O documento discute o cálculo das propriedades eletrônicas e ópticas de um poço quântico duplo usando a técnica da matriz de transferência. As propriedades a serem avaliadas incluem o coeficiente de transmissão, energia própria e densidade de estados. Métodos numéricos como matriz de transferência, método da matriz de propagação e elementos finitos são considerados para o cálculo.

1. Course: Quantum Electronics
Arpan Deyasi
Calculation of
Transmission Coefficient using Transfer
Matrix Technique
1
Arpan Deyasi,RCCIIT
1/17/2021

2. 1/17/2021 Arpan Deyasi,RCCIIT 2
Single Quantum Well

3. 1/17/2021 Arpan Deyasi,RCCIIT 3
Properties to be evaluated:
Electronic Properties
1. Transmission Coefficient
2. Eigen Energy
3. Density of States
Optical Properties
1. Absorption Coefficient
2. Oscillator Strength

4. Numerical Techniques may be
considered for Calculation
Transfer Matrix Technique (TMT)
Propagation Matrix Method (PMM)
Perturbation Method
WKB Approximation
Finite Element Method (FEM)
Finite Difference Time Domain Method (FDTD)
1/17/2021 4Arpan Deyasi,RCCIIT

5. 1/17/2021 Arpan Deyasi,RCCIIT 5
Q. Which have better accuracy?
Q. Which are faster for calculation?
We have to optimize between them

6. 1/17/2021 Arpan Deyasi,RCCIIT 6
FDTD and FEM are most accurate as per the literatures
TMT & PMM are faster which incorporate fast principle

7. 1/17/2021 Arpan Deyasi,RCCIIT 7
Today we will start the calculation of
Electronic Properties
using Transfer Matrix Technique
We will consider
Double Quantum Well structure
for our theoretical work

8. 1/17/2021 Arpan Deyasi,RCCIIT 8
Z
Z=0
Z=a
Z=a+b
Z=2a+b
a ab
DQWTB structure
A1 A2 A3 A4 A5
B1 B2 B3 B4
B5
I II III IV V

9. 1/17/2021 Arpan Deyasi,RCCIIT 9
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
−
=


10. 1/17/2021 Arpan Deyasi,RCCIIT 10
Solution of Schrödinger Equation in different regions
1 1 1 1exp( ) exp( )I A i z B i zψ κ κ= + −
3 1 3 1exp( ) exp( )III A i z B i zψ κ κ= + −
5 1 5 1exp( ) exp( )V A i z B i zψ κ κ= + −
2 2 2 2exp( ) exp( )II A i z B i zψ κ κ= + −
4 2 4 2exp( ) exp( )IV A i z B i zψ κ κ= + −

11. 1/17/2021 Arpan Deyasi,RCCIIT 11
Ben-Daniel Duke Boundary Conditions
I IIΨ =Ψ
* *
1 1I II
I II
d d
dz dzm m
Ψ Ψ
=
A little modification is required in second boundary condition. Why?

12. 1/17/2021 Arpan Deyasi,RCCIIT 12
Both κ1 and κ2 are functions of m*
I IId d
dz dz
Ψ Ψ
=
So to avoid dual effect of m*, we will modify the 2nd condition as

13. 1/17/2021 Arpan Deyasi,RCCIIT 13
at Z = 0 (1st interface)
I IIψ ψ=
1 1 1 1
2 2 2 2
exp( ) exp( )
exp( ) exp( )
A i z B i z
A i z B i z
κ κ
κ κ
+ −
= + −
1 1 2 2A B A B+ = +

14. 1/17/2021 Arpan Deyasi,RCCIIT 14
at Z = 0 (1st interface)
' 'I IIψ ψ=
1 1 1 1 1 1
2 2 2 2 2 2
exp( ) exp( )
exp( ) exp( )
i A i z i B i z
i A i z i B i z
κ κ κ κ
κ κ κ κ
− −
= − −
1 1 1 1 2 2 2 2A B A Bκ κ κ κ− = −

15. 1/17/2021 Arpan Deyasi,RCCIIT 15
at Z = 0 (1st interface)
1 1 2 2A B A B+ = +
1 1 1 1 2 2 2 2A B A Bκ κ κ κ− = −
In matrix
notation
1 2
1 1 1 2 2 2
1 1 1 1A A
B Bκ κ κ κ
     
=      − −     

16. 1/17/2021 Arpan Deyasi,RCCIIT 16
at Z = 0 (1st interface)
1 2
1 1 1 2 2 2
1 1 1 1A A
B Bκ κ κ κ
     
=      − −     
1 2
1 2
1 2
A A
M M
B B
   
=   
   

17. 1/17/2021 Arpan Deyasi,RCCIIT 17
at Z = a (2nd interface)
II IIIψ ψ=
2 2 2 2
3 1 3 1
exp( ) exp( )
exp( ) exp( )
A i z B i z
A i z B i z
κ κ
κ κ
+ −
= + −
2 2 2 2
3 1 3 1
exp( ) exp( )
exp( ) exp( )
A i a B i a
A i a B i a
κ κ
κ κ
+ −
= + −

18. 1/17/2021 Arpan Deyasi,RCCIIT 18
at Z = a (2nd interface)
' 'II IIIψ ψ=
2 2 2 2 2 2
1 3 1 1 3 1
exp( ) exp( )
exp( ) exp( )
i A i z i B i z
i A i z i B i z
κ κ κ κ
κ κ κ κ
− −
= − −
2 2 2 2 2 2
1 3 1 1 3 1
exp( ) exp( )
exp( ) exp( )
A i a B i a
A i a B i a
κ κ κ κ
κ κ κ κ
− −
= − −

19. 1/17/2021 Arpan Deyasi,RCCIIT 19
at Z = a (2nd interface)
2 2 2 2 3 1 3 1exp( ) exp( ) exp( ) exp( )A i a B i a A i a B i aκ κ κ κ+ −= + −
2 2 2 2 2 2
1 3 1 1 3 1
exp( ) exp( )
exp( ) exp( )
A i a B i a
A i a B i a
κ κ κ κ
κ κ κ κ
− −
= − −
In matrix
notation
2 2 2
2 2 2 2 2
31 1
31 1 1 1
exp( ) exp( )
exp( ) exp( )
exp( ) exp( )
exp( ) exp( )
i a i a A
i a i a B
Ai a i a
Bi a i a
κ κ
κ κ κ κ
κ κ
κ κ κ κ
−  
  − −  
− −   
=   − − −  

20. 1/17/2021 Arpan Deyasi,RCCIIT 20
at Z = a (2nd interface)
2 2 2
2 2 2 2 2
31 1
31 1 1 1
exp( ) exp( )
exp( ) exp( )
exp( ) exp( )
exp( ) exp( )
i a i a A
i a i a B
Ai a i a
Bi a i a
κ κ
κ κ κ κ
κ κ
κ κ κ κ
−  
  − −  
− −   
=   − − −  
32
3 4
32
AA
M M
BB
  
=   
   

21. 1/17/2021 Arpan Deyasi,RCCIIT 21
at Z = (a+b) (3rd interface)
III IVψ ψ=
3 1 3 1
4 2 4 2
exp( ) exp( )
exp( ) exp( )
A i z B i z
A i z B i z
κ κ
κ κ
+ −
= + −
3 1 3 1
4 2 4 2
exp( ( ) exp( ( ))
exp( ( )) exp( ( ))
A i a b B i a b
A i a b B i a b
κ κ
κ κ
+ + − +
+ + − +

22. 1/17/2021 Arpan Deyasi,RCCIIT 22
' 'III IVψ ψ=
1 3 1 1 3 1
2 4 2 2 4 2
exp( ) exp( )
exp( ) exp( )
i A i z i B i z
i A i z i B i z
κ κ κ κ
κ κ κ κ
− −
= − −
1 3 1 1 3 1
2 4 2 2 4 2
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
A i a b B i a b
A i a b B i a b
κ κ κ κ
κ κ κ κ
+ − − +
+ − − +
at Z = (a+b) (3rd interface)

23. 1/17/2021 Arpan Deyasi,RCCIIT 23
In
matrix
notation
at Z = (a+b) (3rd interface)
31 1
31 1 1 1
2 2 4
2 2 2 2 4
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
Ai a b i a b
Bi a b i a b
i a b i a b A
i a b i a b B
κ κ
κ κ κ κ
κ κ
κ κ κ κ
+ − +   
  + − − +  
+ − +  
=   + − − +  
3 1 3 1
4 2 4 2
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
A i a b B i a b
A i a b B i a b
κ κ
κ κ
+ + − +
= + + − +
1 3 1 1 3 1
2 4 2 2 4 2
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
A i a b B i a b
A i a b B i a b
κ κ κ κ
κ κ κ κ
+ − − +
+ − − +

24. 1/17/2021 Arpan Deyasi,RCCIIT 24
at Z = (a+b) (3rd interface)
3 4
5 6
3 4
A A
M M
B B
   
=   
  
31 1
31 1 1 1
2 2 4
2 2 2 2 4
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
exp( ( )) exp( ( ))
Ai a b i a b
Bi a b i a b
i a b i a b A
i a b i a b B
κ κ
κ κ κ κ
κ κ
κ κ κ κ
+ − +   
  + − − +  
+ − +  
=   + − − +  

25. 1/17/2021 Arpan Deyasi,RCCIIT 25
at Z = (2a+b) (4th interface)
IV Vψ ψ=
4 2 4 2
5 1 5 1
exp( ) exp( )
exp( ) exp( )
A i z B i z
A i z B i z
κ κ
κ κ
+ −
= + −
4 2 4 2
5 1 5 1
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
A i a b B i a b
A i a b B i a b
κ κ
κ κ
+ − − +
+ + − +

26. 1/17/2021 Arpan Deyasi,RCCIIT 26
at Z = (2a+b) (3rd interface)
' 'IV Vψ ψ=
2 4 2 2 4 2
1 5 1 1 5 1
exp( ) exp( )
exp( ) exp( )
i A i z i B i z
i A i z i B i z
κ κ κ κ
κ κ κ κ
− −
= − −
2 4 2 2 4 2
1 5 1 1 5 1
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
A i a b B i a b
A i a b B i a b
κ κ κ κ
κ κ κ κ
+ − − +
+ − − +

27. 1/17/2021 Arpan Deyasi,RCCIIT 27
at Z = (2a+b) (3rd interface)
( ) ( )
( ) ( )
4 2 4 2
5 1 5 1
exp (2 ) exp (2 )
exp (2 ) exp (2 )
A i a b B i a b
A i a b B i a b
κ κ
κ κ
+ − − +
+ + − +
2 4 2 2 4 2
1 5 1 1 5 1
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
A i a b B i a b
A i a b B i a b
κ κ κ κ
κ κ κ κ
+ − − +
+ − − +
In
matrix
notation
2 2 4
2 2 2 2 4
51 1
51 1 1 1
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
i a b i a b A
i a b i a b B
Ai a b i a b
Bi a b i a b
κ κ
κ κ κ κ
κ κ
κ κ κ κ
+ − +  
  + − − +  
+ − +   
=   + − − +  

28. 1/17/2021 Arpan Deyasi,RCCIIT 28
at Z = (2a+b) (3rd interface)
54
7 8
54
AA
M M
BB
  
=   
   
2 2 4
2 2 2 2 4
51 1
51 1 1 1
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
exp( (2 )) exp( (2 ))
i a b i a b A
i a b i a b B
Ai a b i a b
Bi a b i a b
κ κ
κ κ κ κ
κ κ
κ κ κ κ
+ − +  
  + − − +  
+ − +   
=   + − − +  

29. 1/17/2021 Arpan Deyasi,RCCIIT 29
54
7 8
54
AA
M M
BB
  
=   
   
1 54
7 8
54
AA
M M
BB
−   
=   
   

30. 1/17/2021 Arpan Deyasi,RCCIIT 30
3 4
5 6
3 4
A A
M M
B B
   
=   
  
13 4
5 6
3 4
A A
M M
B B
−   
=   
  
1 13 5
5 6 7 8
3 5
A A
M M M M
B B
− −   
=   
   

31. 1/17/2021 Arpan Deyasi,RCCIIT 31
32
3 4
32
AA
M M
BB
  
=   
   
1 32
3 4
32
AA
M M
BB
−   
=   
   
1 1 1 52
3 4 5 6 7 8
52
AA
M M M M M M
BB
− − −   
=   
   

32. 1/17/2021 Arpan Deyasi,RCCIIT 32
1 2
1 2
1 2
A A
M M
B B
   
=   
   
11 2
1 2
1 2
A A
M M
B B
−   
=   
   
1 1 1 1 51
1 2 3 4 5 6 7 8
51
AA
M M M M M M M M
BB
− − − −   
=   
   

33. 1/17/2021 Arpan Deyasi,RCCIIT 33
51
51
AA
M
BB
  
=   
   
51 11 12
51 21 22
AA M M
BB M M
    
=     
    
1 1 1 1 51
1 2 3 4 5 6 7 8
51
AA
M M M M M M M M
BB
− − − −   
=   
   

34. 1/17/2021 Arpan Deyasi,RCCIIT 34
1 11 5 12 5A M A M B= +
1 21 5 22 5B M A M B= +
51 11 12
51 21 22
AA M M
BB M M
    
=     
    

35. 1/17/2021 Arpan Deyasi,RCCIIT 35
M11 M12
M21 M22
A1
B1
A5
B5
M12 is the transmission coefficient
when the wave is traversing from port 2 to port 1
and port 1 is terminated by matched load

36. 1/17/2021 Arpan Deyasi,RCCIIT 36
M12 = 0 for practical device
1 11 5 12 5A M A M B= +
1
11
5
A
M
A
=
( )
2
5
*
1 11 11
1A
T E
A M M
 
= = 
 

37. 1/17/2021 Arpan Deyasi,RCCIIT 37
Graphical representation of Transmission Coefficient
E0
E
T(E)
E1 E2 E3