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1

EE130 Lecture 21, Slide 1Spring 2003

Lecture #21

OUTLINEThe MOS Capacitor• Electrostatics

Reading: Course Reader


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)

2


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|


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

3


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 ×=≅=


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 −=

4


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


Special Case: Equal Work Functions

What happens when the work function is

different?

ΦM = ΦS

5


General Case: Different Work Functions


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

6


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


Biasing Conditions for p-type Si

VG = VFB VG < VFB VT > VG > VFB

increase VG increase VG

7


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


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

8


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)


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…

9


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 −=−=


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

10


[ ][ ]

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


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

11


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 −+=


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

12


ox

invT

ox

inv

ox

BsABFB

ox

invdepBFB

oxSFBG

CQV

CQ

CqN

V

CQQ

V

VVV

−=

−++=

+−+=

++=

)2(22

)(2

ψεψ

ψ

ψ

)( TGoxinv VVCQ −−=∴

Inversion Layer Charge Density (p-type Si)

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