## Abstract

A graphene electro-absorption optical modulator based on double-stripe silicon nitride waveguide is proposed and analyzed. By embedding four graphene layers in the double-stripe silicon nitride waveguide and the graphene layers co-electrode design, the total metal-graphene contact resistance can be reduced 50% and as high as 30.6GHz modulation bandwidth can be achieved theoretically. The calculated extinction ratio and figure of merit are 0.1658dB/um and 9.7, respectively. And the required switching voltage from its minimum to maximum absorption state is 3.8180V and 780.50fJ/bit power consuming can be achieved. The proposed modulator can remedy the lack of high speed modulator on the passive silicon nitride waveguide.

© 2017 Optical Society of America

## 1. Introduction

Graphene, a single sheet of carbon atoms arranged in a two-dimensional honeycomb lattice, was isolated experimentally by mechanical exfoliation in 2004 [1]. It has many extraordinary properties, such as the extremely high electron mobility of more than 200000cm^{2}/(V·s) [2]. In an un-doped layer of graphene, it has a universal linear absorption coefficient, which values 2.3% over a broad spectrum range from visible to mid-infrared(IR) [3]. Since the Fermi energy of graphene is quite sensitive to the occupation of electronic states, it can be significantly shifted by slightly variations of carrier density, which can be tuned conveniently by applied voltage [4], leading to the gate-controlled optical absorption in graphene.

Utilizing this effect, various geometries of graphene electro-absorption modulators based on silicon-on-insulator (SOI) have been proposed [2,3,5,6]. In 2011, a graphene electro-absorption modulator based on SOI was first reported by Liu et al. [2]. The graphene modulator experimentally demonstrated a mode power attenuation (MPA) of 0.10dB/*μ*m and modulation speed of 1.2GHz. In 2012, Liu et al. experimentally proposed a double-layer graphene optical modulator, which demonstrated a high MPA of 0.16dB/*μ*m, modulation speed of 1.2GHz, insertion loss of 4dB, and power consumption of 1pJ/bit [5]. In 2014, a graphene electro-absorption modulator based on SOI waveguide with low insertion loss was presented experimentally by Muhammad Mohsin et al. [6]. The graphene modulator showed a relatively low modulation speed of 670MHz, insertion loss of 3.3dB and a figure of merit (FOM) Δα/α = 4.8(ratio of extinction to the insertion loss). So far, graphene modulators based on SOI waveguide have been extensively studied. As we known, the main limitation of SOI waveguide is it’s relatively large propagation loss introduced by the surface roughness (about 2~3dB/cm), which is much larger than the typical 0.1dB/cm propagation loss of silicon nitride (Si_{3}N_{4}) waveguide. However, it should be noted that graphene losses will overshadow most of the propagation loss reduction offered by Si_{3}N_{4} because graphene loss is in dB/um, which is several orders larger than the SOI loss in dB/cm. Therefore the linear propagation losses of the silicon graphene and silicon nitride graphene modulators should be comparable. Besides, Si_{3}N_{4} waveguide can be fabricated using CMOS-compatible technologies at the length of tens of centimeters with high yield, and comparing with the SOI waveguide, it has relatively lower refractive index contrast, which brings the larger fabrication tolerances [7] and better performance in the fiber-coupling [8]. And Si_{3}N_{4} stripe waveguide has demonstrated a propagation loss below 0.1dB/m with a fiber-coupling loss less than 1dB [9]. So the silicon nitride graphene modulator may has the potential of easier fabrication and lower insertion loss. In addition, it is well known that Si_{3}N_{4} has a large bandgap of ~5eV compared to Si (~1.12eV), and due to this large bandgap, Si_{3}N_{4} has a negligible two-photon absorption (TPA) loss compare with the rather large TPA loss in silicon [10, 11]. So coupled to the fact that graphene has saturable absorption property, it is very possible that Si_{3}N_{4} devices can offer very low nonlinear losses when compared to silicon [12, 13]. It can be concluded that Si_{3}N_{4} waveguide shows better performances than the SOI waveguide when it is used to build passive devices. But because the Si_{3}N_{4} has no electro-optic effect, no high speed electro-optical modulators can be achieved on the Si_{3}N_{4} waveguide platform. This intrinsic shortcoming can be overcome by embedding graphene in the Si_{3}N_{4} waveguide. In recent years, graphene-on-silicon nitride - waveguides have also been investigated [12–15]. In 2013, a hybrid structure with graphene layer on top of a partially etched Si_{3}N_{4} rib waveguide with MPA of 0.066dB/*μ*m was presented theoretically and experimentally by N.Gruhler et al. [12]. In 2015, a graphene-on-Si_{3}N_{4} electro-optic modulator based on ring resonator was demonstrated experimentally by Christopher T. Phare et al [14]. This modulator operates with a 30GHz bandwidth and a state-of-the-art modulation efficiency of 15dB per 10V. In 2017, Leili Abdollahi Shiramin et al. [15] theoretically proposed a double layer graphene on silicon nitride (DLG-Si_{3}N_{4}) modulator and a double layer graphene embedded in silicon nitride (DLG-E-Si_{3}N_{4}) modulator, which embed the double layer graphene stacks within the waveguide core. In the DLG-Si_{3}N_{4} configuration, a high MPA of ∼0.12dB/*μ*m of the TE mode can be achieved, and a 200-*µ*m long configuration allows for 23 dB extinction ratio and 2.5GHz modulation bandwidth. In the DLG-E-Si_{3}N_{4} configuration, an even higher MPA of ∼0.26dB/*μ*m of the TE mode can be achieved, and a MPA of ∼0.01dB/*μ*m of the TM mode was also obtained. As far as we know, almost all the presented Si_{3}N_{4}/graphene modulators are based on the single stripe Si_{3}N_{4} waveguide. Comparing to the single stripe Si_{3}N_{4} waveguide, double-stripe Si_{3}N_{4} waveguide will have much more symmetrical mode field distribution and lower polarization dependence. While the Si_{3}N_{4}/graphene modulator using the double-stripe Si_{3}N_{4} waveguides has not been proposed.

In this paper, we proposed a four graphene layers electro-absorption optical modulator based on the double-stripe Si_{3}N_{4} waveguides. And the rest of this paper is organized as follows: In Section 2, the optical and electrical properties of graphene are described. In Section 3, the performances of this modulator were studied by the finite element method (FEM). In the final section, the device’s characteristics were summarized and concluded.

## 2. Gate-variable properties of graphene

In [16], it is proved that graphene need to be treated as an anisotropic material instead of an isotropic material because it is one atom thick and its *π* electrons cause electric conduction in its plane. The out-of plane conductivity σ_{⊥} can be different from the in-plane conductivity σ_{||} and the epsilon-near-zero (ENZ) effect does not exist. For the in-plane optical conductivity σ_{||} of graphene, an analytic expression derived by the random phase approximation is used, which is

_{0}=

*e*4

^{2}/*ћ*= 60.85(

*μS*) is the universal conductivity of graphene (e is the charge of an electron and

*ћ*is the reduced Planck constant),

*E*is thermal energy in eV (

_{th}= k_{B}T*k*is the Boltzmann constant, and T is the temperature),

_{B}*E*is the Fermi energy (that is chemical potential

_{F}*µ*) of graphene in eV,

_{c}*E*is photon energy in eV (

_{ph}= hc/λ*h*is the Planck constant), and

*Es = ћ/τ*is scattering energy in eV for the scattering time τ. Thus, the graphene conductivity as a function of chemical potential is shown in Fig. 1(a).

From the complex optical conductivity, the complex dielectric constant *ε*_{||}(*µ _{c}*) of graphene can be calculated as

where *d _{G}* = 0.7nm is the thickness of monolayer graphene [16–18],

*ε*

_{0}is the vacuum permittivity, and the out-of plane permittivity is assumed to be

*ε*

_{⊥}= 2.5 [16]. Thus the obtained dielectric constant of graphene as a function of chemical potential is shown in Fig. 1(b).

As shown in Fig. 2, the carrier density in graphene can be tuned in an electric-field-gated structure [19]. Subsequently the chemical potential of graphene can be dynamically tuned by the applied voltage. The relationship between the carrier density and applied voltage is N = α(V + V_{0}), where α = ε_{0}ε_{r}/(de) is estimated using a parallel-plate capacitor model, V is applied voltage, V_{0} is the voltage offset caused by natural doping, ε_{r} and d is the relative dielectric constant and thickness of the dielectric spacer respectively, e is the elementary charge. Thus the chemical potential of graphene shifts according to:

_{F}= 10

^{6}m/s is the Fermi velocity, positive (negative) n means electron (hole) doping in graphene [4]. For simplicity, |V + V

_{0}| is defined as the applied voltage below.

## 3. Performances of the graphene modulator

The structure of the double-stripe Si_{3}N_{4} optical waveguide is shown in Fig. 3(a). The optical waveguide contains two identical stripe Si_{3}N_{4} waveguides. In order to ensure that only a single mode is supported in the waveguide at λ = 1550nm, the width and thickness of each Si_{3}N_{4} stripe waveguide was optimized to be 1.2*μ*m and 0.17*μ*m, and the two Si_{3}N_{4} stripe waveguides are separated by a 500nm SiO_{2} interlayer [7]. The advantages of using double-stripe Si_{3}N_{4} waveguides are the balancing between the low propagation loss and compact device size induced by high mode confinement [7, 8, 20]. The chemical potential of graphene can be tuned in an electric-gated structure, but the Si_{3}N_{4} core and the SiO_{2} cladding are not conductive, thus it is difficult to realize a capacitor structure to apply voltage for a single layer graphene-on-Si_{3}N_{4} configuration. Hence, in order to facilitate the addition of electrodes and increasing the optical absorption modulation, only the configurations of double graphene layers and four graphene layers were considered. As shown in Fig. 3(b), two layers of graphene are separated by an insulating dielectric spacer with applied voltage between them, namely the graphene-on-graphene(GOG) structure. Thus, as the voltage is applied, one graphene layer is doped by holes and the other by electrons at the same doping level. The thickness of the dielectric spacer is d, the width and length of graphene are W and L, respectively. In Fig. 3(c), it shows the double graphene layers configuration, where the two graphene layers arranged as the GOG structure are placed on the top surface of bottom Si_{3}N_{4} stripe waveguide. And Fig. 3(d) shows the four graphene layers configuration, where two GOG structures are placed on the top and bottom surfaces of the two Si_{3}N_{4} stripe waveguides, respectively

In the two and four graphene layers configurations, the electric field magnitude |E| distributions of the TE and TM modes are shown in Figs. 4(a)-4(d). As we can see, the modes are well confined around the two Si_{3}N_{4} stripes. And electric field magnitude |E| distributions for both configurations are close to circular shape, which could be well coupled to the fiber mode with a spot size converter. As shown in Figs. 4(e)-4(h), the real part and imaginary part of effective mode index (EMI) can be tuned by varying the chemical potential of the graphene for both configurations. For the TE modes, the real part of EMI first increases then decreases with the chemical potential *μ*_{c}, and achieves maxima at *μ*_{c} = 0.4eV. Note that in the vicinity of μ_{c}≥0.4eV a small change in the chemical potential yields a substantial change in the real value of EMI. This effect can be made into the electro-refraction modulator [21]. On the other side, a small change in the chemical potential in both directions around *μ*_{c} = 0.4eV (0.3eV<*μ*_{c}<0.5eV) yields a substantial change in the value of MPA. This effect can be made into the electro-absorption modulator. And according to the interband transition condition |*μ*_{c}|<*ћ*ω/2≈0.4eV, the large absorption when *μ*_{c}<0.4eV is caused by the interband absorption of the graphene. Due to the symmetrical structure of the four graphene layers configuration, a maximum MPA = 0.196dB/*μ*m and minimum MPA = 0.0072dB/*μ*m were obtained in the four graphene layers configuration, which are about two times of the maximum MPA = 0.1dB/*μ*m and minimum MPA = 0.0036dB/*μ*m obtained in the double graphene layers configuration, respectively. For the TM modes, the real part of EMI also achieves minimum value at *μ*_{c} = 0.4eV. Again the maximum MPA = 0.0040dB/*μ*m in the four graphene layers configuration is doubled comparing with the maximum MPA = 0.0022dB/*μ*m in the two graphene layers configuration. From the results above, by increasing the graphene layer numbers, a higher MPA, which will lead to a smaller footprint and lower power consumption can be achieved. It is clear that the MPA of the TE modes are much larger than the TM modes due to the anisotropic absorption. Therefore, for the rest of our work, we only consider the TE modes.

As shown in Fig. 4(f), the MPA of the TE mode reduces rapidly when the *μ*_{c} is in the region 0.3eV<*μ*_{c}<0.5eV, namely the operating region in our modulator, but it changes slowly in other regions. These results can be explained through the evaluation of the tangential electric field, *E*_{||}, at the graphene layer. Because only this field component has an effect on the in-plane absorption of the graphene. It is known that the power absorbed in a unit area can be expressed as P_{d} = Re[σ_{||}(*μ*_{c})]|*E*_{||}|^{2}/2 [22]. Combining with Eq. (2), it can be expressed as

*E*

_{||}|

^{2}= |

*E*

_{x}|

^{2}+ |

*E*

_{z}|

^{2}, and k = ωε

_{0}d

_{G}is a constant. We can see that the power absorption is roughly proportional to imaginary part of dielectric constant and the square of the electric field magnitude in the graphene. The Im[

*ε*

_{||}(

*µ*)] of graphene is ~7.9 at

_{c}*μ*

_{c}= 0eV and decreases to ~0.29 when

*μ*

_{c}= 0.7eV as shown in Fig. 1(b). Thus, according to Eq. (4), by substituting the Im[

*ε*

_{||}(

*µ*)] and |

_{c}*E*

_{||}|

^{2}values in the graphene of the TE mode in the four graphene layers configuration, a same normalized MPA decreasing trend can be obtained from

*μ*

_{c}= 0eV to

*μ*

_{c}= 0.7eV as shown in Fig. 5.

As shown in Fig. 4, the four graphene layers configuration has a larger EMI and MPA modulation compared to the double graphene layers configuration for both the TE and TM modes, thus giving advantages of smaller footprint and lower power consumptions. Therefore, we only consider the four graphene layers configuration in the following section.

The cross section of the four graphene layers modulator based on double-stripe Si_{3}N_{4} waveguide is shown in Fig. 6. One graphene layer in the lower GOG and one graphene layer in the upper GOG are connected to the signal electrode, and the remaining two graphene layers in the two GOG structures are connected to the ground electrode. Due to this graphene layers co-electrode design, the series resistance of the modulator is reduced by 50% [18]. In the four graphene layers modulator structure, the *RC* bandwidth is limited by the graphene sheet resistance, contact resistance and the capacitance of the parallel-plate capacitor. And the frequency response of the devices can be modeled using the equivalent electrical circuit model shown in Fig. 7.

The series resistance, *R _{S}*, is the sum of the metal-graphene contact resistance (

*R*) and sheet resistance (

_{c}^{G}*R*) of each graphene layer. These parameters are objects of many studies and investigations and we referred to the typical results reported in literatures. In particular, the graphene sheet resistance

_{sh}^{G}*R*is varied between 100 and 300Ω/sq, equivalent with a mobility of 6900 and 2300cm

_{sh}^{G}^{2}/V·s, respectively, when the carrier density on the graphene layer is in the order of 10

^{12}-10

^{13}cm

^{−2}[15,23–25]. As for the contact resistance

*R*, this still represents one of the major challenges in graphene technology. Its value depends on the specific metal and process used to form the contact. Here the graphene contact resistance is assumed to be

_{c}^{G}*R*150Ω·

_{c}^{G}=*μ*m, a value achievable e.g. through edge contacting of graphene [26–28]. And

*C*is the parallel-plate capacitor formed by the two graphene layers with an insulating dielectric spacer, and it can be calculated as below:

_{ox}*S = W*L*is the overlap area of two graphene layers, which are shown in Fig. 3(b). Then, the series resistance (

*R*) and total capacitance (

_{S}*C*) of the graphene modulator are calculated as follows [18]:

_{total}In order to find the minimum distance between the metal electrode and waveguide, so that the optical modes of the waveguide remained undisturbed by the metal contact. We plot the electric field magnitude |E_{||}|^{2} across the waveguide center of the TE mode in the X direction and the MPA of the metal electrodes in the double-stripe Si_{3}N_{4} waveguide as shown in Fig. 8. It is obvious that |E_{||}|^{2} of the TE mode is close to zero, when the distance d_{0} between the electrode and the waveguide core edges is d_{0} >1.4*μ*m as shown in Fig. 8(a). Furthermore, to get a more accurate value d_{0}, we considered the absorption of the metal electrodes without graphene embedding in the waveguide. As shown in Fig. 8(b), the mode absorption of the TE mode is quite small when d_{0} >1.4*μ*m. Specifically, the MPA of the metal electrodes is as small as 0.00077dB/*μ*m with d_{0} = 1.7*μ*m. At this point, the absorption of the metal electrodes can be ignored and the optical modes of the waveguide remained undisturbed by the metal contacts. Thus, in this work, we considered a 1*μ*m width for each metal electrode and 1.7*μ*m distance between the electrodes and the waveguide core edges on both sides. And we assumed *R _{sh}^{G} =* 100Ω/sq and

*R*150Ω·

_{c}^{G}=*μ*m, respectively.

From the above formulas, one can get that the insulating dielectric spacer plays an important role in affecting the modulation bandwidth. The higher relative permittivity of the insulating dielectric spacer, the smaller modulation bandwidth can be obtained. Here a 7 nm-thick SiO_{2} layer was used as the insulating dielectric spacer in the four graphene layers modulator.

As shown in Fig. 9, for the TE mode, we defined the modulator is in “OFF” state with maximum MPA of 0.1830dB/*μ*m at μ_{c} = 0.3eV, which corresponds to the applying voltage of about 2.1476V. Then, the modulator is in “ON” state at μ_{c} = 0.5eV and the minimum MPA of 0.0172dB/*μ*m is achieved with applying voltage of 5.9656V. This corresponds to an extinction ratio of Δα = 0.1658dB/*μ*m and FOM of Δα/α = 9.7 with applying voltage (ΔV) of 3.8180V. A modulator length of 18.09*μ*m is needed to achieve 3dB modulation and the corresponding power consumption (E_{bit} = CΔV^{2}/4) is 780.50fJ/bit. And from Eq. (8) a modulation bandwidth as large as *f _{3dB}* = 30.6GHz can be achieved.

As mentioned above, the material and thickness of insulating dielectric spacer play a role in determining the modulation speed and consumption. This is because the insulating dielectric spacer will affect the capacitance and then the modulation performances such as the switching voltage ΔV, 3dB bandwidth and the energy per bit. Here the SiO_{2} and Al_{2}O_{3} were chosen as the insulating dielectric spacer and the corresponding modulation performances were simulated and shown in the Table 1.

For the modulator with 7nm SiO_{2} insulating dielectric spacer, a maximum 3dB modulation bandwidth of 30.6GHz can be achieved because relative low dielectric constant of the SiO_{2} and large insulating dielectric spacer thickness will reduce the parallel-plate capacitor between two graphene layers. But the required applying voltage ΔV is relative larger comparing to the Al_{2}O_{3} insulating dielectric spacer due to the lower dielectric constant of the SiO_{2}, which will bring the larger power consumption comparing to the modulator with Al_{2}O_{3} insulating dielectric spacer. For the modulator with 7nm Al_{2}O_{3} insulating dielectric spacer, a 3dB modulation bandwidth of 10.5GHz can be achieved. Since the power consumption of modulator is proportional to the applied voltage ΔV in graphene [17], a relative low power consumption of 262.90fJ/bit were obtained. It is demonstrated that the higher dielectric constant of the insulating dielectric spacer, the lower power consumption and smaller 3dB modulation bandwidth can be obtained. It can be concluded that the smaller the dielectric constant and the larger thickness of the insulating dielectric spacer, the larger the modulation bandwidth can be achieved. But the cost is the increased power consumption. So there is a trade-off between the modulation bandwidth and the power consumption.

So far, the graphene-assisted electro-absorption modulators are mainly based on Si and Si_{3}N_{4} waveguides. In general, the extinction ratio(Δα) and FOM of graphene modulator based on Si waveguide is larger than that of Si_{3}N_{4} waveguide based graphene modulator because the larger refractive index contrast of the SOI waveguide will bring much higher mode confinement, which will increase the interactions between the graphene and the optical waveguide mode. For example, an extinction ratio of Δα = 0.31dB/*μ*m and FOM of 12.2 have been obtained for silicon graphene modulator with double graphene layers [17]. And an extinction ratio of about 0.26dB/*μ*m and FOM of about 21 have been achieved for silicon graphene modulator with double graphene layers [15]. Although the silicon graphene modulators have better performances on the extinction ratio and FOM than our silicon nitride graphene modulator, but they may suffer from the intrinsic two-photon absorption which will plague silicon at communications wavelengths [10]. Besides, the nano-scale silicon waveguide width needs much more expensive fabrication costs and complex fiber coupling method than the micro-scale silicon nitride waveguide (1.2*μ*m waveguide width). In the case of graphene-assisted modulators based on Si_{3}N_{4} waveguides, the reported double layer graphene on silicon nitride (DLG-Si_{3}N_{4}) modulator in [15] can achieve a higher FOM of 23, which is more than two times of 9.7 obtained in our modulator because of the lower insertion loss (α) of the two graphene layers than the four graphene layers in our structure. While, due to the four graphene layers configuration and the graphene layers co-electrode design, our modulator has a larger extinction ratio of Δα = 0.1658dB/*μ*m and a higher modulation bandwidth about 30.6GHz, compared to the extinction ratio of 0.115dB/*μ*m and maximum bandwidth about 23GHz of the DLG-Si_{3}N_{4} configuration.

## 4. Conclusion

To summarize, a four graphene layers electro-absorption optical modulator based on double-stripe Si_{3}N_{4} waveguide was presented. The proposed graphene modulator can achieve as high as 30.6GHz modulation bandwidth with 18.09*μ*m long active region to meet the 3dB modulation. And a switching voltage of ΔV = 3.8180V and 780.50fJ/bit power consuming can be achieved. By combing the advantages of the low insertion loss of the double-stripe Si_{3}N_{4} waveguide and the high carrier mobility in graphene, a compact and high speed optical modulator base on the Si_{3}N_{4}/graphene hybrid waveguide can be achieved, which will remedy the lacking of high speed modulator on the Si_{3}N_{4} waveguide platform.

## Funding

This work was supported by the National Science Foundation of Jiangsu Province Grant No. BK 20161429.

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