Graphene Embedded Modulator with Extremely Small Footprint and High Modulation Efficiency

By embedding graphene sheet in the silicon waveguide, the overall effective mode index displays unexpected symmetry and the electrorefraction effect has been significantly enhanced near the epsilon-near-zero point. An eight-layer graphene embeddedMachZehnderModulator has been theoretically demonstratedwith the advantage of ultracompact footprint (4× 2 μm), highmodulation efficiency (1.316V⋅μm), ultrafast modulation speed, and large extinction ratio. Our results may promote various on-chip active components, boosting the utilization of graphene in optical applications.


Introduction
Electrooptical modulator, which transfers electronic signals into high bit-rate photonic data, is the key component in onchip interconnection and integrated optoelectronic circuits [1].By applying an electric field to a material, the real and imaginary part of the refractive indices can be changed.A change in the real part of the refractive index caused by the applied voltage is known as the electrorefraction (ER) effect, whereas a change in the imaginary part of the refractive index is known as the electroabsorption (EA) effect [1].However, these two effects are too weak in pure silicon at the communication wavelengths so that it usually needs an extremely large length to reach the required modulation; for example, a 50 Gbit/s modulator has the length of 1 millimeter [2].The large footprint of optical modulator makes it impossible to be integrated into a single chip.To fill the demands of next generation on-chip communication, minimizing the size and improving the speed of modulator become the urgent goals but remain a challenge.Recently, graphene-based modulators have attracted much attention due to their unprecedented ability to enhance the material's EA effect; thus, they can greatly reduce the modulator length to achieve the same effect [2][3][4][5].In [5], Lu and Zhao reported a graphene embedded modulator having a length of only 1 m.However, limited by the inbuilt drawback from the EA modulator, the extinction ratio is low and background noise cannot be ignored.
In this paper, we point out that graphene can also have significant enhancement to the ER effect of the background material.By embedding duplicated graphene layers into a silicon substrate, we demonstrated that the variation of effective mode index Δ eff can be as large as 0.4057 corresponding to a Mach-Zender modulation arm length of only 1.9 m.Note that Δ eff caused by ER effect is only at the level of 10 −4 for conventional semiconductor modulators [6][7][8].To our knowledge, this is the largest Δ eff and smallest modulation arm length ever reported.In addition, our proposed ER modulator has the advantage of large bandwidth, high modulation speed, large extinction ratio, as well as low noise.

The Symmetry of Refractive Index
Graphene is a single atom thick two-dimensional carbon sheet, whose permittivity can be actively tuned by the chemical potentials (the Fermi level).Figure 1 shows the permittivity  of an infinite graphene sheet obtained from the Kubo formula [9] at the wavelength  = 1550 nm (corresponding photon energy ℎ = 0.802 eV).It can be seen that there is a dip in the curve of permittivity magnitude, and the epsilon-near-zero (ENZ) point is obtained at the chemical potential  0 = 0.513 eV where the absolute value of epsilon is approaching zero.When the chemical potential   <  0 , both the real and imaginary part of the permittivity (Re() and Im()) have the positive sign so that the graphene layer behaves like a dielectric material.When   > 0.52 eV, Re() becomes negative and Im() is approaching zero, this means that the graphene sheet acts like a metallic layer, and it is then fabulous to transfer surface plasmons [10].When the chemical potential is gradually increased, the "dielectric graphene" is gradually transforming into "metallic graphene" at the ENZ point  0 = 0.513 eV; therefore, it can be called as the transition chemical potential.It should be noted that there exists a range (0.48 eV <  < 0.52 eV), where both Re() and Im() are very close to zero so that the ENZ effect appears, and the graphene layer exhibits unexpected new properties [11,12], propelling various active optoelectronic devices [12].Moreover, we have checked the performance of the entire telecom-band (C/L/O/E-band), and it turns out that the ENZ point exists at each telecom band.We choose the particular wavelength around 1550 nm, because it is a region of low loss for optical fibers and therefore of interest for optical communication systems.In the following, we will focus on this special area to configure an optical modulator.
To enhance the graphene-light interaction, the structure with the graphene layer embedded in the center of the silicon waveguide is considered since the rectangle waveguide tends to concentrate light in the center.By applying a drive voltage upon the waveguide, the chemical potential of the graphene layer can be modified, which in turn affects the permittivity of graphene layer.Therefore, the permittivity of the overall waveguide will be influenced, and the effective modal index ( eff ) of the whole waveguide will be also changed.From the previous analysis, it is shown that there is a special ENZ point at  0 = 0.513 eV, where the "epsilon-near-zero" effect takes place (both Re() and Im() of graphene are approaching zero).In Figure 2, the real and imaginary part of effective mode index for the proposed waveguide in the ENZ area are investigated through the rigorous mode analysis from finite element package COMSOL.When   is close to  0 , the  eff does change drastically as expected, inferring the corresponding mode properties that change a lot when the embedded graphene layer transfers from its dielectric state to metallic state.Furthermore, for such silicon-graphenesilicon waveguide, both Re() and Im() have unexpected symmetries towards the ENZ point.As depicted in Figures 2(a)-2(c), Re() displays an odd-symmetry towards the ENZ point, while Im() displays an even-symmetry.As the Im() refers to the propagation loss, the waveguide is expected to have the largest loss at the ENZ point, while there is less propagation loss at the edge side of the curve.The strong variation of loss suggests the chance to build EA modulator, as already reported in [5].However, the properties of oddsymmetry curve for Re() have been barely explored before.There is a significant increase of Re() in the ENZ area.The variation amount for Re() can be as large as 0.81 for eight layers of graphene embedded waveguide in Figure 2(a).Please note that for doped silicon, the Δ eff is very weak (at the level of 10 −4 ) [6][7][8].This means the weak ER effect has been improved by 3 orders of magnitude after multilayer graphene sheets are embedded in the silicon waveguide.The impressive value of large  eff variation is greatly appreciated for the modulators based on ER effect, such as Mach-Zehnder modulator.It can be inferred that the number of graphene layers does not influence the scope of the even and odd symmetry distribution, but can affect the Δ eff amount.The more the numbers of graphene layers there are, the more the enhancements that can be produced and the larger the index variation that can be observed.

The Modulator Configuration.
To configure a MZI modulator, two pairs of modes (modes A and D and modes B and C) are chosen in Figure 2(a) due to their large Δ eff while maintain roughly the same loss.Modes B and C refer to the largest Δ eff , but the propagation losses of the two modes are 16.37 dB/m and 14.24 dB/m following the equation  = 8.86 * Im( eff ), which are larger than the maximum allowed loss (corresponding to the energy decay to 1/e of its initial value, which will be discussed later).Modes A and D have the relatively large Δ eff = 0.4057 but simultaneously have relatively low losses.The losses for modes A and D are 2.24 dB/m and 2.19 dB/m, respectively, which are smaller than the maximum allowed loss.
The normalized electric field distribution for mode A is depicted in Figure 2(b) in 2D form and also in Figure 2(d) in 3D form.In Figure 2(d), there is a Gaussian-like field distribution inside the waveguide-the maximum fields are achieved in the center of the waveguide but decay exponentially away from the center, while the minimum fields are at the edges of the waveguide.One important feature is that the fields in the graphene layers are much stronger Max. (d) Min. Max.
(e) than the fields in the background, as depicted in the insert of Figure 2(b).This is because the refractive index for the graphene layer at   = 0.478 eV is 1.207, which is smaller than the index of background silicon ( = 3.45).Therefore, the silicon-graphene-silicon configuration forms a nanoscale slot waveguide that the magnitude of the electric field is much larger in the low index media [13].On the other hand, Figures 2(c) and 2(e) show the normalized electric field distribution for mode D in 2D and 3D forms, respectively.In Figure 2(e), it can be seen that the fields are not only concentrate in the center but also appear at the edges of the waveguide.Moreover, when   = 0.545 eV, the epsilon of graphene layer is −1.102,where the graphene layer starts to behave like a metallic layer.Therefore, the edge effect appears and the corresponding mode is quite strong at the edge of the waveguide which is the sign of Surface Plasmon Polariton (SPP) mode [14].In short summary, the proposed 8-layer graphene waveguide has two operation states.When the chemical potential is smaller than the ENZ point, light is mostly confined inside the graphene layers, and the whole waveguide is a multislot waveguide which has a hybrid dielectric mode.This kind of modes propagates relatively fast, thus it has relatively lower index.When the chemical potential is larger than  0 , light is concentrated in the center and edge of the waveguide and the whole waveguide has a multihybrid SPP mode.This kind of SPP mode propagates relatively slow, and thus the corresponding mode index is larger.By controlling our waveguide shifts between these two kinds of modes, a large index difference as large as 0.4057 can be produced.

Figure of Merits.
The schematic picture for the proposed MZI modulator is shown in Figures 3(a) and 3(b).In practice, the graphene sheet can be grown in CVD, and transferred on the silicon-on-insulator substrate.The modulation arm has the width of 450 nm, and the distance between the two arms is 1.1 m.To maintain the large Δ eff , the chemical potentials   of the two operation states can be chosen according to the ENZ point: one should be smaller than the transition chemical potential (corresponding to the ENZ point), whereas the other should be larger than the transition chemical potential.These two   should have roughly the same interval distance d to the transition chemical potential  0 .Figure 3(c) investigates the influence of the interval distance between the two chemical potentials (Δ  = 2) to the modulator performance.The critical applied voltage   −  Dirac can be obtained through the following: where   = 0.9 × 10 6 m/s is the Fermi velocity and  = 9 × 10 16 m −2 V −1 is estimated from a single capacitor model.Since the voltage offset caused by natural doping  Dirac is a finite number (e.g., 0.8 V in [3]),   − Dirac can be considered as the applied voltage for simplicity [15].  is the applied voltage to achieve  phase shift which is proportional with the interval distance Δ  .The arm length to achieve  phase shift can be calculated through  = Δ eff ×   which first decreases then increases as Δ  increases.There is an inflection point at Δ  = 0.02, where   and   achieve minimum.This situation corresponds to the points B and C in Figure 2(a).
As explained before, points B and C correspond to the largest loss, and thus cannot be taken here.The information of propagation loss can be evaluated by the maximum allowed length  max , indicating the length where the energy decays to 1/ of its original value.The information of  max is also plotted in Figure 3(c), showing a cross point with the   curve at Δ  = 0.067.We would focus our attention on the condition that   <  max in which case the waveguide has enough power at the output.Under this condition, the highest modulation efficiency     would be achieved at the cross point Δ  = 0.067 that require smallest   and   to achieve  phase shift.Let us now consider  1 =0.478 eV for the "on" state and  2 = 0.545 eV for the "off " state; these two points have relatively low loss but a large Δ eff at 0.4057.Thus the arm length of a MZI modulator to achieve a  phase shift is only 1.9 m according to Δ = Δ eff × , which is three orders of magnitude smaller than the present reported value [6][7][8].The corresponding required applied voltages should be 2.3 V for the "on" state and 3.0 V for the "off " state, indicating a small tuning voltage   of 0.7 V.The modulation efficiency is thus 1.316 V⋅m, which is much smaller if compared with recent value in [16].Furthermore, taking into account of the electrode width of 1 m on each arm side, the overall width of the device is 4 m, and thus the footprint of our proposed modulator is only 4 × 2 m 2 , which is smaller if compared with recent phase modulator [15].This smallest size as well as the CMOS compactable structure indicates its valuable capability to be integrated into photonic circuits in a single chip.
The modulation bandwidth is usually characterized by the cutoff frequency (or the 3 dB bandwidth) which is defined by the frequency when the modulation is reduced to 50% of its maximum value.Unlike the semiconductor modulator which has speed limitation posed by the minority carrier lifetime, the graphene modulator has no carrier limitation.Therefore, it can be simply evaluated by the RC delay  3 dB = 1/2RC, where the effective resistance is 5 Ω and the effective capacitance is 128 fF.Thus, the predicted modulation bandwidth of our eight-layer graphene modulator is 250 GHz.It should be noted that the most recent reported bandwidth of pure silicon modulator is only 50 GHz [15] and the limitation of silicon is 485 GHz [17].Also, the modulation speed is estimated at about 500 Gbit/s through the expression  = 2 W (binary element).
One significant advantage for MZI modulator is that the extinction ratio is large if compared with other modulators [18].In principle, the  phase shift between two arms in MZI modulator leads to absolute "0" output for the "off " state.The absolute "0" output requires the light pulses in the two arms to have the exact same intensity but the opposite sign.These requirements can be satisfied in our modulator; due to the even symmetry for Im(), propagation loss in two arm can be equal, thus light intensities are equal to each other; while the phase different between the two arms Δ = Δ eff ×  can be designed to  as long as we take appropriate   , indicating light at the two arms have opposite signs.Even at the unoptimized situation for points A and D in Figure 2(c) with  1 = 0.478 eV and  2 = 0.545 eV, the extinction ratio still can reach 19.64 dB.We should mention that a much higher extinction ratio can be achieved as long as  1 and  2 are chosen with exactly the same Im( eff ).
In conclusion, by embedding graphene sheet into the silicon waveguide, the Re( eff ) and Im( eff ) display unexpected symmetries towards the ENZ point and the ER effect has been significantly enhanced near the ENZ point.An eight-layer graphene sheets embedded modulator has been proposed based on this effect, with the advantage of ultracompact footprint (2 × 4 m 2 ), extremely short arm length (1.9 m), low drive voltage (3 V), high modulation efficiency (1.316 V⋅m), large modulation bandwidth (250 GHz), as well as high extinction ratio.The proposed graphene modulator has great potentials for future active components, showing significant influence for optical interconnects in future integrated optoelectronic systems.The flexibility of graphene sheets may enable radically different photonic devices.The high modulation efficiency as well as ultrafast speed modulation suggests the valuable perspective for utilizing graphene optics in nanophotonic circuits.

Figure 1 :
Figure 1: The permittivity of an infinite graphene sheet: the green/red/black-dot curve indicates the real part/imaginary part and the absolute value of permittivity.The inset picture indicates an infinite graphene sheet.The chemical potential varies from 0 to 1 eV, and the wavelength is fixed at 1550 nm.

Figure 2 :
Figure 2: The real and imaginary part of effective modal index variation under different chemical potentials for graphene-embedded waveguide: (a) silicon waveguide with one layer graphene embedded, (b) silicon waveguide with two-layer graphene embedded, and (c) silicon waveguide with eight-layer graphene embedded.The insert pictures in (a), (b), and (c) show the schematics for waveguide structures. 0 = 0.513 eV corresponds to the epsilon-near-zero point.The insert pictures of (d) and (e) are the 3D distribution of  field-intensity of  0 = 0.513 eV which corresponds to the ENZ point.

Figure 3 :
Figure 3: The schematic picture of the MZI modulator is shown in (a) and (b).(c), the permittivity of an infinite graphene sheet: the green/red/black-dot curves indicate the real part/imaginary part, and the absolute value of permittivity.The inset picture indicates an infinite graphene sheet.The chemical potential varies from 0 to 1 eV, and the wavelength is fixed at 1550 nm.