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EE130 Lecture 21, Slide 1Spring 2003
Lecture #21
OUTLINEThe MOS Capacitor• Electrostatics
Reading: Course Reader
EE130 Lecture 21, Slide 2Spring 2003
MOS Capacitor Structure• Typical MOS capacitors and
transistors in ICs today employ– heavily doped polycrystalline Si
(“poly-Si”) film as the gate-electrode material
• n+-type, for “n-channel” transistors (NMOS)
• p+-type, for “p-channel” transistors (PMOS)
– 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)
Si
+_
GATExox
VG
MOS capacitor (cross-sectional view)
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EE130 Lecture 21, Slide 3Spring 2003
Bulk Semiconductor Potential ψB
• p-type Si:
• n-type Si:
FiB EbulkEq −≡ )(ψ
0)/ln( >= iAB nNq
kTψ
0)/ln( <−= iDB nNq
kTψ
Ec
EF Ev
EiqψB
EcEF
Ev
Ei
|qψB|
EE130 Lecture 21, Slide 4Spring 2003
How does one arrive at this energy-band diagram?
N+ po
lysi
licon
SiO
2
(a)
Ec
Ev
Ec
Ev
Ec
Ev
Ef
3.1
eV
9eV
3.1
eV
(b)
P-S
ilico
n bo
dy
gate body
MOS Equilibrium Energy-Band Diagram
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EE130 Lecture 21, Slide 5Spring 2003
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:
) (
3 3 εε
surfacetheatSi
c
oxide
cSiSi
ox
Siox dx
dEdx
dEso ×=≅=
EE130 Lecture 21, Slide 6Spring 2003
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 Siinto 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:
FMFSG EEqV −=
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EE130 Lecture 21, Slide 7Spring 2003
Voltage Drops in the MOS System• In general,
where 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)
For example: When VG = VFB, Vox = ψs = 0i.e. there is no band bending
soxFBG VVV ψ++=
)()( surfaceEbulkEq iis −=ψ
EE130 Lecture 21, Slide 8Spring 2003
Special Case: Equal Work Functions
What happens when the work function is
different?
ΦM = ΦS
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EE130 Lecture 21, Slide 9Spring 2003
General Case: Different Work Functions
EE130 Lecture 21, Slide 10Spring 2003
E0 : Vacuum levelE0 – Ef : Work functionE0 – Ec : Electron affinitySi/SiO2 energy barrier
χSiO2=0.95 eV
9 eV
Ec, EF
Ev
Ec
Ev
Ef
3.1 eV qΦs= χSi + (Ec–EF) qΦM χSi
E0
3.1 eV
VFB
N+ poly-SiP-type Si
4.8 eV
Ec
Ev
SiO
2
Flat-Band Condition
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EE130 Lecture 21, Slide 11Spring 2003
MOS Band Diagrams (n-type Si)
• Inversion– VG < VT– Surface
becomes p-type
• Accumulation– VG > VFB– Electrons
accumulate at surface
• Depletion– VG < VFB– Electrons
repelled from surface
Decrease VG (toward more negative values) -> move the gate energy-bands up, relative to the Si
decrease VG decrease VG
EE130 Lecture 21, Slide 12Spring 2003
Biasing Conditions for p-type Si
VG = VFB VG < VFB VT > VG > VFB
increase VG increase VG
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EE130 Lecture 21, Slide 13Spring 2003
Accumulation (n+ poly-Si gate, p-type Si)
Ec
EFSEv
|qψs| is small, ≈ 0
Ec= EFM
Ev
M O S
3.1 eV
4.8 eVp-type Si+_VG
VG < VFB
++ + + + +
- - - - - -GATE
|qVG |
oxFBG VVV +≅
| qVox |
Mobile carriers (holes) accumulate at Si surface
EE130 Lecture 21, Slide 14Spring 2003
FBGox VVV −≅
0)( >−−=⇒ FBGoxacc VVCQ
Accumulation Layer Charge Density
p-type Si
+_VG
VG < VFB
++ + + + +
- - - - - -GATE
Qacc (C/cm2)
From Gauss’ Law:
oxSiOox
oxaccoxoxox
SiOaccox
tCCQtV
Q
/ε where/
ε/
2
2
≡−==
−=
(units: F/cm2)
tox
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EE130 Lecture 21, Slide 15Spring 2003
Depletion (n+ poly-Si gate, p-type Si)
Ec
EFSEv
Ec= EFM
Ev
3.1 eV
4.8 eV
p-type Si+_VG
VT > VG > VFB
-- - - --
+ + + + + +GATE
qVG
qVox
qψs
M O S
Wd
Si surface is depleted of mobile carriers (holes)=> Surface charge is due to ionized dopants (acceptors)
EE130 Lecture 21, Slide 16Spring 2003
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
• Poisson’s equation:
• Integrate twice, to obtain ψS:
)0( dWxqN A ≤≤−≅ρ
)0( εε dWxqNρ
dxd
Si
A
Si
≤≤−≅=
2d2
WqN
Si
As ε
ψ =A
sSi
qNW ψε2
d =⇒To find ψs for a given VG, we need to consider the voltage drops in the MOS system…
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EE130 Lecture 21, Slide 17Spring 2003
Voltage Drops in Depletion (p-type Si)
ox
ssiAsFBoxsFBG C
qNVVVV
ψεψψ
2++=++=
p-type Si
+_VG
-- - - --
+ + + + + +GATE
Qdep (C/cm2)
From Gauss’ Law:
oxdepoxoxox
SiOdepox
CQtV
Q
/
ε/2
−==
−=
Qdep is the integrated charge density in the Si:
sSiAAdep qNWqNQ ψε2d −=−=
EE130 Lecture 21, Slide 18Spring 2003
Surface Potential in Depletion (p-type Si)
• Solving for ψS, we haveox
ssiAsFBG C
qNVV
ψεψ
2++=
−−+= 1)(21
2
2
siA
FBGox
ox
siAs qN
VVCC
qNε
εψ
22
2 1)(212
−−+=
siA
FBGox
ox
siAs qN
VVCC
qNε
εψ
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EE130 Lecture 21, Slide 19Spring 2003
[ ][ ]
Asurface
FiFi
Fiii
Bs
NnEbulkEEsurfaceE
EbulkEsurfaceEbulkE
=⇒−−=−
−=−=⇒
)()()(2)()(
2ψψ
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
EE130 Lecture 21, Slide 20Spring 2003
A
BSidmd
i
ABs
qNWW
nN
qkT
)2(2
ln22
ψε
ψψ
==
==
MOS Band Diagram at Threshold (p-type Si)
Ec
EFSEv
Ec= EFM
Ev
qVG
qVox
qψs
M O S
Wdm qψB
qψF
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EE130 Lecture 21, Slide 21Spring 2003
ox
BSiABFBT C
qNVV
)2(22
ψεψ ++=
Threshold Voltage• For p-type Si:
• For n-type Si:
ox
ssiAsFBoxsFBG C
qNVVVV
ψεψψ
2++=++=
ox
BSiDBFBT C
qNVV
ψεψ
222 −+=
EE130 Lecture 21, Slide 22Spring 2003
A
Bsidmd
Bs
qNWW )2(2
2
ψε
ψψ
=≅
≅
Strong Inversion (p-type Si)
p-type Si+_VG
-- - - --
+ + + + + +GATE
Significant density of mobile electrons at surface(surface is n-type)
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
ρ(x)M O S
x
Wdm