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EE231 Vivek Subramanian Slide 1-1

A quick review of MOS Capacitors


MOS Capacitors

What happens when thework function is different?

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MOS Band Diagram Different Work function


E0 : Vacuum levelE0Ef: Work functionE0Ec : Electron affinitySi/SiO2 energy barrier

sMfbV =

SiO2=0.95 eV

9 eV

Ec,Ef

Ev

Ec

Ev

Ef

3.1 eV qs= Si + (EcEf)

qMSi

E0

3.1 eV

Vfb

N+ -poly-Si P-body

4.8 eV

=4.05eV

Ec

Ev

SiO2

Flat Band Condition

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voltage

acrossthe oxide

voltage across the substrate,

i.e. the band bending in the

substrate, also called surface potential

fbg VV

SOxfbg VVV ++=

VOx

Vg

S

EfEf

What if

substrate charge

Ox

SOx

CQV = accinvdepS QQQQ ++=

Non-Flat-band conditions


oxsfbg VVV ++=

s is negligible

3.1eV

Ec ,Ef

Ev

E0

E

c

E

f

Ev

M O S

qVg

Vox

qs

Surface Accumulation

kTEE

vvfeNp

/)( =

( )fbgOx

OxOxS

SOxfbg

Ox

SOx

VVC

VCQ

VVV

CQV

=

=

++=

=

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Can solve for (Vg) and VOx(Vg)

SOxfbg VVV ++=

Ox

SSa

Ox

depa

Ox

dep

Ox

SOx

C

qN

C

XqN

C

Q

C

QV

2 =

+=

=

=

S

Ec,Ef

Ev

Ec

EfEv

M O S

qVg

depletion

region

qs

Wdep

qVox

----

Depletion


aS Npn == 0

( )

Vn

N

q

kT

EEq

V.

EEEE

i

a

bulkfiB

BS

bulkVfSfC

4.0ln

1

802

)()(

=

=

=

=

VtAt

( )

Ox

BaS

C

Nq

BfbT

OxSfbBSgT

VV

VVVV

222

2

++=

++=== ( )

( )

319

/

/

10

=

=

cm

eNp

eNn

kTEE

V

kTEE

C

Vf

fC

C=D

BA =

(Alternative definition: S,th = B + 0.45V)

0.15V

Threshold (of Inversion)

Ec,Ef

M O S

Ev

Ef

Ei

Ec

A

B

C= q

Ev

D

qVg

=qVt

st

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BSBS 2~2

Ox

dep

BfbT

Ox

invT

Ox

inv

Ox

BaS

Bfb

Ox

invdep

Bfbg

C

QVV

C

QV

C

Q

C

NqV

C

QQVV

+=

=++=

++=

2

222

2

( TgOxinv VVCQ =

Ox

S

inv

BS

V

Q

Q

large

large

large

2

EC

EF

Inversion


s

2B

Vf b Vt

Vg

accumulation depletion inversion

Wdep

Wdmax

Vfb Vt

Vg


(s)1/2

Wdmax

= (2s2/qa)1/2

Review : Basic MOS Capacitor Theory

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Qdep

= qNaW

dep

0

Vfb

Vt

Vg


qNaWdep

Qinv

Vfb

Vt

Vg


slope =Cox

(a)

(b)

Qacc

Vfb

Vt

Vg


(c)

qNaW

dmax

slope =Cox

Qs

0

Vfb

Vt

Vg

accumulationregime

depletionregime

inversionregime

Qinv

slope= Cox

invdepaccs QQQQ ++=

Review : Basic MOS Capacitor Theory

VgVT

C

Vfb

COx

small-signal capacitance:

g

S

dV

dQC =


Advanced MOSCAP Physics and Technology

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Na

( ) ( ) ( )

kT

q

epepxp xkTxq

==

0/

0

( ) ( ) kTxqenxn /0=

x

(x)

(0) = S

SiSiO2

EV

Na, p-type

0

( ) 0=

a

i

Nn

2

Eq. 3-9

EF

So were we too simplistic?

Problem with previous analysis Assumes that there are no free carriers in the depletion region

(depletion approximation)

Obviously, this is not true (else, how could we have inversioncharge?)


A more general and accurate MOSCAP analysis

If we dont assume complete absence of carriers in thedepletion region, we have:

Now, we can proceed by noting that:

( ) ( )

( ) ( )( )daxxS

ad

SS

NNenepq

NnNpqx

dx

d

+=

+==

+

00

2

2

2

2

2

2

1

=dx

d

d

d

dx

d

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Calculation of E(x)

Therefore, we have:

Which is E(x)

( )

( )

[ ]

( )2/1

00

00

0 00

2

00

2

2

2

)1()1(2

)1()1(2

2

2

1

++=

+

=

+

=

+

=

=

enepq

dx

d

enepq

dNNenepq

dx

d

NNenepq

dx

d

d

d

dx

d

S

S

da

S

da

S


Calculation of QS

Now, we can solve this equation at x=0 to find the peakfield. By Gauss law, we can therefore find the totalcharge in the silicon

This equation is valid in all regions of operation, since wehavent made any region-specific assumptions so far

( ) ( )[ ] 2/10 112

LawsGauss'by

++=

==

=

=

SoSS

S

S

S

SSS

SS

S

enepq

dx

d

dx

d

EQ

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Calculation of inversion and depletion charge

From the previous analysis, we can also calculate the various charge

components, namely the inversion layer charge and the depletion layercharge (and, of course, the accumulation charge in accumulation).

For example, in inversion, we find:

Since we know E(x) and n(x), we can solve for the total inversion layercharge.

Similarly, we can also solve for QB, which consists of ionizedacceptors and holes.

==c c

surf

x

dxd

dnqdxxnq

0

I/

)()(Q


Graphical analysis of charge distributions The results of the previous analysis are plotted below:

Accumulation Depletion Onset of Inversion

Specific conclusions:

1. The inversion layer charge is extremely close to the surface

2. There are very few mobile carriers in the depletion region

i.e., our simplistic analysis in the previous section is actually reasonablyaccurate.

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Extraction of C-V characteristics

From our equation for QS, we can also determine more accurate C-V

characteristics that predicted by the previous analysis.

( ) ( )[ ] 2/10 112

++= SoSS

SSS enep

qQ

( ) ( )

( ) ( )[ ]

( ) 0

2/1

0

0

2)(12

11

11

2

nNNnp

NNnpq

enep

enepq

d

dQC

daSSS

daSSS

SoS

oS

S

SS

SS

SS

++

+=

++

++==

C

V gV

-- --

-- -

- --

--

E s

accdepinvS QQQQ ++=


Extraction of C at VT

We can use this equation to determine specific capacitance values

At VG = VT, we have:

Where

This value is different from the value predicted by the simple modelby a factor if 2, since the latter does not include the inversion chargepresent at VT

( ) 02)(12

nNNnp

NNnpqC

daSSS

daSSSS

++

+=

maxd

Sdep

XC

=

OxC2//2

2

2 maxd

S

aSB

S

B

aS

aB

aSS

XqN

Nq

N

NqC

====

a

BSd

qNX

22max =

S

invinv

d

dQC

=

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Extraction of C at VFB

Similarly, at VG = VFB, we find that:

Again, we find that the simplistic model is somewhat inaccurate,since it doesnt include the wiggle around flat-band, which affectsthe charge in the silicon. This effect is small, of course.

S

COx

Cinv Cdep Cacc

EV

x=0

s

( )

a

S

SfbS

qN

VC =lengthDebye

kT/2qofbending

bandforXdep

q

kT=

1


C Ox

C s

C

0

S

C

0Vg

CS(S)

S2B

Cdepacc

S

LD

max

2

d

S

X

max

max

d

S

inv

d

Sdep

XC

XC

=

( )

Ox

SSSfbg

C

QVV

+=

m.equilibriuatarepandni.e.

,ppeg.,,inchanges

torespondpn,i.e.,,Q,Q,Q:LF

0S

depaccinv

= e

MOSCAP LFCV Characteristics

We can plot the variation in CS in the various regions to find theMOSCAP LFCV characteristics:

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MOSCAP LFCV Characteristics Graphically:

CS

S

C Ox

M

S

V g

C Ox

C s

s

S

s

2 Bacc

0

S

SS

d

dQC

=

S

Cg

QS

Substrate cap.

=> Vg


MOSCAP HFCV Characteristics We can perform a similar analysis for HFCV:

S

Vg

C

C

S

COx

CinvCdep Cacc

COx

Cs

( )

( ) ( )

( ) ( )[ ]enough)quicklyrespondnotdoeslayerinversionthe(since0nLet

11

11

2

C

HF?forC

0

2/1

00

00S

S

++

++=

SS

S

S

SS

SS

enep

enepq

S

CS(HF)

HF or DDmax

~d

S

X

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LF: n in equilibrium with s(Vg)

tVg

C

HF

n

t

LF

( ) ( )gacacgbiasbias VnVnn +:

HF:

n:nac=0

n

t

HF

Ox

SBfbg

C

QVV ++= 2

n

t

Deep Depletion:

n:nac=0

t

S

B2

C

Vg

VgLinear ramp

+ ac signal

LF

DD

I.

II

III

nbias (Vgbias)s2 B

Xd Xdmax

Ox

S

Bfbg C

Q

VV ++= 2

s>2 B Xd > Xdmax

HF

DD

nbias=0

Summary of MOSCAP CV Characteristics


The Charge Sheet Model

Problems with the simple model

Inaccurate in depletion

Inaccurate in accumulation

Inaccurate in weak inversion (2B > S > B) Problems with the general model

Requires numerical solution for Q I

The charge sheet model provides a reasonable tradeoffbetween the two. It isnt as accurate in depletion or

accumulation, but these regions arent as important forMOSFET operation

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Main Assumptions

Mobile charge exists beyond the onset of weak inversion

(i.e., QI > 0 forS > B unlike the simple model, which assumes that the inversion chargeis zero forS < 2B

Mobile charge is present in a negligibly thin layer similar to the simple model

Depletion region has a sharp boundary similar to the simple model

The surface potential is not clamped past threshold

unlike the simple model, which assumes that S is clamped at 2Bfor all values of VGpast threshold


Derivation of QS

As in the general model, we have:

To simplify, assume we only care about QS in weak and stronginversion (i.e., S > B). Then:

( )

( )2/1

00

002

2

)1()1(2

++=

+

=

enepq

dx

d

NNenepq

dx

d

S

da

S

( ) ( )[ ]

2/1

0 11

2

++=

SoS

S

S

SS

enep

q

Q

||2

||2

2

depinv

a

iS

SaS QQe

N

nqNQ S +=

+=

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Then, we can determine Qdep as in the simple model, and subtract tofind Qinv

Note that Qinv(VG) still requires an iterative solution

In strong inversion, since the exponent dominates, we can simplify to:

This exponential dependence implies that large changes in Qinv result

from small changes in S, which means that S is essentially clamped,as was assumed in the simple model.

( )

Ox

SSSfbg

SSa

a

iS

Sainv

SSadep

C

QVV

qNeN

nqNQ

qNQ

S

+=

+=

=

22

2

2

2

Derivation of Qdep and Qinv

a

iSinv

N

enqQ

S

22

=


In weak inversion, Qinv < QB. This allows us to simplify Qinv:

This equation clearly incorporates the effect of subthreshold current inMOSFETs, unlike the simple VT equations studied previously (forexample, in EE130)

Vg

1 decade ofincrease

for 60/ mV in Vg

typically 80 mV

log Qinv

ss eN

nNqe

N

nqNQ

a

i

S

aS

aS

iSaSinv

BSB

2

2

2

2

2

112

2

+=

>>

( )

Ox

SSSfbg

C

QVV

+=

S

1 decade ofincrease

for 60 mV in s

log Qinv

kTq Se /

Simplification under weak inversion

From first 2terms of taylor

series

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Oxide Charges

In general, these charges all

modify the threshold voltagebased on their charge centroid

In addition, they may altermobility due to coulombicscattering

=0

2 00

)(1

x

ox

SiO

T dxxxV

( )

Ox

SSfb

Ox

T

Ox

itSSMSg

CQV

dxxx

CQQV

Ox

+=++=

0


Mobile Ions People observed odd shifts in C-Vs

Reason: Mobile charge was moving towards /away from interface, changing charge centroid

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Interface traps

Traps cause sloppy C-V and alsogreatly degrade mobility in channel

EE231 Vivek Subramanian

Telegraphic noise in Id of a smallMOSFET is the signature of asignal interface trap.

When a single trap changes fromempty to filled Ninv=-1.

Id Ninv

ninvWL

Id Ninv + Ninv = - + Ninv

+

=

invd

d

NI

I 1

0 : donor type

Noise due to InterfaceTraps

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Physical ToxElectrical Toxe

Electrical ToxRaised by Inversion

Change Centroid and Gate Depletion


= ginv CdVQ

)1(60Ox

dep

C

CmvS +=

Npoly

CpolyCOx

Cdep

XdepImpact on IV

increase the effective Tox by ~5 -20

decrease of Vg by ~0.2V,

also has impact on CV and S

3/

1111

dpolyox

ox

s

dpoly

ox

ox

polyox

WT

WT

CCC

+=

+=

+=

Polysilicon gate depletion

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How to Reduce Gate-Depletion Effect

Metal gate - process integration issues

Or increase active doping concentration in the gate:

In-situ or POCl3 doped poly-Si gate Not suitable for dual-gate CMOS technology

Higher dosage for ion-implanted poly-Si gate Cost, damage and boron penetration issues

or higher activation temperature

S/D diffusion and boron penetration

Poly-Si1-xGex-gate technology


Sec 3.3 presents the classical analysis based on Poissons equationand Fermi-Dirac function (or Boltzmanns relations)

Quantum confinement in the potential well at the Si/SiO2 interfacecreates discrete subbands of energy levels.

Ref. Stern, self-consistent , Phy, Rev. B. vol. 5, p.4891, 1972

40

E0~60mV, dependingon Nsub and Qinv

SiO2

x

Ec 0,1,2j4

3

24

23/2

=

+ jm

hqE

x

Sj

Assume only ground subband is populated

+

invdepx

Siinv

QQqm

hX

3

116

9

2

2

( ) Simmx 100for9.0 0

Quantum effect (in the inversion layer)

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-Effect on VTS has to be larger than 2Bby, say 60mV depending on Nsub.

mVC

CVieV

Ox

dep

Sgt 100~,1,

+=

Empirical model: Rios, A Physical Compact MOSFET Model ., IEDMP.937, 1995

-Effect on CV

-Effect on IVSimilar to CV, but there is a subtle difference between AC chargecentroid and DC charge centroid.

inv

Siinv

XC

Quantum effect (in the inversion layer)


-4 -3 -2 -1 0 1 2 3 4

2x10-7

4x10-7

6x10-7

8x10-7

1x10-6

Measured Data

Cox, Tox=30A

Classical

QM+PD

Tox=30A

Nsub=5.2x1017cm-3

Npoly=4.5x1019cm-3

Capacitance(F/cm

2)

Gate Voltage(V)

Real MOSCAP CV Characteristics

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