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Waveguide-based electro-absorption modulator performance: comparative analysis

Rubab Amin, Jacob B. Khurgin, and Volker J. Sorger

Open Access Open Access

Abstract

Electro-optic modulators perform a key function for data processing and communication. Rapid growth in data volume and increasing bits per second rates demand increased transmitter and thus modulator performance. Recent years have seen the introduction of new materials and modulator designs to include polaritonic optical modes aimed at achieving advanced performance in terms of speed, energy efficiency, and footprint. Such ad hoc modulator designs, however, leave a universal design for these novel material classes of devices missing. Here we execute a holistic performance analysis for waveguide-based electro-absorption modulators and use the performance metric switching energy per unit bandwidth (speed). We show that the performance is fundamentally determined by the ratio of the differential absorption cross-section of the switching material’s broadening and the waveguide effective mode area. We find that the former shows highest performance for a broad class of materials relying on Pauli-blocking (absorption saturation), such as semiconductor quantum wells, quantum dots, graphene, and other 2D materials, but is quite similar amongst these classes. In this respect these materials are clearly superior to those relying on free carrier absorption, such as Si and ITO. The performance improvement on the material side is fundamentally limited by the oscillator sum rule and thermal broadening of the Fermi-Dirac distribution. We also find that performance scales with modal waveguide confinement. Thus, we find highest energy-bandwidth-ratio modulator designs to be graphene, QD, QW, or 2D material-based plasmonic slot waveguides where the electric field is in-plane with the switching material dimension. We show that this improvement always comes at the expense of increased insertion loss. Incorporating fundamental device physics, design trade-offs, and resulting performance, this analysis aims to guide future experimental modulator explorations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Figures (5)
Fig. 1
Fig. 1 Schematic of the waveguide structure with a biasing scheme to the active region (drive voltage, Vd and current (charge) flow into the active layer, Id), the charge -Q in the active layer and + Q in the gate is induced, and propagation direction (indicated with blue arrow, βeff). The associated electric field in the y-direction is shown. da is the width of the active region and teff is the effective thickness. The relevant coordinate system used in this work is also included to accompany the text.

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Fig. 2
Fig. 2 Schematics of different absorption modulation mechanisms considered in this work for the OFF and ON states are for the attenuated and non-attenuated modulator output powers, respectively (a) quantum dot (QD), (b) quantum well (QW) or graphene, (c) excitons in transition metal dichalcogenides (TMDs), and (d) free carrier based modulation schemes (e.g. Si, ITO).

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Fig. 3
Fig. 3 Absorption modulation results for different material classes, for charge-driven active materials. (a) Normalized absorption, α’ = αneffteff as a function of carrier concentration, n2D (cm−2); (b-c) Optical absorption vs. (b) injected charge, Q' (pCμm−2); and (c) drive voltage, Vd (Volts); respectively. (d-j) Physical device performance parameters to obtain 10dB modulation; (d) Modulator length, L' = L/teff; (e) Switching charge, QSW (C/μm2); (f) Electrical device capacitance, C'g (fF/μm2); (g) Switching energy (energy-per-bit function), U'SW (fJ/μm2); (h) 3-dB modulation speed, f3-dB (THz. μm2); (i) Energy-bandwidth ratio, EBR' (fJ/(Thz. μm4)); and (j) Switching voltage, VSW (Volts) for investigated material classes including quantum dots (QD), single and 3-layer quantum well (SQW, 3QW), Graphene, a transition metal dichalcogenide (TMD) material (WSe2), Silicon, and Indium-Tin-Oxide (ITO). The spacing between the active layer and the gate is dgate = 100 nm and εeff = 10 assumed here.

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Fig. 4
Fig. 4 (a-n) Cross-sectional mode profiles for different structures and material classes using FEM analyses, normalized electric field intensity, |E|2 is shown. (o) Corresponding effective cross-sectional modal area, S’eff; and (p) effective thickness, teff. The Si, ITO and graphene structures are chosen form our previous work [26,27]. WSe2 structures are the same in dimensions as graphene ones just changing the active material to WSe2, In0.52Ga0.48As QWs are chosen having 5 nm thickness and 470 nm width, with GaAs separate confinement heterostructure (SCH) and barrier layers.

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Fig. 5
Fig. 5 (a) 10 dB absorption modulator length, L vs. effective thickness, teff for all the comparable material classes with different broadening. (b-g) Relevant device parameters vs. effective mode areas, S'eff for different effective material broadening, γeff (eV) for all the comparable material classes; (b) Switching charge, QSW (C); (c) Capacitance, C (fF); (d) Switching energy (E/bit function), USW (fJ); (e) 3-dB modulation bandwidth, f3-dB (THz); (f) Energy-bandwidth ratio, EBR (fJ/THz); and (g) Switching voltage, VSW (V). The varying amount of broadening is implicit by the materials shown in the legend corresponding to room temperature, i.e. 300 K, and thermal cooling down to 77 K. The corresponding teff and S'eff for the different modes from the previous section are also marked to aid comparing device performances. dgate = 100 nm. Lowering the gate oxide thickness will increase the energy efficiency through improved electrostatics, but also reduce the modulation speed via increased capacitance. All the different modes considered are marked in (a), whereas the different mode performances are scattered across in (b-g) as the markings correspond to the same x-value in all of them. Best EBR is found for graphene and TMD plasmonic slot waveguide-based modulators.

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Tables (1)

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Table 1 Summary of the differential absorption cross sections for different material classes. For parameter definitions refer to the main text.

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Equations (78)

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S eff = 1 n eff 2 ∫ −∞ ∞ ∫ −∞ ∞ n 2 (x,y)( E x 2 + E y 2 )dxdy / E a0 2
α(ω)= σ a (ω) WF S eff N 2D = σ a (ω) t eff N 2D
t eff = S eff /WF= S eff ' /W
t eff ≈ 1 n eff 2 ∫ −∞ ∞ n 2 (x)( E x 2 + E y 2 )dx / E a0 2
α max (ω)L=σ(ω) N 2D L/ t eff =ln(10)≈2.302
α min (ω)L=σ(ω)( N 2D −δ n 2D )L/ t eff =−ln(0.9)≈0.105
δ n 2D L≈2.2 t eff /σ(ω)
Q sw =eWLδ n 2D =2.2eW t eff /σ(ω)=2.2e S eff /σ(ω)F
V sw = d gate eδ n 2D ε 0 ε eff =2.2 e ε 0 ε eff t eff L d gate σ(ω) ≈ e ε 0 ε eff N 2D d gate
L≈2.302 t eff / N 2D σ(ω)
C= ε 0 ε eff WL/ d gate =2.3 ε 0 ε eff d gate W t eff N 2D σ a =2.3 ε 0 ε eff d gate S eff N 2D σ a F
f 3dB =1/2πRC≈ d gate ε 0 ε eff N 2D σ a F 14.5 S eff R
U sw = 1 2 Q SW V SW ≈ e 2 d gate ε 0 ε eff N 2D S eff σ a (ω)F
EBR= U SW / f 3dB ≈14.5 e 2 R S eff 2 σ a 2 (ω) F 2
σ QD (ω)= π α 0 n eff F cv ℏ γ ( m c −1 − m 0 −1 )= π α 0 n eff F cv ℏ 2 / m 0 ℏγ ( m 0 m c −1 )=1.75× 10 −17 c m 2 F cv ℏγ n eff ( m 0 m c −1 )
σ QD (ω)= π α 0 n eff F cv E P ℏω ℏ 2 / m 0 ℏγ ≈3.5× 10 −16 c m 2 F cv ( ℏγ )( ℏω ) n eff
L QD ≈2.3 n eff t eff π α 0 N QD γ F cv ℏ( m c −1 − m 0 −1 )
σ fc (ω)= 4π α 0 n eff ℏ m c γ ω 2 + γ 2 = 4π α 0 n eff ℏ 2 m 0 (ℏ γ eff ) m 0 m c ≈ 7.02× 10 −17 c m 2 (ℏ γ eff ) × m 0 m c
σ fc.max (ω)= 4π α 0 n eff ℏ m c ω × 1 2 =3.53× 10 −17 c m 2 1 ℏω n eff m 0 m c
α QW = 1 1+ β v π α 0 n eff WF N QW S eff F cv H(ℏω− E g )[1− f c ]
d α QW d n 2D = ∂ α QW ∂ f c ∂ f c ∂ E fc d E f d n 2D =− σ QW (ω) t eff −1
σ QW (ω)= F cv 1+ β v π α 0 n eff π ℏ 2 m c kT exp[ ( E c − E f )/kT ][ 1+exp(− E f /kT) ] { 1+exp[ ( E c − E f )/kT ] } 2
σ QW−max (ω)= F cv 1+ β v π α 0 n eff π ℏ 2 2 m c kT ≈2.75× 10 −17 c m 2 1 kT n eff m 0 m c F cv 1+ β v
α QW, max (ω)L= F cv 1+ β v π α 0 n eff N QW L t eff [1− f c,min ]=ln(10)≈2.302
α QW, min (ω)L= F cv 1+ β v π α 0 n eff N QW L t eff [1− f c,max ]=−ln(0.9)≈0.105
Q SW =eWL( n 2D.max − n 2D,min )=eW×2.2 t eff N Q n eff π α 0 1+ β v F cv ×3.2 m c kT π ℏ 2 N QW ≈2.2e S eff σ QW (ω)F
σ QW (ω)= F cv 1+ β v π α 0 n eff π ℏ 2 m c 3kT ≈5.4× 10 −17 c m 2 1 3kT n eff m 0 m c F cv 1+ β v
σ QW (ω)= 1 1+ β v π 2 α 0 n eff ℏ 2 m c ℏ γ eff F cv ≈5.4× 10 −17 c m 2 1 ℏ γ eff n eff m 0 m c 1 1+ β v F cv
L QW ≈2.3 n eff t eff π α 0 N QW 1+ β v F cv
σ gr (ω)= π 2 α 0 n eff × 1 kT exp[ (ℏω/2− E fc )/kT ] { 1+exp[ (ℏω/2− E fc )/kT ] } 2 ℏ 2 v F 2 2 E f
σ gr (ω)= π 2 α 0 n eff × 1 kT 1 4 ℏ 2 v F 2 ℏω ≈9.7× 10 −17 c m 2 1 n eff kT
σ gr (ω)= π 2 α 0 n a × 1 ℏ γ eff 1 4 ℏ 2 v F 2 ℏω ≈9.7× 10 −17 c m 2 1 ℏ γ eff
σ ex (ω)=2π α 0 ℏ m c γ ex ≈3.53× 10 −17 c m 2 1 ℏ γ ex ( m 0 m c −1 )
α QW = π α 0 n eff t eff F cv 1+ β v N QW ( [ exp( π ℏ 2 n 2D N QW m c kT )−1 ] e − E c kT +1 ) −1
α gr = π α 0 n eff t eff 1 e ℏ v F π n 2D −ℏω/2 kT +1
α QD = π α 0 n eff t eff F cv ℏ 2 N QD m 0 kT kT ℏ γ eff ( m 0 m c −1 )( 1− n 2D N QD )
α ex (ω)= 8 α 0 n eff t eff ℏ 2 a ex −2 m 0 kT kT ℏ γ eff m 0 m c ( 1−π a ex 2 n 2D /4 )
α fc (ω)= 4π α 0 n eff t eff ℏ m c γ ω 2 + γ 2 n 2D
EBR= U SW / f 3dB =π Q SW 2 R
EBR=π Q SW 2 R~π e 2 R S eff '2 × ( m c m 0 ) 2 ( ℏ γ eff ) 2 ×2× 10 17 μ m −4 ≈5× 10 4 R 50Ω S eff '2 μ m 4 × ( m c 0.067 m 0 ) 2 ( ℏ γ eff 100meV ) 2 fJ/THz
P= ∫ −∞ ∞ ∫ −∞ ∞ S ¯ z (x,y)dxdy = 1 2 n eff η 0 ∫ −∞ ∞ ∫ −∞ ∞ n 2 (x,y)( E x 2 + E y 2 )dxdy = n eff E a0 2 2 η 0 S eff
g(ℏω)= 1 πℏ γ Δ ω 2 + γ 2
d N ph (x,y) dt = 2π ℏ e 2 4 E eff 2 r 12 2 g(ℏω) N a (x,y)
dU(x,y) dt =ℏω d N ph (x,y) dt = e 2 r 12 2 E eff 2 (x,y) 2 ω ℏγ γ 2 Δ ω 2 + γ 2 N a (x,y)
dP dz =− ∫ active ∫ −W/2 W/2 dU dt dxdy = − e 2 r 12 2 2 ω ℏγ γ 2 Δ ω 2 + γ 2 ∫ active ∫ −W/2 W/2 N a (x,y) E eff 2 (x,y)dxdy
dP dz = − e 2 r 12 2 2 ω ℏγ γ 2 Δ ω 2 + γ 2 ∫ active ∫ −W/2 W/2 N a (x,y) E eff 2 (x,y)dxdy N a0 E a0 2 N a0 2 η 0 P/ S eff n eff
S a,eff = ∫ active ∫ −W/2 W/2 N a E eff 2 dxdy / E a0 2 N a0
dP dz =− e 2 r 12 2 η 0 n eff ω ℏγ γ 2 Δ ω 2 + γ 2 S a,eff S eff N a0 P=− σ a (ω) S a,eff S eff N a0 P
σ a (ω)= e 2 η 0 ℏ n eff r 12 2 ω γ γ 2 Δ ω 2 + γ 2 = 4π α 0 n eff r 12 2 ω γ L(ω)
α(ω)= σ a (ω) S a,eff S eff N a0
S a,eff = N 2D N a0 WF
F= ∫ active N a (x) N a0 ∫ −W/2 W/2 E eff 2 (x,y) E a0 2 dydx /W ∫ active N a (x) N a0 dx
F= W −1 ∫ −W/2 W/2 E eff 2 ( x a ,y) E a0 2 dy
α(ω)= σ a (ω) WF S eff N 2D = σ a (ω) t eff N 2D
t eff = S eff /WF= S eff ' /W
t eff ≈ 1 n eff 2 ∫ −∞ ∞ n 2 (x)( E x 2 + E y 2 )dx / E a0 2
r 12 = F cv 2 P cv m 0 ω
m 0 m c =1+ 2 P cv 2 m 0 E g ≈ E p E g
σ QD (ω)= π α 0 n eff F cv ℏ γ ( m c −1 − m 0 −1 )= π α 0 n eff F cv ℏ 2 / m 0 ℏγ ( m 0 m c −1 )=1.75× 10 −17 c m 2 F cv ℏγ n eff ( m 0 m c −1 )
ε r (ω,x,y)= ε ∞ − N e (x,y) e 2 ε 0 m c 1 ω 2 +iωγ
dU(x,y) dt =ω ε 0 ε im (ω,x) E eff 2 (x) 2 = N e (x,y) e 2 m c γ ω 2 + γ 2 E 2 (x,y) 2
σ fc (ω)= 4π α 0 n eff ℏ m c γ ω 2 + γ 2 = 4π α 0 n eff ℏ 2 m 0 (ℏ γ eff ) m 0 m c ≈ 7.02× 10 −17 c m 2 (ℏ γ eff ) × m 0 m c
E f =kTln[ exp( π ℏ 2 n 2D N QW m c kT )−1 ]
d n 2D d E f = N QW m c π ℏ 2 1 1+exp(− E f /kT)
d n 2D (y) dt = 2π ℏ e 2 4 E eff 2 ( x QW ,y) r cv 2 ρ 2D (ℏω)[1− f c ( E c )]
d n 2D (y) dt = 2π ℏ e 2 4 E eff 2 1 2 P cv 2 m 0 2 ω 2 F cv N QW m r π ℏ 2 H(ℏω− E g )[1− f c ]
d n 2D (y) dt = 1 1+ β v e 2 4ℏ E eff 2 2ℏω F cv N QW H(ℏω− E g )[1− f c ]
α QW = 1 1+ β v π α 0 n eff WF N QW S eff F cv H(ℏω− E g )[1− f c ]
ρ 2D (ℏω)= 1 2π ω ℏ v F 2
d n 2D (y) dt = 2π ℏ e 2 4 E eff 2 1 2 v F 2 ω 2 1 2π ω ℏ v F 2 [1− f c ]= e 2 E eff 2 8 ℏ 2 ω [1− f c ]
α gr = π α 0 n eff t eff [1− f c ]
d α gr d n 2D = ∂ α QW ∂ f c ∂ f c ∂ E fc d E f d n 2D =− σ gr (ω) t eff −1
σ gr (ω)= π 2 α 0 n eff × 1 kT exp[ (ℏω/2− E fc )/kT ] { 1+exp[ (ℏω/2− E fc )/kT ] } 2 ℏ 2 v F 2 2 E f
a ex = 2π ε eff ε 0 ℏ 2 e 2 m r
α ex (ω)=4 σ ex (ω)/π a ex 2 t ' eff
σ ex (ω)=4π α 0 r 12 2 ω γ ≈2π α 0 ℏ γ ( m c −1 − m 0 −1 )
α(ω, n 2D )=α(ω,0)×(1− n 2D / n bl )= 8 α 0 a ex 2 t ' eff ℏ m c γ ( 1−π a ex 2 n 2D /4 )= 8 α 0 a ex 2 t eff ℏ m c γ − σ ex ' n 2D
σ ex (ω)=2π α 0 ℏ m c γ ex ≈3.53× 10 −17 c m 2 1 ℏ γ ex ( m 0 m c −1 )
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