MOSFET Cross-Section

A MOSFET Transistor

Gate

Source

Drain

Source

Substrate

Gate

Drain

MOSFET Schematic Symbols

Formation of the Channel for an Enhancement MOS Transistor

Water Analogy of a (subthreshold) MOSFET

Channel Current vs. Gate Voltage

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

50

100

150

200

250

300

350

400

450

Gate voltage (V)

Ch

an

ne

l Cu

rre

nt

(A

)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Gate voltage (V)

Ch

an

ne

l Cu

rre

nt

(A)

Above-Threshold Sub-Threshold

In linear scale, we have a quadraticdependence

In log-scale, wehave an exponentialdependence

VT

VT

Ith

MOS Capacitor Picture

MOSFET Channel Picture

MOS Capacitor Picture

MOS Electrostatics

Condition is called flatband --- the voltage when this occurs is called flatband

This state is the baseline operating case --- a capacitive divider has one free parameter

Vfb

MOS Electrostatics

Depletion Condition --- gate charge is terminated by charged ions in the depletion region

Part of this region is often referred to as weak-inversion

MOS Electrostatics

Inversion --- further gate charge is terminated by carriers at the silicon--silicon-dioxide interface

MOS Structure Electrostatics

MOS Capacitor Picture

MOS-Capacitor Regions

Qs = ln( 1 + e )((Vg - VT) - Vs)/UTQs = e

( - Vs)/UT

Depletion ((Vg - VT) - Vs < 0)

Qs = e ((Vg - VT) - Vs)/UT

Inversion ((Vg - VT) - Vs > 0)

Qs = ((Vg - VT) - Vs)/UT

MOS Capacitor Picture

MOSFET Channel Picture

Calculation of Drain Current

No recombination 02

2

dx

ndDn

ndx

dnqDJ

l

nnqD drainsource

n

ddC

SSC

V

V

0 l varies as VG

TSC usource en /

TdC udrain en /

n = Ax + B

RGdx

ndD

dt

dnn

2

20 0 0

eeIIUVVUVV TdgTSg / /

0

MOSFET Current-Voltage Curves

1

/)(0

//)(0

///0

TSG

TdsTSG

TDTSTG

uVV

uVuVV

uVuVuVDS

eI

eeI

eeeII

Saturation

4 Tds UV

eeIIuVVuVV TdgTSg //

0

1 //0

TSdTSg uVVuVVeeI

MOSFET Current-Voltage Curves

1

/)(0

//)(0

///0

TSG

TdSTSG

TDTSTG

uVKV

uVuVKV

uVuVuKVDS

eI

eeI

eeeII

Saturation

4 TdS uV

eeIIuVVuVV TdgTSg //

0

1 //0

TSdTSg uVVuVVeeI

Subthreshold MOSFETs

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.910

-11

10-10

10-9

10-8

10-7

10-6

Gate voltage (V)

Dra

in c

urr

en

t (A

)

= 0.58680 Io = 1.2104fA

In linear scale, we have a quadraticdependence

In log-scale, wehave an exponentialdependence

G

S

D

nFET

G

S

D

B

pFET

Determination of Threshold Voltage

0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Gate voltage (V)

Dra

in c

urre

nt /

sub

thre

shol

d fit

VT = 0.86

Drain Current --- Source Voltage

0.6 0.65 0.7 0.75 0.8 0.85 0.9

10-12

10-11

10-10

10-9

10-8

10-7

Source voltage (V)

Dra

in c

urre

nt (

A)

UT = 25.84mV

= 0.545

Drain Characteristics

Origin of Drain Dependencies

Increasing Vd effects the drain-to-channel region:

• increases depletion width

• increases barrier height

Current versus Drain Voltage

Not flat due to Early effect (channel length modulation)

Id = Id(sat) (1 + (Vd/VA) )

or

Id = Id(sat) eVd/VA

RoutId(sat)

GND

Iout

Early Voltage Length Dependence

Width of depletion regiondepends on doping, not L

Might expect Vo to linearly vary with L

MOSFET Operating Regions

Band-diagrampicture moving from subthreshold toabove-threshold

Surface potentialmoving from depletionto inversion

Qualitative Above-Threshold

I = (K/2) (( (Vg - VT) - Vs )2 - (( (Vg - VT) - Vd )2 )

Above Threshold MOSFET Equations

I = (K/2) ( ((Vg - VT) - Vs)2 - ((Vg - VT ) - Vd) 2 )

Saturation: Qd = 0

I = (K/2) ( ((Vg - VT) - Vs)2

If = 1 (ignoring back-gate effects):

I = (K/2) ( 2(Vgs - VT) Vds - Vds2 )

Detailed MOSFET Derivation

I = Q(x) E(x) Current is constant through the channel (no loss)

( E(x) = Electric Field )

Q(x) = CT ( (Vg - VT) - V(x)) CT = CD + Cox

Qs = CT ( (Vg - VT) - Vs), Qd = CT ( (Vg - VT) - Vd)

E(x) = - = (1 / CT ) d V(x)

dxd Q(x)

dxI = ( / CT ) Q(x) d Q(x)

dx

( = Cox / CT)

Detailed MOSFET Derivation

I = (K/2) (( (Vg - VT) - Vs )2 - (( (Vg - VT) - Vd )2 )

Integrate with respect to length: I = ( / CT ) Q(x) d Q(x)dx

Qs = CT ( (Vg - VT) - Vs)

Qd = CT ( (Vg - VT) - Vd)I L = ( / 2 CT ) Q(x)2 |

I = ( / 2 CT ) ( 1 / L) (Qs2 - Qd

2)

K = Cox (W/L)

I = ( CT / 2 ) ( 1 / L) (( (Vg - VT) - Vs) 2 - ( (Vg - VT) - Vd)2)

MOSFET Equations

I = (K/2) ( (Vg - VT - Vs)2 - (Vg - VT - Vd) 2 )

Saturation: (Qd = 0) I = (K/2) (Vgs - VT )2

When ignoring back-gate effects (we often do): = 1

I = (K/2) ( 2(Vgs - VT) Vds - Vds2 ) Above-Threshold:

Subthreshold:I = Is e (1 – e )

Vgs/UT-Vds/UT

Saturation: (Vds> 4 UT) I = Is e Vgs/UT

(Vd > Vg - VT )

Output Characteristics of the Above-Threshold MOSFET

vDS = vGS - VT

iD /ID0

0.75

1.0

0.5

0.25

00 0.5 1.0 1.5 2.0 2.5

vGS-VTVGS0 - VT

= 0

vGS-VTVGS0 - VT

= 0.5

vGS-VTVGS0 - VT

= 0.707

vGS-VTVGS0 - VT

= 0.867

vGS-VTVGS0 - VT

= 1.0

Channel modulation effects

vDSVGS0 - VT

Active Region Saturation Region

Cutoff Region

iDvDS = vGS - VT

Increasing values of vGS

Saturation RegionActive Region

vDS

Interpretation of large-signal model

MOSFETs

G

S

D

nFET

G

S

D

B

pFET

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

50

100

150

200

250

300

350

Gate voltage (V)

Cur

rent

(A

)

K = 37.861 A/V2

VT = 0.806

Drain Current - Gate Voltage

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Gate voltage (V)

sqrt

(Dra

in c

urr

en

t (A

))

VT = 0.806

K = 37.861 A/V2

Drain Current --- Source Voltage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

Gate voltage (V)

sqrt

(Dra

in c

urr

en

t (

A))

(Vg - VT) = 0.595

K/ = 74.585 A/V2

( = 0.54)

( = 0.7)

An Ohmic MOSFET

I = (K/2) ( ((Vg - VT) - Vs)2 - ((Vg - VT ) - Vd) 2 )

If Vd ~ Vs, (small difference)

(Vd - Vs = 50mV)

I = K (Vg - VT)(Vd - Vs)

A MOSFET in Saturation

Saturation: Qd = 0

I = (K/2) ( ((Vg - VT) - Vs)2

Vs = 0

I = (K/2) (Vg - VT) 2

More Ohmic Region Data

I = (K/2) ( ((Vg - VT) - Vs)2 - ((Vg - VT ) - Vd) 2 )

Take the derivative of I with respect to Vd (Vs = 0 )

dI / d Vd = (K/2)( 0 - (-2) ((Vg - VT ) - Vd) ) = (K/2)((Vg - VT ) - Vd)

Influence of VDS on the Output Characteristics

Current versus Drain Voltage

Not flat due to Early effect (channel length modulation)

Id = Id(sat) (1 + (Vd/VA) )

or

Id = Id(sat) eVd/VA

RoutId(sat)

GND

Iout

Early Voltage Length Dependence

Width of depletion regiondepends on doping, not L

Might expect Vo to linearly vary with L

Small-Signal Modeling

gmV ro

V3

V2V2

V1 +

V

-V3

V2

V1

V3

V2

V1

gm ro

Above VTMOSFET

Sub VTMOSFET

Av

I I

I / UT

2I /(V1-V2 -VT) VA / I

VA / I VA / UT

2VA/(V1-V2 -VT)

Small-Signal Modeling

gmV ro

V3

V2V2

r

V1 +

V

-

V3

V2

V1

V3

V2

V1

gm ror

BJT

Above VTMOSFET

Sub VTMOSFET

Av

(UT ) / I

I I

I / UT

I / UT

2I /(V1-V2 -VT)

VA / I

VA / I

VA / I

VA / UT

VA / UT

2VA/(V1-V2 -VT)

Capacitances in a MOSFET

MOSFET Depletion Capacitors

Overlap Capacitances

Capacitance Modeling

Capacitance Modeling

Velocity Saturation

Ideal Drift (Ohm’s Law)

Si Crystal Velocity Limit

Effect of Velocity Saturation

L = 76 nm MOSFET

VT

Square-lawregion

Small-Signal Modeling (with kappa)

gmV ro

V3

V2V2

V1 +

V

-V3

V2

V1

V3

V2

V1

gm ro

Above VTMOSFET

Sub VTMOSFET

Av

I I

I / UT

2I /(V1-V2 -VT) VA / I

VA / I VA / UT

2VA/(V1-V2 -VT)

Small-Signal Modeling (with kappa)

gmV ro

V3

V2V2

r

V1 +

V

-

V3

V2

V1

V3

V2

V1

gm ror

BJT

Above VTMOSFET

Sub VTMOSFET

Av

(UT ) / I

I I

I / UT

I / UT

2I /(V1-V2 -VT)

VA / I

VA / I

VA / I

VA / UT

VA / UT

2VA/(V1-V2 -VT)