II. Graphene Nanophotonic Modulators for Near-Infrared Applications
2.3. Theoretical Investigation of Inverted-Rib-Type Silicon Waveguides Integrated with
2.3.2. Analysis Results
Figure 2.3.4. Dependence of the Si strip width ws on the IRT Si waveguide. (a) Relation of the propagation loss of the IRT Si waveguide to ws for ta = 10 nm, ts = 80 (red), 100 (black), 120 nm (blue), respectively. (b)-(d) Mode profile of the IRT Si waveguide for ts = 100 nm, ta = 10 nm, and ws = 100, 190, and 400 nm.
The estimated relation of the propagation loss of the IRT Si waveguide to ts is shown in Figure 2.3.5(a) for ta = 10 nm and three ws values ws = 160, 190, and 220 nm. The propagation loss of the waveguide for ws = 190 nm reaches the maximum, 0.422 dB/μm at ts = 60 nm. The Ex distributions of the TE mode of the IRT Si waveguide are shown in Figure 2.3.5(b)-(d) when ws is 190 nm and ts is 30, 60, and 210 respectively. If a-Si slab is too thin (i.e. ts = 30 nm, which is described in Figure 2.3.5(b)), the Si slab confines Ex weakly in the slab, Ex is mainly distributed only around the Si strip. In contrast, as the Si slab becomes thicker, the Ex becomes confined a-Si slab more. As described in Figure 2.3.5(d), Ex is dominantly distributed inside the a-Si slab. Therefore, the interaction between double graphene layers and the waveguide mode is not so strong for the both cases. As shown in Figure 2.3.5(c), Ex is strongly confined in the region where double graphene layers are located, which results in large light- graphene interaction. So, the modulation depth is maximum at ts = 60 nm. From the results in Figure 2.3.4 and 2.3.5, ws and ts can be determined to 190 nm and 160 nm respectively, to enable the IRT Si waveguide with double graphene layers to achieve the largest modulation depth.
Figure 2.3.5. Dependence of the thickness of a-Si slab ts on the IRT Si waveguide. (a) Relation of the propagation loss of the IRT Si waveguide to ts for ta = 10 nm, ws = 160 (red), 190 (black), 220 nm (blue), respectively. (b)-(d) Mode profile of the IRT Si waveguide for ws = 190 nm, ta = 10 nm, and ws = 30, 60, and 210 nm.
Lastly, the dependence of the propagation loss of the IRT Si waveguide on ta is considered;
the calculated relation between the propagation loss and ta is shown in Figure 2.3.6(a) for ws = 190 nm and ts = 60 nm. As ta increases from 10 nm to 50 nm, the propagation loss of the waveguide decreases monolithically. Because when ta is thick, the top graphene layer interacts with weaker Ex. Actually, ta
affects not only the propagation loss of the waveguide mode but also the driving voltage and the electrical bandwidth of the device. The graphene-Al2O3-graphene vertical stack functions as a parallel plate capacitor, the capacitance of the graphene-Al2O3-graphene capacitor Cg is inversely proportional to ta. However, the driving voltage is proportional to ta. Therefore, ta should be increased for the larger electrical bandwidth in terms of capacitance but decreased for the smaller driving voltage. From the result in Figure 2.3.6(a), ta can be adjusted between 10 and 50 nm such that a tradeoff between the driving voltage and the electrical bandwidth can be achieved without any significant influence on the modulation depth. There is only 14.3% decrease of the modulation depth of the waveguide when ta is increased from 10 nm to 50 nm. So, ta is tentatively determined to 10 nm for the largest modulation depth of the IRT Si waveguide with double graphene layers.
Figure 2.3.6. Relation the propagation loss to Al2O3 thickness and the chemical potential of graphene. (a) Propagation loss of the IRT Si waveguide as a function of ta for ws = 190 nm and ts = 60 nm. (b) Propagation loss of the IRT Si waveguide as a function of μc for ws = 190 nm, ts = 60 nm, and ta = 10 nm. By this result, the modulation depth of the modulation is calculated to 0.41 dB/μm.
Based on the determined structural parameters of ws = 190 nm, ts = 60, and ta = 10 nm, the propagation loss of the IRT Si waveguide with double graphene layers was calculated as a function of the chemical potential of graphene μc. The relation of the propagation loss to μc is shown Figure 2.3.6(b).
As mentioned before, the propagation loss tendency against μc is similar to the imaginary part of the in- plane dielectric constant εxx. The propagation loss of the waveguide is large and almost steady around μc = 0.2 eV. However, it becomes small around 0.6 eV. In addition, the driving voltage on the capacitor for μc = 0.6 eV, which is explained later in detail, is 5.2 V, and the electric field in the Al2O3 layer is 0.46 V/nm. The magnitude of the electric field on the capacitor is smaller than the dielectric strength of Al2O3 grown by ALD [56], which means there is no breakdown problem for increasing μc to 0.6 eV.
When the driving voltage is tuned such that μc is changing from 0.2 to 0.6 eV, the propagation loss is modulated from 0.42 to 0.01 dB/μm. Therefore, the modulation depth of the EAM is 0.41 dB/μm which is 2.56 times larger than the modulation depths of the previous graphene-based Si waveguide EAM with double graphene layers [44]. Even if the quality of graphene is degraded, in other words, the relaxation time of the graphene τ is reduced from 0.1 to 0.01 ps, the propagation loss of the waveguide decreases from 0.42 to 0.06 dB/μm when μc increases from 0.2 to 0.6 eV, and the modulation depth of the EAM with degraded graphene layers, is 0.36 dB/μm.
The remain structural parameters for the EAM are the length of the tapering region lt, the width of overlap region of double graphene layers wg, and the spacing between two contact electrodes we. First, the coupling efficiency of the taper was calculated as a function of lt by using FDTD simulation.
The fundamental TE mode of the input waveguide is launched initially. Then the transmission to the TE mode of the IRT Si waveguide except graphene layers was calculated. The relation of the coupling efficiency to lt, which is transmitted power though the IRT Si waveguide, is shown in Figure 2.3.7(a).
The coupling efficiency saturates to -0.25 dB when lt is larger than 4 μm. Therefore, the length for the tapering legion is 4 μm for efficient coupling between the conventional Si waveguide and IRT Si waveguide.
In terms of the modulation depth of the EAM, the width of the overlap region wg should be large enough. However, if the electrical bandwidth is considered, wg need to be small enough. Figure
Figure 2.3.7. (a) Coupling efficiency between the IRT Si waveguide mode and conventional Si waveguide mode. (b) Relation between the propagation loss of the IRT Si waveguide and overlap region width, wg.
The transmission of the EAM is modulated by 3 dB when the length of the IRT Si waveguide lg is 7.3 μm and μc is tuned between 0.2 and 0.6 eV. Then, total device length of the EAM, which is lg+2lt, is 15.3 μm. At that time, the insertion loss including coupling loss from the tapers is 0.6 dB when μc is 0.6 eV. The extinction ratio changes by less than 0.22 dB in the wavelength range between 1.5 and 1.6 μm, since the modulation depth of the EAM changes by less than 0.03 dB/μm at this interval.
Therefore, the graphene-based IRT Si waveguide EAM has a broad optical bandwidth over 100 nm.
For the extinction ratio of 15 dB which is the value of the graphene EOM based on MRR, the length of the IRT Si waveguide is 44.6 μm and the device footprint of the EAM is 3×44.6 μm2. This value is about 50 times smaller than the footprint of graphene EOM based on MRR [47].
The electrical properties of the EAM with an extinction ratio of 3 dB are numerically analyzed as follows. The total capacitance of the EAM Ct is determined by two capacitances. One is the geometrical capacitance Cg from the graphene-Al2O3-graphene parallel plate capacitor, and the other is the quantum capacitance of graphene Cq from the low density of state of graphene. Cg is simply given by Cg = ε0εalgwg/ta. εa is the dielectric constant of Al2O3 in the radio frequency (RF) regime, which is set to 10.3 [16]. Cg is calculated to 133 fF. Then, Cq can be calculated by Equation 2.3.1.
2
q 2 g g
2
g g 2
2 ln 2 1 cosh
2
c B
F B
F
e k T
C l w
v k T e c l w
v
(Equation 2.3.1)
The approximation for the second line in the equation is valid since μc is much larger than kBT. So, Cq
is calculated to 1.38 pF. Then the total capacitance can be evaluated by one geometric capacitance and two quantum capacitance of graphene such that Ct = (1/Cg + 2/Cq)-1 [57] and Ct is 112 fF. In case of the resistance, the total resistance of the EAM Rt is determined by the sheet resistance of graphene rs, and the contact resistance between graphene and contact electrode rc. Rt can be calculated by Rt = [2rc + rs(we + wg)]/lg. The reported values of rs and rc are 125 Ω/sq [9] and 100 Ω∙μm [13], respectively. Then total resistance Rt is equal to 113 Ω. The driving voltage Vd is determined by Cg and Cq which is
total capacitance is reduced such that the larger electrical bandwidth can be achieved with considering little energy consumption increment. For example, if the dimensions of the EAM are ta = 20 nm, wg = 1 μm, and lg = 8.4 μm, then f3dB is evaluated to 46.4 GHz with Eb = 620 fJ/bit.
In conclusion, the EAM based on the IRT Si waveguide has been investigated with simulations and numerical analysis. The IRT Si waveguide is integrated with double graphene layers, which form a graphene-Al2O3-graphene capacitor, in the region where the TE mode of the IRT Si waveguide is strongly confined. The modulation depth of the IRT Si waveguide is 0.41 dB/μm, which is 2.56 times larger than those values of the previous double graphene layers on Si waveguide. The IRT Si waveguide can be fabricated using conventional fabrication processes including CMP, ALD, and PECVD. Also, it has efficient coupling efficiency with the conventional Si waveguides. The device footprint for the extinction ratio of 3 dB is just 3×16.4 μm2. The electrical and optical bandwidth of the EAM is evaluated to 46.4 GHz and 100 nm respectively. Lastly, the energy consumption of the EAM is expected to be smaller than 630 fJ/bit. The proposed EAM would be used as a compact modulator for the silicon photonic integrated circuit and the off-chip optical interconnects in the future.
2.4. Experimental Investigation of the Solid-Electrolyte-Gated Graphene-covered Metal-Insulator-