Lecture #30
OUTLINE
The MOS Capacitor
• Electrostatics
Reading: Chapter 16.3
Spring 2007
EE130 Lecture 30, Slide 1
Bulk Semiconductor Potential, fF
qfF  Ei (bulk )  EF
• p-type Si:
kT
fF 
ln( N A / ni )  0
q
• n-type Si:
kT
fF   ln( N D / ni )  0
q
Spring 2007
EE130 Lecture 30, Slide 2
Ec
EF
EF
qfF
Ei
Ev
Ec
|qfF|
Ei
Ev
Voltage Drops in the MOS System
• In general,
VG  VFB  Vox  fs
where
qVFB = FMS = FM – FS
Vox is the voltage dropped across the oxide
(Vox = total amount of band bending in the oxide)
fs is the voltage dropped in the silicon
(total amount of band bending in the silicon)
qfS  Ei (bulk )  Ei ( surface)
For example: When VG = VFB, Vox = fs = 0
i.e. there is no band bending
Spring 2007
EE130 Lecture 30, Slide 3
MOS Band Diagrams (n-type Si)
Decrease VG (toward more negative values)
-> move the gate energy-bands up, relative to the Si
decrease VG
• Accumulation
– VG > VFB
– Electrons
accumulate at
surface
Spring 2007
decrease VG
• Depletion
– VG < VFB
– Electrons
repelled
from surface
EE130 Lecture 30, Slide 4
• Inversion
– VG < VT
– Surface
becomes
p-type
Biasing Conditions for p-type Si
increase VG
VG = VFB
Spring 2007
VG < VFB
increase VG
VT > VG > VFB
EE130 Lecture 30, Slide 5
Accumulation (n+ poly-Si gate, p-type Si)
M
VG < VFB
3.1 eV
O
S
| qVox |
Ec= EFM
GATE
- - - - - -
Ev
|qVG |
+ + + + + +
VG +_
Ec
p-type Si
4.8 eV
Mobile carriers (holes) accumulate at Si surface
Spring 2007
|qfS| is small,  0
EE130 Lecture 30, Slide 6
EFS
Ev
VG  VFB  Vox
Accumulation Layer Charge Density
Vox  VG  VFB
VG < VFB
From Gauss’ Law:

GATE
- - - - - -
xo
+ + + + + +
VG +_
Qacc (C/cm2)
ox
 Qacc / ε SiO2
Vox 

x  Qacc / Cox
ox o
where Cox  ε SiO2 / xo
(units: F/cm2)
p-type Si
 Qacc  Cox (VG  VFB )  0
Spring 2007
EE130 Lecture 30, Slide 7
Depletion (n+ poly-Si gate, p-type Si)
M
VT > VG > VFB
qVox
O
S
W
Ec
GATE
+ + + + + +
VG +_
- - - - - -
p-type Si
Ec= EFM
Ev
Si surface is depleted of mobile carriers (holes)
=> Surface charge is due to ionized dopants (acceptors)
Spring 2007
qfS
3.1 eV
EE130 Lecture 30, Slide 8
4.8 eV
qVG
EFS
Ev
Depletion Width W (p-type Si)
• Depletion Approximation:
The surface of the Si is depleted of mobile carriers to a depth W.
• The charge density within the depletion region is
  qN A
(0  x  W )
d
ρ
qN A


• Poisson’s equation:
dx ε Si
ε Si
(0  x  W )
• Integrate twice, to obtain fS:
qN A 2
fS 
W
2 Si
Spring 2007
2 SifS
W 
qN A
EE130 Lecture 30, Slide 9
To find fs for a given VG, we
need to consider the voltage
drops in the MOS system…
Voltage Drops in Depletion (p-type Si)
From Gauss’ Law:

GATE
VG
- - - - - -
Qdep (C/cm2)
p-type Si
 Qdep / ε SiO2
Vox   ox xo  Qdep / Cox
+ + + + + +
+
_
ox
Qdep is the integrated
charge density in the Si:
Qdep  qN AW   2qN A SifS
2qN A sifS
VG  VFB  fS  Vox  VFB  fS 
Cox
Spring 2007
EE130 Lecture 30, Slide 10
Surface Potential in Depletion (p-type Si)
2qN A sifS
VG  VFB  fS 
Cox
• Solving for fS, we have
2
qN A si 
2Cox (VG  VFB ) 
 1
fS 
 1
qN A si
2Cox 

qN A si
fS 
2
2Cox
Spring 2007

2Cox (VG  VFB ) 
 1
 1
qN A si


2
EE130 Lecture 30, Slide 11
2
Threshold Condition (VG = VT)
• When VG is increased to the point where fs reaches
2fF, the surface is said to be strongly inverted.
(The surface is n-type to the same degree as the bulk is p-type.)
This is the threshold condition.
VG = VT  fS  2fF
E i (bulk )  Ei ( surface)  2Ei (bulk )  EF 
Ei ( surface)  EF  Ei (bulk )  EF 
 nsurface  N A
Spring 2007
EE130 Lecture 30, Slide 12
MOS Band Diagram at Threshold (p-type Si)
M
kT  N A 

fS  2fF  2 ln 
q  ni 
W  WT 
qVox
2 Si (2f F )
qN A
qfF
Ec= EFM
Ev
Spring 2007
EE130 Lecture 30, Slide 13
O
S
WT
qfF
qfs
Ec
EFS
Ev
qVG
Threshold Voltage
• For p-type Si:
2qN A sifS
VG  VFB  fS  Vox  VFB  fS 
Cox
2qN A Si (2fF )
VT  VFB  2fF 
Cox
• For n-type Si:
VT  VFB  2fF 
Spring 2007
2qN D Si 2fF
Cox
EE130 Lecture 30, Slide 14
Strong Inversion (p-type Si)
As VG is increased above VT, the negative charge in the Si is increased
by adding mobile electrons (rather than by depleting the Si more deeply),
so the depletion width remains ~constant at W= WT
(x)
M O S
WT
GATE
+ + + + + +
VG +_
- - - - - -
x
p-type Si
fS  2fF
Significant density of mobile electrons at surface
(surface is n-type)
Spring 2007
EE130 Lecture 30, Slide 15
2 si (2fF )
W  WT 
qN A
Inversion Layer Charge Density (p-type Si)
VG  VFB  fS  Vox
 VFB  2fF 
(Qdep  Qinv )
Cox
2qN A s (2fF ) Qinv
 VFB  2fF 

Cox
Cox
Qinv
 VT 
Cox
 Qinv  Cox (VG  VT )
Spring 2007
EE130 Lecture 30, Slide 16