can calculate the 3 dB bandwidth and energy consumption.Figures 5(b) and (c) demonstrate how the f 3dB and energy consumption (E bit ) change with graphene length for various TiO 2 dielectric thicknesses.The modulator has a 3 dB bandwidth of approximately 460.42 GHz with low power consumption, around 12.17 fJ/bit for a device 40 μm in length.
Finally, the performance of our proposed modulator aligns well with the specifications of other modulators, as indicated in table 2. The achieved results feature a f 3dB of 460.42 GHz, an E bit of 12.17 fJ/bit, and an L of 40 μm.Our modulator performance metrics are compared with other modulator devices in table 2. This includes a high data transmission capacity, energy efficiency, and compact physical size, making it a promising candidate for various applications.
Furthermore, our work can be developed into an efficient modulator due to its high modulation depth, distinguishing between the 'ON' and 'OFF' states, ensuring clear and highspeed signal recognition.This is crucial for telecommunications, supporting the transmission of vast data volumes.The modulator has low power consumption.Its compact design is ideal for integrating high-density photonic circuits, especially on-chip communication systems.Broadband operation covering a wide wavelength range increases flexibility.Additionally, its polarization independence makes it ideal for applications where the modulator needs to be unaffected by polarization state.An input power slope with a high extinction ratio is observed.This represents the difference in light intensity between the 'ON' and 'OFF' states, ensuring a more accurate signal.The modulator exhibits low insertion loss when integrated into an optical path.With these properties, the modulator is therefore efficient and economical for wide adoption.
The proposed structure can feasibly be fabricated.Recently, Yadav et al [60] demonstrated that a TiO 2 /graphene composite can be clearly separated when placed together.Thus, growing four layers of graphene with TiO 2 spacers is possible.This growth process can be accomplished, as reported by Heidari et al [61], by packaging a dual-graphene stack in a distributed Bragg reflector (DBR).

Conclusions
The proposed enhanced modulation performance was achieved by designing a hybrid plasmonic optical modulator integrating multi-layered graphene with TiO 2 on silicon waveguides.This design ensures compact and efficient modulation by utilizing the unique properties of graphene in conjunction with the subwavelength confinement of hybrid plasmonic waveguides (HPWs).A critical advancement was realized by optimizing the dielectric layer's thickness to 30 nm in the third layer, allowing the modulator to attain a compact footprint and achieving a high modulation depth of 0.42 dB μm −1 .Synergistic interplay between graphene and HPWs augments subwavelength confinement and light-graphene interaction, enhancing modulation efficiency, and achieving a high extinction ratio, 16.39 dB μm −1 , while minimizing the insertion loss to 0.1 dB μm −1 .Remarkably, the modulator demonstrates polarization-insensitive modulation depths, has a bandwidth of 460.42 GHz, and reduces energy consumption to 12.17 fJ/bit.However, this work suggests that modulation efficiency can be improved by designing a hybrid plasmonic light modulator that combines multilayer graphene and TiO 2 on a silicon waveguide.Our result marks another path towards improving optical modulator performance, which benefits on-chip data processing and communication technologies.

Figure 1 .
Figure 1.Schematic of a proposed multi-layer graphene modulator, (a) cross-section of multi-layer graphene optical modulator, (b) the SiO 2 wafer and 250 nm thick Si waveguide with stack repetition of multi-layer graphene overlaying structure.Each layer is isolated with TiO 2 .

− 8 . 5 × 10
Value at λ 0 = 1550 nm dn Ap constant in the driving range, −5 to 4.5 V.The field profile of the waveguide has an inset showing the maximum absorption state in figure 2(b), driven at the neutral point, i.e. zero V D .

Figure 2 .
Figure 2. The Re(n eff ), Im(n eff ) and mode profile and insets for the TM mode at the z-axis and y-axis planes have a variable thickness of (a) 10 nm, (b) 30 nm, (c) 50 nm, and (d) 70 nm, in layer number three.

Figure 3 .
Figure 3. Spectra of the real and imaginary parts of single-layer graphene waveguides.Spectral field profile of the fundamental TM mode absorption at varied V D , the absorption depths at the varied dielectric h (3)TiO2 of 10-70 nm with a W Si value of 600 nm, H Si of 250 nm, T of 300 K and λ 0 of 1550 nm.

Figure 4 .
Figure 4.The electro-optical response of the devices for fundamental TM mode absorption, normalized to device lengths of 40-375 μm.TE mode absorption with increasing waveguide width for (a) modulation depth, (b) modulation efficiency, (c) extinction ratio, and (d) insertion loss.

Figure 5 .
Figure 5.A modulated device using (a) an equivalent circuit (b) energy consumption and (c) the variations of 3 dB bandwidth against modulator length at the assumed set of parameters for varying thicknesses from 10 to 70 nm of h (3)TiO 2 .
[41][42][43]ge of electron number and DN h is hole density changes.The input parameters of the CHARGE solver, which describe the physical and electrical characteristics of the simulated silicon device, are provided by equations (5) and(6), where dn Ap and dn An are values of the refractive index resulting from changes in free-hole and free-electron carrier concentrations, respectively.These are values that indicate the doping profile within the silicon material.This defines the concentration of dopant atoms and their distribution, which affects the charge carrier density.dαApand dα An are values of the changes in absorption resulting from variations in the free-hole and freeelectron carrier concentrations, respectively.dnEpand dn En are values of the refractive-index perturbation of Si produced by various concentrations of free-holes and free-electrons, respectively.dαEpand dα En are values of the absorption perturbation of Si produced by different concentrations of free holes and free electrons, respectively.The perturbation of Si absorption affects the movement and behavior of charge carriers.These are the Soref and Bennett coefficient parameters in the silicon index charge determination of this model[41][42][43].The CHARGE input parameters were implemented for simulation, as shown in table 1.
where Δn denotes the refractive index change, Δα is an absorption coefficient variation in cm −1 ,