Graphene Electro-Optical Switch Modulator by Adjusting Propagation Length Based on Hybrid Plasmonic Waveguide in Infrared Band

A modulator is the core of many optoelectronic applications such as communication and sensing. However, a traditional modulator can hardly reach high modulation depth. In order to achieve the higher modulation depth, a graphene electro-optical switch modulator is proposed by adjusting propagation length in the near infrared band. The switch modulator is designed based on a hybrid plasmonic waveguide structure, which is comprised of an SiO2 substrate, graphene–Si–graphene heterostructure, Ag nanowire and SiO2 cladding. The propagation length of the hybrid plasmonic waveguide varies from 0.14 μm to 20.43 μm by the voltage tunability of graphene in 1550 nm incident light. A modulator with a length of 3 μm is designed based on the hybrid waveguide and it achieves about 100% modulation depth. The lower energy loss (~1.71 fJ/bit) and larger 3 dB bandwidth (~83.91 GHz) are attractive for its application in a photoelectric integration field. In addition, the excellent robustness (error of modulation effects lower than 8.84%) is practical in the fabrication process. Most importantly, by using the method of adjusting propagation length, other types of graphene modulators can also achieve about 100% modulation depth.


Introduction
Graphene, a two-dimensional material, was obtained by mechanical peeling in 2004 [1]. Due to the excellent optical and electrical properties, graphene has a great potential in the field of photodetectors [2,3], waveguides [4,5], optical modulators [6,7], supercapacitors [8,9], sensors [10,11] and so on. Scientists have found that the absorption rate of graphene with intrinsic monolayer thickness is about 2.3%, which is 50 times higher than gallium arsenide with same thickness and can be easily tuned by changing graphene conductivity in different gating voltage [12,13]. The mainly reason is that strong light-matter interaction in graphene can be controlled effectively by tuning the Fermi level (E F ) with electrostatic gating. Based on this excellent tunability of the absorption property [14] through electrostatic gating, graphene can be used for designing electro-optical devices and has been widely applied in electro-optical modulators [15][16][17][18][19][20]. Graphene modulators have shown relative high modulation depth in different structures [19,20].
To provide subwavelength confinement and realize light manipulation on nanoscale devices, plasmonic waveguides based on surface plasmon polarizations (SPPs) wave have been researched a lot [21][22][23][24]. To further reduce the high optical loss and increase propagation length, hybrid plasmonic waveguides, constructed by surface plasmon waveguides and silicon waveguides [25], have been proposed. Since then, many kinds of hybrid plasmon waveguide structure have been researched to Sensors 2020, 20 further enhance the waveguide performance [26][27][28][29]. However, there are only few works concerned with the modulation properties based on a hybrid plasmonic waveguide and the lack ways to achieve deeper modulation performance. For the excellent tunability of graphene conductivity, devices based on graphene have a potential in the subwavelength waveguides [30] and modulators [31][32][33].
Although there are many studies of the graphene-based modulator in recent years, there is no solution to obtain high modulation performance by the traditional method. In addition, for the better application in actual engineering, the higher bandwidth and lower energy consumption of modulators are also needed. In order to solve these problems, a novel graphene electro-optical switch modulator based on silver nanowire (GESMBOSN), by adjusting propagation length with a hybrid plasmonic waveguide, is proposed in the near infrared band (~1550 nm). It is worth mentioning that the tunability of optical propagation length in graphene waveguides has been ignored in the past research of graphene modulators. Using tunablity of propagation length to design graphene switch modulators, the proposed switch modulator successfully achieves about 100% modulation depth. The modulator also realizes lower energy loss and larger 3dB bandwidth in the near infrared band. Besides, the excellent robustness provides more convenient conditions for actual production. Most importantly, as a new method to design graphene switch modulators, other different structure's graphene switch modulators can also apply this new method to achieve~100% modulation depth. Figure 1 shows the three-dimensional and sectional model of the structure. The modulator is composed of a hybrid waveguide structure, which is constructed of an SiO 2 substrate, graphene-Si-graphene heterostructure and Ag nanowire. Furthermore, the graphene in this modulator is above and below the Si. Moreover, the whole structure is surrounded by SiO 2 cladding, which include the gap between upper graphene and Ag.

Materials and Methods
Sensors 2020, 20, x 2 of 11 researched to further enhance the waveguide performance [26][27][28][29]. However, there are only few works concerned with the modulation properties based on a hybrid plasmonic waveguide and the lack ways to achieve deeper modulation performance. For the excellent tunability of graphene conductivity, devices based on graphene have a potential in the subwavelength waveguides [30] and modulators [31][32][33]. The modulation depth can be achieved at 0.03 dB/um [34] and 0.6 dB/μm [35] for different graphene hybrid plasmonic waveguide modulators.
Although there are many studies of the graphene-based modulator in recent years, there is no solution to obtain high modulation performance by the traditional method. In addition, for the better application in actual engineering, the higher bandwidth and lower energy consumption of modulators are also needed. In order to solve these problems, a novel graphene electro-optical switch modulator based on silver nanowire (GESMBOSN), by adjusting propagation length with a hybrid plasmonic waveguide, is proposed in the near infrared band (~1550 nm). It is worth mentioning that the tunability of optical propagation length in graphene waveguides has been ignored in the past research of graphene modulators. Using tunablity of propagation length to design graphene switch modulators, the proposed switch modulator successfully achieves about 100% modulation depth. The modulator also realizes lower energy loss and larger 3dB bandwidth in the near infrared band. Besides, the excellent robustness provides more convenient conditions for actual production. Most importantly, as a new method to design graphene switch modulators, other different structure's graphene switch modulators can also apply this new method to achieve ~100% modulation depth. Figure 1 shows the three-dimensional and sectional model of the structure. The modulator is composed of a hybrid waveguide structure, which is constructed of an SiO2 substrate, graphene-Si-graphene heterostructure and Ag nanowire. Furthermore, the graphene in this modulator is above and below the Si. Moreover, the whole structure is surrounded by SiO2 cladding, which include the gap between upper graphene and Ag. In this structure, hSiO2 denotes the height of SiO2 substrate and its value is 100 nm. R denotes the radius of Ag nanowire and the range of R is from 10 nm to 100 nm. The thickness of single-layer graphene is 0.7 nm and hSi denotes the height of Si (hSi = 10 nm). Furthermore, g denotes the gap between Ag nanowire and graphene, whose range is from 1 nm to 10 nm. Using the relationship between voltages and propagation length of this switch modulator, its length can be obtained and achieve about 100% modulation depth, which will be discussed in detail in Figures 6 and 7.

Materials and Methods
For the good optical properties of graphene, its Fermi level can be controlled by external voltages [36]. The relationship between carrier density and Fermi level is calculated by EF = ħVf(π·n0) 1/2 , where ħ is the reduced Planck constant. Vf denotes the Fermi velocity. For graphene, its Fermi velocity is 1.1 × 10 6 m/s. By definition, n0 is the carrier concentration and its value can be calculated In this structure, h SiO2 denotes the height of SiO 2 substrate and its value is 100 nm. R denotes the radius of Ag nanowire and the range of R is from 10 nm to 100 nm. The thickness of single-layer graphene is 0.7 nm and h Si denotes the height of Si (h Si = 10 nm). Furthermore, g denotes the gap between Ag nanowire and graphene, whose range is from 1 nm to 10 nm. Using the relationship between voltages and propagation length of this switch modulator, its length can be obtained and achieve about 100% modulation depth, which will be discussed in detail in Figures 6 and 7.
For the good optical properties of graphene, its Fermi level can be controlled by external voltages [36]. The relationship between carrier density and Fermi level is calculated by E F =hV f (π·n 0 ) 1/2 , whereh is the reduced Planck constant. V f denotes the Fermi velocity. For graphene, its Fermi velocity Sensors 2020, 20, 2864 3 of 11 is 1.1 × 10 6 m/s. By definition, n 0 is the carrier concentration and its value can be calculated by n 0 = ε 0 ε r (U + U 0 )/(d·e). U is the voltage applied to graphene. U 0 is related to the Fermi level of graphene when no voltage is applied, and its value is usually taken as 0 V. ε 0 refers to the dielectric constant in vacuum, and the general value is about 8.854 × 10 −12 F/m. ε r denotes the dielectric constant of substrate materials and e is electron charge; d presents the thickness of substrates, and whose value is equal to h is .
At temperature T = 296 K, the relationship between the Fermi level and electrical conductivity of graphene is given by Equation (1) [31].
where σ is the conductivity of monolayer graphene. σ is consisted by intraband conductivity σ intra and interband conductivity σ inter . σ intra denotes to the electrical conductivity generated by in-band free carrier transitions. σ inter refers to the electrical conductivity generated by free carrier transition between the bands. ω is the angular frequency of incident light and the incident wavelength is 1550 nm at here. τ 1 and τ 2 are the intraband and interband relaxation time of graphene, and the values are 10 fs and 1.2 ps, respectively [31,37]. In addition, σ 0 is equal to πe 2 /2h. Figure 2 shows the image of conductivity as a function of the Fermi level in 1550 nm incident wavelength.
of substrate materials and e is electron charge; d presents the thickness of substrates, and whose value is equal to his. At temperature T=296 K, the relationship between the Fermi level and electrical conductivity of graphene is given by Equation (1) [31].
where σ is the conductivity of monolayer graphene. σ is consisted by intraband conductivity σintra and interband conductivity σinter. σintra denotes to the electrical conductivity generated by in-band free carrier transitions. σinter refers to the electrical conductivity generated by free carrier transition between the bands. ω is the angular frequency of incident light and the incident wavelength is 1550 nm at here. τ1 and τ2 are the intraband and interband relaxation time of graphene, and the values are 10 fs and 1.2 ps, respectively [31,37]. In addition, σ0 is equal to πe 2 /2ħ. Figure 2 shows the image of conductivity as a function of the Fermi level in 1550 nm incident wavelength. The relationship between conductivity and permittivity can be given by Equation (2): where dG denotes the thickness of single-layer graphene. According to Equation (2), the real part of the dielectric constant decides the imaginary part of the conductivity, while the imaginary part of the dielectric constant decides the real part of conductivity. The refractive index can be given by n = (ε(ω)) 1/2 . According to this way, the relationship about voltage and the refractive index of graphene can be shown in Figure 3.  The propagation length reflects the device's transmission properties, which is the length that the electric field decays to 1/e of itself. Larger propagation length means the device has better The relationship between conductivity and permittivity can be given by Equation (2): where d G denotes the thickness of single-layer graphene. According to Equation (2), the real part of the dielectric constant decides the imaginary part of the conductivity, while the imaginary part of the dielectric constant decides the real part of conductivity. The refractive index can be given by n = (ε(ω)) 1/2 . According to this way, the relationship about voltage and the refractive index of graphene can be shown in Figure 3.

of 11
Sensors 2020, 20, x 4 of 11 transmission ability. It is generally believed that the electric field intensity beyond the propagation length cannot be detected. The propagation length is calculated by Lm = λ/[4πIm(Neff)] [27,38], where λ denotes the incident wavelength and Neff refers to the effective refractive index of the modulator. The normalized effective mode area Aeff/A0 can be given by Equation (3) [38]: where Wm refers to the total energy density and max {W(x,y)} represents the maximum energy density of the structure. Smaller Aeff/A0 means stronger optical confinement in the devices. The consumed energy can be given by E = 1/4CVp 2 . C represents the total capacitance of the modulator, which is mainly composed of the quantum capacitance of graphene and the equivalent capacitance between graphene and Si. Vp refers to the peak-to-peak value of external voltage. The 3dB bandwidth of the modulator can be given by f3dB = 1/2πτRC = 1/2πRtotalC, where Rtotal refers to the resistance of the whole modulator. It is composed of the graphene resistance and the contact resistance between graphene and metal electrode. The resistance of graphene can be generally ignored because of the small value (about tens of ohms) [39]. The modal properties of the presented modulator are investigated numerically by using the finite element method package in RF module of COMSOL Multiphysics 5.4 software [40]. The mode analysis with scattering boundary is used to simulate the open boundary condition and to calculate relative propagation length and the normalized effective mode area of the structure. In this simulation, the all device size is set 400 nm × 400 nm in x and y direction. The incident wavelength λ is 1550 nm. The relative permittivities of Ag, Si and SiO2 at 1550 nm incident wavelength are − 129 + 3.3i, 2.25 and 12.25, respectively [27].

Results and Discussion
In order to optimize the size of modulator, the normalized effective mode area and propagation length need to be calculated. By discussing the values of the normalized effective mode area and propagation, the optimal size of gap g and radius R can be decided.
It can be concluded from Figure 4 that the change trend of Aeff/A0 becomes larger with the increase of g. The main reason is that when g is large, the interface area between dielectric Si and The propagation length reflects the device's transmission properties, which is the length that the electric field decays to 1/e of itself. Larger propagation length means the device has better transmission ability. It is generally believed that the electric field intensity beyond the propagation length cannot be detected. The propagation length is calculated by L m = λ/[4πIm(N eff )] [27,38], where λ denotes the incident wavelength and N eff refers to the effective refractive index of the modulator.
The normalized effective mode area A eff /A 0 can be given by Equation (3) [38]: where W m refers to the total energy density and max {W(x,y)} represents the maximum energy density of the structure. Smaller A eff /A 0 means stronger optical confinement in the devices. The consumed energy can be given by E = 1/4CV p 2 . C represents the total capacitance of the modulator, which is mainly composed of the quantum capacitance of graphene and the equivalent capacitance between graphene and Si. V p refers to the peak-to-peak value of external voltage. The 3dB bandwidth of the modulator can be given by f 3dB = 1/2πτ RC = 1/2πR total C, where R total refers to the resistance of the whole modulator. It is composed of the graphene resistance and the contact resistance between graphene and metal electrode. The resistance of graphene can be generally ignored because of the small value (about tens of ohms) [39]. The modal properties of the presented modulator are investigated numerically by using the finite element method package in RF module of COMSOL Multiphysics 5.4 software [40]. The mode analysis with scattering boundary is used to simulate the open boundary condition and to calculate relative propagation length and the normalized effective mode area of the structure. In this simulation, the all device size is set 400 nm × 400 nm in x and y direction. The incident wavelength λ is 1550 nm. The relative permittivities of Ag, Si and SiO 2 at 1550 nm incident wavelength are − 129 + 3.3i, 2.25 and 12.25, respectively [27].

Results and Discussion
In order to optimize the size of modulator, the normalized effective mode area and propagation length need to be calculated. By discussing the values of the normalized effective mode area and propagation, the optimal size of gap g and radius R can be decided.
It can be concluded from Figure 4 that the change trend of A eff /A 0 becomes larger with the increase of g. The main reason is that when g is large, the interface area between dielectric Si and Ag nanowire is not large enough to confine the light into a very small scale. The L m becomes larger with the increase Sensors 2020, 20, 2864 5 of 11 of g. The main reason is that the increase of g can bring the smaller interface area between dielectric Si and Ag nanowire, which will decrease the energy loss of incident light and bring the increase of L m . Figure 5 shows that L m will increase with the larger R. The key factor is that the larger radius will cause a smaller loss of incident light and improve the propagation length. By contrast, larger R reduces the degree of light localization and results in a larger A eff /A 0 . In fact, the results shows that larger L m and smaller A eff /A 0 are contradictory in the devices. In order to obtain the larger L m and keep A eff /A 0 as small as possible, the g is set to 1 nm and R is set to 50 nm. By tuning the refractive index of graphene, N eff will be changed and the propagation length can be adjusted for this switch modulator.
Sensors 2020, 20, x 5 of 11 Ag nanowire is not large enough to confine the light into a very small scale. The Lm becomes larger with the increase of g. The main reason is that the increase of g can bring the smaller interface area between dielectric Si and Ag nanowire, which will decrease the energy loss of incident light and bring the increase of Lm. Figure 5 shows that Lm will increase with the larger R. The key factor is that the larger radius will cause a smaller loss of incident light and improve the propagation length. By contrast, larger R reduces the degree of light localization and results in a larger Aeff/A0. In fact, the results shows that larger Lm and smaller Aeff/A0 are contradictory in the devices. In order to obtain the larger Lm and keep Aeff/A0 as small as possible, the g is set to 1 nm and R is set to 50 nm. By tuning the refractive index of graphene, Neff will be changed and the propagation length can be adjusted for this switch modulator. According to the result in Figure 6, the 1.5 V (corresponding to 19.93 μm propagation length) is used as the light passing voltage and the 2.3 V (corresponding to 0.14 μm propagation length) is used as the light cut-off voltage. Considering the factor for keeping device sizes as small as possible and the results in Figure 6, the modulator length is set to 3 μm and about 100% modulation depth can be obtained. In this case, the modulation function of this switch modulator can be achieved. Figure 7 shows the working principle of this switch modulator. When different voltages are applied to this switch modulator, different electric field distribution is shown in Figure 7b,d. The high electric field intensity in Figure 7b illustrates that the incident light can pass this modulator. By contrast, the low electric field intensity in Figure 7d illustrates that the incident light cannot pass this modulator. With this design method, voltages are applied to both ends of graphene, which can control the light absorption by tuning graphene, so as to realize the change of propagation length for an electro-optical modulator. As shown in Figures 8-10, the incident light will pass this switch Ag nanowire is not large enough to confine the light into a very small scale. The Lm becomes larger with the increase of g. The main reason is that the increase of g can bring the smaller interface area between dielectric Si and Ag nanowire, which will decrease the energy loss of incident light and bring the increase of Lm. Figure 5 shows that Lm will increase with the larger R. The key factor is that the larger radius will cause a smaller loss of incident light and improve the propagation length. By contrast, larger R reduces the degree of light localization and results in a larger Aeff/A0. In fact, the results shows that larger Lm and smaller Aeff/A0 are contradictory in the devices. In order to obtain the larger Lm and keep Aeff/A0 as small as possible, the g is set to 1 nm and R is set to 50 nm. By tuning the refractive index of graphene, Neff will be changed and the propagation length can be adjusted for this switch modulator. According to the result in Figure 6, the 1.5 V (corresponding to 19.93 μm propagation length) is used as the light passing voltage and the 2.3 V (corresponding to 0.14 μm propagation length) is used as the light cut-off voltage. Considering the factor for keeping device sizes as small as possible and the results in Figure 6, the modulator length is set to 3 μm and about 100% modulation depth can be obtained. In this case, the modulation function of this switch modulator can be achieved. Figure 7 shows the working principle of this switch modulator. When different voltages are applied to this switch modulator, different electric field distribution is shown in Figure 7b,d. The high electric field intensity in Figure 7b illustrates that the incident light can pass this modulator. By contrast, the low electric field intensity in Figure 7d illustrates that the incident light cannot pass this modulator. With this design method, voltages are applied to both ends of graphene, which can control the light absorption by tuning graphene, so as to realize the change of propagation length for an electro-optical modulator. As shown in Figures 8-10, the incident light will pass this switch According to the result in Figure 6, the 1.5 V (corresponding to 19.93 µm propagation length) is used as the light passing voltage and the 2.3 V (corresponding to 0.14 µm propagation length) is used as the light cut-off voltage. Considering the factor for keeping device sizes as small as possible and the results in Figure 6, the modulator length is set to 3 µm and about 100% modulation depth can be obtained. In this case, the modulation function of this switch modulator can be achieved. Figure 7 shows the working principle of this switch modulator. When different voltages are applied to this switch modulator, different electric field distribution is shown in Figure 7b,d. The high electric field intensity in Figure 7b illustrates that the incident light can pass this modulator. By contrast, the low electric field intensity in Figure 7d illustrates that the incident light cannot pass this modulator. With this design method, voltages are applied to both ends of graphene, which can control the light absorption by tuning graphene, so as to realize the change of propagation length for an electro-optical modulator. As shown in Figures 8-10, the incident light will pass this switch modulator at 1.5 V.
By contrast, the incident light will disappear in this switch modulator at 2.3 V. In this way, the function of signal modulation can be realized.
Sensors 2020, 20, x 6 of 11 modulator at 1.5 V. By contrast, the incident light will disappear in this switch modulator at 2.3 V. In this way, the function of signal modulation can be realized.       The total capacitance is mainly composed of graphene quantum capacitance and equivalent capacitance between graphene and Si. After calculation, the quantum capacitance of graphene is ~0.098 F/m and the equivalent capacitance between graphene and Si is ~0.0108 F/m. Due to the width and length of this switch modulator is 400 nm and 3 μm, its total capacitance is 10.66 fF.   The total capacitance is mainly composed of graphene quantum capacitance and equivalent capacitance between graphene and Si. After calculation, the quantum capacitance of graphene is ~0.098 F/m and the equivalent capacitance between graphene and Si is ~0.0108 F/m. Due to the width and length of this switch modulator is 400 nm and 3 μm, its total capacitance is 10.66 fF.   The total capacitance is mainly composed of graphene quantum capacitance and equivalent capacitance between graphene and Si. After calculation, the quantum capacitance of graphene is ~0.098 F/m and the equivalent capacitance between graphene and Si is ~0.0108 F/m. Due to the width and length of this switch modulator is 400 nm and 3 μm, its total capacitance is 10.66 fF. The total capacitance is mainly composed of graphene quantum capacitance and equivalent capacitance between graphene and Si. After calculation, the quantum capacitance of graphene is 0.098 F/m and the equivalent capacitance between graphene and Si is~0.0108 F/m. Due to the width and length of this switch modulator is 400 nm and 3 µm, its total capacitance is 10.66 fF. Moreover, Sensors 2020, 20, 2864 8 of 11 1.5 V and 2.3 V voltages are used to control the on and off for this switch modulator. According to Equation E = 1/4CV p 2 , the energy loss of the whole switch modulator is~1.71 fJ/bit. The total resistance mainly includes graphene resistance and the contact resistance between graphene and metal electrode. Because the resistance of graphene is too small (about tens of ohms) and can be ignored, the total resistance is 177.93 ohms. According to Equation f 3dB = 1/2πτ RC = 1/2πR total C, the 3 dB bandwidth of this switch modulator is 83.91 GHz.

Fabrication Error Tolerance and Manufacturing Process
It can be observed from Figure 11a that the change of g only has a little influence on the propagation length. The maximum error of propagation length is given by (max(L m ) − min(L m ))/max(L m ). The maximum error of this switch modulator in 1.5 V is about 2.89% (g = 1 corresponds to the~19.926 µm propagation length and g = 4 corresponds the~20.502 µm propagation length). For the error of propagation length in 2.3 V, maximum error of this switch modulator is about 0.58% (g = 1 corresponds to the~0.1396 µm propagation length and g = 4 corresponds to the~0.1404 µm propagation length). As shown in Figure 11b, the maximum error of this switch modulator in 1.5 V is about 8.84% (R = 50 corresponds the~19.926 µm propagation length and R = 54 corresponds to the~21.688 µm propagation length). For the error of propagation length in 2.3 V, maximum error of this switch modulator is about 5.74% (R = 50 corresponds the~0.1396 µm propagation length and R = 54 corresponds to the~0.1477 µm propagation length). The results prove that this switch modulator has excellent robustness.
Sensors 2020, 20, x 8 of 11 Moreover, 1.5 V and 2.3 V voltages are used to control the on and off for this switch modulator. According to Equation E = 1/4CVp 2 , the energy loss of the whole switch modulator is ~1.71 fJ/bit. The total resistance mainly includes graphene resistance and the contact resistance between graphene and metal electrode. Because the resistance of graphene is too small (about tens of ohms) and can be ignored, the total resistance is 177.93 ohms. According to Equation f3dB = 1/2πτRC = 1/2πRtotalC, the 3 dB bandwidth of this switch modulator is 83.91 GHz.

Fabrication Error Tolerance and Manufacturing Process
It can be observed from Figure 11a that the change of g only has a little influence on the propagation length. The maximum error of propagation length is given by (max(Lm)min(Lm))/max(Lm). The maximum error of this switch modulator in 1.5V is about 2.89% (g = 1 corresponds to the ~19.926 μm propagation length and g = 4 corresponds the ~20.502 μm propagation length). For the error of propagation length in 2.3 V, maximum error of this switch modulator is about 0.58% (g = 1 corresponds to the ~0.1396 μm propagation length and g = 4 corresponds to the ~0.1404 μm propagation length). As shown in Figure 11b, the maximum error of this switch modulator in 1.5 V is about 8.84% (R = 50 corresponds the ~19.926 μm propagation length and R = 54 corresponds to the ~21.688 μm propagation length). For the error of propagation length in 2.3 V, maximum error of this switch modulator is about 5.74% (R = 50 corresponds the ~0.1396 μm propagation length and R = 54 corresponds to the ~0.1477 μm propagation length). The results prove that this switch modulator has excellent robustness. The switch modulator can be manufactured with general commercial SOI wafers. Firstly, two grating couplers can be produced at both ends of the device by deep reactive ion etching (The cycle of grating couplers is set to 780 nm in 1550 nm working length), which can achieve the phase matching and let the incident light enter or output the switch modulator. Then a layer of graphene can be transferred onto 100 nm SiO2. 10 nm thickness of Si upon the graphene layer is deposited as the substrate by atomic layer deposition, which can constrain the electric field distribution inside the device. Another layer of graphene can be transferred in the same way. Then, a SiO2 layer can be transferred onto graphene, which can be used as a cladding material for the gap between Ag and the upper graphene. In addition, the SiO2 layer is also used to grow a silver nanowire. Flipping the entire structure, by ion beam etching an SiO2 layer in a z direction definite size, a cylindrical hole can be etched out. After using the magnetron sputtering technology and masks technology, silver ions can fill empty slots in the cylinder to form the silver nanowire needed. Finally, flipping the entire structure to the original place, SiO2 cladding can be grown on the whole devices [41,42]. In this way, an Ag nanowire can be put on the single sheet of graphene and we can precisely control the gap between silver and graphene. The switch modulator can be manufactured with general commercial SOI wafers. Firstly, two grating couplers can be produced at both ends of the device by deep reactive ion etching (The cycle of grating couplers is set to 780 nm in 1550 nm working length), which can achieve the phase matching and let the incident light enter or output the switch modulator. Then a layer of graphene can be transferred onto 100 nm SiO 2 . 10 nm thickness of Si upon the graphene layer is deposited as the substrate by atomic layer deposition, which can constrain the electric field distribution inside the device. Another layer of graphene can be transferred in the same way. Then, a SiO 2 layer can be transferred onto graphene, which can be used as a cladding material for the gap between Ag and the upper graphene. In addition, the SiO 2 layer is also used to grow a silver nanowire. Flipping the entire structure, by ion beam etching an SiO 2 layer in a z direction definite size, a cylindrical hole can be etched out. After using the magnetron sputtering technology and masks technology, silver ions can fill empty slots in the cylinder to form the silver nanowire needed. Finally, flipping the entire structure to the original place, SiO 2 cladding can be grown on the whole devices [41,42]. In this way, an Ag nanowire can be put on the single sheet of graphene and we can precisely control the gap between silver and graphene.

Conclusions
In this paper, a novel graphene electro-optical switch modulator based on adjusting propagation length is proposed in the near infrared band. It successfully combines the advantages of hybrid waveguide and the tunability of graphene to adjust the propagation length, which achieves~100% modulation depth for the switch modulator in theory. The numerical simulations at the wavelength of 1550 nm show that GESMBOSN has relatively low loss (~1.71 fJ/bit) and very large 3 dB bandwidth (~83.91 GHz). In addition, the wonderful robustness of this switch modulator is also conducive to the actual application. Most importantly, with this new method to design a graphene switch modulator, other graphene switch modulators can also have the potential to arrive at about 100% modulation depth.