Lecture 16
OUTLINE
• The MOS Capacitor (cont’d)
– Electrostatics
Reading: Pierret 16.3; Hu 5.2-5.5
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
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|qfS| is small,  0
Lecture 16, Slide 2
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
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Lecture 16, Slide 3
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)
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qfS
3.1 eV
Lecture 16, Slide 4
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
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2 SifS
W 
qN A
Lecture 16, Slide 5
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
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Lecture 16, Slide 6
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 7
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.)
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Lecture 16, Slide 8
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
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Lecture 16, Slide 9
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 10
C. C. Hu, Modern Semiconductor Devices for ICs, Figure 5-8
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)
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Lecture 16, Slide 11
2 si (2fF )
W  WT 
qN A
R. F. Pierret, Semiconductor Device Fundamentals, p. 575
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 )
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Lecture 16, Slide 12
fS and W vs. VG
(p-type Si)
2fF
fS:
2
qN A si 
2Cox (VG  VFB ) 
 1
fs 
 1
2
qN A si
2Cox 

0
WT 
0
accumulation
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(for VFB  VG  VT )
VG
accumulation V depletion V inversion
FB
T
W:
2
2ε Si (2fF )
qN A
2
2 SifS  Si 
2Cox (VG  VFB ) 
 1
W

 1 (for VFB  VG  VT )
qN A
Cox 
qN A si


VFB depletion VT inversion
Lecture 16, Slide 13
VG
Total Charge Density in Si, Qs
(p-type Si)
Qacc  Cox (VG  VFB )
depletion
0
accumulation
VFB
accumulation
inversion
VT
depletion
inversion
0
VFB
VG accumulation
depletion
inversion
VT
VG
Qdep  qN AW
accumulation
Qs  Qacc  Qdep  Qinv
VG
depletion
inversion
0
VFB
0
VT
VG
VT
Qinv
slope = -Cox
Qinv  Cox (VG  VT )
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VFB
Lecture 16, Slide 14