Calculation of absorption coefficient in bulk and quantum well structure for interband transition

1. Course: Quantum Electronics
Arpan Deyasi
Absorption Coefficient of Bulk
Semiconductor and Quantum Well
for Interband Transition
1
Arpan Deyasi, RCCIIT
1/17/2021

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Optical Absorption in Semiconductor
E.M wave incident on a solid material undergoes
1. partially reflection at vacuum
2. partially absorbed by the solid itself

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Absorption process in bulk semiconductor
Band-to-Band: Electron in V.B band absorbs a photon with enough
energy to be excited to the C.B, leaving a hole behind

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Absorption process in bulk semiconductor
Band-to-Exciton: Electron in V.B absorbs almost enough
energy to be excited to the C.B. The electron and hole it
leaves behind remain electrically "bound" together, much
like the electron and proton of a hydrogen atom.

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Absorption process in bulk semiconductor
Band-to-Impurity/Impurity to Band: Electron absorbs a
photon that excites it from the V.B to an empty impurity
atom, or from an occupied impurity atom to the C.B.

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Absorption process in bulk semiconductor
Free carrier: Electron in C.B., or hole in V. B., absorbs a
photon and is excited to a higher energy level within the
same set of bands (i.e, conduction or valence)

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Absorption process in quantum well
Intra-band: Transitions can occur only between even and odd
index levels and are only operative for light polarized parallel to
the direction of quantization.

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Absorption process in quantum well
Inter-band: Inter-band transitions can occur between
conduction and valence bands, or between different valence
bands (light-hole, heavy-hole, and spin-off).

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( )
( ) ( )
dI z
I z
dz
α ω= −
Absorption Coefficient in Bulk Semiconductor
for Interband Transition
In bulk semiconductor,
absorption coefficient

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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
Rate of energy absorption
. ( )
d
A dI z
dt
ξ
= −
. ( ). . ( )
d
A I z dz
dt
ξ
α ω=
A: cross-section of the semiconductor

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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
Energy absorption is also defined as
. t
v c
d
T
dt
ξ
ω →= 
total transition probability
from valence to conduction band

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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
. ( ). . ( ) . t
v cA I z dz Tα ω ω →= 
.
( )
. ( ).
t
v cT
A I z dz
ω
α ω →
=


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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
For propagation of sinusoidal electromagnetic wave
2
2
0
0
. ( )
( )
2
rn
I z
c
ω ω
µ
= Λ
Λ0: amplitude of em wave
nr: real part of refractive index

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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
0
22
0
2 . . .
( )
. . . ( ).
t
v c
r
c T
A dz n
µ ω
α ω
ω ω
→
=
Λ

0
22
0
2 . . .
( )
. . ( ).
t
v c
r
c T
V n
µ ω
α ω
ω ω
→
=
Λ

2
2
0 0
2 .
( )
. . . . ( ).
t
v c
r
T
V c n
α ω
ω ε ω
→
=
Λ


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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
Transition probability of an electron from
valence band to conduction band
22
. ( )c vv c c v k kT k H k E E
π
δ ω→ = < > − − 

After calculating Hamiltonian
2
2 2
0*
0
. ( )
2 c vv c k k
q
T M E E
m
π
δ ω→= Λ − − 


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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
M: transfer of momentum from valence band to conduction band
Total transition probability
2 (1 )t kv kc
v c FD FD
c v
T f f→= −∑∑
Factor ‘2’ included for Pauli’s spin degeneracy

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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
2
2 2 3
022
0
( )
8 c v
t
v c k k
q V
T M E E d k
m
δ ω
π
→= Λ − −∫ 

3/2*2
2 2
02 22
0
2
.2
8
t r
v c g
q V m
T M E
m
π ω
π
→
 
=Λ − 
 



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Absorption Coefficient in Bulk Semiconductor
for Interband Transition
3/2*2
2 2
02 2
0 0
2 1
( ) .
2 . . .
r
g
r
q m
M E
c n m
α ω ω
ωπε
 
= Λ − 
 


 
α
E

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Absorption Coefficient in Quantum Well
for Interband Transition
For 2D system, reduced DoS has to be considered
3/2*
2 2
1 2
2
r
g
m
Eω
π
 
− 
 


*
2
,
( )
r m n
v c nm
m n
m
g g E
L
ω
π
Θ −∑ 


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Absorption Coefficient in Quantum Well
for Interband Transition
n m
nm g c vE E E E= + +
<gv
m|gc
n>: overlap integral between envelope
functions of C.B and V.B
L: width of quantum well

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Absorption Coefficient in Quantum Well
for Interband Transition
*2 2
2
0 0
. .
( ) ( )
2 . . . . .
r
nm
r
q N m M
E
c n L m
ω
α ω ω
ωε
= Θ −

Nω: no. of quantum well

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Absorption Coefficient in Quantum Well
for Interband Transition
Define oscillator strength
2
0
2
vc
M
f
m ω
=

*2
,0 0
. .
( ) ( )
4 . . . . .
r
vc nm
m nr
q N m
f E
c n L m
ω
α ω ω
ε
Θ −∑ 


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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