## Abstract

The suspended triple-layer graphene modulator is firstly investigated theoretically. We find there appear two modulation depths for electro-absorption modulation. The light-graphene interaction is enhanced to its maximum by our designed waveguide structure. The highest modulation depth for electro-absorption modulation can be 0.834 dB/µm, causing a 3-dB footprint of only 0.94 µm^{2}. For electro-refractive modulation, there appear several 100% modulations with a much smaller π-phase shift length of only 11.3 µm. This modulator also shows great potential for high-speed modulation with a prediction value of 759.85 GHz, while the switch energy can be as low as 0.61 fJ/bit with low applied voltage. Moreover, the verification simulation by COMSOL is also presented, which shows very good agreement with our calculation results, and the figure of merit (defined as the ratio of modulation depth to insertion loss) of this modulator can be 2105. We believe these results can pave the way to design practical high-speed, compact-footprint, and high-efficiency devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical modulator is a key device in many real applications. The amplitude of the carrier optical wave will be modified by the information signal, and it is used for information deliver. Traditional silicon-based optical modulator has been developed to its fundamental limits for modulation speed, footprint, and efficiency. The graphene-based optical modulator was firstly reported in 2011 [1]. Then papers [2–16,20] follow this research to improve properties of graphene-based optical modulators. For these modulators, different structures are researched, such as monolayer [2,3], double-layer [4–7], dual-graphene-on-graphene configuration (four-layer) [8,9], multi-layer [10,11], graphene-covered-microfiber [12,13], dielectric-loaded plasmonic structure [14]. The modulation speed can be 35 GHz [15], the modulation depth can be 100% [16], the operation bandwidth can be 15 THz [14], and the footprint can be only 0.18*0.3 µm^{2} [14]. These good results are based on the much broader and higher tunable absorption [17] and ultrahigh carrier mobility [18,19] of graphene.

However, the compromise between modulation speed and modulation efficiency is always an issue which stops a better modulator: When the modulation speed is higher, the capacitor of the modulator should be lower, then the thickness of the insulator between the two electrodes should be thicker and the active area smaller. This will make the modulation efficiency much lower for higher applied voltage, demanding smaller footprint, lower modulation depth.

Recently, we put forward a much better suspending self-biasing graphene modulator [20]. By suspending the modulator, we design it to have much higher modulation speed, as high as 559.2 GHz. The footprint for electro-absorption modulation is only 1.5 µm^{2}. And the modulation depth can also be 100% for electro-refractive modulation. The most important one is the figure of merit (FOM) (defined as the ratio of modulation depth to insertion loss) can be to ∼ 2700. These sounding results are contributed by three reasons: First, the modulator can be designed to realize the highest light-graphene interaction; Second, the suspending structure make the channel mobility reduction caused by graphene-dielectric interaction much lower; Third, the suspending of the modulator makes the insertion loss much lower. This suspended self-biasing graphene modulator has been developed to very near the fundamental limits [21]. However, the applied voltage is still large and the modulation depth for the electro-absorption modulation is still low. Moreover, the light-graphene interaction can be enhanced further by suspending triple graphene layers, which has not been researched yet.

In this paper, we report a new graphene modulator which is called as suspended triple-layer graphene modulator. In this case, three graphene layers are sandwiched by two insulator slabs, and the modulator is suspended. The three graphene layers are biased simultaneously by each other, which makes the middle layer double doped as each of the two side layers. The light-graphene interaction will be enhanced significantly for there are three graphene layers and the middle layer is always at the highest mode energy distribution center. Moreover, two insulator slabs make the confinement of the mode better and the parallel connection of the two capacitors make the total resistance lower, and the applied voltage will also be lower for thinner insulator slab at the best light-graphene interaction.

In section 2, we derive the mode profile equations and the dispersion equation of TE mode, and analyze the waveguide structure model. In section 3.1, we analyze the mode characteristics in this waveguide. There appear two modulation depths and two mode index peaks, which are caused by the middle graphene layer and the two side graphene layers, respectively. Then we design the waveguide to realize the highest light-graphene interaction for the highest modulation depth at the optimized insulator thickness. The reason for highest light-graphene interaction are given by analyzing the mode profiles. In section 3.2, we give the performance of this modulator. The transmittance for electro-absorption type shows two modulations. For the electro-refractive type, the π-phase shift length is only 11.3 µm, and it shows several 100% modulations. The modulation speed can be as high as 759.85 GHz. Moreover, the applied voltage and energy consumption is lower than the suspending self-biasing graphene modulator. In section 3.3, the verification full-wave simulation by COMSOL is given, the results from COMSOL are matched very well with our calculation. We believe this modulator will be very useful for many applications and the results we get will be very useful for designing other modulators and optical devices.

## 2. Model and dispersion equation

The waveguide structure is shown in Fig. 1 (a). Three graphene layers are sandwiched by two insulator slabs which have the same material and a width of *w*. The thickness of a graphene layer is *δ* = 0.33 nm. The waveguide is suspended in the air for three reasons: First, by suspending, the mode will be confined in the graphene center, the mode confinement will also be better, and the light-graphene interaction will be enhanced; Second, the mode profile will be totally symmetrical, and the insertion loss will be lower; Third, the waveguide will be away from any other disturbance which effects the quality of the device, so the mobility of graphene will be better.

The voltage is applied between the middle layer and the two side layers. If the middle layer is connected to the negative electrode of the voltage, the electrons from middle graphene layer will move to the two side layers, as shown in Fig. 1 (b).

The middle graphene layer will be doped with holes, and the two side layers will be doped with electrons. Moreover, the doping degree in the middle layer will be two times of doping degree of each side layer. If each of the two side layers has a chemical potential of *μ*_{c}, the middle layer will have a chemical potential of $\sqrt 2 {\mu _c}$ according to the theory in Ref. [22]. The equivalent circuit of this modulator is shown in Fig. 1 (c).

From Fig. 1 (c), we can know the equivalent circuit of this modulator is two modulators parallel connected to each other and biased simultaneously. So the total resistance and capacitance will be:

And the modulation speed will be:*RC*. Moreover, the light-graphene interaction will be enhanced further because the mode amplitude will be the peak on the middle layer for the symmetrical mode profile. The contradiction between modulation speed and modulation efficiency will be reduced further.

Figure 2 illustrates a possible way to fabricate the suspended triple-layer graphene modulator. First a trench is made in the semiconductor wafer by etching (a); then an annealed graphene is transferred [23] on the trench (b); after that a thick insulator slab should be deposited on the suspended first layer graphene (c); then another annealed graphene is transferred on the top of the first insulator slab (d); the other insulator slab is then deposited on the top (e); at last the top side graphene layer is transferred.

According to the well-known two-dimensional analysis [24], the dimension of the waveguide in *y*-direction can be assumed as large enough for not disturbing the mode configuration. The suspended triple-layer graphene waveguide may support transverse magnetic (TM) and transverse electric (TE) modes simultaneously. For TM mode, the light-graphene interaction is much more complex than that of TE mode, because it has two components of the electric field (*E*_{x} and *E*_{z}). The one-atom-layer thick graphene’s periodicity is in the two-dimensional lattice plane, and it reveals anisotropic material properties: the in-plane permittivity (${\varepsilon _ \textrm{g} }$) can be actively tuned by the Fermi level, and the out-of-plane permittivity (${\varepsilon _ \bot }$) which does not vary with the external parameters. Only the electric field can interact with graphene, so the dispersion relation of TM mode cannot be derived directly. However, for TE mode, it only has an in-plane component *E*_{y} which interacts with the graphene [10]. We discuss the symmetric TE mode here. The mode profile of this modulator can be assumed as [25]:

*A*

_{1}∼

*A*

_{10}are unsolved mode coefficients in different regions, ${h_1} = {({\varepsilon _g}k_0^2 - {\beta ^2})^{1/2}}$, ${h_2} = {({\varepsilon _1}k_0^2 - {\beta ^2})^{1/2}}$, ${h_3} = {({\beta ^2} - {\varepsilon _g}k_0^2)^{1/2}}$, and ${h_4} = {({\beta ^2} - {\varepsilon _2}k_0^2)^{1/2}}$. The wave vector in vacuum is ${k_0} = 2{\pi }/\lambda$. ${\varepsilon _ \textrm{g} }$, ${\varepsilon _ {1}}$ or ${\varepsilon _ {2}}$ is relative permittivity of graphene (The equivalent permittivity of graphene can be written as ${\varepsilon _\textrm{g}} = i{\sigma _\textrm{g}}/\omega {\varepsilon _0}\delta$ [26], where

*δ*is the thickness of monolayer graphene,

*ω*is the angular frequency of carrier wave, and

*ɛ*

_{0}is the permittivity of vacuum. In the optical range, the surface conductivity

*σ*

_{g}of graphene can be predicted by the Kubo formula [27].), insulator slab or the outside air, respectively. The complex propagation constant is

*β*=

*β*

_{1}+ i

*β*

_{2}, where

*β*

_{1}is related to the effective refractive index (

*N*

_{eff}=

*β*

_{1}/

*k*

_{0}), and

*β*

_{2}is related to the mode power attenuation (MPA)

*α*= 20

*β*

_{2}*log

_{10}(e). According to Eq. (4), the longitudinal magnetic field of the modes is derived by using ${H_z}(x) = \frac{j}{{\omega \mu }}\frac{\partial }{{\partial x}}{E_y}(x)$. Based on the continuities of tangential field components at the interfaces, the dispersion equation is further derived as follows [28–30]: where ${P_1} = [1 + \frac{{{h_2}}}{{{h_3}}}\tan ({h_2}w) - \frac{{{h_3}}}{{{h_4}}} - \frac{{{h_2}}}{{{h_4}}}\tan ({h_2}w)]{e^{ - {h_3}\delta }}$, ${P_2} = [1 - \frac{{{h_2}}}{{{h_3}}}\tan ({h_2}w) + \frac{{{h_3}}}{{{h_4}}} - \frac{{{h_2}}}{{{h_4}}}\tan ({h_2}w)]{e^{{h_3}\delta }}$, ${P_3} = [ - \frac{{{h_3}}}{{{h_4}}}\frac{{{h_1}}}{{{h_2}}}\tan ({h_2}w) + \frac{{{h_1}}}{{{h_4}}} + \frac{{{h_1}}}{{{h_2}}}\tan ({h_2}w) - \frac{{{h_1}}}{{{h_3}}}]{e^{ - {h_3}\delta }}$, and ${P_4} = [\frac{{{h_3}}}{{{h_4}}}\frac{{{h_1}}}{{{h_2}}}\tan ({h_2}w) + \frac{{{h_1}}}{{{h_4}}} + \frac{{{h_1}}}{{{h_2}}}\tan ({h_2}w) + \frac{{{h_1}}}{{{h_3}}}]{e^{{h_3}\delta }}$.

However, in this modulator, both TE mode and TM mode may be stimulated. These two kinds of modes have totally orthogonality electric and magnetic field polarity. In order to stimulate pure TE mode in experimental system, we only need to set the optical laser source to make sure the mode electric polarity is in the plane of the waveguide in *y*-axis direction. In this way, the TM mode will be forbidden for the totally orthogonality polarity.

## 3. Results and discussions

#### 3.1 Physics of the mode

First we choose Aluminum Oxide (*ɛ*_{1} = 3.06 at *λ* = 1.55 µm [31]) as the insulator material. When *w* = 100 nm, we calculate Eq. (5) and get the MPA (*α*) and effective refractive index (*N*_{eff}) of the mode as a function of *µ*_{c} (the chemical potential of each side graphene layer) at *λ* = 1.55 µm, as shown in Fig. 3:

We can see the MPA have two abrupt turn points. The first turn point happens at 0.2835 eV: the MPA changes from 0.453 dB/µm to 0.274 dB/µm causing a normalized modulation depth Δ*α* of 0.179 dB/µm. At this point, the chemical potential of the middle graphene layer is 0.401 eV, which is exactly half of the photon energy. Before this point the interband absorption of both the middle graphene layer and two side graphene layers happen, the MPA is very high. After this point the absorption of the middle layer turns to intraband absorption which causes the first modulation depth. So the first modulation depth is caused by the middle graphene layer, which is 0.179 dB/µm.

The second turn point happens at 0.401 eV: the MPA changes from 0.274 dB/µm to 0.0005 dB/µm causing a normalized modulation depth of 0.274 dB/µm. This modulation depth is caused by the two side layers which we call as modulation depth 1 (Δ*α*_{1}). Modulation depth 1 is larger than the first modulation depth, which is because both the two side layers turn from interband absorption to intraband absorption. There is a total normalized modulation depth which is 0.453 dB/µm, we call it as modulation depth 2 (Δ*α*_{2}).

There are two peaks for effective refractive index (*N*_{eff}) which happen at exactly the two MPA turn points respectively. The maximum Δ*N*_{eff} is 0.036. Both the modulation depths and Δ*N*_{eff} are much better than those of the self-biasing graphene modulator [20] which we reported before.

We get both modulation depth 1 and 2 as a function of the insulator thickness *w*, as shown in Fig. 4 (a). The changing law of modulation depth to *w* is similar as that of the self-biasing graphene modulator. There are Δ*α* peaks happening at the optimized insulator thickness *w*_{opt}, which shows the highest light-matter interaction. The maximum Δ*α*_{1} are 0.286 dB/µm, 0.506 dB/µm, 0.539 dB/µm, and the corresponding *w*_{opt1} are 77.8 nm, 20.9 nm, 16.5 nm, for *ɛ*_{1} = 3.06, 16 (a possible middle value at wavelength 1.55 µm), 22 (permittivity of Ta_{2}O_{5} [32] at wavelength 1.55 µm), respectively. The maximum Δ*α*_{2} are 0.460 dB/µm, 0.786 dB/µm, 0.834 dB/µm, and the corresponding *w*_{opt2} are 85.6 nm, 22.4 nm, 17.6 nm, for *ɛ*_{1} = 3.06, 16, 22, respectively. The highest modulation depth (0.834 dB/µm) is much higher than that of the self-biasing graphene modulator which is only 0.54 dB/µm [20] at the same conditions. So the 3-dB footprint of this modulator will be only 0.26 µm (half-maximum width of the mode profile) *3.597 µm (3 dB/(0.834 dB/ µm)) = 0.94 µm^{2}.

The maximum modulation depth of Δ*α*_{1} and Δ*α*_{2} and the corresponding *w*_{opt1} and *w*_{opt2} are gotten as a function of *ɛ*_{1}, as shown in Fig. 4 (b). These two modulation depths increase monotonously as *ɛ*_{1} increases. *w*_{opt1} and *w*_{opt2} decrease monotonously as *ɛ*_{1} increases. And *w*_{opt2} is always slightly larger than *w*_{opt1}. When *ɛ*_{1 }= 22, we find *w*_{opt1 }= 16.5 nm and *w*_{opt2 }= 17.6 nm.

For TE mode, the electric field *E*_{y} is in the plane of graphene, so the amplitude of *E*_{y} which interacts with graphene will determine the interaction strength of one photon, and the confinement of the mode will determine the photon number which interact with graphene. We have obtained the normalized mode profile as a function of vertical axis *x* when *ɛ*_{1 }= 22, and *w* = 17 nm (*w*_{opt}), 5 nm, and 50 nm, as shown in Fig. 5 (a):

The mode confinement increases when *w* is changing from 5 nm to 50 nm. So the interaction photon number with graphene will increase as *w* is changing from 5 nm to 50 nm. After our calculation, we find that the mode amplitude in the middle graphene layer is always 1. So the mode confinement will determine the light-graphene interaction for the middle graphene layer. This is why *w*_{opt2} is always slightly larger than *w*_{opt1} (modulation depth 2 is caused by the middle graphene layer).

We also get the mode amplitude in each of the two side graphene layers, as shown in Fig. 5 (b). We can see that as *w* increases, the mode amplitude in the side graphene layers decreases (it is 0.995, 0.954 and 0.760 for *w* = 5 nm, 17 nm and 50 nm, respectively). So there is a trade-off between mode confinement and mode amplitude in graphene to realize the highest light-graphene interaction. For different insulator materials, we also have found that when the permittivity is higher, the mode confinement is absolutely better and the mode amplitude in graphene is also larger. So high-κ materials [33] are absolutely better for higher light-graphene interaction.

#### 3.2 Modulation performance

We have obtained the mode power transmittance spectrum of the modulator at the side-layer chemical potentials of 0.3 eV, 0.4 eV and 0.5 eV, with a propagation length of *L* = 5 µm when *ɛ*_{1} = 22, *w*_{opt} = 17 nm, as shown in Fig. 6:

From Fig. 6, we can see there are 2 sharp changes of the transmittance for each chemical potential. The first one happens at $\lambda = \pi \hbar c/{\mu _c}$, the second happens at $\lambda = \pi \hbar c/\sqrt 2 {\mu _c}$, both of which are at the two MPA turn points, respectively. This will be very important for two signal modulation for controlling the modulation depth.

After 5 µm propagating, a modulation depth of 61.66% can be achieved at *λ* = 1.55 µm. The insertion loss can be calculated as [5]:

*T*

_{max}= 0.999618), and the FOM of this modulator is also ultrahigh ∼ 1265.

After calculation, we find that the highest Δ*N*_{eff} happens at *ɛ*_{1} = 22, *w*_{opt} = 17 nm can be 0.0685, which is much larger than the state-of-art value of 0.043 [20]. So the π-phase shift arm length for this kind of M-Z modulator can be only 11.3 µm. It is much lower than the state-of-art value of 18.0 µm [20]. For the M-Z modulator, we make the side-layer chemical potential of the reference arm fixed at *µ*_{c} = 1.0 eV and add voltage signal on the modulation arm. The side-layer chemical potential of the modulation arm can be changed from 0.2 eV to 1 eV. The normalized power transmittance *T* (Δ*µ*_{c}) of this M-Z modulator can be expressed as follows [31]:

_{0}, α

_{1}is the MPA of the reference arm and the modulation arm, respectively. We calculate Eq. (8) when

*L*= 12 µm (one π-phase shift), 34 µm (three π-phase shifts), 68 µm (six π-phase shifts), and plot the transmittance

*T*(Δ

*μ*

_{c}) of this M-Z modulator in Fig. 7:

From Fig. 7, we can see when Δ*µ*_{c} is lower than 0.6 eV, there is one 100% modulation for *L* = 12 µm, two 100% modulations for *L* = 34 µm, three 100% modulations for *L* = 68 µm. These 100% modulations are cause by the two side graphene layers which cause the second *N*_{eff} peak as shown in Fig. 3. For this Δ*N*_{eff}, the insertion loss is low for it is modulation depth 1 (after this point all the three graphene layers will only cause intraband absorption) which is in operation. The waveguide maximum allowed length *L*_{max} (*L*_{max} = 1/α) is much longer (on the scale of ∼ cm), and we can always make sure there is enough energy at the output port.

We can also see that when Δ*µ*_{c} is larger than 0.6 eV, there is no 100% modulation, and the output energy is much lower. This is because now it is modulation depth 2 (after this point only the middle graphene layer will only cause intraband absorption) which is in operation, and the insertion loss is very high (α = 0.539 dB/µm), and *L*_{max} = 1/α = 1.86 µm is very low. The choosing of *L* > 1.86 µm makes almost no energy can get out from the modulation arm, and only the reference arm can put out energy which is 25% of the total energy. The small nadir in this region when *L* = 12 µm is caused by the first *N*_{eff} peak.

The 3-dB modulation bandwidth can be calculated by Eq. (3), where *R* is the total resistance caused by one graphene layer. According to Ref. [5,20,34,35], we choose *R* = 165 Ω is reasonable. *C* = *ɛ*_{1}*ɛ*_{0}*S*/*w* is one capacitance of the active area of the modulator. Here we choose that *S* = 0.5 µm *5 µm = 2.5 µm^{2}, where 0.5 µm is the width of the modulator in *y*-axis [36], 5 µm is the length of the modulator. So the capacitance is *C* = 28.63 fF for *ɛ*_{1} = 22, *w*_{opt} = 17 nm, then a 3-dB modulation bandwidth of 22.46 GHz is obtained, which is larger than that of Ref. [20] (16.45 GHz). We find that the modulation bandwidth can be 759.85 GHz for *ɛ*_{1} = 3.06 (Al_{2}O_{3}), *w*_{opt} = 80 nm, which is much larger than 559.2 GHz reported in Ref. [20].

The side-layer chemical potential *μ*_{c} can be tuned by applying voltage *V*_{g}:

*μ*

_{c}, as shown in Fig. 8 (a).

We can see the applied voltage is always smaller than 2.3 V for *ɛ*_{1} = 22, *w*_{opt} =17 nm when the chemical potential is changing from -1 eV to 1 eV. For *ɛ*_{1} = 10, *w*_{opt} = 30 nm, the applied voltage should be larger (< 9.0 V).

We also have obtained the optimized (*w* = *w*_{opt}) applied voltage and the corresponding energy consumption as a function of insulator permittivity *ɛ*_{1} when *μ*_{c} = 0.3 eV, as shown in Fig. 8 (b). It shows that the optimized voltage and energy consumption will be smaller when the permittivity is larger. The applied voltage is on the scale of 0.2 V ∼ 2.8 V, and the energy consumption is on the scale of 0.61 fJ/bit ∼ 8.3 fJ/bit when *ɛ*_{1} is changing from 22 to 5. The energy consumption is lower than the values (larger than 1.23 fJ/bit) reported in our former work [20].

#### 3.3 Full-wave simulation by COMSOL

Our results have been verified by full-wave simulation using commercial software (COMSOL). In the simulation, we set *ɛ*_{1} = 22, the thickness of graphene is 0.33 nm, double of insulator thickness is h-core = 0.034 µm with a solving area of h-cladding = 2 µm and *L* = 5 µm. The mode profile in the longitudinal transmission direction is obtained, as shown in Fig. 9. We can see the decay of the mode amplitude is negligible for *λ* = 2.5 µm and *µ*_{c} = 0.3 eV, however it is significant for *λ* = 0.6 µm and *µ*_{c} = 0.3 eV.

We also retrieve the scattering parameter *S*_{21} as a function of the carrier wavelength, as shown in Fig. 10.

By comparing Figs. 10 and 6, we can see the results obtained from these two different methods are matched exactly well for *µ*_{c} = 0.3 eV, 0.4 eV and 0.5 eV. The difference from *S*_{21} and the transmittance *T* is only 0.001. *S*_{21} is slightly larger, because of the small reflection which has been considered in the COMSOL simulation. In this case the insertion loss is of ∼ 0.002 dB. We also get *S*_{21} when *µ*_{c} = 0.1 eV, and 0.2 eV. There is no abrupt turn point for *µ*_{c} = 0.1 eV and only one for *µ*_{c} = 0.2 eV, because the considered wavelength scale is only up to 2.5 µm whose photon energy is 0.50 eV. The total modulation depth at *λ* = 1.55 µm is 4.21 dB, and we can achieve a FOM of 2105. This value is much larger than the value by calculation, because here we have considered the *S*_{21} of *µ*_{c} = 0.1 eV, and 0.2 eV which will cause modulation depth 2 (happen at *µ*_{c} = 0.2835 eV when *λ* = 1.55 µm).

## 4. Summary

We have reported a very good modulator called as suspended triple-layer graphene modulator. The model structure and mode physics are analyzed in detail. We find there appear two modulation depths for electro-absorption modulation. After our design, the total modulation depth can be 0.834 dB/µm and a 3-dB footprint is only 0.94 µm^{2}. The light-graphene interaction will be enhanced at the optimized insulator thickness *w*_{opt} and the mechanism is analyzed by the mode field distribution in the waveguide. For electro-refractive modulation, we have achieved a very high (highest up to date) change of mode index (Δ*N*_{eff}) of 0.0685 causing a π-phase shift length of only 11.3 µm. Several 100% modulations are found in the M-Z modulator. The most sounding result is that we predict a very high modulation speed of as high as 759.85 GHz, while the switch energy can be as low as 0.61 fJ/bit with low applied voltage (< 9.0 V). The verification simulation by COMSOL is also shown as a comparison, which has very good agreement with our calculation results. The FOM of this modulator can be 2105. We believe these results are very useful for design graphene-based devices.

## Funding

Engineering and Physical Sciences Research Council (EPSRC) (EP/K01711X/1, FP7-ICT-604391).

## Acknowledgments

The authors would like to acknowledge financial supports from the Engineering and Physical Sciences Research Council (EPSRC) on Grant “Graphene Flexible Electronics and Optoelectronics” (EP/K01711X/1) and the EU Graphene Flagship (FP7-ICT-604391).

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