Lecture #21

OUTLINE
The MOS Capacitor
• Electrostatics

Reading: Course Reader

Spring 2003 EE130 Lecture 21, Slide 1

MOS Capacitor Structure

• Typical MOS capacitors and
MOS capacitor (cross-sectional view) transistors in ICs today employ
– heavily doped polycrystalline Si
(“poly-Si”) film as the gate-
electrode material
GATE • n+-type, for “n-channel”
xox transistors (NMOS)
• p+-type, for “p-channel”
VG +_ transistors (PMOS)
– SiO2 as the gate dielectric
Si
• 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

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 Bulk Semiconductor Potential ψB
qψ B ≡ Ei (bulk ) − E F
• p-type Si: Ec

kT
ψB = ln( N A / ni ) > 0 EF
qψ B
Ei

q Ev

• n-type Si: EF
Ec

kT |qψB|
ψ B = − ln( N D / ni ) < 0 Ei

q Ev

Spring 2003 EE130 Lecture 21, Slide 3

MOS Equilibrium Energy-Band Diagram

Ec
3.1 eV
3.1 eV

Ec
Ef
N +polysilicon

Ec
P-Silicon body

Ev
9eV

Ev
SiO2

gate body

Ev

(a) (b)
How does one arrive at this energy-band diagram?
Spring 2003 EE130 Lecture 21, Slide 4

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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:
ε Si dEc dEc
ox = Si ≅3 Si so = 3×
ε ox dx oxide dx Si ( at the surface )

Spring 2003 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

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 Voltage Drops in the MOS System
• In general,
VG = VFB + Vox + ψ s
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)
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

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 General Case: Different Work Functions

Spring 2003 EE130 Lecture 21, Slide 9

Flat-Band Condition
E0
χSiO2 =0.95 eV
Ec
qΦM
3.1 eV 3.1 eV χSi qΦs = χSi + (Ec –EF)

Ec, EF Ec
VFB
Ef
Ev
Ev
9 eV
P-type Si
N+ poly-Si SiO
E0 : Vacuum level
4.8 eV
E0 – Ef : Work function
E0 – Ec : Electron affinity
Si/SiO2 energy barrier Ev

Spring 2003 EE130 Lecture221, Slide 10

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 MOS Band Diagrams (n-type Si)
Decrease VG (toward more negative values)
-> move the gate energy-bands up, relative to the Si
decrease VG decrease VG

• Accumulation • Depletion • Inversion

– VG > VFB – VG < VFB – VG < VT
– Electrons – Electrons – Surface
accumulate at repelled becomes
surface from surface p-type

Spring 2003 EE130 Lecture 21, Slide 11

Biasing Conditions for p-type Si

increase VG increase VG

VG = VFB VG < VFB VT > VG > VFB

Spring 2003 EE130 Lecture 21, Slide 12

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 Accumulation (n+ poly-Si gate, p-type Si)
M O S
VG < VFB
3.1 eV | qVox |

Ec= EFM

GATE Ev
- - - - - - |qVG |
|qψs| is small, ≈ 0
+ + + + + +

VG +_ Ec
p-type Si 4.8 eV EFS
Ev

Mobile carriers (holes) accumulate at Si surface

VG ≅ VFB + Vox
Spring 2003 EE130 Lecture 21, Slide 13

Accumulation Layer Charge Density

VG < VFB Vox ≅ VG − VFB

From Gauss’ Law:

ox = −Qacc / ε SiO2
GATE
- - - - - -

+ + + + + +
tox Vox = t = −Qacc / Cox
ox ox
+
VG _ Qacc (C/cm2) where Cox ≡ ε SiO2 / tox
p-type Si (units: F/cm2)

⇒ Qacc = −Cox (VG − VFB ) > 0

Spring 2003 EE130 Lecture 21, Slide 14

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 Depletion (n+ poly-Si gate, p-type Si)
M O S
VT > VG > VFB
qVox
Wd
Ec

GATE EFS
+ + + + + + 3.1 eV qψ s Ev
- - - - - - qVG
+
VG _ 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 (0 ≤ x ≤ Wd )
d ρ qN A
• Poisson’s equation: = ≅− (0 ≤ x ≤ Wd )
dx ε Si ε Si
• Integrate twice, to obtain ψS:

qN A 2 2ε Siψ s To find ψs for a given VG, we

ψs = Wd ⇒ Wd = need to consider the voltage
2ε Si qN A drops in the MOS system…

Spring 2003 EE130 Lecture 21, Slide 16

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 Voltage Drops in Depletion (p-type Si)
From Gauss’ Law:
GATE ox = −Qdep / ε SiO2
+ + + + + +
- - - - - - Vox = t = −Qdep / Cox
ox ox
VG +_
Qdep (C/cm2)
Qdep is the integrated
p-type Si
charge density in the Si:
Qdep = − qN AWd = − 2qN Aε Siψ s

2 qN Aε siψ s
VG = VFB + ψ s + Vox = VFB + ψ s +
Cox
Spring 2003 EE130 Lecture 21, Slide 17

Surface Potential in Depletion (p-type Si)

2 qN Aε siψ s
VG = VFB + ψ s +
Cox
• Solving for ψS, we have

qN Aε si  2Cox (VG − VFB ) 

2
ψs =  1+ − 1
2Cox  qN Aε si 
2
qN Aε si  2Cox (VG − VFB ) 
2
ψs =  1+ − 1
2Cox
2
 qN Aε si 
Spring 2003 EE130 Lecture 21, Slide 18

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 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 O S
kT  N A 
ψ s = 2ψ B = 2 ln 
q  ni  qVox Wdm
qψ B
Ec
2ε Si ( 2ψ B )
Wd = Wdm = qψ F EFS
qN A qψ s Ev
qVG

Ec= EFM

Ev

Spring 2003 EE130 Lecture 21, Slide 20

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 Threshold Voltage
• For p-type Si:
2 qN Aε siψ s
VG = VFB + ψ s + Vox = VFB + ψ s +
Cox
2qN Aε Si ( 2ψ B )
VT = VFB + 2ψ B +
Cox

• For n-type Si:

2qN Dε Si 2ψ B
VT = VFB + 2ψ B −
Cox
Spring 2003 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 ρ(x)
GATE M O S
+ + + + + +
- - - - - -

VG +_ x
p-type Si

ψ s ≅ 2ψ B
Significant density of mobile electrons at surface 2ε si ( 2ψ B )
(surface is n-type) Wd ≅ Wdm =
qN A
Spring 2003 EE130 Lecture 21, Slide 22

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Inversion Layer Charge Density (p-type Si)
VG = VFB + ψ S + Vox
(Qdep + Qinv )
= VFB + 2ψ B −
Cox
2qN Aε s (2ψ B ) Qinv
= VFB + 2ψ B + −
Cox Cox
Qinv
= VT −
Cox

∴ Qinv = −Cox (VG − VT )

Spring 2003 EE130 Lecture 21, Slide 23

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