Switching energy limits of waveguide-coupled graphene-on-graphene optical modulators

Steven J. Koester; Huan Li; Mo Li

doi:10.1364/OE.20.020330

Optics Express
Vol. 20,
Issue 18,
pp. 20330-20341
(2012)
•https://doi.org/10.1364/OE.20.020330

Switching energy limits of waveguide-coupled graphene-on-graphene optical modulators

Steven J. Koester, Huan Li, and Mo Li

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Steven J. Koester, Huan Li, and Mo Li, "Switching energy limits of waveguide-coupled graphene-on-graphene optical modulators," Opt. Express 20, 20330-20341 (2012)

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Abstract

The fundamental switching energy limitations for waveguide coupled graphene-on-graphene optical modulators are described. The minimum energy is calculated under the constraints of fixed insertion loss and extinction ratio. Analytical relations for the switching energy both for realistic structures and in the quantum capacitance limit are derived and compared with numerical simulations. The results show that sub-femtojoule per bit switching energies and peak-to-peak voltages less than 0.1 V are achievable in graphene-on-graphene optical modulators using the constraint of 3 dB extinction ratio and 3 dB insertion loss. The quantum-capacitance limited switching energy for a single TE-mode modulator geometry is found to be < 0.5 fJ/bit at λ = 1.55 μm, and the dependences of the minimum energy on the waveguide geometry, wavelength, and graphene location are investigated. The low switching energy is a result of the very strong optical absorption in graphene, and the extremely-small operating voltages needed as the device approaches the quantum capacitance regime.

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Fig. 1 Schematic diagram of graphene-on-graphene modulator. The device consists of a Si waveguide core with SiO₂ cladding on all sides. A voltage is applied between the graphene sheets which modulates the Fermi-level in both layers. In this work, it is assumed that at zero bias, the Fermi-level positions in the two graphene sheets are at ± hc/2λ. The graphene sheets are assumed to be separated by a thin dielectric layer and can be positioned either at the top of the waveguide core (as shown above) or embedded within the center of the waveguide core.

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Fig. 2 Simulation results showing mode profiles of the in-plane electric field magnitude, |E_t|², for three different graphene/waveguide structures. Graphene-on-graphene located (a) in the middle of the core of a fundamental TE mode waveguide, (b) on top of the core of a fundamental TE mode waveguide, and (c) on top of the core of a fundamental TM mode waveguide.

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Fig. 3 Plot of global waveguide optimization for waveguide geometries shown in Fig. 2. The optimization is shown for graphene-on-graphene located (a) in the middle of the core of a fundamental TE mode waveguide, (b) on top of the core of a fundamental TE mode waveguide, and (c) on top of the core of a fundamental TM mode waveguide. In all of the plots, the switching energy normalized to wavelength, E/λ, is plotted on the left axis vs. the normalized width, w, for several values of normalized waveguide thickness, t. In each of the graphs, the absolute value of E can be calculated by multiplying the normalized value on the left axis by λ. On the right axis, E has been plotted for λ = 1.55 μm. In each case, the minimum switching energy, E_min, can be determined from the global optimum point, which occurs for specific values of w and t. In all cases, ER = L = 3 dB.

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Fig. 4 (a) Plot of minimum energy/bit vs. wavelength for a graphene-on-graphene optical modulator for the three different graphene/waveguide configurations considered in this work. In this plot, ER = L = 3 dB. (b) Minimum energy/bit vs. extinction ratio, ER, at λ = 1.55 μm and L = 3 dB.

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Fig. 5 Plot of (a) peak-to-peak voltage and (b) switching energy per bit vs. equivalent oxide thickness (EOT) for graphene-on-graphene optical modulator. For these plots, the graphene is located on the top of a single-TE-mode Si waveguide (Case 2 as described above). In all cases, ER = L = 3 dB.

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Fig. 6 (a) Plot of bandwidth vs. EOT for different wavelengths. (b) Energy-delay product vs. wavelength. Note the results are independent of EOT. In both plots, ER = L = 3 dB.

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Tables (2)

Table I Optimized waveguide parameters used in minimum energy calculations.

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Table II Performance comparison of graphene-on-graphene modulators to literature values for Si/SiGe optical modulators. The graphene modulator parameters were chosen for a single-TE-mode Si waveguide (Case 2) and EOT = 1 nm.

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Equations (24)

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E = \frac{1}{4} C_{m} V_{p p}^{2},

C_{m} = \frac{0.5 c_{o x} c_{Q}}{c_{o x} + 0.5 c_{Q}} \cdot (W_{m} L_{m}),

Δ V = 2 m \frac{k T}{e},

V_{p p} = 2 Δ V + V_{o x} = 4 m \frac{k T}{e} \cdot (1 + \frac{c_{Q}}{2 c_{o x}}),

c_{Q} = \frac{8 e^{2} π k T}{h^{2} v_{F}^{2}} \cdot \ln [2 (1 + \cos h (\frac{h c}{2 k T λ}))],

c_{Q} \approx \frac{4 π e^{2} c}{h v_{F}^{2} λ} .

c_{o x} = \frac{3.9 ε_{0}}{E O T},

L_{m} = \frac{γ}{α_{0}},

L = - 10 \log (T_{\max}) .

T_{\max} = \exp (- \frac{2 γ}{1 + e^{m}}) .

E R = 10 \log (T_{\max} / T {}_{\min}),

T_{\max} / T_{\min} = e^{2 γ β},

β = \frac{e^{m} - 1}{e^{m} + 1} .

γ = \frac{3}{2} \ln (2),

m = \ln (2) .

V_{p p} = 4 \cdot \ln (2) \cdot \frac{k T}{e},

C_{m} = \frac{c_{Q}}{2} \cdot W_{m} \cdot \frac{3 \ln (2)}{2 α_{0}} .

C_{m} = \frac{3 π e^{2} c}{h v_{F}^{2} λ} \cdot W_{m} \cdot \frac{\ln (2)}{α_{0}} .

E \approx \frac{4 c}{v_{F}^{2}} \cdot {(\frac{k T}{e})}^{2} \cdot \frac{π e^{2}}{h α_{0}} \cdot \frac{W_{m}}{λ} .

E \approx 8 \cdot \frac{c λ}{v_{F}^{2}} \cdot {(\frac{k T}{e})}^{2} \cdot \frac{w}{a},

α_{0} = \frac{a}{λ} \cdot \frac{π e^{2}}{2 h},

W_{m} = w λ .

E \approx 1.33 \times 10^{3} \cdot λ \cdot (\frac{w}{a}) fJ,

n \times (H_{1} - H_{2}) = σ [n \times (E \times n)] .

Case #	*Optimal w = W_m/λ*	*Optimal t = T_Si/λ*	Minimum Switching Energy E_min
Case #	*Optimal w = W_m/λ*	*Optimal t = T_Si/λ*	Normalized (fJ/μm)	1.55 μm (fJ)	3.5 μm (fJ)
1	0.236	0.123	0.311	0.482	1.09
2	0.298	0.075	0.508	0.788	1.78
3	0.147	0.195	0.460	0.713	1.61

Device	V_pp (V)	C_m (fF)	E/bit (fJ)	ER (dB)	Ref.
Si disk	1	12	3	3.2	20
SiGe FKE	3	11	25	8	21
Ge FKE	4	25	100	7.5	22
Ge/SiGe QCSE	1	24	6	5	23
Ge/SiGe QCSE	1	3	0.75	3	24
This work	0.15	289	1.7	3	–

Case #	*Optimal w = W_m/λ*	*Optimal t = T_Si/λ*	Minimum Switching Energy E_min
Case #	*Optimal w = W_m/λ*	*Optimal t = T_Si/λ*	Normalized (fJ/μm)	1.55 μm (fJ)	3.5 μm (fJ)
1	0.236	0.123	0.311	0.482	1.09
2	0.298	0.075	0.508	0.788	1.78
3	0.147	0.195	0.460	0.713	1.61

Device	V_pp (V)	C_m (fF)	E/bit (fJ)	ER (dB)	Ref.
Si disk	1	12	3	3.2	20
SiGe FKE	3	11	25	8	21
Ge FKE	4	25	100	7.5	22
Ge/SiGe QCSE	1	24	6	5	23
Ge/SiGe QCSE	1	3	0.75	3	24
This work	0.15	289	1.7	3	–

Abstract

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Tables (2)

Equations (24)

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