1. Introduction

Owing to the exceptional optical and electrical properties of graphene, it has garnered significant attention and turned into one of the hotspots in the rapidly growing field of photonic integrated circuitry (PIC). As one-atom-thick sp2 bonded carbon atoms arranged in a hexagonal lattice, graphene has become distinguished by defying conventional semiconductors leveraging unparalleled gapless and linear band structure of massless Dirac fermions [1–4]. The unique characteristics arise from the confinement of electrons in a single layer of atoms and the low density of states near the Dirac point, making the chemical potential highly susceptible to carrier density. Influenced by Fermi energy, a graphene sheet strongly interacts with light, ranging from near-infrared to visible. This absorption can be actively tuned by extra charge accumulation on graphene sheets via electrical gating and/or chemical doping [5]. Such ability has been harnessed extensively in optical applications, including mode lock lasers [6], waveguides [7], switches [8], modulators [9], logic gates [10,11], photodetectors [12,13], and so on.

Optical switching and modulation play a key role in integrated optics, where ultrafast optical signal processing and ultrashort optical pulses are required. A myriad of works has been devoted to theoretically and experimentally investigate different configurations of electro-optical (EO) modulators based on the Pockels effect in crystals [14,15]. Nonetheless, the bulkiness and low EO effects of conventional crystals still inhibit the exploitation of these structures in chip-scale optical components. The former downside has been resolved in silicon modulators. But, the linear EO effect is absent in bulk unstrained silicon crystals [16,17], resulting in limited modulation bandwidth and extinction ratio (ER) of the device. Although incorporating Si waveguides with n-i-p ring resonators on a silicon-on-insulator (SOI) substrate has shown a noticeable reduction of the size (12 µm), the cavity photon lifetime of 33 ps still implies a narrow bandwidth. Also, their performance is highly sensitive to temperature and fabrication errors [18]. In contrast, the electro-absorption phenomenon in waveguide-integrated bulk and quantum-confined compound semiconductors have exhibited broader bandwidth and lower power consumption [19]. However, high total insertion loss (IL) (3.7 dB for GeSi) deteriorates the modulator performance.

Over the past two decades, propagation of evanescently confined light on the metal-dielectric interface, namely surface plasmon polariton (SPP), has attracted interest due to interfacial subwavelength confinement of light and enhanced light-matter interaction [20]. The extreme localization of propagating light well below the diffraction limit offers promising modulation and switching functionality [21]. Nonetheless, it suffers from high ohmic loss as light penetrates in the metal layer and attenuates in the order of several dB/µm. Novel waveguide structures have been proposed to achieve Si-compatible plasmonic devices, mostly relying on ITO and TiN [22,23]. A high-speed compact EO modulator has reached an ER of 10 dB with IL < 3 dB. This structure takes advantage of selectively coupling light into the lossy plasmonic medium in off-state, where attenuation is required, and routing it back to the Si waveguide in on-state [24]. Nevertheless, it requires a precise fabrication process.

Another approach to directly tackle the large loss in plasmonic components is the employment of graphene as an active plasmonic material, supporting graphene surface plasmons (GSPs) with longer propagation lengths due to a low electron scattering rate. Besides, the high tunability of GSPs in the terahertz (THz) region introduces a promising pathway for biomolecular sensing and communications in this barely explored spectrum [25,26]. As a fundamental part of most photonic devices, various waveguides based on graphene nanoribbons have been reported [26–28]. However, they suffer from plasmon damping originating from the edge scattering effect and bandgap opening [29]. Thus, plasmonic waveguides using pattern-free graphene monolayer have been proposed to reduce the propagation loss in the orders of 2∼6 dB/µm over a wide range of wavelengths [30]. It is worth noting that the high electron mobility in graphene, stemming from high relaxation time, can be drastically reduced in samples supported by a substrate. This behavior has been attributed to extrinsic factors such as surface roughness [31], surface interfacial polar optical phonons [32], and surface charge traps [32], as the mobility drops from 200000 cm²/V.s to under 10000 cm²/V.s for unsuspended graphene sheets. Tantalizing waveguiding and cross-coupling characteristics of suspended graphene sheets have been demonstrated in a structure consisting of a monolayer graphene sheet supported by two SiO₂ ridges and one Si ridge in the middle, 10 nm below the graphene sheet. In this configuration, the propagation length is 25 times more than that of the unsuspended counterpart, and an ER of 10.52 dB has been achieved for a cross-coupling setup [33]. Despite having illustrated promising results, no effort has been made to apply this structure in EO modulation in mid-infrared (MIR) and THz regions. Thus, it is highly favorable to place emphasis on designing modulators and switches profiting from the high carrier mobility of suspended graphene configuration. At the same time, the absorption of these devices can be adjusted easily by applying an external gate voltage. Despite the wide range of appealing applications in the MIR and THz regions, these spectra have been less exploited in recent years. Therefore, it is highly desirable to design efficient and compact silicon-based photonic devices operating in these regimes [34]. Hence, we design a novel resonant plasmonic modulator based on suspended graphene waveguides to benefit from its advantages, like low loss, high propagation length, and tunability.

The rest of the paper is organized as follows. The basic characteristics of a single suspended graphene waveguide are investigated in Section 2. Section 3 is dedicated to introducing the proposed modulator. The effects of geometric parameters and gate voltage variations on the transmission spectrum are studied in this section. Finally, the conclusion is expressed in Section 4.

2. Principles of basic suspended graphene waveguide

Figure 1 illustrates the 3D schematic and cross-section of a single suspended graphene waveguide as the basic component of our modulator design. The graphene sheet lying on the x-y plane is supported by two gold (Au) ridges located at both ends of the structure in the y-direction and a single Si ridge is extruded in the middle. The height of the two gold ridges is (t₂ + t₁) to maintain the t₁ gap between the graphene sheet and the Si ridge. The width of all ridges is W. A gold layer is buried under a 20-nm-thick Si spacer layer to act as an electrode.

 

Fig. 1. (a) 3D schematic illustration of the suspended graphene waveguide. (b) Cross-section (y-z plane) view of the structure with geometrical parameters and bias connections

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The fabrication feasibility of such a structure with SiO₂ ridges instead of gold has been reported by [33]. To avoid the existence of resist residues and surface contaminations, the lithography-free technique promises high-quality graphene with carrier mobility as high as 120000 cm²/V.s [35]. As further discussed, the surface conductivity adjustment of the graphene sheet is performed by applying a voltage to the gate electrode connected to the gold ridges. The gold layer under the Si spacer is also connected to the ground. A thin layer of Pd between the graphene and gold electrodes can reduce the contact resistance below 100 ohms without a noticeable effect on the performance of the waveguide [36]. The Si and gold permittivities are taken from Palik [37] and CRC [38], respectively. The 0.34-nm-thick graphene sheet is modeled as a two-sided surface conductivity σ (ω, µ_c, Г, T), where ω, µ_c, Г, and T denote angular frequency, chemical potential, scattering rate, and temperature, respectively. The surface conductivity of graphene is expressed by the Kubo formula as [39]:

 

Fig. 2. Chemical potential variations versus applied gate voltage for different values of t₁. The solid black curves are fitted by the parallel plate capacitor model.

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The more the t₁ decreases, the more the charges accumulate on the graphene over the Si ridge, resulting in the greater chemical potential. However, the smaller gap leads to the collapse of the device by larger attractive electrostatic force in high voltages and also lower frequency response due to a greater capacitance. According to the parallel plate capacitor model, the chemical potential can be described through ${\mu _c}(V) = \hbar {v_F}\sqrt {\pi {a_0}|{V - {V_0}} |}$ wherein V₀ is caused by natural doping. Using curve fitting, the parameter a₀ for different values of t₁ = 5 nm, 10 nm, and 20 nm are approximated to be 8.68 × 10¹⁵ m^-2V^-1, 6.02 × 10¹⁵ m^-2V^-1, 3.73 × 10¹⁵ m^-2V^-1, respectively. Also, the corresponding V₀ values are -8 V, -15.7 V, and -24.3 V, respectively. Although the change in the conductivity of graphene occurs locally, and especially right above the Si ridge, for the sake of simplicity, we assume that the chemical potential of the entire graphene sheet varies monolithically as the guided wave is tightly confined laterally within the gap in our simulations as seen in Fig. 3(a). To obtain the propagation properties, the real and imaginary parts of the effective index are calculated. Figures 3(b)-(f) demonstrate the variation of the real and imaginary parts of the effective index versus wavelength, geometric parameters, and chemical potential. In all cases, the perturbation in the refractive index of the Si ridge due to carrier density distribution is taken into account [43]. Figure 3(b) shows that by decreasing wavelength, the N_eff increases, leading to higher confinement of field into the gap region. Since the K_eff is directly proportional to the ohmic losses within the graphene, the propagation loss increases by increasing the K_eff. So, the propagation length is higher in shorter wavelengths.

 

Fig. 3. (a) Normalized electric field magnitude for the fundamental TM mode around the Si ridge at two different wavelengths (W = 80 nm and G = 500 nm). The real and imaginary parts of the effective index are demonstrated as a function of (b) wavelength, (c) width of the Si ridge, (d) distance between the Si ridge and the gold ridge, (e) gap between Si ridge and graphene sheet, and (f) chemical potential.

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Figure 3(c) represents the changes in effective indices components of the fundamental mode as a function of Si ridge width (W). As the W Increases, the traveling wave is more confined due to the higher values of N_eff. However, decreasing the W guarantees both the excitement of only the fundamental mode and also higher propagation length. Furthermore, taking fabrication limitations into account, the W of 80 nm is wide enough to ensure an acceptable loss as well as acting as a single-mode waveguide in the desired spectrum window.

As seen in Fig. 3(d), for values of parameter G lower than 200 nm, the gold ridges start to interact effectively with the plasmonic mode, leading to an exponential rise in the K_eff and IL correspondingly. Figure 3(e) depicts the waveguiding characteristics versus the gap between the graphene and the Si ridge (t₁). As the gap decreases, both N_eff and K_eff increase due to increased interaction between the Si ridge and propagating SPPs. The wider gap results in lower propagation loss, less confinement, and smaller capacitance. Choosing t₁ = 10 nm gives a sufficient propagation length of 18.5 µm and acceptable variation in chemical potential with low gate voltage changes.

As illustrated in Fig. 2, the graphene chemical potential can be controlled by applying a bias voltage. When the gate voltage increases, the chemical potential drops accordingly, leading to a significant rise in the K_eff and the propagation loss, correspondingly (Fig. 3(f)). The advantage of suspended graphene plasmonic waveguide over the previously reported unsuspended ones is more conspicuous here, as the propagation loss can be as low as 1700 dB/cm, which corresponds to about 18 µm propagation length at the chemical potential of 0.7 eV. Applying ten volts to the gate electrode results in a decline of the chemical potential to about 0.4 eV and propagation length to 10.4 µm.

3. Suspended graphene electro-optical modulator

The modulation unit of our proposed suspended graphene EO modulator is illustrated schematically in Fig. 4. The graphene SPPs are excited by radiation of a TM-polarized wave to the input waveguide and propagate toward the + x direction, according to the top view of the structure shown in Fig. 4(b). The symmetric coupling scheme is devised to split the input wave equally to the two arms separated by the G₁ distance and afterward couple it back to the output waveguide. The length of the top and bottom waveguides (L₂) is assumed to be 2 µm to ensure that there is no transmission between the input and output waveguides. However, these waveguides are not too large to attenuate the coupled wave and lengthen the modulation unit. The graphene sheet is held by two gold ridges located at a distance of G₂ from the top and bottom waveguides.

 

Fig. 4. (a) 3D schematic of the modulator unit. (b) Top (y-x plane) view of the proposed structure. Dimensions are not in scale.

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Since the modulation behavior of the structure is highly affected by changes in coupling length, the link between this parameter with gate voltage and spatial separation needs to be carefully analyzed. The coupling scheme is based on directional coupling, which consists of two symmetrical waveguides separated by a gap. As the separation gap decreases, the interaction between two fundamental modes increases, giving rise to symmetrical and anti-symmetrical supermodes traveling with different velocities. The coupling length can be calculated by the equation of L_c = λ₀/2|n⁺ - n^-| where n⁺ and n^- are effective indices of the symmetrical and anti-symmetrical modes, respectively.

The real part of the effective indices and the coupling length are calculated for different values of the gap. As shown in Fig. 5(a), it is clear that the mode coupling increases strongly as the interaction between two modes enhances in near proximities. This increase in the difference between the real parts of the refractive indices of two modes manifests itself in smaller coupling lengths. Besides, the dependence of coupling length upon gate voltage is shown in Fig. 5(b). When the gap is equal to 70 nm, L_c is about 650 nm at the chemical potential of 0.7 eV, which promises an ultra-compact coupling unit. In the higher negative voltages, the higher chemical potential leads to less confinement of traveling SPPs, resulting in a smaller coupling length.

 

Fig. 5. Coupling length and the real parts of the effective indices for symmetric (dashed blue curve) and anti-symmetric (solid blue curve) supermodes as a function of (a) gap size and (b) gate voltage.

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To realize the modulation behavior of the device and its highly tunable transmission, the dependence of the geometric parameters, labeled in Fig. 4(b), and applied voltage on the transmission spectrum is investigated. For this purpose, we utilize the three-dimensional finite-difference time-domain (3D FDTD) calculations in which the mode source excites SPPs with propagating wavevector along the + x direction. Moreover, the guided light decays in the perfectly matched boundary layers located at both ends of the device in the x-direction. The proposed approach is based on taking advantage of low IL of suspended graphene waveguide in the on-state and resonances in a Fabry-Perot-like cavity while interferences with standing waves in the y-direction reduce the effective coupling. This approach promises a small footprint, low IL, and high ER simultaneously. As demonstrated in Fig. 1, the refractive index of the graphene sheet is controlled by applying a voltage difference between the gold ridges and the gold layer located beneath the Si spacer. Unlike non-resonant modulation mechanisms such as all-plasmonic Mach-Zehnder modulator [44], when the wave is routed to two pathways with different controllable effective indices, this mechanism avails the inherently narrowband nature of resonant cavity and ohmic losses mostly appeared in off-state, significantly reducing the IL in on-state.

Here, the transmission spectra are calculated for different geometric parameters. In all cases, the parameters W, t₁, L₂, and chemical potential are considered to be 80 nm, 10 nm, 2000 nm, and 0.7 eV, respectively, unless it is mentioned. To begin with, the transmitted power is captured at the end of the modulator for different values of G₁, which indicates the distance between the middle waveguides (input and output ports) and coupling waveguides (top and bottom arms). Figure 6 demonstrates the transmittance at a wide range of wavelengths for t₂ = G₂ = L₁ = 90 nm. The transmittance spectra show that there are at least two great dips in all cases. However, for smaller values of G₁, the resonant dips are more prominent, especially at the lower wavelengths. For instance, setting G₁ = 50 nm results in two dips with the transmittance of -24 dB and -20 dB at the wavelengths of λ = 7.67 µm and λ = 8.11 µm, respectively. Furthermore, the transmittance plummets from -1.8 dB at λ = 7.55 µm to -24 dB at λ = 7.67 µm, giving the ER = 10log (P_on-state/P_off-state) = 22.2 dB, which is about 10 dB higher than that of the previously reported plasmonic-assisted resonant modulator [24]. On the other hand, increasing G₁ leads to a lower ER. It should be noted that according to the previous calculations for coupling length, the smaller value of G₁ results in an efficient coupling at the lower wavelengths.

 

Fig. 6. Transmission spectra of the modulator for different values of G₁.

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According to Fig. 7, when the gold ridges distance from the top and bottom waveguides, the transmission dips are shifted towards the higher wavelengths, and the free spectral range is widened, as expected for a typical resonance cavity. To determine the effects of G₂ on the transmittance spectra independently, the parameters G₁ = 90, t₂ = 90, and L₁ = 500 nm are taken to be fixed, and G₂ is altered from 50 nm to 130 nm by 20 nm increments. The highest ER of 19.6 dB is achieved for G₂ = 90 nm at λ = 8.91 µm. For G₂ values more than 150 nm, the resonance behavior disappears, and no dip is observed in the transmission spectrum. This is because the gold ridges are too far from the coupling waveguides that the strongly confined light in the waveguides has no interaction with these ridges.

 

Fig. 7. Transmission spectra of the modulator for different values of G₂.

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Apart from the spacing of the gold ridges, the effects of other geometric parameters in resonant wavelengths and ER values are noticeable without deteriorating the modulation performance. Figure 8 illustrates that as L₁ increases, the number of dips also increases. This behavior is attributed to the fact that the wave propagation conditions towards the output waveguide strongly depend on the standing wave pattern formed in the waveguides, wavelength, and L₁. As L₁ increases, the number of optical power transitions between the middle waveguides and coupling waveguides increases, resulting in different patterns of the standing waves wrought in the coupling waveguides. Depending on the pattern, constructive or destructive interferences occur. Furthermore, by increasing L₁, the number of wavelengths that satisfy the resonance condition increases. Therefore, more dips are observed in the transmission spectrum.

 

Fig. 8. Transmission spectra of the modulator for different values of L₁.

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As abovementioned, the electric field distribution is highly confined both laterally and vertically above the Si ridges. Consequently, for a simple waveguide setup, the propagation characteristics are highly sensitive to t₁ (Fig. 3(e)) while the effects of t₂ variation are negligible. Needless to say that to maintain the t₁ distance unchanged, the height of the gold ridges must vary correspondingly as t₂ changes. However, the effect of t₂ variations must be considered since the transmission spectrum changes noticeably. Figure 9 shows the transmission spectra for different values of t₂ while the other geometric parameters including G₁ = G₂ = 90 nm, L₁ = 500 nm, and t₁ = 10 nm are fixed. The simulation results exhibit the highest ER of 23 dB for t₂ = 70 nm and λ = 9.088 µm. As a result, t₂ = 70 nm is selected for the following simulations.

 

Fig. 9. Transmission spectra of the modulator for different values of t₂.

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To have a better understanding of resonant behavior and signal extinction, the y- and z-components of the electric field distribution (E_y and E_z) are monitored 5 nm above the Si ridge. The geometric parameters are selected to be G₁ = G₂ = 90, L₁ = 500, and t₂ = 70 nm. For these values, the transmission spectrum has two dips at the wavelengths of 9.088 µm and 9.711 µm. For better qualitative comparison, the E_y and E_z distributions are normalized separately to their own maximum intensity. As shown in Fig. 10(a), the resonance cavity gives rise to the y-component of the electric field at the resonant frequency around the coupling regions. But, such a field distribution is not observed for λ = 10 µm. As seen in Fig. 10(b), the propagating waves are completely out of phase in the region where the coupling waveguides and the output waveguide overlaps at λ = 9.088 µm, resulting in destructive interference. Therefore, the transmission is reduced to -25 dB at this wavelength. Since the phase difference in λ = 9.711 µm is slightly smaller, the transmission is a little greater, equal to -11 dB. However, the efficient coupling at λ = 10 µm leads to the maximum transmission of -1.5 dB.

 

Fig. 10. (a) The y-component and (b) the z-component of the electric field distribution for three different wavelengths over the x-y plane.

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Compared to the single waveguide, the modulator structure has higher IL in the on-state due to the proximity of the coupling waveguides to the lossy gold ridges. Additionally, the coupling losses are not negligible in the latter structure, which are absent in the single waveguide. To reduce the losses, it is possible to merge the end region of the coupling waveguides by two bent Si waveguides and, afterward, connect to the output waveguide. In this condition, the coupling losses are decreased at the cost of reducing the ER because the formation of the cavity at both sides of the coupling waveguides diminishes the transmission at the resonance wavelength.

In order to highlight the modulation mechanism, the transmission over the wavelength range of 7 µm to 11 µm is calculated under different bias voltages. As shown in Fig. 11, the transmission is highly susceptible to the effective index of graphene SPPs, showcasing the potentiality of this novel structure to be used as resonant switches and controllable notch filters. As the chemical potential changes from the predefined 0.7 eV to lower values by applying a positive voltage to the gold ridges, the dips are shifted to higher wavelengths. Applying a voltage as low as 5 volts causes an about 464 nm shift in the resonant peaks, giving the sensitivity of Δλ_res/ΔV_g = 92.8 nm/V, where Δλ_res and ΔV_g are the resonance wavelength and gate voltage variations, respectively. Also, the proposed structure exhibits a relatively high quality factor (Q-factor) defined as λ_res/Δλ_FWHM, where FWHM stands for full-width at half-maximum at resonance peaks. The Q-factor of the modulator is as high as 105 and 98 at λ_res-1 = 9.088 µm and λ_res-2 = 9.711 µm. The relatively high sensitivity to gate voltage offers low power consumption, higher working frequency, and mechanical stability since low voltages are required to change the resonant peak.

 

Fig. 11. Transmission spectra of the modulator as a function of wavelength for different values of gate voltage.

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Ultimately, we calculate the total capacitance (C_T) and total ohmic resistance (R) of the device in order to find the modulation bandwidth as a crucial characteristic of the modulator. The gate capacitance (C_G) of the device can be obtained utilizing the parallel plate model described in section 2. According to the fitted curve for t₁ = 10 nm and t₂ = 70 nm, the gate capacitance is approximately 0.096 µF/cm². The other contributing capacitance connected in series with the C_G is regarded as quantum capacitance (C_Q) expressed by [45]:

4. Conclusion

In this work, we proposed a novel high ER, compact, and tunable EO modulator based on the suspended graphene plasmonic waveguides. Taking advantage of intrinsically low loss pattern-free suspended graphene waveguide and bypassing the lossy cavity in off-state, the IL is considerably reduced to about 1.3 dB. By analyzing the steady-state charge distribution, the relationship between the graphene chemical potential and gate voltage was extracted. According to the simulation results obtained by the 3D FDTD method, the transmission spectrum exhibits a great dependency on the geometric parameters, achieving the ER as high as 22 dB. The resonance wavelength is strongly dependent on the gate voltage, resulting in a relatively high sensitivity of 92.8 nm/V. Due to the very low ohmic resistance and suitable capacitance, the modulation bandwidth is acquired as high as 71 GHz. Also, the footprint of the whole structure is less than 3 µm². Apart from modulation applications, the introduced structure reveals remarkable controllable characteristics that can be exploited in other integrated photonic devices such as notch filters and switches operating at the MIR regime.