Charge-potential profile of MOS Capacitor

1. Charge-Potential
Characteristics of
MOS Capacitor
Arpan Deyasi
Dept of ECE, RCCIIT, Kolkata, India
11/3/2020 1Arpan Deyasi, RCCIIT, India

2. 11/3/2020 Arpan Deyasi, RCCIIT, India 2
VM>0 & large
Al SiO2
Si(p)
1
2
point of
inversion
3
ΨF
ΨS
RecapitulationofInversionCondition

3. 11/3/2020 Arpan Deyasi, RCCIIT, India 3
VM>0 & large
Al SiO2
Si(p)
1
2
point of
inversion
3
ΨF
ΨS
ClassificationofInversion
2ΨF >Ψs>ΨF
Weak inversion

4. 11/3/2020 Arpan Deyasi, RCCIIT, India 4
VM>0 & large
Al SiO2
Si(p)
1
2
point of
inversion
3
ΨF
ΨS
ClassificationofInversion
Ψs≥2ΨF
Strong inversion

5. Mathematical representation of band bending
Band bending is defined
Ψ(z) = ΨI(z) - ΨI(z ∞)
Assumption
Ψ(z)>0 for downward bending
Ψ(z)<0 for downward bending
Boundary conditions
Ψ(z)=0 in bulk semiconductor
Ψ(z=0) = ΨS at SiO2-Si interface
11/3/2020 5Arpan Deyasi, RCCIIT, India

6. Poisson’s Equation for the Device
[ ]
2
2
( ) ( ) ( ) ( )D A
d q
p z n z N z N z
dz
ψ
ε
=− − + −
In bulk substrate, p(z)=NA
2
( ) i
A
n
n z
N
=
11/3/2020 6Arpan Deyasi, RCCIIT, India

7. Charge Neutrality Condition
( ) ( ) ( ) ( ) 0D Ap z n z N z N z− + − =
( )
2
( ) i
D A A
A
n
N z N z N
N
− = −
11/3/2020 7Arpan Deyasi, RCCIIT, India

8. Charge Variation
( ) exp ( )i F I
q
p z n
kT
 
= Ψ − Ψ  
( ) exp ( )i F
q
p z n
kT
 
= Ψ − Ψ  
11/3/2020 8Arpan Deyasi, RCCIIT, India

9. Charge Variation
( ) exp expi F
q q
p z n
kT kT
   
= Ψ −Ψ      
( ) expA
q
p z N
kT
 
= −Ψ  
11/3/2020 9Arpan Deyasi, RCCIIT, India

10. Charge Variation
( ) exp ( )i I F
q
n z n
kT
 
= Ψ − Ψ  
( ) exp ( )i F
q
n z n
kT
 
= Ψ − Ψ  
11/3/2020 10Arpan Deyasi, RCCIIT, India

11. Charge Variation
( ) exp expi F
q q
n z n
kT kT
   
= −Ψ Ψ      
2
( ) expi
A
n q
n z
N kT
   
= Ψ     
11/3/2020 11Arpan Deyasi, RCCIIT, India

12. Poisson’s Equation for the Device
2
2 22
exp
exp
A A
i i
A A
q
N N
kTd q
dz n nq
N kT N
ψ
ε
  
−Ψ −    = −
    
− Ψ +        
11/3/2020 12Arpan Deyasi, RCCIIT, India

13. Poisson’s Equation for the Device
2
2 2
exp 1
exp 1
A
i
A
q
N
kTd q
dz n q
N kT
ψ
ε
   
−Ψ −      = −
    − Ψ −      
11/3/2020 13Arpan Deyasi, RCCIIT, India

14. Solution of Poisson’s Equation
Step 2: Multiply both sides by (dΨ/dz)
Step 1: Integrate from bulk (Ψ=0)
to surface [(dΨ/dz)=0]
11/3/2020 14Arpan Deyasi, RCCIIT, India

15. Solution of Poisson’s Equation
2
2 2
0 0
exp 1
exp 1
d Adz
i
A
q
N
kTd q
dz n q
N kT
ψ
ε
Ψ
Ψ
   
−Ψ −      = −
    − Ψ −      
∫ ∫
11/3/2020 15Arpan Deyasi, RCCIIT, India

16. 11/3/2020 Arpan Deyasi, RCCIIT, India 16
0
2
0
exp 1
exp 1
d
dz
A
i
A
d d q
dz dz
q
N
kT
n q
N kT
ψ
ε
Ψ
Ψ
 
=− × 
 
   
−Ψ −      
    − Ψ −      
∫
∫
Solution of Poisson’s Equation

17. Solution of Poisson’s Equation
0
2
0
exp 1
exp 1
d
dz
d Adz
i
A
d d d
dz dz dz
q
N
kTq d
dzn q
N kT
ψ ψ
ψ
ε
Ψ
Ψ
   
× =   
   
   
−Ψ −        − ×       − Ψ −      
∫
∫
11/3/2020 17Arpan Deyasi, RCCIIT, India

18. Solution of Poisson’s Equation
0
2
0
exp 1
exp 1
d
dz
A
i
A
d d
d
dz dz
q
N
kTq
d
n q
N kT
ψ ψ
ψ
ε
Ψ
Ψ
   
=   
   
   
−Ψ −      − ×
    − Ψ −      
∫
∫
11/3/2020 18Arpan Deyasi, RCCIIT, India

19. Expression of Electric Field
[ ]
2
2
2
2
2
( )
exp 1
exp 1
A
i
A
d kTN
z
dz
q q
kT kT
n q q
kT kTN
ψ
ξ
ε
 
= = × 
 
  Ψ Ψ    
− + −     
     
  Ψ Ψ    
 + − −    
      
11/3/2020 19Arpan Deyasi, RCCIIT, India

20. 11/3/2020 20Arpan Deyasi, RCCIIT, India
Boundary condition for Electric Field
at z=0 (insulator-semiconductor interface)
[i] Ψ = ΨS
[ii] ξ = ξS

21. 11/3/2020 Arpan Deyasi, RCCIIT, India 21
Surface Electric Field
0.5
2
2
2
exp 1
exp 1
A
S
S S
i S S
A
kTN
q q
kT kT
n q q
kT kTN
ξ
ε
= ×
  Ψ Ψ    
− + −     
     
  Ψ Ψ    
 + − −    
      

22. 11/3/2020 Arpan Deyasi, RCCIIT, India 22
Surface Charge
From Gauss’ law S SQ εξ= −
0.5
2
2
2
exp 1
exp 1
S S A
S S
i S S
A
Q kTN
q q
kT kT
n q q
kT kTN
εξ ε=− = ×
  Ψ Ψ    
− + −     
     
  Ψ Ψ    
 + − −    
      

23. 11/3/2020 Arpan Deyasi, RCCIIT, India 23
Graphical representation of Charge Profile
CASE-I
0
0
S
SQ
Ψ =
=
at flat-band
condition
ΨS
QS

24. 11/3/2020 Arpan Deyasi, RCCIIT, India 24
Graphical representation of Charge Profile
CASE-II
0SΨ <
accumulation
condition
exp
2
S
S
q
Q
kT
Ψ 
∝ − 
 
ΨS
QS

25. 11/3/2020 Arpan Deyasi, RCCIIT, India 25
Graphical representation of Charge Profile
CASE-III
depletion
condition
0&S smallΨ >
1/2
S SQ ∝ Ψ
ΨS
QS

26. 11/3/2020 Arpan Deyasi, RCCIIT, India 26
Graphical representation of Charge Profile
CASE-IV
inversion
condition
ΨS
QS
0&SΨ > large
exp
2
S
S
q
Q
kT
Ψ 
∝  
 

27. 11/3/2020 Arpan Deyasi, RCCIIT, India 27
ΨS
QS
Graphical representation of Charge Profile
accumulation
depletion
weak
inversion
strong
inversion
ΨF
ΨF
Distance from
flat-band point
to point of
inversion is ΨF
Distance from
point of
inversion to inversion
transition is also ΨF