Broadband Graphene/TiO2 Optical Modulator Based on Hybrid Plasmonic Waveguide for Ultrafast Switching and Low-Voltage State

This work presents a novel contribution to graphene/TiO2 electro-optical modulators based on silicon-on-silica waveguide with a hybrid plasmonic waveguide to achieve ultrafast switching and low-voltage states. Waveguide structure consists of a rectangular silicon core covered by a high relative permittivity TiO2 dielectric layer with two layers of graphene, air-clad, and silica lower cladding. Effective refractive indices can be tailored to support the propagation of the transverse magnetic mode with a suitable design related to an electro-absorption modulator for simulation results. Modulation depth and bandwidth were enhanced by the waveguide width and dielectric thickness, respectively. Maximum and minimum absorption depths at the driving voltage states can determine modulators. The simulation produced the highest efficient modulator with high speed at 3 dB bandwidth of 93.7 GHz using a low energy consumption of 210.6 fJ/bit, a small footprint (24 μm2), and a broad operating spectrum range from 1310 to 1550 nm. This is because the physical process acts according to the modulator at the Fermi energy of graphene and the structure of the waveguide. These modulators can have practical applications due to their distinctive advantages, including a small device footprint, low voltage operation, ultrafast modulation switching across a broad wavelength range, and low-energy operation.


I. INTRODUCTION
I N THE Internet era, the computer chip industry is rapidly growing due to the proliferation of computer technology.It is necessary to reduce the size of computer chips and embedded chips to integrate them with external devices.The speed, efficiency, stability, and usability of silicon-based optical modulators are important features [1], [2].The most promising optical and electronic integration technologies are for the development of optical communication systems.This is because electrical impulses are converted into photonic data at high bit rates [3].Additionally, the most remarkable part of these devices is the development of low-cost, high-performance solutions that are compatible with complementary metal oxide semiconductor (CMOS) technology for ultra-fast optical modulators [4].However, miniaturization of optical devices, modulation speed, limited availability of bandwidth, and high losses persist.These are significant issues that impede integration on a very wide scale.Substituting light signals for electronic signals is the best solution.This is feasible with optical devices on a chip.There must be continuous innovation to concurrently develop alternative devices.Accordingly, there is a continuing demand for performance enhancements and the creation of new equipment alternatives.The graphene-based electro-absorption modulator device was described by Ming Liu et al. [5], who accomplished a modulation where monolayer graphene is coated onto a semiconductor wafer by actively tuning the Fermi level of a graphene monolayer.Graphene is a two-dimensional substance with exceptional electrical and optical properties [6].It was found that a monolayer graphene sheet can achieve a carrier mobility of more than 200,000 cm 2 /V•s [7].Additionally, suspending graphene can reduce the contact resistance (between graphene and metal) and the graphene sheet resistance due to its increased mobility.This results in a higher-quality device.As a result of its outstanding optical and electrical properties, graphene applications are the subject of intense study [8].As a monolayer of hexagonal lattice-organized carbon atoms, graphene possesses ultrahigh electron mobility, strong light interactions, outstanding thermal conductivity, giant optical nonlinearity, and high flexibility [9].Moreover, its gapless, linear band structure makes it a broadband absorber from the visible to the infrared spectrum [10].Graphene optical absorptions can be effectively altered by manipulating the Fermi level through the gate voltage [5], [11].These unique characteristics make graphene a prospective option for an active material to create innovative optical modulators [12], [13], [14].They have also been found beneficial for numerous broadband and ultra-compact applications such as polarizers [13], [14] photodetectors [15], [16], modulators [1], [17], [18], [19], and sensors [20], [21], [22].Typically, the material and thickness of a dielectric layer, the structure of a waveguide, and the size of the graphene directly affect the operating characteristics, resulting in a tradeoff between absorption depth and bandwidth.Consequently, it is crucial to design modulator structures rationally.
Recently, there was a report for graphene electro-optic modulators with a 65 nm interlayer Al 2 O 3 dielectric layer of a ring-waveguide generated a 3dB bandwidth of 30 GHz and an energy consumption of 800 fJ/bit with the external drive voltage from 0 V to 15 V [23].Also, the graphene optical modulators were presented with a 20-nm-thick Al 2 O 3 dielectric layer of the silicon waveguide, resulting in a high modulation speed of up to 35 GHz and an energy consumption of 1400 fJ/bit with an external drive voltage from −40 V to 40 V [24].An integrated ultra-high-performance graphene optical modulator with a 20nm-thick Al 2 O 3 dielectric layer of a waveguide achieved by the 3 dB bandwidth of 60 GHz speed, a low energy consumption of 2.25 fJ/bit, with a low drive voltage for −4 V to 0 V [25].These show that the current state-of-the-art speed range for these graphene optical modulators is related to the dielectric thickness, the waveguide structure, and the drive voltage.
However, the previous reports on the performance of graphene optical modulators were limited by the Al 2 O 3 dielectric layer with speeds ranging between 30-60 GHz.Therefore, ongoing advancements are essential to enhance further performance on graphene-based modulators with demonstrating speeds exceeding 30 GHz, underscoring the need for continual development in this field.In this research, a graphene/TiO 2 electro-optical modulator based on a silicon-on-silica (SOS) waveguide is presented with a hybrid plasmonic waveguide structure, which combines the properties of plasmonic and photonic waveguides.We have also verified performance with different TiO 2 dielectric thicknesses on broadband optical bandwidth in the range from 1310 to 1550 nm.The electro-optical modulator structure stimulation can be estimated by the two-dimensional (2D) FDE solver, the three-dimensional (3D) EME solver, and the CHARGE solver, employing ANSYS Lumerical software in simulation design.The simulation results lead to an ultra-high-performance graphene optical modulator.

II. DEVICE STRUCTURE AND OPERATING PRINCIPLE
A schematic diagram of the graphene/TiO 2 optical modulator based on SOS waveguides is illustrated in Fig. 1.The modulator utilizes an SOS waveguide in conjunction with a hybrid plasmonic waveguide.It combines the properties of plasmonic and photonic waveguides as principle mentioned in [26].In the photonic mode, light signals or electronic waves are partially conducted within a dielectric waveguide (silicon) and partially within a plasmonic structure (graphene/TiO 2 ) using a plasmon, which relies on surface plasmon polaritons to enhance the modulator's performance [26], [27].This combination enables ultrafast switching and low-voltage states.For TiO 2 material, a dielectric in the capacitive stack with a high relative permittivity of 80 [28] is applied to trap charges on graphene [29].The SOS waveguide is comprised of a silicon (Si) core with width (W Si ) varying from 300 nm to 600 nm and height (H Si ) of 250 nm.A TiO 2 layer of thickness (t TiO2 ) is deposited on the surface of the waveguide core, with air as the top cladding and silica as the bottom cladding.W D is dielectric width, which varies according to the thickness of t TiO2 .This device structure was calculated at TiO 2 layer thicknesses ranging from 10 to 90 nm.Two graphene sheets were placed on the core of the waveguide that covers the TiO 2 layer, one on top and one on the side.The graphene layer thickness was found to be 0.1 nm.This device corresponds to that of a recently reported experiment with an optical module [5].The waveguide consists of a graphic sheet slab (W g-slab ) with a width of 500 nm.The purpose of the waveguide etched is to establish a connection between the core and the gold (Au) electrodes, which have a width of 50 nm and a height of 250 nm, while a platinum (Pt) counter electrode is used to reduce access resistance at a width of 500 nm and a height of 25 nm.Furthermore, the minimum distance between the core and the electrodes must be correctly positioned away from the Si core.Consequently, it does not interfere with the optical mode within the waveguide [5].
The dispersion of such a SOS waveguide is related to the density of free carriers in the semiconductor, which affects a change in the refractive index.For a modulator, the refractive index of Silicon has been previously described [30].It relates the  [30], [31], [32] refractive index change and the absorption coefficient variation at a pump wavelength (λ 0 ), as follows Δn = dn Ap (ΔN e ) dn Ep + dn An (ΔN h ) dn En (1) where ΔN e and ΔN h are electron and hole density changes, respectively.The values dn Ap , dn An , dα Ap , dα An , dn Ep , dn En , dα Ep , and dα En are the Soref and Bennett coefficients, shown in Table I [30], [31], [32].
The surface conductivity (σ) of monolayer graphene, described by the Kubo formula, is applied to model the two graphene layers on top and left wall of the SOS waveguide [33], [34], [35], [36], which is given by: where ω is angular frequency, Γ is a phenomenological scattering rate (or relaxation time τ = 1/2Γ) that is assumed to be independent of energy Fermi level ), and v is the constant velocity of the graphene sheet (v = 10 8 cm/s) [37], [38], T is temperature, j are imaginary numbers, e is the charge of an electron, h is a reduced Planck's constant ( h = h/2π), and k B is the Boltzmann's constant.
The first and second terms in (4) are referred to the intraand inter-band contributions, respectively.In the mid-infrared wavelength regions, the inter-band conductivity can be neglected [35], [37].The intra-band conductivity approximation of the graphene layer is evaluated using a Boltzmann-Drude expression and Fermi-Dirac statistics, where μ c >>T [37].Therefore, the surface conductivity of the single graphene layer is given by: The electric permittivity (ε || ) of the monolayer graphene can be modeled using a uniaxial anisotropic permittivity [35], [36], [37], which is given by: where d g is a thickness of the monolayer graphene, ε 0 is the permittivity of vacuum, and ε r is the background relative permittivity and in expressed as n = √ εμ.This electro-optical modulator performance is based on a combination of optical and electrical considerations, in which the structure is analyzed with simulations using two-dimensional (2D) FDE, three-dimensional (3D) EME and the CHARGE solver to define voltage or surface recombination boundary conditions in an electrical simulation using ANSYS Lumerical software [38].The modulated propagation losses of the waveguide depend on the material properties, waveguide geometry and external drive voltage.The chemical potential of the graphene/TiO 2 modulator can be controlled by applying an external drive voltage and a dielectric in the capacitive stack [39].Modulation is investigated by tuning the Fermi level of the graphene.With a biased gate voltage of the modulator, the shifted Fermi level modifies its optical absorption rate and the optical response of the waveguide.In the electrical mode, the waveguide core contains electron-hole density material characteristics depending on the drive voltage.The charge mode uses semiconductor 3D density, which was applied to model semiconductor materials and waveguide geometries.The 3D electron density (n 3D ) of the graphene can be calculated by [5], [32], [38]: ) where m * is the electron or hole effective mass, E is the electron energy.However, the resulting electron density of the graphene layer also resolves the mismatch between n 3D and the actual electron density calculated by the 2D density of states when the Fermi level is below the Dirac point (E F ≤ 0.05 eV) and above the Dirac point (E F > 0.05 eV.The fitting parameters use a simple scaling factor, and an electron (hole) effective mass of 1.768 and 0.4614, respectively [38], [40].
The n 3D model closely matches the actual electron density.As a function of the drive voltage, the charge variation in the silicon waveguide affects the μ c of monolayer graphene.In optical mode, the geometry of the waveguide is identical to that specified by the CHARGE solver.The acceptor concentration within the silicon waveguide is 10 18 cm −3 [38], [39].Regarding the drive voltage, the electron-hole density is loaded.The same density was obtained from the CHARGE solver, which was calculated using the designed waveguide and the chemical potential of the two graphene sheets on the top and the side of the device.A cross-sectional field profile of the waveguide mode and propagation of the long waveguide are simulated for the modulator.These processes can be applied to achieve an electro-optical modulator in the form of the propagation losses of the waveguide for each drive voltage [5], [39].

III. SIMULATION RESULTS AND DISCUSSION
The proposed electro-optical modulator, shown in Fig. 1, is examined by varying the optical absorption rate via a gate voltage between the graphene layer and the waveguide core.The TiO 2 material with a high relative permittivity of 80 is used as a dielectric material in the capacitive stack.The thickness of the TiO 2 layer is 10 to 90 nm when tuned for optimization.TiO 2 is used to connect the core of the Si waveguide under the graphene layer to trap charges on the graphene.Single layers of graphene are located on the top and left of the waveguide core, with a scattering rate of 15 meV and a temperature of 300 K [38], [39].
The charge chemical potential and core electron-hole density are applied in relation to a drive voltage.Si waveguide core dimensions are W Si = 400 nm and H Si = 250 nm.To apply the electric permittivity in (5) and the Soref and Bennett relation in ( 1) and ( 2), the refractive index of a single graphene layer and Si are calculated.The refractive indices of SiO 2 and TiO 2 are 1.55 and 2.50, respectively [40].
The effective refractive indices (n eff ) for such a waveguide structure as a function of various drive voltages, from −5.5 to 4.5 V, are obtained for the fundamental transverse electric (TE) and transverse magnetic (TM) modes at a pump wavelength of 1550 nm using an FDE solver.This is shown in Fig. 2(a) and (b), respectively.The real part (Re(n eff )) of the high changed only slightly from ∼2.106 to 2.115 and ∼1.675 to 1.685 for the TE and TM modes, respectively.Nonetheless, the imaginary part (Im(n eff )) can change dramatically and stabilize up to 1.967 and 1.651 in the range of −1 to 2 V for the TE and TM modes.Fig. 2(c) shows the cross-sectional field profile mode, which demonstrates that the waveguide can support the propagation of the optical mode and the interfaced field between the waveguides.Graphene can be altered to maximize the field using different drive voltages.However, the field profiles of the mode do not change significantly for drive voltages of −2.5 V, 0 V, and 3 V.Fig. 2(d) shows the optical propagation along the waveguide length.The simulation results indicate that the optical absorption related to its imaginary part can be changed dramatically for drive voltages of −2.5 V, 0 V, and 3 V.
Fig. 3(a) and (b) present the transmission characteristics of waveguides for the fundamental TE and TM modes, respectively, where the waveguide widths are W Si of 300 nm, 350 nm, 400 nm, 450 nm, 500 nm, 550 nm, and 600 nm, and the drive voltage is at the neutral point (V D = 0) due to maximal absorption states.Field profiles of the waveguides are shown in the inset pictures.The simulation results demonstrate that the waveguides with W Si values of 300 nm, 350 nm, 550 nm, and 600 nm, cannot support propagation of the optical TE mode due to a low Re(n eff ).However, TM mode absorption increases with waveguide width.The TM modes for all the designed waveguides can support propagation of the optical mode because reducing the transverse dimensions of the waveguide core corresponds to propagation for the TM mode.Increasing waveguide width leads to an increased n eff and field profile distribution, which is mainly concentrated in the waveguides.Moreover, the Im(n eff ) of the waveguides for the TM mode depends on increasing waveguide width, which is related to optical absorption.Absorption is the result of a combination of both the waveguide geometry and material properties.Absorption is the result of a combination of both the waveguide geometry and material properties.This property has a resemblance to a hybrid plasmonic waveguide [26], width W Si = 300 nm, 350 nm, 400 nm, 450 nm, 500 nm, 550 nm, and 600 nm, height H Si = 250 nm, t TiO2 = 10 nm, λ = 1550 nm, T =300 K, a scattering rate of 15 meV, and a drive voltage of 0 V. which accomplishes light confinement through the coupling of light steered by two different waveguides between a dielectric waveguide (Si) and a plasmonic waveguide (graphene/TiO 2 ).It is created by leaving a tiny gap between a metal surface and a material with a high refractive index.Our graphene/TiO 2 SOS waveguide is designed using graphene for the metal surface.However, graphene is not affected by changing the optical mode of the waveguide.Its light is confined to the TiO 2 dielectric gab and the surface plasmon graphene due to the high effective refractive index region [41], [42], as shown in the inset of Fig. 3(b).Their mode profile for the fundamental TM mode interacts with the TiO 2 dielectric and graphene layer more than in the waveguide core due to its high refractive index.This is different from the fundamental TE mode, where most of the light is concentrated in the core because the waveguide height is much smaller than the waveguide width [41], [42].As a result, we focus on the modulators for the fundamental TM mode.
As shown in Fig. 4(a), the electro-optical modulator devices are simulated by driving different voltages over the graphene layer with FDE.The designed waveguides are functions of the waveguide widths of 300 nm, 400 nm, 500 nm, and 600 nm while the parameters, H Si , t TiO 2 , λ 0 , T, and the scattering rate, are 250 nm, 10 nm, 1550 nm, 300 K and 15 meV, respectively.The absorption depth can be described by the ratio of the maximum to the minimum absorption or the absorption coefficient in decibels per micro unit length (dB/μm) is α 100% − α 5% [43], [44].The TM mode absorption of the devices as a function of the voltage ranging from −5 to 4.5 V is shown in Fig. 4(a) for the different waveguide widths.The 3D field propagation along the waveguides is used to generate the absorption states using eigenmode expansion (EME) mode.By varying the width of 300 nm, 400 nm, 500 nm, and 600 nm, the maximum absorption state is 0.0581 dB/m, 0.0874 dB/m, 0.1011 dB/m, and 0.1063 dB/m, respectively, without the drive voltage.The minimum absorption state of all the waveguides is zero at the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
drive voltage of −1.5 V and 2.8 V.It can be seen that these absorption depths for waveguide structures increase with the waveguide width but do not affect absorption at higher voltages.These results correspond to the imaginary part n eff for the designed waveguides in Fig. 3.This is related to application of a drive voltage on the graphene layer and TiO 2 with the high permittivity [8], [36], [45].These voltages cause a change in its chemical potential, equivalent to shifting the Fermi level, resulting in such an absorption depth.At low drive voltages, in the range of −1.5 to 2.8 V, the Fermi level (E F (V D )) is close to the Dirac point (E F (V D ) < hν 0 /2) and inter-band transitions occur by photon absorption (hn/2).Electrons transit from an occupied state below the Fermi level to an unoccupied state in a higher band by the incoming photons (hν 0 /2).At negative voltages, less than −1.5 V, the Fermi level is less than half the photon energy (−hν/2).Meanwhile, at positive voltages, higher than 3 V, the Femi level is greater than half the photon energy (hν/2) and all states of electrons are filled.In both of these cases, no inter-band transitions are allowed [5].Therefore, the modulator's absorption is suppressed.
The physical effect of the absorption depth (α) is related to the Fermi energy level of graphene and the waveguide structures corresponding to the imaginary part n eff for the waveguides in Fig. 2(b).To improve these devices, modulator efficiency is defined as (α 100 -α 5% ) × L/(V α100% -V α5% ), where L is the device length [46].Fig. 4(b) displays the modulator efficiencies, which strongly depend on the waveguide width.A 600 nm width reduces its efficiency, compared to the other modulators, and increases with modulator length.Furthermore, the extinction ratio (ER) and insertion loss (IL) were calculated as (α 100% /α 5% ) × L and (α 5% − α 100% ) × L, respectively [43], [44], and are shown in Fig. 4(c) and (d).The obtained ER values correspond with the calculated efficiency, but the IL values show the opposite behavior, in which a 300 nm width exhibits higher excess loss for a waveguide length.
Fig. 5(a) shows the modulator depth at different drive voltages for a TiO 2 dielectric layer with t TiO2 values of 10 nm, 30 nm, 50 nm, 70 nm, and 90 nm, respectively.The simulation results show that their absorption depths for the fundamental TM mode are about 0.1063 dB/μm, 0.1029 dB/μm, 0.0958 dB/μm, 0.0901 dB/μm, and 0.0625 dB/μm.respectively, which decrease with increasing t TiO2 .Such dielectric thicknesses affect the absorption distribution at higher voltages with an increasing full width at the half maximum (FWHM) value, which is related to the equivalent capacitance per unit area of the devices [44].Moreover, modulator efficiencies decrease with the dielectric thickness, but increase with the modulator length, as shown in Fig. 5(b).The obtained ER values correspond with these efficiencies, but the IL values exhibit the opposite behavior, in which a 10 nm dielectric thickness exhibits higher excess loss for a given waveguide length, as shown in Fig. 5(c) and (d), respectively.
The optical response of the device at a wavelength of 1310 nm was investigated and the same modulators were applied to simulate absorption at various drive voltages using the optimized dielectric thickness and the refractive index of Si in Table I.The simulation results show that the overall absorption depths are higher than at a wavelength of 1550 nm and decreased with increasing dielectric thickness, as shown in Fig. 6(a) and (b), respectively.These results are related to a higher overlap of the optical mode between the graphene layer and the designed waveguide.Furthermore, thicker dielectric leads to a higher FWHM in Fig. 6(b).The relationship between the Fermi energy of graphene and the drive voltages indicates an important role in the absorption depth and FWHM at higher voltages according to where ν F is the Fermi velocity (ν F = 10 6 m/s), ε TiO2 is the relative dielectric constant of the spacer, V is the external drive voltage, V 0 is the voltage offset caused by natural doping (V 0 = 0 for pure graphene), e is the elementary charge [44], [46], [47].
As can be seen, the Fermi energy level of graphene can be shifted by increasing the drive voltage and decreased by increasing the dielectric thickness, which affects the effective refractive index of the waveguides related to the dielectric constant (ε r ) of graphene.Consequently, the waveguide width and dielectric thickness can be used to control modulator absorption and bandwidth, respectively.
From the simulation results, the drive voltage causes a change in the chemical potential of the graphene, equivalent to shifting the Fermi level.It affects the relationship between the absorption depth.However, we were able to achieve low-voltage states and ultrafast switching.For switching states, the "OFF" state indicates that light is absorbed, while the "ON" state is the opposite.As shown in Fig. 4, at a 1550 nm wavelength, the most efficient modulator is with W Si = 600 nm and t TiO 2 = 10 nm.The "OFF" and "ON" states are absorption at low drive voltages, in the range of 1.5 to 2.8 V with transmission at high voltage (<−1.5 V and >3 V), respectively.The modulator efficiency is 1 dB/V, the extinction ratio is 5 dB, and the insertion loss is −0.025 dB with an L of 40 μm.Furthermore, at a wavelength of 1310 nm, the switching voltage is the same, but the absorption is greater, as depicted in Fig. 6.Such results indicate that these optical modulators can operate between 1310 nm and 1550 nm.In a manner similar to [44], [45], modulator performance is calculated using the electrical circuit model shown in Fig. 7(a).
Using the electrical circuit model, the results shown in Fig. 7(a) are similar to previously published reports [46], [47], [48], where R total is the total resistance of the modulator, C total Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II COMPARISON OF EXTRACTED PARAMETERS IN THIS WORK AND IN DIFFERENT MODULATOR
is the total capacitance of the modulator, and V pp is the drive voltage difference between the switching states.The parallel resistance in (10) and capacitance in (11) of the modulator are calculated as where R g = R sq (W Si +H Si +W g-slap )/L is the distance between the metallic electrode and capacitor [46], [47], C GI is the capacitance between the graphene layer and waveguide core, and Although the structure is similar to the experimental modulator for the graphene-based broadband optical modulator [5], the efficiency has explored the proposed modulator with graphene/TiO 2 optical modulator.The results are outstanding compared to previous reports, as shown in Table II.Our graphene/TiO 2 modulators are designed with different TiO 2 dielectric thicknesses of 10 nm, 30 nm, 50 nm, 70 nm, and 90 nm with dimensions.The modulator bandwidth is also significantly increased for smaller devices (24 μm 2 ) with 10 nm thick TiO 2 considered the best performer with f 3dB = 32.1 GHz, E bit = 94.63 fJ/bit, and L = 20 μm.High relative allowance for a length of 40 μm, f 3dB = 45.4GHz, and E bit = 102.13fJ/bit with a low applied voltage state ranging from −5.5 V to 0 V.The proposed work is considered to provide higher bandwidth with lower energy consumption, and the drive voltage is minimal compared to those reported by Dalir et al. [23] and Heidari et al. [24].Although Heidari et al. [25] reported an f 3dB efficiency of up to 60 GHz, in our proposed work, modifying the TiO 2 layer thickness to 90 nm can achieve a peak performance modulator bandwidth of f 3 dB up to 93.7 GHz and increasing from 10 nm thickness of TiO 2 to 2 times.This indicates the prominence of tunable modulators.These modulators have also been validated using electro-optical models for simulation.Graphene and silicon at a pump wavelength of 1310 nm, the absorption state can resemble a 3 dB bandwidth and energy consumption.As a result, such a modulator can operate over a broad optical bandwidth in the range of 1310 to 1550 nm.

IV. CONCLUSION
The graphene/TiO 2 electro-optical modulator based on a silicon-on-silica (SOS) waveguide and a hybrid plasmonic waveguide is created by a TiO 2 dielectric with a high relative permittivity in the capacitive stack.We achieved low-voltage states and ultrafast switching by varying the waveguide width and dielectric layer thickness.Our highest performance used the 600 nm width waveguide, which can generate the maximum absorption state of 0.106 dB/μm without the drive voltage and the minimum absorption state of zero at the drive voltage of −1.5 V and 2.8 V.These absorption states relate to the OFF (maximum) and ON (minimum) switching states of a modulator.
(1.2) The 40 μm long modulator at the pump wavelength of 1550 nm can generate a 3 dB bandwidth of 45.4 GHz and low energy of 102.13 fJ/bit with a low voltage state at 0 V (OFF) and −1.5 V (ON).Moreover, a 90 nm TiO 2 dielectric thickness of the modulator can generate a 3 dB bandwidth of 93.7 GHz (>2 octaves) and an energy consumption of 210.6 fJ/bit, with the low voltage state from 0 V to −5.5 V. Additionally, the modulators were investigated at the pump wavelength of 1310 nm, the obtained 3 dB bandwidth and the energy consumption remain similar to the previous results.Consequently, these modulators can work over a broad optical bandwidth from 1310 to 1550 nm.The physical process behind the modulators is related to the Fermi energy of graphene and the waveguide structure.As the obtained ultrafast switching, low energy consumption, and low voltage state are essential, the proposed scheme would offer the most significant benefits in high speed and small footprint (24 μm 2 ) for on-chip electro-optical modulators.

Fig. 2 .
Fig. 2. Effective refractive indexes, where the solid curve is the real part, and the dotted curve is the imaginary part for the fundamental.(a) TE mode (horizontally polarized) (b) TM mode (vertically polarized) using the waveguides with W Si = 400 nm, H Si = 250 nm and t TiO2 = 10 nm, where λ = 1550 nm, T = 300 K, the scattering rate is 15 meV, and waveguide spectral field profile for the fundamental TE and TM modes (c) the cross-sectional field profile of the mode (d) spectrum absorption over the length of the waveguide for driving voltages of −2.5, 0 and 3 V.

Fig. 3 .
Fig. 3. Mode profiles based on the effective refractive indices and the transmission relationship, where the (solid curve) real part and (dotted curve) imaginary part, for the fundamental.(a) TE mode and (b) TM mode with varying waveguidewidth W Si = 300 nm, 350 nm, 400 nm, 450 nm, 500 nm, 550 nm, and 600 nm, height H Si = 250 nm, t TiO2 = 10 nm, λ = 1550 nm, T =300 K, a scattering rate of 15 meV, and a drive voltage of 0 V.

Fig. 5 .
Fig. 5. Electro-optical response of the devices for fundamental TE mode absorption at varied drive voltages, normalized to a device length of 40 µm.TE mode absorption for (a) absorption (b) modulation efficient (c) extinction ratio and (d) insertion loss as dielectric thickness t TiO2 = 10 nm, 30 nm, 50 nm, 70 nm, and 90 nm, waveguide width W Si = 600 nm, height H Si = 250 nm, λ = 1550 nm, T =300 K, and scattering rate of 15 meV are varied.

Fig. 6 .
Fig. 6.Absorption depths of (a) electro-optical response of the devices for fundamental TM mode absorption at varied drive voltages, normalized to a device length of 40 µm, TE mode absorption for absorption with variousness (a) waveguide widths of 300 nm, 400 nm, 500 nm, and 600 nm (b) dielectric thicknesses of 10 nm, 30 nm, 50 nm, 70 nm, and 90 nm using height H Si = 250 nm, λ = 1310 nm, T = 300 K, and scattering rate of 15 meV.

Fig. 7 .
Fig. 7. Modulated device using an equivalent circuit.(a) An equivalent circuit, when R g is graphene sheet resistance, R c is contact resistance between graphene and electrode, C GI is oxide flat capacitance between single graphene layer and waveguide core, R z is load resistance, R sub is substrate resistance, and C air is capacitance between the electrodes through the air, C ox is the bandwidth of silicon oxide capacitance.(b) Modulation bandwidth.(c) Energy consumption for varying TiO 2 dielectric thicknesses (t TiO2 ).
The parameter values are as follows: R c = 200 Ωμm[47], R sq = 300 Ω/sq[49], R sub = 400 Ω, R z = 50 Ω, C ox = 120 fF, and C air = 12 fF[50].With these values, Fig.7(b) and (c) can be used to illustrate how the 3 dB bandwidth and energy consumption vary with graphene length for different TiO 2 dielectric thicknesses.The modulator bandwidth increases with TiO 2 thicknesses.Graphene length increases due to decreased capacitance C GI and resistance R g of the modulators, but their power consumption increases because of the difference in the drive voltages between the switching states at the higher voltage.A TiO 2 thickness of under 90 nm, the proposed modulator has a large 3 dB bandwidth of ∼93.7 GHz and low power consumption of 210.6 fJ/bit with a 40 μm length.

TABLE I SOREF
AND BENNETT FITTING COEFFICIENTS