Broadband graphene-on-silicon modulator with orthogonal hybrid plasmonic waveguides

1 Introduction

Electro-optic modulator, converting electronic signals into high-bit-rate photonic data, is a crucial device in integrated optoelectronics and has been widely applied in optical interconnect [1], [2], environmental monitoring [3], and optical sensing [4], etc. The signals can be modulated to the phase, amplitude, and polarization of lights by electro-absorption effect [5], magneto-optic effect [6], and thermo-optic effect [7]. Due to the fast response speed and low power consumption, electro-absorption modulators that exchange the electrical and optical signals via adjusting the optical absorption of materials with external voltages has attracted extensive attention [8], [9]. According to the electro-absorption materials, these modulators can be classified into two categories: III–V material based-modulators and IV material based-modulators [10], [11], [12], [13], [14]. III–V based-modulators are the most common modulators and have been extensively used in photoelectric information processing because of the high modulation efficiency, better temperature stability, and low bias. But the large footprint, high cost and fabrication difficulty make it hard in optoelectronic integrated circuits [10], [11]. Recently, IV material based-modulators, such as silicon-based modulators, attracted research interests for the high integration of photoelectric devices. However, the silicon-based modulator always has a low modulation rate and a large footprint because of the weak electro-absorption effect in silicon [13], [14]. Thus, how to design a silicon-based modulator with ultra-compact device footprint, low energy consumption, high modulation speed, and broad working bandwidth is always a challenge in on-chip optical communications and interconnects.

To decrease the footprint and improve the modulation efficiency, many silicon-based modulators with novel structures have been proposed, such as high-Q resonators [15], [16], hybrid semiconductors [17], [18], and surface plasmon polariton structures [19]. The high-Q optical resonator may increase the modulation depth and reduce the footprint, but it suffers from narrow working bandwidth and poor thermal stability [15]. Integrating semiconductors into the silicon-based modulator may partially resolve these issues, but the difficult fabrication process will enlarge the size of the modulator to micrometers [20], [21]. Surface plasmon polariton modulators can improve the modulation efficiency and reduce the footprint that makes the modulator compact, but the large loss and low modulation speed are inevitable because of the additional losses induced by surface plasmon resonance [19]. One of the most enticing promises that delimit further development of silicon-based modulators might come from the lack of suitable optical materials which have high electric-to-optic conversion efficiency.

Graphene, a two-dimensional nanomaterial composed of carbon atoms with hexagonal lattices, has aroused growing interest by virtue of its excellent electrical, optical, thermal, and mechanical properties, which may have great potential in designing graphene-based optoelectronic devices [22], [23], [24], [25], [26]. Because of the zero-band-gap structure, graphene has a wide response range, and its optical absorption ranges from the visible to the infrared band [27], [28]. Electrons in graphene behave as massless particles, and the mobility is up to 200,000 cm² V⁻¹ s⁻¹ at room temperature, implying that the graphene-based devices may have a large theoretical modulation bandwidth [29], [30]. Furthermore, the complex conductivity of graphene can be easily adjusted by chemical and electrostatic doping, which can be realized by applying a gate bias voltage to the graphene layers [31]. These exceptional electrical and optical properties, excellent thermal conductivity, stability, and carrier mobility, make graphene a priority electro-optic material for electro-absorption modulators [32], [33], [34]. In 2011, Liu et al. proposed the first graphene-on-silicon modulator (GOSM) by placing single-layer graphene on a silicon waveguide, and the 3 dB modulation bandwidth up to 1 GHz and modulation depth of 0.18 dB/μm are achieved [24]. Subsequently, various structures have been proposed to enhance the modulation efficiency of GOSM, such as silicon-based slot waveguides [35], double-layer graphene waveguides [36], and Mach-Zehnder interferometers [37], etc. However, because the optical mode field is much larger than the thickness of graphene, the overlap between graphene and the optical field is quite small, resulting in a weak light-graphene interaction and limiting the further enhancement of modulation efficiency. Researchers have devoted to improving the efficiency of light-graphene interaction, including micro-ring resonances [38], [39], localized plasmon resonances [40], and plasmonic waveguides [41], [42]. For the micro-ring resonance modulators, the interaction only happens in a few micro-meter resonator footprints, but the modulation depth is usually restricted to ∼10 dB [43]. Moreover, it always suffers from working bandwidth limitations and temperature instability. Localized plasmon resonance modulators may reduce the sensitivity to temperature and increase the modulation rate to several hundred gigahertz, but the working bandwidth is limited [44]. Using the field confinement of subwavelength structure, plasmonic waveguide modulators have a high light–matter interaction, but the requirement of a few micro-meters active graphene footprints will cause a high capacitance and energy consumption [45]. Thus, it is still a challenge to achieve broadband GOSM with high modulation depth, small footprint, and low energy consumption.

Hybrid plasmonic waveguide (HPW) with great potential in confining optical mode field at subwavelength scale has been applied in many fields, such as optical absorber [46], optical sensor [47], and optical tweezers [48]. Its sandwich structure composes a metal layer, a high index material layer, and a low index material layer, and the optical mode field is highly confined in the low-index layer due to the high index difference between the layers [49], [50]. Recently, HPW has been used in GOSM to improve modulation efficiency, but the requirement of larger active graphene footprint remains a constraint [51]. In this work, we propose a broadband GOSM with orthogonal HPWs. The HPWs are arranged in the two orthogonal directions on the cross-section of the modulator to confine optical mode fields and lower the requirement of the active area of graphene, which can reduce the capacitance and energy consumption. After optimizing the parameters, this modulator has a modulation depth of 26.20 dB/μm for transverse magnetic and achieves a 3 dB modulation bandwidth of 462.77 GHz with only 0.33 μm² active region and 2.82 fJ/bit energy consumption at 1.55 μm. Besides, further studies also suggest that this modulator has a wide working wavelength ranging from 1.3 to 2 μm. By virtue of the excellent modulation performance, this GOSM is expected to be applied in optical interconnects, photonic integrated chip, and other fields.

2 Design of the GOSM with orthogonal HPWs

The 3D schematic configuration and the cross-sectional view of the proposed GOSM are displayed in Figures 1(a) and (b), respectively. The modulator is composed of a graphene-hBN-graphene layer, two high index silicon (Si) layers, and two metallic silver (Ag) layers, which are inserted into a silica (SiO₂) waveguide acting as the protective and substrate layer. The two Si and two Ag layers with the same width w and thickness t₁ are interleaved in the modulator and separated by a low index SiO₂ layer (the thickness is t₂) to construct orthogonal HPWs. This orthogonal structure can compress the light field twice on the cross section of modulator to achieve the strong confinement of optical mode fields due to the high index difference between the layers. The two graphene layers are separated by hBN and placed in the middle of the modulator, serving as the active medium for light absorption. The gold is used as the electrode to introduce external electric fields. Because the optical absorption of graphene relates to the chemical potential μ_c (Fermi level) which can be adjusted by external voltages, we can modulate optical signals by regulating the voltage on gold electrodes. The refractive index of the materials, Ag, Si, and SiO₂, are measured by Palik and can be obtained from the FDTD Solutions directly [52]. At the working wavelength of 1.55 μm, the relative permittivity of hBN (ε_hBN) is 2.94 measured by Woessner, which can be used to prevent graphene from being chemically doped by other materials and enhanced the charge mobility [53].

Figure 1:

Structure of the GOSM with orthogonal HPWs. (a) Three-dimensional schematic illustration and (b) cross-section of the modulator.

Light coupling input/output of the modulator can be realized by conventional waveguide grating couplers, which couples lights through the Bragg diffraction of grating if the light incidents on the grating with a specific angle [54]. After lights are coupled into the modulator, the orthogonal HPWs confines the optical mode field to the size of subwavelength to enhance the light-graphene interaction, and the optical absorption changes with the shift of the chemical potential μ_c adjusted by external voltages. Figure 2 (a) shows the real and imaginary part changes of the effective refractive index (n_eff) when the chemical potential changes from 0.2 to 0.8 eV. As shown in Figure 2(a), the imaginary part of n_eff increases sharply first and then decreases, which has a maximum at the chemical potential of μ_c = 0.51 eV and a minimum at the chemical potential of μ_c = 0.41 eV, suggesting that the proposed modulator achieves the maximum and minimum absorption at μ_c = 0.51 eV and μ_c = 0.41 eV. Because the real part difference of n_eff between μ_c = 0.51 eV and μ_c = 0.41 eV is 0.027, the phase change of the incident light is small and can be ignored. Thus, μ_c = 0.51 eV and μ_c = 0.41 eV are chosen as the ‘OFF’ state and ‘ON’ states at the working wavelength of 1.55 μm. Figure 2(b) is the corresponding optical absorption curve, the light field distribution of ‘ON’ and ‘OFF’ states in the illustration shows the light field is well confined to the subwavelength region, which can enhance the light-graphene interaction to improve the performance of modulator. Hence, the loss of modulator mainly includes the insertion loss caused by the optical absorption at ‘ON’ states and the coupling loss of light, the coupling loss of light consists the coupling loss of waveguide grating couplers and the connection loss between the coupler and the modulator. The connection loss can be calculated by 10log10(S1/S2), where S₁ is the light input areas of the modulator, and S₂ is the light output areas of the waveguide grating coupler. If the coupler is exactly aligned with the modulator and S2=S1=2wt1, the connection loss can be ignored [55]. Because the coupling efficiency of conventional waveguide grating couplers can higher than 50%, the coupling loss of modulator may below 6 dB, including coupling input loss and coupling output loss [55], [56].

Figure 2:

(a) Real and Imaginary part of n_eff. (b) Attenuation constants at different μ_c with the width of w = 100 nm and the thickness of t₁ = 20 nm.

Here, graphene is modeled as an anisotropic material because of the single layer of carbon atoms and the π electrons, which make the electric conduction can only achieve in the plane [1]. Thus, the conductivity of graphene can be divided into the in-plane conductivity σ_|| and the out of-plane conductivity σ_⊥. The in-plane conductivity σ_|| is parallel to the graphene plane, which can be approximately calculated by the Kubo formalisms as follows [51], [57], [58]:

where e is the charge of electrons, k_B is the Boltzmann constant, T is the temperature, ♄ is the reduced Planck constant, ω is the angular frequency, and τ is the interband relaxation time.

The permittivity used for the out-of-plane direction in graphene ε_⊥(ω) is 2.5, while the in-plane relative permittivity ε_||(ω) of graphene as the function of chemical potential can be calculated by [37], [59]:

where ε₀ is the permittivity in vacuum, and d is the thickness of graphene. The calculation results with λ = 1550 nm, T = 300 K, τ = 0.1 ps are shown in Figure 3. We can see that the permittivity ε(ω) dramatically changes as the increase of chemical potential, which means that the effective mode index (n_eff) of the modulator can be adjusted by altering the chemical potential of graphene.

Figure 3:

Permittivity of graphene as the function of chemical potential.

More interesting, the chemical potential of graphene will shift with external voltages and can be expressed as [1], [31]:

where vF = 0.9ℒ106 m/s is the Fermi velocity, V_g is the applied basis voltage, V₀ is the voltage offset, d is the thickness of hBN, e is the charge of electrons, and ε_hBN is the permittivity of hBN. Thus, the n_eff of the modulator can be adjusted by the applied voltage V_g, which can be used to modulate optical signals.

To quantify the performance of the modulator, we define the n_eff, attenuation constant (α), and modulation depth (MD), as follows [60], [61]:

where, λ₀ is the wavelength in free space, β is the propagation constant, Im(n_eff) is the imaginary part of n_eff. α (ON) and α (OFF) are the values of α at ‘ON’ and ‘OFF’ states respectively.

3 Simulation results and analysis

3.1 Effect of the geometric parameters of HPWs

The HPW usually composes of three individual layers: a high-index layer, a metal layer, and a low-index layer. Here, we will discuss the effects of the thickness t₁ and the width w of the high-index layer and metallic layer (the high-index layer and metallic layer have the same width and thickness), which is critical in determining the n_eff of modulator. In the simulations, the thickness of graphene and hBN are respectively fixed as 0.7 nm (the thickness of single graphene layer [1], [37], [62], [63], [64]) and 5 nm, while the width and the thickness of hBN are respectively set as 100 and 5 nm. Figure 4(a) shows the imaginary part changes of n_eff with the increase of t₁ and μ_c (w is fixed to 100 nm). With the t₁ increases from 10 to 55 nm, the imaginary part of n_eff increases sharply and then decreases with a peak value at t₁ = 12 nm because the large t₁ will induce a weak optical mode confinement. When the t₁ is set as 20 nm, the n_eff changes with the increase of w and μ_c, as presented in Figure 4(b). Similar to Figure 4(a), the imaginary part of n_eff also has a peak value at w = 30 nm, but the changes are smaller than t₁, which means that the influence of t₁ on the refractive index is much greater than w. To comprehensively consider the effect of the thickness t₁ and width w on the modulation performance, we calculate the attenuation constants α at ‘ON’ and ‘OFF’ states. Figure 5(a) and (b) show the shift of α at the chemical potential of μ_c = 0.41 eV and μ_c = 0.51 eV with the increase of t₁ and w, which can be used to calculate modulation depth. As shown in Figure 6, the maximum modulation depth is 16.55 dB/μm, where w = 240 nm, and t₁ = 12 nm.

Figure 4:

Imaginary part of n_eff (a) at different t₁ and μ_c, when w = 100 nm and λ = 1550 nm (b) at different w and μ_c, when t₁ = 20 nm and λ = 1550 nm.

Figure 5:

Shifts of (a) α at μ_c = 0.41 eV, (b) α at μ_c = 0.51 eV with the increase of t₁ and w.

Figure 6:

Shifts of modulation depth with the increase of t₁ and w.

Besides, the thickness of the low-index SiO₂ layer (t₂) also has a positive effect on the performance. Here, we set w = 240 nm, t₁ = 12 nm and analyze the effect of the thickness of SiO₂ layer (t₂) on α and modulation depth. As presented in Figure 7(a), with the t₂ increases from 13 to 25 nm, α(OFF) increases sharply first and then decreases with a peak value of 37.60 dB/μm at t₂ = 16 nm, while α(ON) increases first and then decreases slowly. Thus, the maximum modulation depth is calculated to be 26.12 dB/μm at t₂ = 16 nm, as shown in Figure 7(b).

Figure 7:

(a) Changes of α at ‘ON’ and ‘OFF’ states, and (b) modulation depth with the increase of t₂.

3.2 Effect of the width of graphene

As the active medium of GOSM, the geometric parameters of graphene will directly affect the performance because the optical signals are modulated by adjusting the permittivity of graphene with external voltages. Here, the width of graphene will be discussed, and the results are shown in Figure 8. The thickness of the Si, SiO₂ and graphene are set as 12, 16 and 0.7 nm respectively, while the width of Si is fixed as 240 nm. Figures 8(a) and (b) show the variation curves of α and modulation depth with the increase of the width of graphene. As shown in Figure 8(a), with the width of graphene increases from 10 to 210 nm, α(OFF) increases first and then tends to be a stable value of 37.72 dB/μm, while α(ON) has an equilibrium value of 11.50 dB/μm. These indicating that the enhancement of modulation depth is very limited as the width greater than 110 nm, as shown in Figure 8(b). When the width of graphene is set to be 110 nm, the modulation depth of 26.20 dB/μm is obtained.

Figure 8:

(a) Changes of α at ‘ON’ and ‘OFF’ states, and (b) modulation depth with the increase of the width of graphene.

3.3 Performance of the modulator

To study the modulation performance, we discuss the operating voltage, 3 dB modulation bandwidth, and energy consumption of the modulator at the working wavelength of 1.55 μm. In the simulations, Si layers have a thickness of 12 nm and a width of 240 nm, and the SiO₂ has a thickness of 16 nm, while the graphene has a width of 110 nm. As mentioned before, the chemical potential can be flexibly regulated by external voltages, and the voltage difference of ΔV = 2.56 V can switch the ‘ON’ (μ_c = 0.41 eV) and ‘OFF’ (μ_c = 0.51 eV) states. To estimate 3 dB modulation bandwidth and energy consumption, the modulator is regarded as an equivalent graphene-hBN-graphene capacitor model [24], [58]. So, the 3 dB modulation bandwidth and energy consumption can be calculated by f3dB=1/2πRC and Ebit=1/4CΔV2, respectively [44], [51], [61]. Here, C is the capacitance of capacitor, which can be obtained from C=ε0εhBNS/d. ε₀ is the permittivity of vacuum, d is the thickness of hBN, and S is the footprint of modulator (the overlap area of two graphene layers separated by hBN). Due to the length of the active region in the modulator is 3 μm, the S of 0.33 μm² (0.11 × 3 μm) and the C of 1.72 fF are obtained. R is the resistance, which mainly comes from gold-graphene contact resistance R_c and graphene sheet resistance R_g. Because the graphene sheet resistance is very small and can be safely neglected for the tiny footprint, we chose R=Rc=200Ω as the contact resistance to facilitate the comparison with recently reported GOSMs [51]. Hence, the 3 dB modulation bandwidth and energy consumption are calculated to be 462.77 GHz and 2.82 fJ/bit, respectively.

Besides, the characteristic of working wavelengths also plays a crucial role in evaluating the performance of modulators. To verify the broadband property, we investigate the wavelength-dependent response of the modulator. Figure 9(a–d) show the characteristics of modulation depth, voltage difference, energy consumption, and 3 dB modulation bandwidth with the working wavelength ranges from 1.3 to 2 μm, respectively. As shown in Figure 9, the modulation depth, voltage difference, and energy consumption decrease with the increase of working wavelengths, while the 3 dB modulation bandwidth increases. These changes mainly depend on the permittivity of graphene. As the working wavelength increases from 1.3 to 2 μm, the modulation depth decreases from 38.40 to 8.58 dB/μm, which still higher than conventional modulators and satisfies the needs of practical applications. The voltage difference and the energy consumption decrease from 3.40 to 1.39 V, and 4.97 to 0.83 fJ/bit respectively because the working point of graphene will shift towards smaller voltage, which means that the optical signals at higher wavelength requires less energy for modulation. Unlike modulation depth, voltage difference, and energy consumption, the 3 dB modulation bandwidth mainly relates to the permittivity of hBN. Thus, the variation of 3 dB modulation bandwidth is very small because the permittivity of hBN is approximately maintained a constant over the whole wavelength range. These results indicate that the proposed modulator has an excellent modulation performance and a wide wavelength tolerance, which are greatly superior to conventional GOSMs.

Figure 9:

Broadband working characteristics of the modulator. (a) Modulation depth, (b) voltage difference between ‘OFF’ and ‘ON’, (c) energy consumption, and (d) 3 dB modulation bandwidth.

4 Conclusion

In conclusion, we have proposed a GOSM based on orthogonal HPWs at near-infrared wavelengths. With the strong optical mode field confinement of HPWs, the light-graphene interaction is significantly improved, which directly affect the modulation performance. The results show that the proposed modulator has a modulation depth of 26.20 dB/μm and a 3 dB modulation bandwidth of 462.77 GHz with only 0.33 μm² footprint and 2.82 fJ/bit energy consumption at the working wavelength of 1.55 μm. Even working in a wide wavelength range from 1.3 to 2 μm, the modulator also has excellent modulation performance with the voltage difference is below 3.40 V, the modulation depth is over 8.58 dB/μm, the 3 dB modulation bandwidth is over 462.42 GHz, and the energy consumption is below 4.97 fJ/bit. Compared with conventional GOSMs, this modulator presents priority characteristics in operation bandwidth, energy consumption, and 3 dB modulation bandwidth, which may have great potential prospects in various integrated interconnects and optoelectronic devices.