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Course: Quantum Electronics
Arpan Deyasi
Topic: Absorption Coefficient of
Quantum Well for Intraband Transition
1
Arpan Deyasi,RCCIIT
1/17/2021
1/17/2021 2Arpan Deyasi,RCCIIT
Absorption Coefficient in Quantum Well
for Intraband Transition
1/17/2021 3Arpan Deyasi,RCCIIT
Absorption Coefficient in Quantum Well
for Intraband Transition
For two consecutive levels, wavefunctions are
2
( ) cosn
n z
z
L L
π
ψ
 
=  
 
2
( ) sinm
m z
z
L L
π
ψ
 
=  
 
1/17/2021 4Arpan Deyasi,RCCIIT
Absorption Coefficient in Quantum Well
for Intraband Transition
Absorption coefficient for quantum well inside
conduction band is
2 2
*2
, ( )0
2 . . . 1
( ) .
. . . .
( ) ( )
z
i j i jr
i j
j i FD FD
q N
z i p j
c n m V
E E f f
ωπ
α
ε ω
δ ω
≠
×
× − − × −
∑



1/17/2021 5Arpan Deyasi,RCCIIT
Absorption Coefficient in Quantum Well
for Intraband Transition
‘i’ th state is associated with wavevetor ‘ki’
‘j’ th state is associated with wavevetor ‘kj’
We also assume corresponding integers are ‘n’ and ‘m’
1/17/2021 6Arpan Deyasi,RCCIIT
Absorption Coefficient in Quantum Well
for Intraband Transition
22
z i z ji p j nk p mk=
2
/2
2
/2
2
cos
2
sin
L
z
L
n z
L L
i p j
d m z
i dz
dz L L
π
π−
  
×  
  = −
    
 −   
     
∫

1/17/2021 Arpan Deyasi,RCCIIT 7
Absorption Coefficient in Quantum Well
for Intraband Transition
if n = 1, m =2 (two lowest consecutive states)
2 2
2
8 64
3 9
i
LHS
L L
=− =
 
1/17/2021 Arpan Deyasi,RCCIIT 8
Absorption Coefficient in Quantum Well
for Intraband Transition
2 2
*2 2
0
, ( )
2 . . . 1 64
( )
. . . . 9
( ) ( )
r
i j
j i FD FD
i j i j
q N
c n V m L
E E f f
ωπ
α ω
ε ω
δ ω
≠
= × × ×
− − × −∑
 


1/17/2021 Arpan Deyasi,RCCIIT 9
Absorption Coefficient in Quantum Well
for Intraband Transition
Oscillator strength between two consecutive states
are given by
2
2 8
* 3
ijO
m Lω
 
= × 
 


1/17/2021 Arpan Deyasi,RCCIIT 10
Absorption Coefficient in Quantum Well
for Intraband Transition
2
*
0
. . .
( ) ( )
. . . .
ij
r
q N
O E
c n m
ωπ
α ω δ ω
ε
= × × ∆ −



1/17/2021 Arpan Deyasi,RCCIIT 11
Absorption Coefficient in Quantum Well
for Intraband Transition
To fit with practical data, Lorentzian lineshape
function is incorporated
2
*
0
2 2
. . .
( )
. . . .
( )
[( )
ij
r
q N
O
c n m
E
E
ωπ
α ω
ε
δ ω
π ω
= × ×
 Γ
∆ − × 
∆ − +Γ 




1/17/2021 12Arpan Deyasi,RCCIIT
References
O.Manaresh, “Semiconductor Heterojunctions
and Nanostructures”, MH
https://ocw.mit.edu/courses/electrical-engineering-and-computer-
science/6-772-compound-semiconductor-devices-spring-2003/lecture-
notes/Lecture15.pdf

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Absorption coefficient of quantum well structure

  • 1. Course: Quantum Electronics Arpan Deyasi Topic: Absorption Coefficient of Quantum Well for Intraband Transition 1 Arpan Deyasi,RCCIIT 1/17/2021
  • 2. 1/17/2021 2Arpan Deyasi,RCCIIT Absorption Coefficient in Quantum Well for Intraband Transition
  • 3. 1/17/2021 3Arpan Deyasi,RCCIIT Absorption Coefficient in Quantum Well for Intraband Transition For two consecutive levels, wavefunctions are 2 ( ) cosn n z z L L π ψ   =     2 ( ) sinm m z z L L π ψ   =    
  • 4. 1/17/2021 4Arpan Deyasi,RCCIIT Absorption Coefficient in Quantum Well for Intraband Transition Absorption coefficient for quantum well inside conduction band is 2 2 *2 , ( )0 2 . . . 1 ( ) . . . . . ( ) ( ) z i j i jr i j j i FD FD q N z i p j c n m V E E f f ωπ α ε ω δ ω ≠ × × − − × − ∑   
  • 5. 1/17/2021 5Arpan Deyasi,RCCIIT Absorption Coefficient in Quantum Well for Intraband Transition ‘i’ th state is associated with wavevetor ‘ki’ ‘j’ th state is associated with wavevetor ‘kj’ We also assume corresponding integers are ‘n’ and ‘m’
  • 6. 1/17/2021 6Arpan Deyasi,RCCIIT Absorption Coefficient in Quantum Well for Intraband Transition 22 z i z ji p j nk p mk= 2 /2 2 /2 2 cos 2 sin L z L n z L L i p j d m z i dz dz L L π π−    ×     = −       −          ∫ 
  • 7. 1/17/2021 Arpan Deyasi,RCCIIT 7 Absorption Coefficient in Quantum Well for Intraband Transition if n = 1, m =2 (two lowest consecutive states) 2 2 2 8 64 3 9 i LHS L L =− =  
  • 8. 1/17/2021 Arpan Deyasi,RCCIIT 8 Absorption Coefficient in Quantum Well for Intraband Transition 2 2 *2 2 0 , ( ) 2 . . . 1 64 ( ) . . . . 9 ( ) ( ) r i j j i FD FD i j i j q N c n V m L E E f f ωπ α ω ε ω δ ω ≠ = × × × − − × −∑    
  • 9. 1/17/2021 Arpan Deyasi,RCCIIT 9 Absorption Coefficient in Quantum Well for Intraband Transition Oscillator strength between two consecutive states are given by 2 2 8 * 3 ijO m Lω   = ×     
  • 10. 1/17/2021 Arpan Deyasi,RCCIIT 10 Absorption Coefficient in Quantum Well for Intraband Transition 2 * 0 . . . ( ) ( ) . . . . ij r q N O E c n m ωπ α ω δ ω ε = × × ∆ −   
  • 11. 1/17/2021 Arpan Deyasi,RCCIIT 11 Absorption Coefficient in Quantum Well for Intraband Transition To fit with practical data, Lorentzian lineshape function is incorporated 2 * 0 2 2 . . . ( ) . . . . ( ) [( ) ij r q N O c n m E E ωπ α ω ε δ ω π ω = × ×  Γ ∆ − ×  ∆ − +Γ     
  • 12. 1/17/2021 12Arpan Deyasi,RCCIIT References O.Manaresh, “Semiconductor Heterojunctions and Nanostructures”, MH https://ocw.mit.edu/courses/electrical-engineering-and-computer- science/6-772-compound-semiconductor-devices-spring-2003/lecture- notes/Lecture15.pdf