This document discusses using the plane wave matching method (PMM) to calculate the transmission coefficient for a quantum well structure with varying potentials. It proposes using a generalized wave vector and wave equations that incorporate the simultaneous variation of barrier and well potentials. The method involves defining a junction matrix at each interface and a step matrix for each layer to calculate the propagation matrix, which relates the wavefunction coefficients between interfaces. This allows modeling more realistic quantum well structures with non-step potentials.

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

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

3. Drawbacks of Transfer Matrix Technique
1/17/2021 3Arpan Deyasi, RCCIIT
Step potential well: Ideal case
Real quantum well can’t have step potential profile,
in fact, different complex potential profiles are
considered for application purpose

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Solution
We have to consider a mathematical technique
where variation of potential at each point of
both barrier and well layers can be incorporated
V = 0
V = V0

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Change required in Mathematical Formulation
*
2 2
2 ( )wm z E
κ =

*
0
1 2
2 ( )( )bm z E V
κ
−
=

for well for barrier

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We have to make a generalized wave-vector
for both barrier and well
Change required in Mathematical Formulation
Why?
Then we will be able to incorporate simultaneous
variation of barrier and well potentials

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Generalized wave-vector
*
2
2 ( )( )j j
j
m z E V
κ
−
=

calculated at jth point

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Z
DQWTB structure
j j+1

9. 1/17/2021 9Arpan Deyasi, RCCIIT
Generalized wave equations
exp( ) exp( )j j j j jA i z B i zψ κ κ= + −
1 1 1 1 1exp( ) exp( )j j j j jC i z D i zψ κ κ+ + + + += + −

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Generalized boundary conditions
1j j+Ψ =Ψ
1j jd d
dz dz
+Ψ Ψ
=

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At interface
1j j+Ψ =Ψ
1 1 1 1
exp( ) exp( )
exp( ) exp( )
j j j j
j j j j
A i z B i z
C i z D i z
κ κ
κ κ+ + + +
+ −
= + −

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At interface
1' 'j j+Ψ =Ψ
1 1 1 1 1 1
exp( ) exp( )
exp( ) exp( )
j j j j j j
j j j j j j
i A i z i B i z
i C i z i D i z
κ κ κ κ
κ κ κ κ+ + + + + +
− −
= − −
1 1 1 1 1 1
exp( ) exp( )
exp( ) exp( )
j j j j j j
j j j j j j
A i z B i z
C i z D i z
κ κ κ κ
κ κ κ κ+ + + + + +
− −
= − −

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At interface
1
1 1
1
1 1
exp( ) exp( )
exp( )
exp( )
j j j j
j
j j
j
j
j j
j
A i z B i z
C i z
D i z
κ κ
κ
κ
κ
κ
κ
κ
+
+ +
+
+ +
− −
 
=   
 
 
− −  
 

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At interface
1 1
1 1 1 1
exp( ) exp( )
exp( ) exp( )
j j j j
j j
j j j j
j j
A i z B i z
C i z D i z
κ κ
κ κ
κ κ
κ κ
+ +
+ + + +
− −
   
= − −      
   
1 1 1 1
exp( ) exp( )
exp( ) exp( )
j j j j
j j j j
A i z B i z
C i z D i z
κ κ
κ κ+ + + +
+ −
= + −
In matrix
notation
1
1 1
1
1 1
1 1
1 1
j j
j j
j j
j j
A C
B D
κ κ
κ κ
+
+ +
+
 
      
=      −−      

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1
1 1
1
1 1
1 1
1 1
j j
j j
j j
j j
A C
B D
κ κ
κ κ
+
+ +
+
 
      
=      −−      
At interface
1
1
1 1
1
1 1
1 1
1 1
j j
j j
j j
j j
A C
B D
κ κ
κ κ
−
+
+ +
+
 
      
=      −−      

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At interface
1
1 1
1
1 1
1 11
1 12
j j
j j
j j
j j
A C
B D
κ κ
κ κ
+
+ +
+
 
      
=      −−      
1 1
1
11 1
1 1
1
2
1 1
j j
j jj j
j jj j
j j
A C
B D
κ κ
κ κ
κ κ
κ κ
+ +
+
++ +
    
+ −              
=    
       
− +       
     

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1 1
1 1
1 1
1
2
1 1
j j
j j
j
j j
j j
P
κ κ
κ κ
κ κ
κ κ
+ +
+ +
    
+ −           
=  
    
− +       
     
At interface
Let
Pj : junction matrix

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Z
DQWTB structure
j j+1Lj

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Significance of Lj
travelling from ‘j’ to ‘j+1’ with a distance ‘Lj’
matches the positive coefficients as well as
negative coefficients

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1exp( )j j j jA i L Cκ +=
For Lj
1exp( )j j j jB i L Dκ +− =
In matrix
notation
1
1
exp( ) 0
0 exp( )
j j j j
j j j j
i L A C
i L B D
κ
κ
+
+
    
=    −    

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For Lj
1
1
exp( ) 0
0 exp( )
j j j j
j j j j
A i L C
B i L D
κ
κ
+
+
−    
=    
    
exp( ) 0
0 exp( )
j j
L
j j
i L
P
i L
κ
κ
− 
=  
 
Let
PL : step matrix

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Q: How to calculate propagation matrix?
It is the Cartesian product of
Junction matrix with Step matrix

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Propagation Matrix calculation
1 1
1 1
1 1
1
2
1 1
j j
j j
j
j j
j j
P
κ κ
κ κ
κ κ
κ κ
+ +
+ +
    
+ −           
=  
    
− +       
     
exp( ) 0
0 exp( )
j j
L
j j
i L
P
i L
κ
κ
− 
=  
 

24. 1/17/2021 Arpan Deyasi, RCCIIT 24
Propagation Matrix calculation
j LP P P= ×
1 1
1 1
1 1
1
2
1 1
exp( ) 0
0 exp( )
j j
j j
j j
j j
j j
j j
P
i L
i L
κ κ
κ κ
κ κ
κ κ
κ
κ
+ +
+ +
    
+ −           
=  
    
− +       
     
− 
× 
 

25. 1/17/2021 Arpan Deyasi, RCCIIT 25
Propagation Matrix calculation
1 1
1 1
exp( ) 1 exp( ) 1
1
2
exp( ) 1 exp( ) 1
j j
j j j j
j j
j j
j j j j
j j
i L i L
P
i L i L
κ κ
κ κ
κ κ
κ κ
κ κ
κ κ
+ +
+ +
    
− + − −           
=  
    
− +       
     
11 12
21 22
P P
P
P P
 
=  
 

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Propagation Matrix calculation
1
1
j j
j j
A C
P
B D
+
+
   
=   
   
111 12
121 22
j j
j j
A CP P
B DP P
+
+
    
=    
    

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Propagation Matrix calculation
11 1 12 1j j jA P C P D+ += +
21 1 22 1j j jB P C P D+ += +
111 12
121 22
j j
j j
A CP P
B DP P
+
+
    
=    
    

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P11 P12
P21 P22
Aj
Bj
Cj+1
Dj+1
P12 is the transmission coefficient
when the wave is traversing from port 2 to port 1
and port 1 is terminated by matched load

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P12 = 0 for practical device
11
1
j
j
A
P
C +
=
( )
2
1
*
11 11
1j
j
C
T E
A P P
+
 
= =  
 
11 1 12 1j j jA P C P D+ += +

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