Quantum Well – mathematical calculation for real structure

1. Course: Quantum Electronics-II
Lecture: 2
Arpan Deyasi
Topic: Quantum Well –
A Deviation from Ideal Structure
5/9/2020 1
Arpan Deyasi, designed for Dept of
Elec.Sc, A.P.C.C on 9th May 2020

2. Recapitulation: ‘Particle in a Box’ problem
2 2
* 2
( ) ( ) ( ) ( )
2
d
V z z E z z
m dz
ψ
ψ ψ− + =

for 0<z<L, V(z) = 0
for z≤0 & z≥L, V(z) = ∞
L
2 2 2
* 2
2
n
n
E
m L
π
=

5/9/2020 2
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

3. Q1: What is the material outside of the well?
Q2: How can the potential barrier be infinite?
Is it practically realizable?
Q3: How the electron effective mass inside and outside the
material be same?
Q4: How step potential configuration be
physically realizable?
Queries
5/9/2020 3
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

4. how a quantum well is really formed
5/9/2020 4
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

5. how a quantum well is really formed
5/9/2020 5
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

6. Exciton
how carriers
are trapped
5/9/2020 6
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

7. Electron motion in a Quantum well
Two dimensions are unrestricted
Only one dimension is restricted
5/9/2020 7
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

8. 5/9/2020
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020
8
Electron motion in a Quantum wire
Only one dimension are unrestricted
Two dimensions are restricted

9. 5/9/2020
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020
9
Electron motion in a Quantum dot
 All the three dimensions are restricted

10. Introducing Schrödinger Equation
2 2
* 2
( )
- ( ) ( ) ( ) ( )
2m ( )
d z
V z z E z z
z dz
ψ
ψ ψ+ =

2
1 ( )
- ( ) ( ) ( ) ( )
2 *( )
d d z
V z z E z z
dz m z dz
ψ
ψ ψ
  
+ =  
  

5/9/2020 10
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

11. Introducing Schrödinger Equation
for V=0
2
1 ( )
- ( ) ( )
2 *( )w
d d z
E z z
dz m z dz
ψ
ψ
  
=  
  

for V=V0
2
0
1 ( )
- ( ) ( ) ( )
2 *( )b
d d z
V z E z z
dz m z dz
ψ
ψ ψ
  
+ =  
  

5/9/2020 11
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

12. Introducing Schrödinger Equation
for V=0
*2
2 2
2 ( )( )
( ) ( ) 0wm zd z
E z z
dz
ψ
ψ+ =

*
1 2
2 ( )wm z E
κ =

2
2
12
( )
( ) ( ) 0
d z
z z
dz
ψ
κ ψ+ =
material parameter
5/9/2020 12
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

13. Introducing Schrödinger Equation
for V=V0
( )
*2
02 2
2 ( )( )
( ) ( ) 0bm zd z
E z V z
dz
ψ
ψ+ − =

[ ]*
0
2 2
2 ( )bm z E V
κ
−
=

2
2
22
( )
( ) ( ) 0
d z
z z
dz
ψ
κ ψ+ =
material parameter
5/9/2020 13
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

14. Introducing Schrödinger Equation
κ1: wave-vector of quantum well,
i.e., for lower bandgap material
κ2: wave-vector of quantum barrier,
i.e., for higher bandgap material
κ1 = f(mw
*) ------ material parameter dependence
κ2 = f(mb
*) ------ material parameter dependence
5/9/2020 14
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

15. Where is the physics of semiconductor device?
( )
2 2
2
*
1 .......
2
E E E
m
κ
α β+ + + =

band non-parabolicity
of second order
band non-parabolicity
of first order
5/9/2020 15
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

16. Two-Dimensional Electron Gas (2DEG)
Wave nature
Electrons in CB
are considered
Quantum well
Electrons confined in
conduction band of
quantum well
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A.P.C.C on 9th May 2020

17. Multiple Quantum Well/Superlattice
5/9/2020 17
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

18. Q. What is the difference between
MQW & Superlattice?
Superlattice is a type of Multiple Quantum Well structure where
wavefunctions between adjacent quantum wells
can help carrier transfer through
quantum-mechanical tunneling process
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Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020

19. Superlattice
MQW
but not
superlattice
5/9/2020 19
Arpan Deyasi, designed for Dept of Elec.Sc,
A.P.C.C on 9th May 2020
Thickness of the sandwich
barrier is critical for
tunneling process