Early Voltage in MOSFETs

ECE 584, Summer 2002 Brad NobleChapter 3 Slides

1

Early Voltage in MOSFETs• Due to channel length modulation:

E

DSTHGSPNDS

THGSPN

DSTHGSoxnDS

V

VVVSKI

L

LVVSK

VVVL

WCI

12

1

12

1

12

1

2

2

2


2

Saturation Voltage• Vpinchoff = VDS,sat = VGS – VTH

– Separates resistive from saturation region

• The drain current is given by

• Solving for VDS,sat:

• Good to solve for quiescent voltage-current.

PNoxnL

WNTHTN

TNGSNDS

SKCVV

VVI

and

NFETfor ,2

21

N

DSTNGSsatDS

IVVV

2

,


3

Ex: Find VDS,sat for an NFET


4

Variations in VTH Across Channel

• We assume VTH is constant across channel

THIS IS NOT TRUE!

• Depletion region is thick at S and thin at D.

Fox

depmsTH

C

QV 2

Cox

Cdep

inversionlayer

Gate oxidecapacitance

Depletion cap,function of x

5.11factor slope ldsubthresho2

2

ox

dep

TNGSN

DS

C

Cn

VVn

I


5

Notes on PFETs• PFETs typically have a shape factor 3 or 4

times larger than NFETs

• Body effect can be eliminated in PFETs by tying the n-well to VDD

– Need 6m spacing between n-wells to isolate.– Dr. Engel always does this on input devices,

not always elsewhere.


6

Weak Inversion• What really happens if VGS < VTN?

• In digital design, IDS = 0.

• We call it “weak inversion” or W.I.

• IDS is primarily due to Idrift in strong inversion and Idiffusion in weak inversion.

V1

V950

TH

GS

V

.V


7

Modes of Inversion• IDS = Idrift + Idiffusion

• If VGS > VTN the channel has been inverted.

• To be more precise, we can say the channel has been “strongly inverted” (S.I.) due to an abundance of carriers in the channel.

• Inversion is independent of whether the FET is in the linear or saturation region.


8

Weak Inversion Idiffusion

• Drain is more reverse biased than source:

• To find Idiff, compute gradient

• Because no carriers are lost as they travel from S to D, current is the same for all x and gradient is not a function of x.

• Note: This is not really true due to recombination, but its close!

kT

VVqNN SGO

S

exp0

kT

VVqNN SGO

D

exp0

dxdN


9

W.I. Surface Potential

oxd

d

oxd

oxs

CjCj

Cj

CC

C

11

1

oxC

dC

GV

potentialsurface

S

factor slopeldsubthresho ,5.11

ox

d

C

Cn

device law lExponentia

Uexp1

Uexp

nUexp

0

T

DS

T

S

T

GDDS SII

v-v-v


10

Deriving Weak Inversion IDS

T

S

T

D

T

GSSD

L

NN

L

NN

dx

dN

Uexp

Uexp

Uexp0 v-v-v

iprelationshEinstein thea.k.a. ,

coeff.diffusion is where,

:unit widthper current The

q

kTD

Ddx

dNqD

W

I

nn

nnDS

T

D

T

S

T

GDDS I

L

WI

Uexp

Uexp

nUexp

0

v-v-v


11

W.I. FET As Exp. Law Dev.• S must be big for device to be useful.

• If VDS = 100mV, can be neglected.

• For W.I. vDS,Sat 100mV

• Looks like a BJT

T

DS

Uexp

v-

mV100for ,U

expnU

exp0

DS

T

S

T

GDDS VSII

v-v

T

BESC II

Uexp

v

BEv

Ci


12

Inversion Coefficient• Let

• Shape factor as a function of :

Lets you chose shape to match inversion mode.

2U2 t coefficieninversion

T

DS

n

i o

< 0.1 Weakly Inverted (W.I.)

> 10 Strongly Inverted (S.I.)

0.1 < < 10 Moderately Inverted (M.I.)

2U2 TPN

DS

Kn

iS

o


13

Ex. Using Inversion Coeff.

W.I. 049.0

(100))(26mV)2(1.5)(100

1uA ,100 uA,1

2

SiO

DS

M.I. 9.4

(100))(26mV)2(1.5)(100

100uA ,100 uA,100

2

SiO

DS

S.I. 49

(100))(26mV)2(1.5)(100

1mA ,100 mA,1

2

SiO

DS


14

Small Signal Analysis

GSV

gsGS vv

Bias DCChange SmallVoltage Total

O

GSv O

BSv

O

DSv

O

DSi

GSO V GSvVoltageQuiescent


15

Ex: Quiescent PointV75.00 TV

V81EV

59.02V

A50 PNK

m

mS

10μ

μ100

O

GSv O

BSv

O

DSv

O

DSi

EFFPN

VSK

O

OOO DS

SBGSDS

vvvvi T0 122

2

1 2

mA34375.181

5175.0350

10

100

2

1 2

2

V

A

m

mO

DSi

Question: How many digits are significant?

V3O

GSv

V5O

DSv

V0O

BSv


16

Small Signal Model Limits• Suppose the previous circuit is the input

device of an amplifier.

• Small-signal model holds as long as the deviations are small qkT


17

Taylor Series Expansion• Taking a Taylor expansion of one variable:

SBSB

SBDS

DS

DSGS

GS

DS

DSDSiii

ii vv

vv

vv

0000

mg dsg mbg

gsvv-vvGSGSGS O

dsvvDS sbvv

SB

DSDSiii O

DS

202

100 ))(())(()()( xxxfxxxfxfxf

Approx.Linear


18

Small Signal Model Params

dsE

DS

DS

DSds

rV

iig

O 10

v

1typically ,22

where0

O

SBF

mSB

SBmb g

ig

vv

OO

O

DSmEDSEGS

DSm igVi

V

ig 2 , largefor ,12

0

DSv

v

dsmrgGain


19

Example: Small Signal Analysis

μA57.30)26)(0166.0()26)(1592.1(

2

:analysis signal small Using

μA65.31 mA 3754.1

:) work!of (lotsequation full Using

V026.5 V,0 ,)by (up V026.3Let

250 V,81 mA,34375.1

mVmSmVmS

2V

μA

dsE

DSgsdsdsgsm

DSBSGS

E

V

iiggi

ii

qkT

Vi

O

O

O

vvvv

vvv

DS

DS

DS

DSDS


20

Small Signal Low-Freq Model

gsmg v dsmbg v dsr

dsi

gsv

small

signals

S.I.

Sat

1.5factor slope ldsubthresho where,2

nn

ig

ODS

m

OO

ds

E

ds

Eds

i

LV

i

Vr


21

Ex: Find gm and rO

μm10

μm4

μA10

L

W

iO

DS

M10μA10

μm10μmV10

μS235.1

uA10μA/V1001042

so, If saturated? and S.I.it Is

2

ODS

ENds

m

i

LVr

g


22

Transconductance: W.I. & M.I.• What is gm for a weakly inverted FET?

• What is gm for a moderately inverted FET?

T

DS

GS

DSm

n

iig

O

U

0

v

exp1

where,U

0

T

DS

GS

DSm

n

iig

O

v

modes allfor E

DSds

V

ig

O

Not

in te

xtbo

oks!

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Early Voltage in MOSFETs

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