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Electromagnetic simulation of quantum well structures

Shouyuan Shi, Ge Jin, and Dennis W. Prather

Open Access Open Access

Abstract

We present an auxiliary differential equation Finite-difference Time-domain (ADE-FDTD) approach to numerically model the wave propagation within a gain or absorbing medium such as quantum well structures. Start from traditional quantum electronics theory, the macroscopic susceptibility of the semiconductor is derived and expressed by a multiple-Lorentz-like model based on Prony’s method. With the auxiliary differential equation method each Lorentz-like model can be simulated in the time domain and the induced polarization is then determined by summing all the models. By incorporating the induced polarization into the time-domain Maxwell’s equations, electromagnetic wave propagation in the quantum well medium can be accurately modeled using the FDTD method.

©2006 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. Band structure of the semiconductor.

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Fig. 2.
Fig. 2. FDTD grid sketch map

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Fig. 3.
Fig. 3. Comparison of the real part and image part of χ(ω) as computed by Prony’s method and quantum electronics theory with injection current density N= 6×1018/ cm-3.

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Fig. 4.
Fig. 4. Comparison of the field absorption factor and phase shift factor as computed by ADE-FDTD and analytical result with injection current density a) N= 1×1018/ cm-3 , b) N= 6×1018/ cm-3 , and c) N= 12×1018/ cm-3.

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Fig. 5.
Fig. 5. (a) Multiple quantum well structure, (b) Snapshots of pulse shape propagating in the MQW structure , (c) Absorption factor and phase shift of the MQW structure.

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Tables (1)

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Table 1. Numerical Values Used in the validation Calculations

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Equations (45)

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df e ( k ) dt = − j γE ( t ) ħ [ ρ eh ( k ) − ρ eh * ( k ) ] − f e ( k ) − f e 0 ( k ) τ e
df h ( k ) dt = − j γE ( t ) ħ [ ρ eh * ( k ) − ρ eh ( k ) ] − f h ( k ) − f h 0 ( k ) τ h
dρ eh ( k ) dt = − j γE ( t ) ħ [ f e ( k ) + f h ( k ) − 1 ] − j E k ħ ρ eh ( k ) − ρ eh ( k ) T 2
f e 0 ( k ) = 1 exp ( E e ( k ) − F e k b T ) + 1
f h 0 ( k ) = 1 exp ( E h ( k ) − F h k b T ) + 1
f e ( k ) = f e 0 ( k ) − τ e τ e + τ h [ f e 0 ( k ) + f h 0 ( k ) − 1 ] L ( ħ ω − E k ) I I s 1 + L ( ħ ω − E k ) I I s
f h ( k ) = f h 0 ( k ) − τ h τ e + τ h [ f e 0 ( k ) + f h 0 ( k ) − 1 ] L ( ħ ω − E k ) I I s 1 + L ( ħ ω − E k ) I I s
ρ eh = − jγ [ E ( ω ) ⊗ ( f e ( k ) + f h ( k ) − 1 ) ] j ( ωħ + E k ) + ħ T 2
P ( t ) = Tr [ ργ ] = γ ( ρ eh + ρ eh * )
P ( t ) = ε 0 χ ( t ) ⊗ E ( t )
χ ( ω ) = γ ε 0 E ( ω ) ( ρ eh + ρ eh * )
χ ( ω ) ≈ γ 2 ε 0 ∑ k 1 V [ f e 0 ( k ) + f h 0 ( k ) − 1 ] [ 1 j ( ωħ − E k ) + ħ T 2 − 1 j ( ωħ − E k ) + ħ T 2 ]
D ( E ′ ) = ∑ n m n 2 πħ 2 H ( E ′ − E n )
D r , in ( E ′ ) = m r , i 2 πħ 2 H ( E ′ − E g , in ) , i = h , l ; n = 1,2 , ⋯
m r , i = m n m v m c + m v
E g , in = E g + E c , n + E v , in
χ ( ω ) ≈ ∑ i , n γ i 2 ε 0 ∫ D n ( E ) ( f e 0 + f h 0 − 1 ) [ 1 j ( ωħ − E k ) + ħ T 2 − 1 j ( ωħ + E k ) + ħ T 2 ] dE
γ i 2 = e 2 m 0 2 ω 2 M b 2 δ i
M b 2 = 2 ξ m 0 E g
δ i = { 3 4 [ 1 + E c , n + E v , hn ε − E g ] , i = h 1 4 [ 5 − 3 ( E c , n + E v , ln ) ε − E g ] , i = l
χ ( ω ) ≈ ∑ i , n e 2 M b 2 m 0 2 ω 2 ε 0 ∫ δ i D n ( E ) ( f e 0 + f h 0 − 1 ) • [ 1 j ( ωħ − E k ) + ħ T 2 − 1 j ( ωħ + E k ) + ħ T 2 ] dE
{ tan ( 2 m E n ħ 2 d z 2 ) = V 0 − E n E n , for even mode ( n = 2,4,6 , ⋯ ) cot ( 2 m E n ħ 2 d z 2 ) = − V 0 − E n E n , for odd mode ( n = 1,3,5 , ⋯ )
{ ∇ × E = − μ ∂ H ∂ t ∇ × H = ε ∂ E ∂ t + ∂ P ∂ t
{ μ ∂ H y ∂ t = − ∂ E x ∂ z ε ∂ E x ∂ t + ∂ P ∂ t = − ∂ H y ∂ z
χ ( t ) ≈ 2 γ 2 ħ ε 0 ∑ i , n ∫ δ i D n ( ε ) ( f e 0 + f h 0 − 1 ) sin ( ε ħ t ) exp ( − ω T t ) U ( t ) dε
J ( t ) = ∂ P ( t ) ∂ t = ε 0 ∂ χ ( t ) ∂ t ⊗ E ( t ) = σ ( t ) ⊗ E ( t )
σ ( t ) ≈ − 2 ħ ∑ i , n γ 2 ∫ δ i D n ( ε ) ( f e 0 + f h 0 − 1 ) [ ε ħ cos ( ε ħ t ) − ω T sin ( ε ħ t ) ] • exp ( − ω T t ) U ( t ) dε
σ ( t ) = ∑ i = 1 P C i D i = ∑ i = 1 P C i exp [ ( α i + jω i ) t ] U ( t )
σ ( ω ) = 2 ∑ i = 1 P 2 jωA i − ( A i α i + B i ω i ) ( α i 2 + ω i 2 ) − 2 jωα i − ω 2 = ∑ i = 1 P 2 σ i ( ω )
σ p + E 1 σ p − 1 + E 2 σ p − 2 + ⋯ + E p σ 1 = 0
σ p + 1 + E 1 σ p + E 2 σ p − 1 + ⋯ + E p σ 2 = 0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮
σ N + E 1 σ N − 1 + E 2 σ N − 2 + ⋯ + E p σ N − p = 0
D p + E 1 D p − 1 + E 2 D p − 2 + ⋯ + E p − 1 D 1 + E p = 0
C 1 + C 2 + ⋯ + C p = σ 1
D 1 C 1 + D 2 C 2 + ⋯ + D p C p = σ 2
( D 1 ) 2 C 1 + ( D 2 ) 2 C 2 + ⋯ + ( D p ) 2 C p + σ 3
⋮ ⋮ ⋱ ⋮ ⋮
( D 1 ) N − 1 C 1 + ( D 2 ) N − 2 C 2 + ⋯ + ( D p ) N − 1 C p = σ N
J ( ω ) = ∑ i = 1 P 2 J i ( ω ) = ∑ i = 1 P 2 σ i ( ω ) E ( ω )
{ ∂ 2 J i ( t ) ∂ 2 t − 2 α i ∂ J i ( t ) ∂ t + ( α i 2 + ω i 2 ) J i ( t ) = 2 A i ∂ E x ( t ) ∂ t − 2 ( A i α i + B i ω i ) E x ( t ) J ( t ) = ∑ i = 1 P 2 J i ( t )
E x n + 1 2 ( I ) = E x n − 1 2 ( I ) − Δ t Δ zε ( I ) [ H y n ( I + 1 2 ) − H y n ( I − 1 2 ) ] − Δ t ε ( I ) J n ( I )
H y n + 1 ( I + 1 2 ) = H y n ( I + 1 2 ) − Δ t Δ zμ [ E x n ( I + 1 ) − E x n ( I ) ]
J i n + 1 ( I ) = 2 − ( α i 2 + ω i 2 ) Δ t 2 1 − α i Δ t J i n ( I ) − 1 + α i Δ t 1 − α i Δ t J i n − 1 ( I )
− ( A i α i + B i ω i ) Δ t 2 − 2 A i Δ t 1 − α i Δ t E x n + 1 2 ( I ) − ( A i α i + B i ω i ) Δ t 2 + 2 A i Δ t 1 − α i Δ t E x n − 1 2 ( I )
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