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Lecture #21
OUTLINE
The MOS Capacitor
• Electrostatics
Reading: Course Reader
Spring 2003
EE130 Lecture 21, Slide 1
MOS Capacitor Structure
• Typical MOS capacitors and
transistors in ICs today employ
MOS capacitor (cross-sectional view)
– heavily doped polycrystalline Si
(“poly-Si”) film as the gateelectrode material
GATE
• n+-type, for “n-channel”
transistors (NMOS)
• p+-type, for “p-channel”
transistors (PMOS)
xox
VG +_
Si
– SiO2 as the gate dielectric
• band gap = 9 eV
• εr,SiO2 = 3.9
– Si as the semiconductor material
• p-type, for “n-channel”
transistors (NMOS)
• n-type, for “p-channel”
transistors (PMOS)
Spring 2003
EE130 Lecture 21, Slide 2
1
Bulk Semiconductor Potential ψB
qψ B ≡ Ei (bulk ) − E F
• p-type Si:
kT
ψB =
ln( N A / ni ) > 0
q
Ec
EF
• n-type Si:
kT
ψ B = − ln( N D / ni ) < 0
q
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EF
qψ B
Ei
Ev
Ec
|qψB|
Ei
Ev
EE130 Lecture 21, Slide 3
MOS Equilibrium Energy-Band Diagram
Ef
Ec
9eV
P-Silicon body
SiO2
N +polysilicon
3.1 eV
3.1 eV
Ec
Ev
gate
Ec
Ev
body
Ev
(a)
(b)
How does one arrive at this energy-band diagram?
Spring 2003
EE130 Lecture 21, Slide 4
2
Guidelines for Drawing MOS Band Diagrams
• Fermi level EF is flat (constant with distance x) in the Si
– Since no current flows in the x direction, we can assume that
equilibrium conditions prevail
• Band bending is linear in the oxide
– No charge in the oxide => d /dx = 0 so is constant
=> dEc/dx is constant
• From Gauss’ Law, we know that the electric field
strength in the Si at the surface, Si, is related to the
electric field strength in the oxide, ox:
ox
Spring 2003
=
ε Si
ε ox
Si
≅3
Si
so
dEc
dx
= 3×
oxide
dEc
dx
Si ( at the surface )
EE130 Lecture 21, Slide 5
MOS Band-Diagram Guidelines (cont.)
• The barrier height for conduction-band electron flow
from the Si into SiO2 is 3.1 eV
– This is equal to the electron-affinity difference (χSi and χSiO2)
• The barrier height for valence-band hole flow from the Si
into SiO2 is 4.8 eV
• The vertical distance between the Fermi level in the
metal, EFM, and the Fermi level in the Si, EFS, is equal to
the applied gate voltage:
qVG = E FS − E FM
Spring 2003
EE130 Lecture 21, Slide 6
3
Voltage Drops in the MOS System
• In general,
where
VG = VFB + Vox + ψ s
qVFB = φMS = φM – φS
Vox is the voltage dropped across the oxide
(Vox = total amount of band bending in the oxide)
ψs is the voltage dropped in the silicon
(total amount of band bending in the silicon)
qψ s = Ei (bulk ) − Ei ( surface)
For example: When VG = VFB, Vox = ψs = 0
i.e. there is no band bending
Spring 2003
EE130 Lecture 21, Slide 7
Special Case: Equal Work Functions
ΦM = ΦS
What happens when
the work function is
different?
Spring 2003
EE130 Lecture 21, Slide 8
4
General Case: Different Work Functions
Spring 2003
EE130 Lecture 21, Slide 9
Flat-Band Condition
χSiO2 =0.95 eV
E0
Ec
qΦM
Ec, EF
3.1 eV χSi
3.1 eV
Ec
VFB
Ef
Ev
Ev
9 eV
N+ poly-Si
E0 : Vacuum level
E0 – Ef : Work function
E0 – Ec : Electron affinity
Si/SiO2 energy barrier
Spring 2003
qΦs = χSi + (Ec –EF)
SiO
P-type Si
4.8 eV
Ev
EE130 Lecture221, Slide 10
5
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
decrease VG
• Depletion
– VG < VFB
– Electrons
accumulate at
– Electrons
repelled
from surface
surface
Spring 2003
• Inversion
– VG < VT
– Surface
becomes
p-type
EE130 Lecture 21, Slide 11
Biasing Conditions for p-type Si
increase VG
VG = VFB
Spring 2003
VG < VFB
increase VG
VT > VG > VFB
EE130 Lecture 21, Slide 12
6
Accumulation (n+ poly-Si gate, p-type Si)
M
VG < VFB
O
S
3.1 eV
| qVox |
Ec= EFM
GATE
Ev
- - - - - -
|qVG |
|qψs| is small, ≈ 0
+ + + + + +
VG +_
Ec
p-type Si
4.8 eV
VG ≅ VFB + Vox
Mobile carriers (holes) accumulate at Si surface
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EFS
Ev
EE130 Lecture 21, Slide 13
Accumulation Layer Charge Density
Vox ≅ VG − VFB
VG < VFB
From Gauss’ Law:
ox
GATE
- - - - - -
VG
tox
+ + + + + +
+
_
Qacc (C/cm2)
= −Qacc / ε SiO2
Vox =
t = −Qacc / Cox
ox ox
where Cox ≡ ε SiO2 / tox
p-type Si
(units: F/cm2)
⇒ Qacc = −Cox (VG − VFB ) > 0
Spring 2003
EE130 Lecture 21, Slide 14
7
Depletion (n+ poly-Si gate, p-type Si)
M
VT > VG > VFB
O
qVox
S
Wd
Ec
GATE
VG
qψ s
3.1 eV
+ + + + + +
- - - - - -
+
_
qVG
EFS
Ev
Ec= EFM
p-type Si
Ev
4.8 eV
Si surface is depleted of mobile carriers (holes)
=> Surface charge is due to ionized dopants (acceptors)
Spring 2003
EE130 Lecture 21, Slide 15
Depletion Width Wd (p-type Si)
• Depletion Approximation:
The surface of the Si is depleted of mobile carriers to a depth Wd.
• The charge density within the depletion region is
ρ ≅ − qN A
• Poisson’s equation:
(0 ≤ x ≤ Wd )
d
ρ
qN A
=
≅−
dx ε Si
ε Si
(0 ≤ x ≤ Wd )
• Integrate twice, to obtain ψS:
ψs =
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qN A 2
Wd ⇒ Wd =
2ε Si
2ε Siψ s
qN A
To find ψs for a given VG, we
need to consider the voltage
drops in the MOS system…
EE130 Lecture 21, Slide 16
8
Voltage Drops in Depletion (p-type Si)
From Gauss’ Law:
VG +_
= −Qdep / ε SiO2
GATE
ox
- - - - - -
Vox =
+ + + + + +
Qdep (C/cm2)
p-type Si
t = −Qdep / Cox
ox ox
Qdep is the integrated
charge density in the Si:
Qdep = − qN AWd = − 2qN Aε Siψ s
VG = VFB + ψ s + Vox = VFB + ψ s +
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2 qN Aε siψ s
Cox
EE130 Lecture 21, Slide 17
Surface Potential in Depletion (p-type Si)
VG = VFB + ψ s +
2 qN Aε siψ s
Cox
• Solving for ψS, we have
ψs =
2
qN Aε si 
2Cox (VG − VFB ) 
− 1
 1+
qN Aε si
2Cox 

qN Aε si
ψs =
2
2Cox
Spring 2003
2

2Cox (VG − VFB ) 
− 1
 1+
qN Aε si


2
EE130 Lecture 21, Slide 18
9
Threshold Condition (VG = VT)
• When VG is increased to the point where ψs reaches
2ψΒ, 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 ⇒ ψ s = 2ψ B
E i (bulk ) − Ei ( surface) = 2[Ei (bulk ) − E F ]
Ei ( surface ) − EF = −[Ei (bulk ) − E F ]
⇒ nsurface = N A
Spring 2003
EE130 Lecture 21, Slide 19
MOS Band Diagram at Threshold (p-type Si)
M
ψ s = 2ψ B = 2
Wd = Wdm =
kT  N A 

ln
q  ni 
qVox
2ε Si ( 2ψ B )
qN A
qψ F
O
S
Wdm
qψ B
qψ s
Ec
EFS
Ev
qVG
Ec= EFM
Ev
Spring 2003
EE130 Lecture 21, Slide 20
10
Threshold Voltage
• For p-type Si:
VG = VFB + ψ s + Vox = VFB + ψ s +
VT = VFB + 2ψ B +
2 qN Aε siψ s
Cox
2qN Aε Si ( 2ψ B )
Cox
• For n-type Si:
VT = VFB + 2ψ B −
Spring 2003
2qN Dε Si 2ψ B
Cox
EE130 Lecture 21, Slide 21
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 d= Wdm
Wdm
GATE
ρ(x)
M O S
+ + + + + +
VG +_
- - - - - -
x
p-type Si
ψ s ≅ 2ψ B
Significant density of mobile electrons at surface
(surface is n-type)
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Wd ≅ Wdm =
2ε si ( 2ψ B )
qN A
EE130 Lecture 21, Slide 22
11
Inversion Layer Charge Density (p-type Si)
VG = VFB + ψ S + Vox
= VFB + 2ψ B −
= VFB + 2ψ B +
= VT −
∴
Spring 2003
(Qdep + Qinv )
Cox
2qN Aε s (2ψ B ) Qinv
−
Cox
Cox
Qinv
Cox
Qinv = −Cox (VG − VT )
EE130 Lecture 21, Slide 23
12