The document discusses resonant tunneling through a double quantum well system under an applied electric field. It first presents the mathematical formulations for resonant tunneling under unbiased and biased conditions. It then describes the condition for resonant tunneling, which is when the thickness of the middle barrier allows resonant tunneling between adjacent quantum wells. Finally, it shows calculations of the conductance for the system in the limits of low temperature and bias, relating it to the quantum unit of conductance.

1. Resonant Tunneling
Arpan Deyasi
Kolkata, India

2. Resonant tunneling at unbiased condition
Barrier

3. Resonant tunneling at unbiased condition

4. Mathematical Formulation
Under unbiased condition
2 2
* 2
2
d
V E
m dz

 
− + =
If electric field ‘ξ’ is applied
( )
2 2
* 2
2
d
V q z E
m dz

  
− + − =

5. Resonant tunneling at biased condition
misalignment
ξ1

6. Resonant tunneling at biased condition
ξ2> ξ1

7.  
0
2 ( ), ( ) ( )
2
L L
d
I q f E v T

   


= 
Condition for Resonant Tunneling
Consider double quantum well system under application of electric field
Assume thickness of middle barrier is such that resonant tunneling occurs
between adjacent quantum wells
Current due to electrons flowing from left-to-right

8. T(κ): transmission probability
v(κ): velocity of electron wave when energy levels of adjacent
quantum wells are matched
μ(L): Fermi level of left quantum well
f: Fermi function
Condition for Resonant Tunneling

9. Condition for Resonant Tunneling
From parabolic dispersion relation
2 2
*
2
E
m

=
2
*
dE
d m


=
dE
v
d
=

10. Condition for Resonant Tunneling
Equation of current becomes
 
0
2 ( ), ( ) ( )
2 ( )
L L
dE
I q f E v E T E
v E
 


= 
 
0
( ), ( )
L L
dE
I q f E T E
 


= 

11. Condition for Resonant Tunneling
 
2 ( ), ( )
L
L L
U
dE
I q f E T E
h
 

= 
UL: lower limit of Fermi function at left side

12. Condition for Resonant Tunneling
Current due to electrons flowing from right-to-left
 
2 ( ), ( )
R
R R
U
dE
I q f E T E
h
 

= 
UR: lower limit of Fermi function at right side

13. Condition for Resonant Tunneling
Net current (assuming flowing from left-to-right)
   
( )
2
( ), ( ), ( )
L
net L R
L R
U
I I I
q
f E f E T E dE
h
   

= −
= −

Resonant tunneling equation from Esaki and Tsu

14. Calculation of Conductance
For constant applied bias, device is operated at lower temperature
electrons become highly degenerate
Make applied bias very low
Fermi function can be expanded in Taylor series form

15. Calculation of Conductance
   
( ), ( ),
L R
f E f E
   
−
1 1
, ,
2 2
f E qV f E qV
 
   
 + − −
   
   

16. 1 ( , )
( )
2
1 ( , )
( )
2
f E
f E qV
f E
f E qV







 
= + +
 

 

 
− − −
 

 
Calculation of Conductance
( , )
f E
qV



=


17. Calculation of Conductance
Substituting in the expression of Esaki and Tsu
2 ( , )
( )
L
net
U
q f E
I qV T E dE
h




=


2
2 ( , )
( )
L
net
U
q V f E
I T E dE
h




=



18. Calculation of Conductance
Conductance
2
( , )
2 ( )
L
net
U
I q f E
G T E dE
V h




= =



19. Quantum unit of Conductance
The term (q2/h) is called quantum unit of conductance
Q.U.C = 38.7 μS
Corresponding resistance = 25.8 KΩ

20. Calculation of Conductance
If temperature is very low
( , )
( )
f E
E




→


21. 2
2 ( ) ( )
L
U
q
G E T E dE
h


= 
Calculation of Conductance
2
2 ( )
q
G T E
h
=

22. Calculation of Conductance
If ( ) 1
T E →
Corresponding resistance = 12.9 KΩ