Calculation of density of states for bulk semiconductor and quantum structures

1. Course: Quantum Electronics
Arpan Deyasi
Topic: Density of states of Bulk and Quantum
Structures
1
Arpan Deyasi, RCCIIT
1/17/2021

2. 1/17/2021 2Arpan Deyasi, RCCIIT
What is DoS?
Number of available energy states
per unit energy interval
per unit dimension
in real space
EC
EC + dEC
EV
EV + dEV
dEV
dEC
E
k

3. 1/17/2021 Arpan Deyasi, RCCIIT 3
What do we mean by ‘dimension’?
For ‘bulk’, it is ‘volume’
For ‘quantum well’, it is ‘area/surface’
For ‘quantum wire’, it is ‘line/length’
For ‘quantum dot’, it is a ‘point/dot’

4. 1/17/2021 4Arpan Deyasi, RCCIIT
Energy band diagram is drawn in E-k plane
‘k’ is wave-vector, not a physical quantity
No of electrons is measured by magnitude of current
So we must know the density of electrons
in real space instead of k-space

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Fermi sphere

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Fermi surface

7. 1/17/2021 7Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
Let’s start with Bloch theorem
( , , ) ( , , )x y zx y z x L y L z Lψ ψ= + + +
Consider a 3D semiconductor with dimensions Lx, Ly, Lz

8. 1/17/2021 8Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
For validity of wave function
2
2
2
x x x
y y y
z z z
k L n
k L n
k L n
π
π
π
=
=
=
Volume in k-space
3
(2 ) x y z
x y z
x y z
n n n
k k k
L L L
π
=

9. 1/17/2021 9Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
Let
1x y zn n n= = =
x y zL L L L= = =
Volume of unit cell in k-space
3
3
(2 )
kV
L
π
=

10. 1/17/2021 10Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
Volume of Fermi sphere in k-space
34
3
FV kπ=
Volume of semiconductor in real space
3
RV L=

11. 1/17/2021 11Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
number of energy states in real space
1 1
F
k R
N V
V V
= × ×
3
3
3 3
4 1
3 8
L
N k
L
π
π
= × ×
3
2
( )
6
k
N N k
π
= =

12. 1/17/2021 12Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
Introducing Pauli’s exclusion principle
3 3
2 2
( ) 2
6 3
k k
N k
π π
=× =
2
2
( )
k
N k
k π
∂
=
∂

13. 1/17/2021 13Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
From parabolic dispersion relation
2 2
*
2
k
E
m
=

2
*
E k
k m
∂
=
∂

2
* *
2 * 2E m E E
k m m
∂
= =
∂




14. 1/17/2021 14Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
*
1
2
k m
E E
∂
=
∂ 
N N k
E k E
∂ ∂ ∂
= ×
∂ ∂ ∂
2 *
2
1
2
N k m
E Eπ
∂
= ×
∂ 

15. 1/17/2021 15Arpan Deyasi, RCCIIT
DoS for bulk semiconductor
*
2 2
2 * 1
2
N m E m
E Eπ
∂
= ×
∂  
3/2*
2 2
1 2
2
N m
E
E π
 ∂
= × 
∂  
For a particular material
N
E
E
∂
∝
∂
E
ρ(E)

16. 1/17/2021 16Arpan Deyasi, RCCIIT
DoS for Quantum Well
Area in k-space
2
2
(2 )
kA
L
π
=
Area of Fermi sphere in k-space
Area of semiconductor in real space
2
FA kπ=
2
RA L=

17. 1/17/2021 17Arpan Deyasi, RCCIIT
DoS for Quantum Well
number of energy states in real space
1 1
F
k R
N A
A A
= × ×
2
2
2 2
1
4
L
N k
L
π
π
= × ×
2
( )
4
k
N N k
π
= =

18. 1/17/2021 18Arpan Deyasi, RCCIIT
DoS for Quantum Well
Introducing Pauli’s exclusion principle
2 2
( ) 2
4 2
k k
N k
π π
=× =
( )
k
N k
k π
∂
=
∂

19. 1/17/2021 19Arpan Deyasi, RCCIIT
DoS for Quantum Well
From parabolic dispersion relation
*
1
2
k m
E E
∂
=
∂ 
N N k
E k E
∂ ∂ ∂
= ×
∂ ∂ ∂
*
1
2
N k m
E Eπ
∂
= ×
∂ 

20. 1/17/2021 20Arpan Deyasi, RCCIIT
DoS for Quantum Well
* *
2 1
2
N m E m
E Eπ
∂
= ×
∂  
*
2
N m
E π
∂
=
∂ 
DoS is independent of energy?

21. 1/17/2021 21Arpan Deyasi, RCCIIT
DoS for Quantum Well
The result is obtained for a particular sub-band
Considering all the sub-bands
*
2
1
( )
n
i
i
N m
E E
E π =
∂
= Θ −
∂
∑
E
ρ(E)
E1 E2 E3 Ei-1 Ei Ei+1

22. 1/17/2021 22Arpan Deyasi, RCCIIT
DoS for Quantum Wire
Length in k-space
Length of Fermi sphere in k-space
Length of semiconductor in real space
2
kL
L
π
=
2FL k=
RL L=

23. 1/17/2021 23Arpan Deyasi, RCCIIT
DoS for Quantum Wire
number of energy states in real space
1 1
F
k R
N L
L L
= × ×
1
2
2
L
N k
Lπ
= × ×
k
N
π
=

24. 1/17/2021 24Arpan Deyasi, RCCIIT
DoS for Quantum Wire
Introducing Pauli’s exclusion principle
2
k
N
π
=
2
( )N k
k π
∂
=
∂

25. 1/17/2021 25Arpan Deyasi, RCCIIT
DoS for Quantum Wire
From parabolic dispersion relation
*
1
2
k m
E E
∂
=
∂ 
N N k
E k E
∂ ∂ ∂
= ×
∂ ∂ ∂
*
2 1
2
N m
E Eπ
∂
= ×
∂ 

26. 1/17/2021 26Arpan Deyasi, RCCIIT
DoS for Quantum Wire
2 * 1N m
E Eπ
∂  
=  
∂  
E
ρ(E)
E1 E2 E3 E4 E5

27. 1/17/2021 27Arpan Deyasi, RCCIIT
DoS for Quantum Dot
E4E3E2E1
ρ(E)
E E
ρ(E)
E1 E2 E3 E4