Mechanism analysis of terahertz graphene electro-optic modulator with plasma structure

Jia-Bin LI; WANG-Xiao-Hua; Wen-Jie WANG

doi:10.11972/j.issn.1001-9014.2021.02.001

Abstract

Based on the theory of plasmon resonance, we present a photon plasma ridge waveguide consisting of a metal/graphene/Al₂O₃/graphene stacking structure. The modulation performance is optimized through the design of the waveguide and plasma structures by analyzing the microscopic mechanism of photoelectric reaction. A low-insertion-loss near-infrared electro-optic modulator with a modulation rate of 730 GHz, energy consumption of 0.7 fJ/bit, and 3-dB bandwidth of 3.66 THz is achieved.

Keywords

characteristic optimization graphene-based electro-optic modulator mechanism analysis plasma structure

Introduction

Electro-optic modulator（EOM），as the core component of the optical fiber communication and integrated optics，has received wide attention recently. The demand for EOM with higher rate，smaller size，lower power consumption，and easy integration increases dramatically due to the rapid development of the optical fiber communication network and the application of optical interconnection on chip. At present，the material that is used to improve the performance of EOM is graphene，graphene-based EOM has the following advantages：graphene has a strong inter-band optical transition ability，which enhances the interaction between graphene and light，and the modulation performance will be improved；graphene-based EOM has higher modulation rates due to the fact that graphene has extremely high electron mobility ^［1-5］；graphene has wide operating wavelength bandwidth，and its operating frequency covers almost the entire optical communication working bandwidth，even to the mid-infrared，far-infrared，terahertz，and microwave range ^［6］；the photoconductivity of graphene is adjustable；the fabrication process is compatible with existing CMOS process technology^［7-8］.

In the past few years，researchers at home and abroad had conducted extensive research on graphene-based EOM，many research groups had reported excellent results ^［9-16］. In 2016，Zhang's team proposed an improved graphene-based light modulator structure with a 3dB modulation bandwidth of up to 35 GHz，showing good thermal stability ^［17］. In 2017，Lin et al. designed a graphene-based EOM that can achieve a modulation depth of 8 dB/mm in the wavelength range of 2.05 m to 2.45 m ^［18］. In 2018，Liu et al. proposed a graphene/alumina multilayer stacked EOM based on photonic crystal nanobeams，with transmission and absorption modulation depths of 11.25 nm/eV and 12.50 nm/eV，respectively^［19］. In 2018，Hu et al. designed a graphene-based polarization insensitive sensor modulator with insertion loss of less than 2 dB at the wavelength range of 35 nm（including c-band）^［20］.

In recent years，several research groups at home and abroad have carried out research on graphene-based plasma EOM. In 2017，Peng et al. proposed a novel graphene EOM based on the hybrid plasma effect，which can improve the modulation efficiency to 1.23 dB/m，introduce insertion loss of only 0.32 dB in the 3.6-m-long modulation region，and the modulation bandwidth is up to 480 GHz ^［21］. In 2019，Wang et al. designed a graphene/hexagonal boron nitride structure-based plasma EOM at the sub-wavelength scale，the results show that the modulation depth is up to 4 dB/μm，the transmission loss is 2.6 dB，the modulation speed is up to 39 GHz，and the energy loss is 21.2 fJ /bit ^［22］.

However，monolayer graphene only has a light absorption rate of 2.3%，its optical response is weak. The interaction between light and matter in the graphene-based EOM is so weak that it is difficult to continuously improve the modulation efficiency and the reduce the device size. Surface plasmas have strong local property，which exhibit strong field restraint and field enhancement. By covering the surface of the graphene-based EOM with plasma structure，the interaction between the mode field and graphene can be enhanced ^［23-25］，the modulation performance of EOM that can be operated in the terahertz，infrared，and visible light band is improved ^{［26–28］}.

In this study，we investigate a graphene-based electro-optic modulator with plasma structure using the finite-element method implemented in the multi-physics software COMSOL. By designing the waveguide and plasma structure，the properties of EOM will be improved. The modulation mechanism of EOM by structure parameters will be analyzed through the microscopic mechanism of photoelectric reaction.

1 Device design and theoretical analysis

The structure of the electro-optic modulator is shown in Fig. 1（a），mainly composed of Si waveguide，graphene/Al₂O₃/graphene stacking structure，plasma structure，and SiO₂ substrate. The width and height of the waveguide are 400 nm and 200 nm，respectively. Two graphene layers are placed on top of the Si waveguide，which is isolated by Al₂O₃. The two graphene layers are extended from the opposite sides of the Si waveguide for metallization. A few nanometers of Ag，as the plasma structure，is placed on top of graphene layer. The thicknesses of the graphene and Al₂O₃ are 0.7 nm and 10 nm，respectively. The free-space wavelength is λ = 1 550 nm. We assume that the electrodes are far away from the Si waveguide，so that they do not influence the optical mode.

$Structures of the electro-optic modulator. The length of the modulator is 20 μm, the width and height of the substrate are 200 nm and 2 μm, respectively. The refractive index of SiO2, Si, Al2O3 are 1.444, 3.42, and 1.595, respectively (a) single plasma structure, (b) double plasma structures, (c) three plasma structures, (d) four plasma structures$

Figure 1.Structures of the electro-optic modulator. The length of the modulator is 20 μm, the width and height of the substrate are 200 nm and 2 μm, respectively. The refractive index of SiO₂, Si, Al₂O₃ are 1.444, 3.42, and 1.595, respectively (a) single plasma structure, (b) double plasma structures, (c) three plasma structures, (d) four plasma structures

According to the dispersion relation of surface plasma waves，surface plasma waves can be excited only when the wave vector of the surface plasma is larger than the light wave vector in the medium and the tangential wave vector satisfies the matching condition. The wave vector of the surface plasma wave in the x direction can be expressed as：

k_{x} = k_{0} \sqrt[]{\frac{ε_{m} ε_{d}}{ε_{m} + ε_{d}}}

,（1）

where ɛ_mis the real part of the dielectric constant of metallic plasma structure，ɛ_dis the dielectric constant of the medium.

For the metallic plasma structure and medium，ɛ_m and ɛ_d are greater than 1，ɛ_m is greater than ɛ_d. The wave vector of the surface plasma wave in the x direction can be expressed as：

k_{x} = k_{0} \sqrt[]{\frac{ε_{m} ε_{d}}{ε_{m} + ε_{d}}}

k_{0}

.（2）

When light is emitted from the medium into the interface between the plasma structure and the medium，the wave vector in the x direction can be expressed as：

{\tilde{k}}_{x} = k_{0} n_{x} s i n θ

k_{0}

.（3）

From Eqs.2-3 we can see that the matching principle of wave vector can be satisfied under certain conditions，the wave vectors can be coupled，and the surface plasma waves can be excited.

To quantify the mode field characteristics and enhancement of the equidistant waveguide，the normalized mode field area is defined as ^［35-37］：

\frac{A_{e f f}}{A_{0}}

（μ=0 eV）,（4）

A_{0} = \frac{λ^{2}}{4}

,（5）

A_{e f f} = \iint \frac{W (r) d^{2} r}{m a x (W (r))}

,（6）

W (r, ω) = \frac{1}{2} R e \{\frac{d (ω ε (r, ω))}{d ω}\} {|E (r, ω)|}^{2} + \frac{1}{2} μ_{0} {|H (r, ω)|}^{2}

,（7）

where A₀ is the area of the diffraction limit in free space，A_eff is the ratio of the total energy to the maximum energy density，W（r）is electromagnetic field density，E（r）is the electric field density，H（r）is the magnetic density.

The smaller the normalized mode field area of the modulator，the greater the energy density of the electromagnetic field can be，facilitating the light-graphene interaction.

Based on the device structure，the modulation speed（f）of the modulator can be expressed as：

f = \frac{1}{2 π R C}

,（8）

where R and C represent the resistance and capacitance of the modulator，respectively. R is composed of R_c（the resistance between the metal and graphene layer）and R_g（the resistance of the graphene sheet），while C is composed of the quantum capacitance in each graphene sheet C_Q and oxide plate capacitance C_p connected in series.

The quantum capacitance per unit area C_Q can be expressed as ^［32］：

C_{Q} = \frac{2 e^{2} k T}{π {(ℏ ν_{F})}^{2}} l n [2 (1 + c o s h \frac{E_{F}}{k T})]

,（9）

where e is the electron charge，k is the Boltzmann constant，T is the temperature，ħ is the reduced Planck constant，v_F is the Fermi velocity，and E_F is the Fermi level.

By utilizing the simple parallel plate capacitor model，C_p can be calculated as：

C_{p} = \frac{ε_{0} ε_{x}}{E O T}

,（10）

where ε₀ is the dielectric constant of vacuum，ɛ_x is the relative permittivity of the medium，and ETO is the equivalent thickness of Al₂O₃ between the two-layer graphene.

The graphene sheet resistance R_g can be calculated as：

R_{g} = \frac{L_{g}}{σ w_{g} d}

,（11）

where σ is the conductivity of graphene，d is the thickness of graphene，W_g is the width of graphene，and L_g is the length of graphene.

The contact resistance between the metal and graphene R_ccan be expressed by the Landauer formula as ^［33］：

R_{c} = \frac{h}{4 e^{2}} \cdot \frac{1}{T M}

,（12）

where M is the number of conduction modes in graphene and T is the carrier transmission probability. The number of quantum modes M can be expressed as：

M = \frac{Δ E_{F} W_{g}}{π ℏ ν_{F} L_{g}}

,（13）

where ∆E_Fis the gap between the Fermi level and Dirac energy.

The energy consumption E can be expressed as：

E = \frac{1}{4} C V_{p p}^{2}

,（14）

where V_pp is the peak–to–peak voltage.

2 Results and discussion

The performance of modulator can be enhanced due to the interaction between graphene and light，the optical properties of graphene affect the performance of modulator to a large extent. In the graphene-based modulator，even a small variation in the refractive index of graphene may lead to a large shift of the resonant wavelength，the switching “on/off” of the modulator depends on the optical refractive index of graphene. The inter-band transition of graphene can be controlled by tuning the Fermi level（E_F）by electrical gating，optical excitation，or chemical doping ^{［29–31］}，which has a large influence on its refractive index. Here，the refractive index of graphene is modulated by adding a voltage on graphene. The refractive index of graphene as a function of the voltage that is shown in Ref. 34 is cited in our research. The graphene exhibits the maximum real part of the refractive index at approximately 4 V，the refractive index transforms from dielectric to metal at approximately 6 V. In this study，voltages of 2.8 V，4.2 V，5.4 V，and 6.6 V corresponding to the refractive indices of graphene of 2.5，4.5，1.33，and 0.17 respectively are selected to tune the properties of EOM. The voltage of 6.6 V is considered to be the “OFF” state point.

The metal plasma structure in nanoscale has a unique capability to concentrate light at the deep-sub-wavelength scale. The strong in-plane electric field confinement of the plasmonic mode brings a high in-plane electric field density at the edges of plasmonic structure and thus enhances interaction with graphene. The surface plasmon polaritons can be excited effectively at the interface between metal and medium only if the plasma wave at the interface satisfies the matching conditions that are shown in Eqs. 2-3，one of the mainly structure factors to match these conditions is the height of the metal plasma structure. Figure 2（a）shows the transmission spectra of modulator for different heights of the metal plasma structure. Modulation depths that are larger than 35 dB are observed in all the curves. The modulation depths corresponding to different heights of metal plasma structure differ slightly between each other，except for the case with 5 nm height of metal plasma structure；the resonant wavelength exhibits a red-shift trend with respect to the height of the metal plasma structure. Figure 2（b）shows the calculated 3-dB bandwidth of modulator for different heights of the metal plasma structure. It is seen that the metal plasma structure of 5 nm height corresponds to the minimum 3-dB bandwidth，the metal plasma structure of 3 nm height corresponds to the maximum 3-dB bandwidth. As the height of the metal plasma structure increases，the curve of 3-dB bandwidth presents a non-monotonic trend. This is because that the decrease of thickness of the metal plasma structure will squeeze the localized plasmon to be near the location of the graphene，and apparently the strong field localization will enhance the light-graphene interaction，the 3-dB bandwidth of modulator increases gradually with the decrease of thickness of the metal plasma structure. When the thickness of the metal plasma structure is less than 3 nm，the conditions for plasmon excitation cannot be completely satisfied，the 3dB-bandwidth gradually decreases with the further reduction of the thickness of the metal plasma structure. By balancing the modulation depth and 3-dB bandwidth，the height of the plasma structure is set to 3 nm.

Figure 2.Modulation performances of modulator for different heights of plasma structure (a) transmission spectra, (b) 3-dB bandwidth

Bias voltage /V	R /Ω	C/fF	f /GHz	E /（fJ/bit）
2.8	155	1.943	527	7.01
4.2	127	1.949	645	2.81
5.4	112	1.952	730	0.70

Table 1. Modulation speeds and energy consumptions of the modulator under different bias voltages

View all Tables

It is known that only the electric field components parallel to graphene contribute to the light absorption property of graphene. This selectivity requests special structure that can strongly confine in-plane electric field components in graphene. Since the plasma structure can enhance in-plane electric field density at the interface between graphene and plasma structure，the performance of the modulator will be improved. This means that how to further enhance the electric field is a key problem. Based on the structure of the electro-optic modulator in Fig. 1（a），the metal plasma structure is divided into two part as shown in Fig. 1（b）. The gap between two plasma structures is tens or hundreds of nanometers wide，which is much smaller than the wavelength. This kind of modulator structure supports higher in-plane electric field Bloch mode due to the fact that the electric field is highly confined in the gap laterally，the interference of a narrow resonance with a broad continuum of states leads to the enhancement of optical transmission. By calculating the Eqs.4-7 we find that the normalized mode field area of the double-plasma based modulator is smaller than that of single-plasma based modulator，implying strong interaction between light and graphene. Figure 3（a）shows the transmission spectra of modulator for different widths of gap between the two plasma structures. Modulation depths that are larger than 30 dB are observed in all the curves. The modulation depths corresponding to different widths of gap differ slightly between each other as the width of gap is smaller than 160 nm. The resonant wavelength exhibits a blue-shift trend with respect to the width of gap between the two plasma structures. Figure 3（b）shows that the calculated 3-dB bandwidth increases first and then decreases with the increase of gap width，the gap width of 140 nm corresponds to the maximum 3-dB bandwidth. The results above imply that the plasmon resonances in the horizontal direction are excited by a horizontal electric field in the presence of the gap between the two plasma structures，which enhanced the light–matter interaction and improved the transmission performance of the modulator. The horizontal electric fields from plasmon resonance of the two plasma structures are superimposed and enhanced by interference，gradually increasing in-plane electric field density. The condition for the interference enhancement of electric field is no longer fully satisfied when the width of gap of the two plasma structures is greater than 140 nm，the 3-dB bandwidth decreases as the gap width increases. By balancing the modulation depth and 3-dB bandwidth，the width of gap of the two plasma structures is set to 140 nm.

Figure 3.Modulation performances of modulator for different gap widths between the two plasma structures (a) Transmission spectra, (b) 3-dB bandwidth

Further，we investigate the electro-optic modulators with three and four plasma structures as shown in Figs. 1（c-d）. Figure 4（a）shows the transmission spectra of three-plasma based modulator for different widths of gap between the two plasma structures. It is seen that the modulation depths that are larger than 25 dB are observed in all the curves，the modulation depths corresponding to different gap widths varies greatly. The resonant wavelength exhibits a blue-shift trend with respect to the width of gap between the two plasma structures. Figure 4（b）shows the calculated 3-dB bandwidth of three-plasma based modulator for different widths of gap between the two plasma structures. The 3-dB bandwidth exhibits disordered variations. This demonstrates that the light-matter interaction could be enhanced in some cases by the segmented horizontal electric field or weakened in other cases. We can infer that the condition for the interference enhancement of electric field is satisfied when the width of gap between the two plasma structures is set to some values，but not when the gap width is set to other values. The gap width of 100 nm between the two plasma structures（the width of plasma structure is 66.67 nm）in the three-plasma based modulator corresponds to the maximum 3-dB bandwidth of 3.5 THz，which is larger than that in the double-plasma based modulator.

Figure 4.Modulation performances of modulator for different gap widths between adjacent plasma structures. The plasma structure is divided into three equal parts and the two gaps are of equal width (a) transmission spectra, (b) 3-dB bandwidth

Figure 5 shows the transmission spectra and the calculated 3-dB bandwidth of four-plasma based modulator for different widths of gap between the two plasma structures，transmission behavior and the trend of 3-dB bandwidth curve that are like that of three-plasma based modulator that is shown in Fig. 4 are observed. A maximum 3-dB bandwidth（corresponding to a gap width of 75 nm or plasma structure width of 43.75 nm）of 3.66 THz is observed，which is larger than that in the three-plasma based modulator. The results above demonstrate that the in-plane electric field parallel to graphene can be enhanced by adjusting the width of gap between the two plasma structures due to the degree to which the condition for the interference enhancement of electric field is satisfied，thus the performance of the modulator is improved. It is foreseen that the 3-dB bandwidth will increase further by adjusting the amount of plasma structure and the width of gap between the two plasma structures.

Figure 5.Modulation performances of modulator for different gap widths between adjacent plasma structures. The plasma structure is divided into four equal parts and the three gaps are of equal width (a) transmission spectra,: (b) 3-dB bandwidth

We consider the single- and double-plasma based modulators as examples，whose transmissions as a function of the wavelength under different voltages are shown in Fig. 6. For the single-plasma based modulator，the resonant wavelength is almost unchanged，approximately 1 705.6 nm. Based on this resonant wavelength，the transmission of light at the “OFF” state（voltages of 6.6 V corresponding to refractive indices of graphene of 0.17）is approximately 20 dB. For the double-plasma based modulator，the resonant peak exhibits a significant blue-shift trend owing to the increase in the refractive index of graphene. Wide band between the resonant wavelength at the “on” state and the resonant wavelength at the “OFF” state is observed，the modulation range of the modulator has been expanded greatly. The transmission of light at the “OFF” state is smaller than 10 dB based on the different resonant wavelengths. The weak transmission of light at the “OFF” state corresponding to the resonant wavelength represents the excellent performance of modulator in some ways due to the small transmission loss of light. Figure 7 shows that the changes of 3-dB bandwidth of modulator with the refractive index of graphene，the maximum 3-dB bandwidths are observed at a bias voltage of 4.2 V（corresponding to the maximum refractive index of graphene）in single- and double-plasma based modulators. Furthermore，the 3-dB bandwidths of the double-plasma based modulator are bigger than that of single-plasma based modulator structure in different bias voltage due to the degree of interference enhancement of the electric field paralleled to graphene.

Figure 6.Transmission spectra as a function of the wavelength corresponding to different bias voltages (a) single-plasma based modulator structure, (b) double- plasma based modulator structure

Figure 7.3-dB bandwidths of single- and double-plasma modulators under different bias voltages

According to the Eqs.8-14，the equivalent resistance R，equivalent capacitance C，modulation speed f，and the energy consumption E are calculated，as shown in Table 1（all calculations are based on the optimal structure and parameters discussed above）. Based on the Eq.10，The calculation of capacitance includes the effect of carrier transport characteristics of heterojunction interface on the dielectric constant. Considering various scattering mechanisms in carrier transport，the calculation result of traditional drift-diffusion model has a relatively big error. In this paper，Monte Carlo method is used to characterize dielectric materials with non-mutational refractive index function，so as to optimize the calculation of equivalent capacitance.

From Table 1 we can see that strong correlation between the modulation speed and voltage is observed，the modulation speed increases with the voltage；the maximum modulation speed is calculated to be 730 GHz under the bias voltage of 5.4 V. The energy consumption E significantly varies with the voltage；the smallest energy consumption is calculated to be 0.7 fJ/bit under the bias voltage of 5.4 V，corresponding to the smallest difference from the “OFF”-state voltage. Figure 8 shows that slightly different 3-dB bandwidths are observed at bias voltages of 4.2 V and 5.4 V. By balancing the energy consumption and 3-dB bandwidth，we reveal that the bias voltage of 5.4 V is appropriate.

3 Conclusion

Based on the concept of localized plasmon resonance，we presented the photon plasma ridge waveguide consisting of the metal/graphene/Al₂O₃/graphene stacking structure. The plasma structure that can improve the modulation performance of modulator was designed and the light–matter interaction was enhanced owing to the plasmon resonances in the horizontal direction that is excited by the horizontal electric field. The optimization of the gap between adjacent plasmon structures showed that a larger 3-dB bandwidth could be achieved by increasing the number of plasmon structures. Furthermore，multi-plasma structures effectively reduced the transmission loss. According to the simulation results using the multi-physics software COMSOL，a terahertz near-infrared EOM with a modulation rate of 730 GHZ and energy consumption of 0.7 fJ/bit was achieved. The demonstrated excellent performances satisfy the application requirements of modern photonic integrated circuits.

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