Calculation of DoS and effective DoS

1. Course: Electronic Devices
paper code: EC301
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
8/4/2020 1Arpan Deyasi, RCCIIT
Topic: Density of States

2. 8/4/2020 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

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What do we mean by ‘dimension’?
For ‘bulk’, it is ‘volume’

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Energy band diagram is drawn in E-k plane
‘k’ is wave-vector, not a physical quantity
No of electrons determines 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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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

7. 8/4/2020 7Arpan 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
=

8. 8/4/2020 8Arpan 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
=

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DoS for bulk semiconductor
Volume of Fermi sphere in k-space
34
3
FV kπ=
Volume of semiconductor in real space
3
RV L=

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DoS for bulk semiconductor
number of energy states in k-space
1
F
k
N V
V
= ×
number of energy states in real space
1 1
F
k R
N V
V V
= × ×

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DoS for bulk semiconductor
3
3
3 3
4 1
3 8
L
N k
L
π
π
= × ×
3
2
( )
6
k
N N k
π
= =

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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. 8/4/2020 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
∂
= =
∂



*
2
2m E
k =


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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. 8/4/2020 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)
*
2
2
2m E
k =


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DoS for bulk semiconductor
* 3/ 2
3
4
( ) (2 )N E dE m EdE
h
π
= ×
For electrons in CB
* 3/ 2
3
4
( ) (2 )e e CN E dE m E E dE
h
π
=× −
E
k
* 3/ 2
3
4
( ) (2 )e e CN E m E E
h
π
=× −
EC
EC + dEC

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DoS for bulk semiconductor
* 3/ 2
3
4
( ) (2 )h h VN E m E E
h
π
=× −
For holes in VB
* 3/ 2
3
4
( ) (2 )h h VN E dE m E EdE
h
π
=× −
E
k
EV
EV + dEV
dEV

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Effective Density of States
Carrier concentration of electron in CB
( ) ( ) ( )e en E N E f E=
Under equilibrium, for a given band
0 ( ) ( )e en N E f E dE= ∫

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Effective Density of States
* 3/ 2
3
0
4
(2 )
1
1 exp
e C
F
m E E
h
n
dE
E E
kT
π 
× − × 
 
  
=   
  
−   +       
∫

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Effective Density of States
* 3/ 2
3
0
4
(2 )
1
1 exp
e C
F
m E E
h
n
dE
E E
kT
π 
× − × 
 
  
=   
  
−   +       
∫

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3/2
0 2
2 *
2 expe C Fm kT E E
n
h kT
π  −    
−    
    
Effective Density of States
0 exp C F
C
E E
n N
kT
 −  
= −  
  
Effective DoS of electrons in CB

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Effective Density of States
Carrier concentration of holes in VB
( ) ( ) ( )h hp E N E f E=
Under equilibrium, for a given band
0 ( ) ( )h hp N E f E dE= ∫

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Effective Density of States
* 3/ 2
3
0
4
(2 )
1
1 exp
h V
F
m E E
h
p
dE
E E
kT
π 
× − × 
 
  
=   
  
−   +       
∫

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3/2
0 2
2 *
2 exph F Vm kT E E
p
h kT
π  −    
−    
    
Effective Density of States
0 exp F V
V
E E
p N
kT
 −  
= −  
  
Effective DoS of holes in VB