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 µm2. 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 [216,20] 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 [47], 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 µm2 [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 µm2. 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.

 

Fig. 1. (a) The structure of suspended triple-layer graphene waveguide modulator. (b) The movement of carriers when the triple-layer graphene modulator is biased. (c) The equivalent circuit of the suspended triple-layer graphene modulator.

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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 μc, 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:

$$\frac{1}{{{R_{total}}}} = \frac{1}{{1.5R}} + \frac{1}{{1.5R}} = \frac{4}{{3R}}$$
$${C_{total}} = 2C$$
And the modulation speed will be:
$${f_{\textrm{3dB}}} = \frac{1}{{2\pi {R_{\textrm{total}}}{C_{\textrm{total}}}}} = \frac{1}{{3\pi RC}}$$
The modulation speed is larger than that of the suspended self-biasing graphene modulator which is 1/4πRC. 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.

 

Fig. 2. Shows the fabrication steps of the device. The fabrication starts from the silicon waveguide with a trench in the middle caused by etching, as shown in (a). In (b), a prepared graphene sheet is mechanically transferred. (c) Deposit of a thick insulator. (d) Mechanically transfer the middle graphene layer. (e) Deposit of the other thick insulator. (f) Mechanically transfer the up side graphene layer.

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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 (Ex and Ez). 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 Ey which interacts with the graphene [10]. We discuss the symmetric TE mode here. The mode profile of this modulator can be assumed as [25]:

$${E_y}(x) = \left\{ \begin{array}{ll} {A_{10}}{e^{ - {h_4}(x - 3\delta /2 - w)}} & x \ge 3\delta /2 + w\\ {A_6}{e^{ - {h_3}(x - \delta /2 - w)}} + {A_7}{e^{ - {h_3}(x - \delta /2 - w)}}& \delta /2 + w \le x \le 3\delta /2 + w\\ {A_2}\cos {h_2}(x - \delta /2) + {A_3}\sin {h_2}(x - \delta /2)& \delta /2 \le x \le \delta /2 + w\\ {A_1}\cos ({h_1}x)& - \delta /2 \le x \le \delta /2\\ {A_4}\cos {h_2}(x + \delta /2) + {A_5}\sin {h_2}(x + \delta /2)& - \delta /2 - w \le x \le - \delta /2\quad \\ {A_8}{e^{ - {h_3}(x + \delta /2 + w)}} + {A_9}{e^{ - {h_3}(x + \delta /2 + w)}}& - 3\delta /2 - w \le x \le - \delta /2 - w\\ {A_{10}}{e^{ - {h_4}(x + 3\delta /2 + w)}}& x \le - 3\delta /2 - w \end{array} \right., $$
where A1A10 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 ɛ0 is the permittivity of vacuum. In the optical range, the surface conductivity σg of graphene can be predicted by the Kubo formula [27].), insulator slab or the outside air, respectively. The complex propagation constant is β = β1 + iβ2, where β1 is related to the effective refractive index (Neff = β1/k0), and β2 is related to the mode power attenuation (MPA) α = 20β2*log10(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 [2830]:
$$\tan ({h_1}\delta /2) = \frac{{{P_1} + {P_2}}}{{{P_3} + {P_4}}}, $$
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 (ɛ1 = 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 (Neff) of the mode as a function of µc (the chemical potential of each side graphene layer) at λ = 1.55 µm, as shown in Fig. 3:

 

Fig. 3. MPA (black line) and mode index (red dash line) of TE mode in the suspended triple-layer graphene modulator as a function of chemical potential µc of the side graphene layers.

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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 Δα 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 (Δα1). 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 (Δα2).

There are two peaks for effective refractive index (Neff) which happen at exactly the two MPA turn points respectively. The maximum ΔNeff is 0.036. Both the modulation depths and ΔNeff 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 Δα peaks happening at the optimized insulator thickness wopt, which shows the highest light-matter interaction. The maximum Δα1 are 0.286 dB/µm, 0.506 dB/µm, 0.539 dB/µm, and the corresponding wopt1 are 77.8 nm, 20.9 nm, 16.5 nm, for ɛ1 = 3.06, 16 (a possible middle value at wavelength 1.55 µm), 22 (permittivity of Ta2O5 [32] at wavelength 1.55 µm), respectively. The maximum Δα2 are 0.460 dB/µm, 0.786 dB/µm, 0.834 dB/µm, and the corresponding wopt2 are 85.6 nm, 22.4 nm, 17.6 nm, for ɛ1 = 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 µm2.

 

Fig. 4. (a) The modulation depths (both Δα1 and Δα2) change as a function of insulator thickness w at λ =1.55 µm for different dielectric: ɛ1 = 3.06, 16, and 22, respectively. (b) The highest modulation depth (black line) for different insulator permittivity ɛ1 and the corresponding optimized dielectric thickness (red line) for both modulation depth 1 and 2, respectively.

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The maximum modulation depth of Δα1 and Δα2 and the corresponding wopt1 and wopt2 are gotten as a function of ɛ1, as shown in Fig. 4 (b). These two modulation depths increase monotonously as ɛ1 increases. wopt1 and wopt2 decrease monotonously as ɛ1 increases. And wopt2 is always slightly larger than wopt1. When ɛ1 = 22, we find wopt1 = 16.5 nm and wopt2 = 17.6 nm.

For TE mode, the electric field Ey is in the plane of graphene, so the amplitude of Ey 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 ɛ1 = 22, and w = 17 nm (wopt), 5 nm, and 50 nm, as shown in Fig. 5 (a):

 

Fig. 5. (a) The mode profile as a function of coordinate x when µc = 0.3 eV at λ = 1.55 m, ɛ1 = 22, and w = 17 nm (black line), 5 nm (red dashed line) and 50 nm (blue dotted line). (b) The corresponding mode amplitude in the side graphene layers.

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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 wopt2 is always slightly larger than wopt1 (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 ɛ1 = 22, wopt = 17 nm, as shown in Fig. 6:

 

Fig. 6. The mode power transmittance of the modulator as a function of incident wavelength when µc = 0.3 eV (black line), 0.4 eV (red dashed line) and 0.5 eV (blue dotted line).

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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]:

$$\alpha = - 10\textrm{Log}[{T_{\max }}]$$
And the figure of merit (FOM) can be expressed as [14]:
$$\textrm{FOM} = \Delta \alpha /\alpha$$
We get an insertion loss of only 0.00166 dB (Tmax = 0.999618), and the FOM of this modulator is also ultrahigh ∼ 1265.

After calculation, we find that the highest ΔNeff happens at ɛ1 = 22, wopt = 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 µc = 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 Tµc) of this M-Z modulator can be expressed as follows [31]:

$$T(\Delta {\mu _c}) = \frac{1}{4} \times [\exp ( - {\alpha _0}L) + \exp ( - {\alpha _1}L) + 2\exp ( - \frac{{{\alpha _0}L + {\alpha _1}L}}{2})\cos (\Delta \phi )]$$
where $\Delta \phi = \frac{{2\pi }}{\lambda }\Delta {N_{eff}}L$, α0, α1 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 Tμc) of this M-Z modulator in Fig. 7:

 

Fig. 7. The transmittance of the M-Z modulator as a function of the difference of the side-layer chemical potential (Δµc) between two arms when L = 12 µm (black line), 34 µm (red dashed line), and 68 µm (blue dotted line).

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From Fig. 7, we can see when Δµc 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 Neff peak as shown in Fig. 3. For this ΔNeff, 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 Lmax (Lmax = 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 Δµc 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 Lmax = 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 Neff 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 = ɛ1ɛ0S/w is one capacitance of the active area of the modulator. Here we choose that S = 0.5 µm *5 µm = 2.5 µm2, 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 ɛ1 = 22, wopt = 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 ɛ1 = 3.06 (Al2O3), wopt = 80 nm, which is much larger than 559.2 GHz reported in Ref. [20].

The side-layer chemical potential μc can be tuned by applying voltage Vg:

$${V_g} = \frac{{\textrm{e}w}}{{{\pi }{\varepsilon _1}{\varepsilon _0}{\hbar ^2}{v_F}^2}}\mathop \smallint \limits_0^\infty \varsigma [{({\textrm{e}^{\frac{{\varsigma - {\mu _c}}}{{{k_B}\textrm{T}}}}} + 1)^{ - 1}} - {({\textrm{e}^{\frac{{\varsigma + {\mu _c}}}{{{k_B}\textrm{T}}}}} + 1)^{ - 1}}\textrm{]d}\varsigma , $$
where $\varsigma $ is energy. We choose an average Fermi velocity of ${v_F} = 1.5 \times {10^6}\textrm{m/s}$. When the chemical potential is changing from −1 eV to 1 eV, we calculate Eq. (9) and obtain the applied voltage as a function of side-layer chemical potential μc, as shown in Fig. 8 (a).

 

Fig. 8. (a) The applied voltage as a function of side-layer chemical potential μc: black line: ɛ1 = 10, wopt = 30 nm, red line: ɛ1 = 22, wopt = 17 nm; (b) The applied voltage and corresponding energy consumption as a function of permittivity ɛ1 of the insulator when w is at wopt and μc = 0.3 eV.

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We can see the applied voltage is always smaller than 2.3 V for ɛ1 = 22, wopt =17 nm when the chemical potential is changing from -1 eV to 1 eV. For ɛ1 = 10, wopt = 30 nm, the applied voltage should be larger (< 9.0 V).

We also have obtained the optimized (w = wopt) applied voltage and the corresponding energy consumption as a function of insulator permittivity ɛ1 when μc = 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 ɛ1 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 ɛ1 = 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 µc = 0.3 eV, however it is significant for λ = 0.6 µm and µc = 0.3 eV.

 

Fig. 9. The mode profile in the longitudinal transmission direction at different wavelength: λ = 2.5 µm (a), λ = 1.5 µm (b), λ = 0.6 µm (c) when ɛ1 = 22, μc = 0.3 eV. Simulated by COMSOL.

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We also retrieve the scattering parameter S21 as a function of the carrier wavelength, as shown in Fig. 10.

 

Fig. 10. The scattering parameter S21 as a function of wavelength at μc = 0.1 eV (black solid line), 0.2 eV (red dashed line), 0.3 eV (blue dotted line), 0.4 eV (green dashed dotted line), and 0.5 eV (pink dashed dotted dotted line). Simulated by COMSOL.

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By comparing Figs. 10 and 6, we can see the results obtained from these two different methods are matched exactly well for µc = 0.3 eV, 0.4 eV and 0.5 eV. The difference from S21 and the transmittance T is only 0.001. S21 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 S21 when µc = 0.1 eV, and 0.2 eV. There is no abrupt turn point for µc = 0.1 eV and only one for µc = 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 S21 of µc = 0.1 eV, and 0.2 eV which will cause modulation depth 2 (happen at µc = 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 µm2. The light-graphene interaction will be enhanced at the optimized insulator thickness wopt 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 (ΔNeff) 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).

References

1. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011). [CrossRef]  

2. Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016). [CrossRef]  

3. Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014). [CrossRef]  

4. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012). [CrossRef]  

5. S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012). [CrossRef]  

6. Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017). [CrossRef]  

7. X. Hu and J. Wang, “High figure of merit graphene modulator based on long-range hybrid plasmonic slot waveguide,” IEEE J. Quantum Electron. 53(3), 1–8 (2017). [CrossRef]  

8. S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014). [CrossRef]  

9. M. Fan, H. Yang, P. Zheng, G. Hu, B. Yun, and Y. Cui, “Multilayer graphene electro-absorption optical modulator based on double-stripe silicon nitride waveguide,” Opt. Express 25(18), 21619–21629 (2017). [CrossRef]  

10. R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013). [CrossRef]  

11. L. Yang, T. Hu, A. Shen, C. Pei, B. Yang, T. Dai, H. Yu, Y. Li, X. Jiang, and J. Yang, “Ultracompact optical modulator based on graphene-silica metamaterial,” Opt. Lett. 39(7), 1909–1912 (2014). [CrossRef]  

12. Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013). [CrossRef]  

13. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014). [CrossRef]  

14. J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013). [CrossRef]  

15. H. Dalir, Y. Xia, Y. Wang, and X. Zhang, “Athermal Broadband Graphene Optical Modulator with 35 GHz Speed,” ACS Photon. 3(9), 1564–1568 (2016). [CrossRef]  

16. G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015). [CrossRef]  

17. Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014). [CrossRef]  

18. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008). [CrossRef]  

19. X. Du, I. Skachko, A. Barker, and E. Andrei, “Approaching ballistic transport in suspended graphene,” Nat. Nanotechnol. 3(8), 491–495 (2008). [CrossRef]  

20. J. Liu and Y. Liu, “Ultrafast suspended self-biasing graphene modulator with ultrahigh figure of merit,” Opt. Commun. 427, 439–446 (2018). [CrossRef]  

21. M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014). [CrossRef]  

22. R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016). [CrossRef]  

23. X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009). [CrossRef]  

24. R. Mendis and D. Grischkowsky, “THz interconnect with low-loss and low-group velocity dispersion,” IEEE Microw. Wireless Compon. Lett. 11(11), 444–446 (2001). [CrossRef]  

25. A. Yariv, Optical Electronics in Modern Communications (Oxford University, 2007).

26. N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012). [CrossRef]  

27. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007). [CrossRef]  

28. J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal-clad antiresonant reflecting hollow waveguides,” Appl. Opt. 54(14), 4549–4555 (2015). [CrossRef]  

29. J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal film coated double-dielectric-slab waveguides and the tunable filter application,” Opt. Commun. 351, 103–108 (2015). [CrossRef]  

30. J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave in double-metal-film waveguides and its application of analysis wavelength,” Appl. Opt. 54(28), 8406–8411 (2015). [CrossRef]  

31. C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012). [CrossRef]  

32. C.-C. Lee, S. Suzuki, W. Xie, and T. R. Schibli, “Broadband graphene electro-optic modulators with sub-wavelength thickness,” Opt. Express 20(5), 5264–5269 (2012). [CrossRef]  

33. G. D. Wilk, R. M. Wallace, and J. M. Anthony, “High-κ gate dielectrics: Current status and materials properties considerations,” J. Appl. Phys. 89(10), 5243–5275 (2001). [CrossRef]  

34. C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015). [CrossRef]  

35. F. Xia, V. Perebeinos, Y. Lin, Y. Wu, and P. Avouris, “The origins and limits of metal–graphene junction resistance,” Nat. Nanotechnol. 6(3), 179–184 (2011). [CrossRef]  

36. G. Kovacevica and S. Yamashita, “Design optimizations for a high-speed two-layer graphene optical modulator on silicon,” IEICE Electron. Express 13(14), 20160499 (2016). [CrossRef]  

References

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  • |

  1. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
    [Crossref]
  2. Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
    [Crossref]
  3. Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
    [Crossref]
  4. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
    [Crossref]
  5. S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
    [Crossref]
  6. Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
    [Crossref]
  7. X. Hu and J. Wang, “High figure of merit graphene modulator based on long-range hybrid plasmonic slot waveguide,” IEEE J. Quantum Electron. 53(3), 1–8 (2017).
    [Crossref]
  8. S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014).
    [Crossref]
  9. M. Fan, H. Yang, P. Zheng, G. Hu, B. Yun, and Y. Cui, “Multilayer graphene electro-absorption optical modulator based on double-stripe silicon nitride waveguide,” Opt. Express 25(18), 21619–21629 (2017).
    [Crossref]
  10. R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
    [Crossref]
  11. L. Yang, T. Hu, A. Shen, C. Pei, B. Yang, T. Dai, H. Yu, Y. Li, X. Jiang, and J. Yang, “Ultracompact optical modulator based on graphene-silica metamaterial,” Opt. Lett. 39(7), 1909–1912 (2014).
    [Crossref]
  12. Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013).
    [Crossref]
  13. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
    [Crossref]
  14. J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
    [Crossref]
  15. H. Dalir, Y. Xia, Y. Wang, and X. Zhang, “Athermal Broadband Graphene Optical Modulator with 35 GHz Speed,” ACS Photon. 3(9), 1564–1568 (2016).
    [Crossref]
  16. G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
    [Crossref]
  17. Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
    [Crossref]
  18. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
    [Crossref]
  19. X. Du, I. Skachko, A. Barker, and E. Andrei, “Approaching ballistic transport in suspended graphene,” Nat. Nanotechnol. 3(8), 491–495 (2008).
    [Crossref]
  20. J. Liu and Y. Liu, “Ultrafast suspended self-biasing graphene modulator with ultrahigh figure of merit,” Opt. Commun. 427, 439–446 (2018).
    [Crossref]
  21. M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
    [Crossref]
  22. R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
    [Crossref]
  23. X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
    [Crossref]
  24. R. Mendis and D. Grischkowsky, “THz interconnect with low-loss and low-group velocity dispersion,” IEEE Microw. Wireless Compon. Lett. 11(11), 444–446 (2001).
    [Crossref]
  25. A. Yariv, Optical Electronics in Modern Communications (Oxford University, 2007).
  26. N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
    [Crossref]
  27. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
    [Crossref]
  28. J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal-clad antiresonant reflecting hollow waveguides,” Appl. Opt. 54(14), 4549–4555 (2015).
    [Crossref]
  29. J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal film coated double-dielectric-slab waveguides and the tunable filter application,” Opt. Commun. 351, 103–108 (2015).
    [Crossref]
  30. J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave in double-metal-film waveguides and its application of analysis wavelength,” Appl. Opt. 54(28), 8406–8411 (2015).
    [Crossref]
  31. C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
    [Crossref]
  32. C.-C. Lee, S. Suzuki, W. Xie, and T. R. Schibli, “Broadband graphene electro-optic modulators with sub-wavelength thickness,” Opt. Express 20(5), 5264–5269 (2012).
    [Crossref]
  33. G. D. Wilk, R. M. Wallace, and J. M. Anthony, “High-κ gate dielectrics: Current status and materials properties considerations,” J. Appl. Phys. 89(10), 5243–5275 (2001).
    [Crossref]
  34. C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
    [Crossref]
  35. F. Xia, V. Perebeinos, Y. Lin, Y. Wu, and P. Avouris, “The origins and limits of metal–graphene junction resistance,” Nat. Nanotechnol. 6(3), 179–184 (2011).
    [Crossref]
  36. G. Kovacevica and S. Yamashita, “Design optimizations for a high-speed two-layer graphene optical modulator on silicon,” IEICE Electron. Express 13(14), 20160499 (2016).
    [Crossref]

2018 (1)

J. Liu and Y. Liu, “Ultrafast suspended self-biasing graphene modulator with ultrahigh figure of merit,” Opt. Commun. 427, 439–446 (2018).
[Crossref]

2017 (3)

Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
[Crossref]

X. Hu and J. Wang, “High figure of merit graphene modulator based on long-range hybrid plasmonic slot waveguide,” IEEE J. Quantum Electron. 53(3), 1–8 (2017).
[Crossref]

M. Fan, H. Yang, P. Zheng, G. Hu, B. Yun, and Y. Cui, “Multilayer graphene electro-absorption optical modulator based on double-stripe silicon nitride waveguide,” Opt. Express 25(18), 21619–21629 (2017).
[Crossref]

2016 (4)

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

H. Dalir, Y. Xia, Y. Wang, and X. Zhang, “Athermal Broadband Graphene Optical Modulator with 35 GHz Speed,” ACS Photon. 3(9), 1564–1568 (2016).
[Crossref]

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
[Crossref]

G. Kovacevica and S. Yamashita, “Design optimizations for a high-speed two-layer graphene optical modulator on silicon,” IEICE Electron. Express 13(14), 20160499 (2016).
[Crossref]

2015 (5)

C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal-clad antiresonant reflecting hollow waveguides,” Appl. Opt. 54(14), 4549–4555 (2015).
[Crossref]

J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal film coated double-dielectric-slab waveguides and the tunable filter application,” Opt. Commun. 351, 103–108 (2015).
[Crossref]

J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave in double-metal-film waveguides and its application of analysis wavelength,” Appl. Opt. 54(28), 8406–8411 (2015).
[Crossref]

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

2014 (6)

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

L. Yang, T. Hu, A. Shen, C. Pei, B. Yang, T. Dai, H. Yu, Y. Li, X. Jiang, and J. Yang, “Ultracompact optical modulator based on graphene-silica metamaterial,” Opt. Lett. 39(7), 1909–1912 (2014).
[Crossref]

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref]

S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014).
[Crossref]

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

2013 (3)

R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
[Crossref]

J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
[Crossref]

Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013).
[Crossref]

2012 (5)

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref]

S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
[Crossref]

C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
[Crossref]

C.-C. Lee, S. Suzuki, W. Xie, and T. R. Schibli, “Broadband graphene electro-optic modulators with sub-wavelength thickness,” Opt. Express 20(5), 5264–5269 (2012).
[Crossref]

N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
[Crossref]

2011 (2)

F. Xia, V. Perebeinos, Y. Lin, Y. Wu, and P. Avouris, “The origins and limits of metal–graphene junction resistance,” Nat. Nanotechnol. 6(3), 179–184 (2011).
[Crossref]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref]

2009 (1)

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

2008 (2)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

X. Du, I. Skachko, A. Barker, and E. Andrei, “Approaching ballistic transport in suspended graphene,” Nat. Nanotechnol. 3(8), 491–495 (2008).
[Crossref]

2007 (1)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

2001 (2)

R. Mendis and D. Grischkowsky, “THz interconnect with low-loss and low-group velocity dispersion,” IEEE Microw. Wireless Compon. Lett. 11(11), 444–446 (2001).
[Crossref]

G. D. Wilk, R. M. Wallace, and J. M. Anthony, “High-κ gate dielectrics: Current status and materials properties considerations,” J. Appl. Phys. 89(10), 5243–5275 (2001).
[Crossref]

Absil, P.

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Ajayan, P. M.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

An, J.

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Andrei, E.

X. Du, I. Skachko, A. Barker, and E. Andrei, “Approaching ballistic transport in suspended graphene,” Nat. Nanotechnol. 3(8), 491–495 (2008).
[Crossref]

Anthony, J. M.

G. D. Wilk, R. M. Wallace, and J. M. Anthony, “High-κ gate dielectrics: Current status and materials properties considerations,” J. Appl. Phys. 89(10), 5243–5275 (2001).
[Crossref]

Asselberghs, I.

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Avouris, P.

F. Xia, V. Perebeinos, Y. Lin, Y. Wu, and P. Avouris, “The origins and limits of metal–graphene junction resistance,” Nat. Nanotechnol. 6(3), 179–184 (2011).
[Crossref]

Banerjee, S. K.

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Bao, J.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

Barker, A.

X. Du, I. Skachko, A. Barker, and E. Andrei, “Approaching ballistic transport in suspended graphene,” Nat. Nanotechnol. 3(8), 491–495 (2008).
[Crossref]

Bolotin, K. I.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

Boltasseva, A.

N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
[Crossref]

Bozhevolnyi, S. I.

Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
[Crossref]

Brems, S.

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Cai, W.

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Campenhout, J. V.

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Capasso, F.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Cardenas, J.

C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Chen, B.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

Chen, H.

R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
[Crossref]

Chen, Y. P.

N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
[Crossref]

Chung, T. F.

N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
[Crossref]

Colombo, L.

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Cui, Y.

Dai, T.

Dalir, H.

H. Dalir, Y. Xia, Y. Wang, and X. Zhang, “Athermal Broadband Graphene Optical Modulator with 35 GHz Speed,” ACS Photon. 3(9), 1564–1568 (2016).
[Crossref]

Davies, A. G.

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Ding, Y.

Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
[Crossref]

Du, W.

R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
[Crossref]

Du, X.

X. Du, I. Skachko, A. Barker, and E. Andrei, “Approaching ballistic transport in suspended graphene,” Nat. Nanotechnol. 3(8), 491–495 (2008).
[Crossref]

Emani, N. K.

N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
[Crossref]

Fallahi, A.

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

Fan, M.

Fang, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

Fang, Z.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

Feng, M.

Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013).
[Crossref]

Fudenberg, G.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

Geng, B.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref]

Gosciniak, J.

J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
[Crossref]

Grischkowsky, D.

R. Mendis and D. Grischkowsky, “THz interconnect with low-loss and low-group velocity dispersion,” IEEE Microw. Wireless Compon. Lett. 11(11), 444–446 (2001).
[Crossref]

Guan, X.

Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
[Crossref]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Halas, N. J.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

Hao, R.

R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
[Crossref]

Harada, Y.

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
[Crossref]

Hibino, H.

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
[Crossref]

Hone, J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

Hori, Y.

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
[Crossref]

Hu, G.

Hu, H.

Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
[Crossref]

Hu, T.

Hu, X.

X. Hu and J. Wang, “High figure of merit graphene modulator based on long-range hybrid plasmonic slot waveguide,” IEEE J. Quantum Electron. 53(3), 1–8 (2017).
[Crossref]

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Hu, Y.

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Hu, Z.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

Huyghebaert, C.

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Javier García de Abajo, F.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

Jiang, W.-S.

Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013).
[Crossref]

Jiang, X.

Jiang, Z.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

Jin, X.

R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
[Crossref]

Jin, Y.

Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref]

Jung, I.

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Kats, M. A.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref]

Kildishev, A. V.

N. K. Emani, T. F. Chung, X. J. Ni, A. V. Kildishev, Y. P. Chen, and A. Boltasseva, “Electrically Tunable Damping of Plasmonic Resonances with Graphene,” Nano Lett. 12(10), 5202–5206 (2012).
[Crossref]

Kim, P.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

Kim, S.

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Klima, M.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
[Crossref]

Kobayashi, Y.

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
[Crossref]

Koester, S. J.

S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
[Crossref]

Kong, J.

Y. Yao, R. Shankar, M. A. Kats, Y. Song, J. Kong, M. Loncar, and F. Capasso, “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014).
[Crossref]

Kou, R.

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
[Crossref]

Kovacevica, G.

G. Kovacevica and S. Yamashita, “Design optimizations for a high-speed two-layer graphene optical modulator on silicon,” IEICE Electron. Express 13(14), 20160499 (2016).
[Crossref]

Lee, C.-C.

Lee, Y. H. D.

C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Li, E.

R. Hao, W. Du, H. Chen, X. Jin, L. Yang, and E. Li, “Ultra-compact optical modulator by graphene induced electro-refraction effect,” Appl. Phys. Lett. 103(6), 061116 (2013).
[Crossref]

Li, L. H.

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Li, M.

S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
[Crossref]

Li, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

Li, X.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref]

X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science 324(5932), 1312–1314 (2009).
[Crossref]

Li, Y.

Liang, G.

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Liang, H.

Liang, H. K.

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Lin, Y.

F. Xia, V. Perebeinos, Y. Lin, Y. Wu, and P. Avouris, “The origins and limits of metal–graphene junction resistance,” Nat. Nanotechnol. 6(3), 179–184 (2011).
[Crossref]

Linfield, E. H.

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Lipson, M.

C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Liu, J.

J. Liu and Y. Liu, “Ultrafast suspended self-biasing graphene modulator with ultrahigh figure of merit,” Opt. Commun. 427, 439–446 (2018).
[Crossref]

J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave in double-metal-film waveguides and its application of analysis wavelength,” Appl. Opt. 54(28), 8406–8411 (2015).
[Crossref]

J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal film coated double-dielectric-slab waveguides and the tunable filter application,” Opt. Commun. 351, 103–108 (2015).
[Crossref]

J. Liu, H. Liang, M. Zhang, and H. Su, “THz wave transmission within the metal-clad antiresonant reflecting hollow waveguides,” Appl. Opt. 54(14), 4549–4555 (2015).
[Crossref]

Liu, M.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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Liu, W.

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H. Dalir, Y. Xia, Y. Wang, and X. Zhang, “Athermal Broadband Graphene Optical Modulator with 35 GHz Speed,” ACS Photon. 3(9), 1564–1568 (2016).
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Zhou, W.-Y.

Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013).
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Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, N. A. Mortensen, and S. Xiao, “Efficient graphene based electro-optical modulator enabled by interfacing plasmonic slot and silicon waveguides,” Nanoscale 9(40), 15576–15581 (2017).
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Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. Javier García de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
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ACS Photon. (2)

H. Dalir, Y. Xia, Y. Wang, and X. Zhang, “Athermal Broadband Graphene Optical Modulator with 35 GHz Speed,” ACS Photon. 3(9), 1564–1568 (2016).
[Crossref]

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated Terahertz Graphene Modulator with 100% Modulation Depth,” ACS Photon. 2(11), 1559–1566 (2015).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

R. Kou, Y. Hori, T. Tsuchizawa, K. Warabi, Y. Kobayashi, Y. Harada, H. Hibino, T. Yamamoto, H. Nakajima, and K. Yamada, “Ultra-fine metal gate operated graphene optical intensity modulator,” Appl. Phys. Lett. 109(25), 251101 (2016).
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S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
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IEEE J. Quantum Electron. (1)

X. Hu and J. Wang, “High figure of merit graphene modulator based on long-range hybrid plasmonic slot waveguide,” IEEE J. Quantum Electron. 53(3), 1–8 (2017).
[Crossref]

IEEE Microw. Wireless Compon. Lett. (1)

R. Mendis and D. Grischkowsky, “THz interconnect with low-loss and low-group velocity dispersion,” IEEE Microw. Wireless Compon. Lett. 11(11), 444–446 (2001).
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IEICE Electron. Express (1)

G. Kovacevica and S. Yamashita, “Design optimizations for a high-speed two-layer graphene optical modulator on silicon,” IEICE Electron. Express 13(14), 20160499 (2016).
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J. Appl. Phys. (1)

G. D. Wilk, R. M. Wallace, and J. M. Anthony, “High-κ gate dielectrics: Current status and materials properties considerations,” J. Appl. Phys. 89(10), 5243–5275 (2001).
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J. Phys. Condens. Matter (1)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
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Laser Photon. Rev. (1)

Y. Hu, M. Pantouvaki, J. V. Campenhout, S. Brems, I. Asselberghs, C. Huyghebaert, P. Absil, and D. V. Thourhout, “Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon,” Laser Photon. Rev. 10(2), 307–316 (2016).
[Crossref]

Laser Phys. Lett. (1)

Z.-B. Liu, M. Feng, W.-S. Jiang, W. Xin, P. Wang, Q.-W. Sheng, Y.-G. Liu, D. N. Wang, W.-Y. Zhou, and J.-G. Tian, “Broadband all-optical modulation using a graphene-covered-microfiber,” Laser Phys. Lett. 10(6), 065901 (2013).
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Nano Lett. (5)

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Figures (10)

Fig. 1.
Fig. 1. (a) The structure of suspended triple-layer graphene waveguide modulator. (b) The movement of carriers when the triple-layer graphene modulator is biased. (c) The equivalent circuit of the suspended triple-layer graphene modulator.
Fig. 2.
Fig. 2. Shows the fabrication steps of the device. The fabrication starts from the silicon waveguide with a trench in the middle caused by etching, as shown in (a). In (b), a prepared graphene sheet is mechanically transferred. (c) Deposit of a thick insulator. (d) Mechanically transfer the middle graphene layer. (e) Deposit of the other thick insulator. (f) Mechanically transfer the up side graphene layer.
Fig. 3.
Fig. 3. MPA (black line) and mode index (red dash line) of TE mode in the suspended triple-layer graphene modulator as a function of chemical potential µc of the side graphene layers.
Fig. 4.
Fig. 4. (a) The modulation depths (both Δα1 and Δα2) change as a function of insulator thickness w at λ =1.55 µm for different dielectric: ɛ1 = 3.06, 16, and 22, respectively. (b) The highest modulation depth (black line) for different insulator permittivity ɛ1 and the corresponding optimized dielectric thickness (red line) for both modulation depth 1 and 2, respectively.
Fig. 5.
Fig. 5. (a) The mode profile as a function of coordinate x when µc = 0.3 eV at λ = 1.55 m, ɛ1 = 22, and w = 17 nm (black line), 5 nm (red dashed line) and 50 nm (blue dotted line). (b) The corresponding mode amplitude in the side graphene layers.
Fig. 6.
Fig. 6. The mode power transmittance of the modulator as a function of incident wavelength when µc = 0.3 eV (black line), 0.4 eV (red dashed line) and 0.5 eV (blue dotted line).
Fig. 7.
Fig. 7. The transmittance of the M-Z modulator as a function of the difference of the side-layer chemical potential (Δµc) between two arms when L = 12 µm (black line), 34 µm (red dashed line), and 68 µm (blue dotted line).
Fig. 8.
Fig. 8. (a) The applied voltage as a function of side-layer chemical potential μc: black line: ɛ1 = 10, wopt = 30 nm, red line: ɛ1 = 22, wopt = 17 nm; (b) The applied voltage and corresponding energy consumption as a function of permittivity ɛ1 of the insulator when w is at wopt and μc = 0.3 eV.
Fig. 9.
Fig. 9. The mode profile in the longitudinal transmission direction at different wavelength: λ = 2.5 µm (a), λ = 1.5 µm (b), λ = 0.6 µm (c) when ɛ1 = 22, μc = 0.3 eV. Simulated by COMSOL.
Fig. 10.
Fig. 10. The scattering parameter S21 as a function of wavelength at μc = 0.1 eV (black solid line), 0.2 eV (red dashed line), 0.3 eV (blue dotted line), 0.4 eV (green dashed dotted line), and 0.5 eV (pink dashed dotted dotted line). Simulated by COMSOL.

Equations (9)

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1 R t o t a l = 1 1.5 R + 1 1.5 R = 4 3 R
C t o t a l = 2 C
f 3dB = 1 2 π R total C total = 1 3 π R C
E y ( x ) = { A 10 e h 4 ( x 3 δ / 2 w ) x 3 δ / 2 + w A 6 e h 3 ( x δ / 2 w ) + A 7 e h 3 ( x δ / 2 w ) δ / 2 + w x 3 δ / 2 + w A 2 cos h 2 ( x δ / 2 ) + A 3 sin h 2 ( x δ / 2 ) δ / 2 x δ / 2 + w A 1 cos ( h 1 x ) δ / 2 x δ / 2 A 4 cos h 2 ( x + δ / 2 ) + A 5 sin h 2 ( x + δ / 2 ) δ / 2 w x δ / 2 A 8 e h 3 ( x + δ / 2 + w ) + A 9 e h 3 ( x + δ / 2 + w ) 3 δ / 2 w x δ / 2 w A 10 e h 4 ( x + 3 δ / 2 + w ) x 3 δ / 2 w ,
tan ( h 1 δ / 2 ) = P 1 + P 2 P 3 + P 4 ,
α = 10 Log [ T max ]
FOM = Δ α / α
T ( Δ μ c ) = 1 4 × [ exp ( α 0 L ) + exp ( α 1 L ) + 2 exp ( α 0 L + α 1 L 2 ) cos ( Δ ϕ ) ]
V g = e w π ε 1 ε 0 2 v F 2 0 ς [ ( e ς μ c k B T + 1 ) 1 ( e ς + μ c k B T + 1 ) 1 ]d ς ,

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