Analysis of conduction band nonparabolicity effect on eigenstates of quantum well structures

1. Course: Quantum Electronics
Arpan Deyasi
Topic: Effect of Band nonparabolicity
on Eigenstates
1
Arpan Deyasi, RCCIIT
1/17/2021

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Z
DQWTB structure

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Graphical representation of Transmission Coefficient
E0
E
T(E)
E1 E2 E3

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Schrödinger Equation for well region
for V=0
2
2
22
( )
( ) ( ) 0
d z
z z
dz
ψ
κ ψ+ =
*
2 2
2 ( )wm z E
κ =

Schrödinger Equation for barrier region
2
2
22
( )
( ) ( ) 0
d z
z z
dz
ψ
κ ψ+ =
for V=V0
( )*
0
1 2
2 ( )bm z E V
κ
−
=


5. Parabolic Dispersion Relation
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Ideal; most of the materials do not obey the relationship
2 2
*
2
E
m
κ
=

E
k

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2 2
2
*
(1 ...)
2
E E E
m
κ
α β+ + + =

Nonparabolic Dispersion Relation
E
k

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α, β,…. are nonparabolic coefficients
Expressions of κ1 & κ2 will change
Magnitudes of band nonparabolic coefficients
control the energy states

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Modified wave-vectors
*
2 2
2 ( ) (1 )wm z E Eα
κ
+
=

for well region
Consider first-order band nonparabolicity
*
0
1 2
2 ( )[ (1 ) ]bm z E E Vα
κ
+ −
=

for barrier region

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Graphical representation of Transmission Coefficient
E1p
E
T(E)
E2p E3p E4p
E1n E2n E3n
E4n
ΔE21n
ΔE21p
ΔE32n
ΔE32p
ΔE31n
ΔE31p
N
P

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Effect of nonparabolic coefficients
1. Energy values will be lowered
2. The effect will be more significant for higher order
states
3. Corresponding intersubband transition energy will be
reduced
4. Operating wavelength of the photodetector will be
enhanced

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Algorithm to calculate transmission coefficient using TMT
S1: Input parameters: ‘a’, ‘b’, ‘mb
*’, ‘mw
*’, ‘V0’
S2: Consider energy range of interest (E<V0)
S3: Calculate wave vectors ‘κ1’ and ‘κ2’
S4: Calculate interface matrices ‘Mi’
S5: Calculate composite interface matrix ‘M’
S6: Calculate transmission coefficient from ‘M11’

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Algorithm to calculate transmission coefficient using PMM
S1: Input parameters: ‘a’, ‘b’, ‘mj
*’, ‘Vj’
S2: Consider energy range of interest (E<V0)
S3: For energy ‘Ej’, calculate wave number ‘κj’ for each
position in the potential ‘Vj’
S5: Consider generalized length ‘Lj’ between two consecutive
points
S4: Calculate junction matrix ‘Pjunc(j)’ between any two
consecutive points inside the structure

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Algorithm to calculate transmission coefficient using PMM
S7: Calculate total propagation matrix ‘Pprop(j)’ as a
Cartesian product of junction matrix and step matrix
S6: Calculate step matrix ‘Pstep(j)’ between any two
consecutive points inside the structure
S8: Repeat it for every points inside the structure
S9: Calculate transmission coefficient from ‘P11’

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Quasi-peak
As per the transmission coefficient profile, eigenenergy states
are obtained from sharp peaks
This implies are energy states are very thin levels
Practically all energy bands consist of multiple energy levels
Henceforth, there should exist finite width of energy peak

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Quasi-peak
Practically, one small peak is associated with every principle
peak with a small finite energy difference
The second peak is called quasi-peak

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E1
E
T(E)
E2 E3
E4
Quasi-peak in Transmission Coefficient profile
E1qp E2qp E3qp E4qp