# A 130 GHz Electro-Optic Ring Modulator with Double-Layer Graphene

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle of Operation

_{crit}= t, no power is transmitted; whereas, when α→1, almost unitary transmission is obtained (at the wavelength of the resonance). The rapid control of α is precisely what one can obtain with graphene, and this is the operation principle that we want to investigate.

_{i}O

_{2}substrate (relative index of 1.444) is shown in Figure 2. The radius of the ring R is 1.2 μm. The gap between the straight waveguide and the ring waveguide g is 80 nm. The width and height of the waveguide are 400 nm and 340 nm, respectively. The graphene-on-graphene structure is placed on the top of the silicon waveguide (relative index of 3.477). The average length of graphene L is 3.5 μm. They are separated by a thin layer of 10 nm Al

_{2}O

_{3}(relative index of 1.765). This two-layer graphene is connected to the P

_{d}electrodes with a width of 100 nm. We assume that the electrodes are far away from the silicon waveguide, so that they will not influence the optical mode.

## 3. Relative Index of Graphene

_{0}is the offset voltage caused by doping, ε

_{0}is the dielectric constant of the vacuum, ε

_{r}is the relative permittivity of the medium, e is the electron charge, and d is the thickness of Al

_{2}O

_{3}between the two-layer graphene.

_{F}is the Fermi velocity.

_{total}is the total optical conductivity of graphene, consisting of inter-band σ′

_{inter}, σ″

_{inter}, and intra-band σ

_{intra}contributions as

_{0}is 60.8 μS, the angular frequency ω is 0.8 eV (corresponding to 1550 nm), T = 296 K, the inter-band conductivity τ

_{1}is 1.2 ps, and the intra-band conductivity τ

_{2}is 10 fs. The conductivity of graphene is calculated as a function of the Fermi level E

_{F}, by using

_{g}is 0.7 nm, and the relative index of graphene is acquired.

## 4. Numerical Simulations and Optimization

- With dielectric graphene (n
_{g}= 1.332 + i0.105, voltage = 5.4 V) n_{eff}= 2.2755 − i6.2283 × 10^{−5} - With metallic graphene (n
_{g}= 0.174 + i1.672, voltage = 6.6 V) n_{eff}= 2.2756 − i4.3248 × 10^{−3}

## 5. Estimation of Modulation Speed and Energy Required

_{c}(the resistance between the metal and graphene layer) and R

_{g}(the resistance of graphene in series). C is composed of the quantum capacitance in either graphene sheet C

_{Q}and the oxide capacitance C

_{p}in series.

_{F}is the Fermi level of graphene. According to the definition of quantum capacitance:

_{F}. It is calculated by Equations (2) and (3) that the working interval of the graphene electro-absorption modulator is ~0.5 eV. From Figure 9, the quantum capacitance of the unit area is about 0.13 F/m

^{2}, the active area of graphene S is 8.04 μm

^{2}, so the C

_{Q}is about 1.045 pF.

_{0}is the dielectric constant of the vacuum, ε

_{r}is the relative permittivity of the medium, and d is the thickness of Al

_{2}O

_{3}between the two-layer graphene, therefore the plate capacitance C

_{p}is 0.996 fF. Finally, the total capacitance of the modulator is calculated to be 0.994 fF by Equation (10).

_{g}of the graphene film itself and the contact resistance R

_{c}formed by the contact of the graphene with the metal electrode. These resistors are connected in series to obtain the equivalent resistance of the modulator

^{6}S/m. Since the conductivity of the graphene film is very large, the resistance of the graphene film is extremely small, about 8 Ω.

_{c}can be expressed by the Landauer formula [27] as:

_{F}is the gap between the Fermi level and the Dirac energy.

_{u}electrode as 184 Ω and P

_{d}electrode as 457 Ω, respectively [29]. Liu et al. measured the total resistances between the graphene film and A

_{g}, P

_{d}, and Al electrodes as 2 kΩ, 2.3 kΩ, and 10 kΩ, respectively [30]. Through a process of metal-catalyzed etching in hydrogen, Leong et al. obtained the contact resistance between graphene film and Ni electrodes as low as 100 Ω [31].

_{pp}is the peak-to-peak value of the 1.2 V modulation voltage. C was obtained to be 0.994 fF. Therefore, the energy consumption was calculated to be about 0.358 fJ/bit. It is far lower than that reported of ~800 fJ/bit in ref. [21] and an average of tens of fJ/bit in ref. [23]. High energy-consumption of the devices not only requires an extra radiator, but it also complicates the design, affects the stability of the device, and affects the on-chip integration of the photoelectric device. Due to the small size, low energy consumption, and high thermal conductivity of graphene and metal electrodes, the device we designed has better thermal stability than other similar structures. It is conducive to high-density integration.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Transmission curve as a function of the ring transmission α for t = 0.8 at resonance. Inset: schematic diagram of the ring structure, with the definition of the straight waveguide and ring single pass transmission, t and α, respectively.

**Figure 4.**(

**a**) Power density of the electric field of the fundamental transverse electric (TE) mode. (

**b**) Transmission curve as a function of the wavelength.

**Figure 5.**Distribution of the light field. The bias voltages are 5.4 V (

**a**) and 6.6 V (

**b**) respectively.

**Figure 6.**Power density of the electric field of the fundamental TE mode. The gap between the electrodes and the waveguide vary from 250 nm (

**a**); 200 nm (

**b**); 150 nm (

**c**); to 100 nm (

**d**).

**Figure 7.**(

**a**) The curve of the 3 dB bandwidth and energy consumption with the radius of the ring R. (

**b**) The curve of the modulation depth and energy consumption with the average length L of the graphene. (

**c**) The curve of the 3 dB bandwidth and energy consumption with the gap between the straight waveguide and the ring waveguide.

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**MDPI and ACS Style**

Wu, L.; Liu, H.; Li, J.; Wang, S.; Qu, S.; Dong, L. A 130 GHz Electro-Optic Ring Modulator with Double-Layer Graphene. *Crystals* **2017**, *7*, 65.
https://doi.org/10.3390/cryst7030065

**AMA Style**

Wu L, Liu H, Li J, Wang S, Qu S, Dong L. A 130 GHz Electro-Optic Ring Modulator with Double-Layer Graphene. *Crystals*. 2017; 7(3):65.
https://doi.org/10.3390/cryst7030065

**Chicago/Turabian Style**

Wu, Lei, Hongxia Liu, Jiabin Li, Shulong Wang, Sheng Qu, and Lu Dong. 2017. "A 130 GHz Electro-Optic Ring Modulator with Double-Layer Graphene" *Crystals* 7, no. 3: 65.
https://doi.org/10.3390/cryst7030065