Lecture 16
OUTLINE
• The MOS Capacitor (cont’d)
– Electrostatics
Reading: Pierret 16.3; Hu 5.2-5.5
Bulk Semiconductor Potential, fF
qfF  Ei (bulk )  EF
• p-type Si:
kT
fF 
ln( N A / ni )  0
q
Ec
EF
qfF
Ev
• n-type Si:
kT
fF   ln( N D / ni )  0
q
EE130/230M Spring 2013
Lecture 16, Slide 2
EF
Ei
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
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Lecture 16, Slide 3
MOS Band Diagrams for n-type Si
Decrease VG toward more negative values
 the gate electron energy increases relative to that in the Si
decrease VG
• Accumulation
– VG > VFB
– Electrons
accumulate at
surface
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decrease VG
• Depletion
– VG < VFB
– Electrons
repelled from
surface
Lecture 16, Slide 4
• Inversion
– VG < VT
– Surface
becomes ptype
MOS Band Diagrams for p-type Si
increase VG
VG = VFB
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VG < VFB
VT > VG > VFB
Lecture 16, Slide 5
increase VG
Accumulation
(n+ poly-Si gate, p-type Si)
M
VG < VFB
3.1 eV
O
S
| qVox |
Ec= EFM
GATE
- - - - - + + + + + +
VG
+
_
Ev
|qVG |
xo
Ec
p-type Si
4.8 eV
Mobile carriers (holes) accumulate at Si surface
EE130/230M Spring 2013
|qfS| is small,  0
Lecture 16, Slide 6
EFS
Ev
VG  VFB  Vox
Accumulation Layer Charge Density
VG < VFB
Vox  VG  VFB
From Gauss’ Law:

GATE
- - - - - + + + + + +
VG
+
_
Qacc (C/cm2)
xo
ox
 Qacc / ε SiO2
Vox 

x  Qacc / Cox
ox o
where Cox  ε SiO2 / xo
p-type Si
(units: F/cm2)
 Qacc  Cox (VG  VFB )  0
EE130/230M Spring 2013
Lecture 16, Slide 7
Depletion
(n+ poly-Si gate, p-type Si)
M
VT > VG > VFB
qVox
O
S
W
Ec
GATE
+ + + + + +
VG
+
_
qfS
3.1 eV
- - - - - -
p-type Si
Ec= EFM
Ev
Si surface is depleted of mobile carriers (holes)
=> Surface charge is due to ionized dopants (acceptors)
EE130/230M Spring 2013
Lecture 16, 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
EE130/230M Spring 2013
2 SifS
W 
qN A
Lecture 16, 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

ox
 Qdep / ε SiO2
Vox   ox xo  Qdep / Cox
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
EE130/230M Spring 2013
Lecture 16, 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
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
2Cox (VG  VFB ) 
 1
 1
qN A si


2
Lecture 16, 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. 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
(The surface is n-type to the same degree as the bulk is p-type.)
EE130/230M Spring 2013
Lecture 16, 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
EE130/230M Spring 2013
Lecture 16, 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 
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2qN D Si 2fF
Cox
Lecture 16, 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)
WT
M O S
GATE
+ + + + + +
VG
+
_
x
- - - - - -
p-type Si
fS  2fF
Significant density of mobile electrons at surface
(surface is n-type)
EE130/230M Spring 2013
Lecture 16, 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 )
EE130/230M Spring 2013
Lecture 16, Slide 16