ECE 663

MOSFET I-Vs

Substrate

Channel Drain

InsulatorGate

Operation of a transistorVSG > 0 n type operation

Positive gate bias attracts electrons into channelChannel now becomes more conductive

More electrons

Source

VSD

VSG

Some important equations in the inversion regime (Depth direction)

VT = ms + 2B + ox

Wdm = [2S(2B)/qNA]

Qinv = -Cox(VG - VT)

ox = Qs/Cox

Qs = qNAWdm

VT = ms + 2B + [4SBqNA]/Cox

Substrate

Channel Drain

InsulatorGate

Source

x

ECE 663

MOSFET Geometry

x

y

z

L

Z

S D

VG

VD

ECE 663

How to include y-dependent potential without doing the whole problem over?

ECE 663

Assume potential V(y) varies slowly along channel, so the x-dependent and y-dependent electrostats are independent (GRADUAL CHANNEL APPROXIMATION)

i.e.,

Ignore ∂Ex/∂y

Potential is separable inx and y

ECE 663

How to include y-dependent potentials?

S = 2B + V(y)

VG = S + [2SSqNA]/Cox

Need VG – V(y) > VT to invert channel at y (V increases threshold)

Since V(y) largest at drain end, that

end reverts from inversion todepletion first (Pinch off)

SATURATION [VDSAT = VG – VT]

j = qninvv = (Qinv/tinv)v

I = jA = jZtinv = ZQinvv

ECE 663

So current:

Qinv = -Cox[VG – VT - V(y)]

v = -effdV(y)/dy

ECE 663

So current:

I = eff ZCox[VG – VT - V(y)]dV(y)/dy

I = eff ZCox[(VG – VT )VD- VD2/2]/L

Continuity implies ∫Idy = IL

ECE 663

But this current behaves like a parabola !!

ID

VD

IDsat

VDsat

I = eff ZCox[(VG – VT )VD- VD2/2]/L

We have assumed inversion in our model (ie, always above pinch-off)

So we just extend the maximum current into saturation…

Easy to check that above current is maximum for VDsat = VG - VT

Substituting, IDsat = (CoxeffZ/2L)(VG-VT)2

What’s Pinch off?

0

0 0

0

VG VG

Now add in the drain voltage to drive a current. Initially you get an increasing current with increasing drain bias

0 VD

VG VG

When you reach VDsat = VG – VT, inversion is disabled at the drain end (pinch-off), but the source end is still inverted The charges still flow, just that you can’t draw more current with higher drain bias, and the current saturates

Square law theory of MOSFETs

I = eff ZCox[(VG – VT )VD- VD2/2]/L, VD < VG - VT

I = eff ZCox(VG – VT )2/2L, VD > VG - VT

J = qnv

n ~ Cox(VG – VT )

v ~ effVD /L

NEW

ECE 663

Ideal Characteristics of n-channel enhancement mode MOSFET

ECE 663

Drain current for REALLY small VD

TGD

DTGinD

DDTGinD

VVV

VVVCLZ

I

VVVVCLZ

I

2

21

Linear operation

Channel Conductance:

)( TGin

VD

DD VVC

LZ

VI

gG

Transconductance:

Din

VG

Dm VC

LZ

VI

gD

ECE 663

In Saturation

• Channel Conductance:

• Transconductance:

2

2 TGinD VVCLZ

satI

0

GVD

DD V

Ig

TGin

VG

Dm VVC

LZ

VI

gD

ECE 663

Equivalent Circuit – Low Frequency AC

• Gate looks like open circuit• S-D output stage looks like current source with channel

conductance

gmdD

G

VG

DD

VD

DD

vgvgi

VVI

VVI

IDG

ECE 663

• Input stage looks like capacitances gate-to-source(gate) and gate-to-drain(overlap)

• Output capacitances ignored -drain-to-source capacitance small

Equivalent Circuit – Higher Frequency AC

ECE 663

• Input circuit:

• Input capacitance is mainly gate capacitance

• Output circuit:

ggateggdgsin vfCjvCCji 2

gmout vgi

gate

m

in

out

fCg

ii

2

Din

VG

Dm VC

LZ

VI

gD

Equivalent Circuit – Higher Frequency AC

ECE 663

Maximum Frequency (not in saturation)

• Ci is capacitance per unit area and Cgate is total capacitance of the gate

• F=fmax when gain=1 (iout/iin=1)

2max

max

22

2

LV

ZLC

CVLZ

f

Cg

f

Dn

i

iDn

gate

m

ZLCC igate

ECE 663

Maximum Frequency (not in saturation)

2max 2 L

Vf Dn

LVv

vL

D /

/

1max

(Inverse transit time)

NEW

ECE 663

Switching Speed, Power Dissipation

ton = CoxZLVD/ION

Trade-off: If Cox too small, Cs and Cd take over and you lose

control of the channel potential (e.g. saturation)

(DRAIN-INDUCED BARRIER LOWERING/DIBL)

If Cox increases, you want to make sure you don’t control

immobile charges (parasitics) which do not contribute tocurrent.

ECE 663

Switching Speed, Power Dissipation

Pdyn = ½ CoxZLVD2f

Pst = IoffVD

ECE 663

CMOS

NOT gate (inverter)

ECE 663

CMOS

NOT gate (inverter)

Positive gate turns nMOS on

Vin = 1 Vout = 0

ECE 663

CMOS

NOT gate (inverter)

Negative gate turns pMOS on

Vin = 0 Vout = 1

ECE 663

So what?

• If we can create a NOT gate we can create other gates (e.g. NAND, EXOR)

ECE 663

So what?

Ring Oscillator

ECE 663

So what?

• More importantly, since one is open and one is shut at steady state, no current except during turn-on/turn-off Low power dissipation

ECE 663

Getting the inverter output

Gain

ON

OFF

ECE 663

0

GVD

DD V

Ig

TGin

VG

Dm VVC

LZ

VI

gD

What’s the gain here?

ECE 663

Signal Restoration

ECE 663

BJT vs MOSFET

• RTL logic vs CMOS logic

• DC Input impedance of MOSFET (at gate end) is infinite Thus, current output can drive many inputs FANOUT

• CMOS static dissipation is low!! ~ IOFFVDD

• Normally BJTs have higher transconductance/current (faster!)

IC = (qni2Dn/WBND)exp(qVBE/kT) ID = CoxW(VG-VT) 2/L

gm = IC/VBE = IC/(kT/q) gm = ID/VG = ID/[(VG-VT)/2]

• Today’s MOSFET ID >> IC due to near ballistic operation

NEW

ECE 663

What if it isn’t ideal?

• If work function differences and oxide charges are present, threshold voltage is shifted just like for MOS capacitor:

• If the substrate is biased wrt the Source (VBS) the threshold voltage is also shifted

i

BAsB

i

fms

i

BAsBFBT

C

qN

CQ

C

qNVV

)2(22

)2(22

i

BSBAsBFBT C

VqNVV

)2(22

ECE 663

Threshold Voltage Control

• Substrate Bias:

i

BSBAsBFBT C

VqNVV

)2(22

BBSBi

AsT

BSTBSTT

VC

qNV

VVVVV

222

)0()(

ECE 663

Threshold Voltage Control-substrate bias

ECE 663

It also affects the I-VVG

The threshold voltage is increased due to the depletion regionthat grows at the drain end because the inversion layer shrinksthere and can’t screen it any more. (Wd > Wdm)

Qinv = -Cox[VG-VT(y)], I = -effZQinvdV(y)/dy

VT(y) = + √2sqNA/Cox

= 2B + V(y)

ECE 663

It also affects the I-V

IL = ∫effZCox[VG – (2B+V) - √2sqNA(2B+V)/Cox]dV

I = (ZeffCox/L)[(VG–2B)VD –VD2/2

-2√2sqNA{(2B+VD)3/2-(2B)3/2}/3Cox]

ECE 663

We can approximately include this…

Include an additional charge term from the depletion layer capacitance controlling V(y)

Q = -Cox[VG-VT]+(Cox + Cd)V(y)

where Cd = s/Wdm

Q = -Cox[VG –VT - MV(y)], M = 1 + Cd/Cox

ID = (ZeffCox/L)[(VG-VT - MVD/2)VD]

ECE 663

Comparison between different models

Square Law Theory

Body Coefficient

Bulk Charge Theory

Still not good below threshold or above saturation

ECE 663

Mobility• Drain current model assumed constant mobility in

channel• Mobility of channel less than bulk – surface scattering• Mobility depends on gate voltage – carriers in inversion

channel are attracted to gate – increased surface scattering – reduced mobility

ECE 663

Mobility dependence on gate voltage

)(10

TG VV

ECE 663

Sub-Threshold Behavior

• For gate voltage less than the threshold – weak inversion

• Diffusion is dominant current mechanism (not drift)

LLnon

qADyn

qADAJI nnDD

)()(

kTVqi

kTqi

DBs

Bs

enLn

enn

/)(

/)(

)(

)0(

ECE 663

Sub-threshold

kTqkTqVkT

inD

sD

B

eeLenqAD

I ///

1

We can approximate s with VG-VT below threshold since all voltage drops across depletion region

kTVVqkTqVkT

inD

TGD

B

eeLenqAD

I ///

1

•Sub-threshold current is exponential function of applied gate voltage•Sub-threshold current gets larger for smaller gates (L)

ECE 663

Subthreshold Characteristic

GD VIS

log1

Subthreshold Swing

Tunneling transistor– Band filter like operation

J Appenzeller et al, PRL ‘04

Ghosh, Rakshit, Datta(Nanoletters, 2004)

(Sconf)min=2.3(kBT/e).(etox/m)

Hodgkin and Huxley, J. Physiol. 116, 449 (1952a)

Subthreshold slope = (60/Z) mV/decade

Much of new research depends on reducing S !

Much of new research depends on reducing S !

• Increase ‘q’ by collective motion (e.g. relay) Ghosh, Rakshit, Datta, NL ‘03

• Effectively reduce N through interactions Salahuddin, Datta • Negative capacitance Salahuddin, Datta

• Non-thermionic switching (T-independent) Appenzeller et al, PRL

• Nonequilibrium switchingLi, Ghosh, Stan

• Impact IonizationPlummer

ECE 663

More complete model – sub-threshold to saturation

• Must include diffusion and drift currents• Still use gradual channel approximation• Yields sub-threshold and saturation behavior for long

channel MOSFETS• Exact Charge Model – numerical integration

D s

B

V

p

p

V

D

nsD

p

nVF

eLL

ZI

0

0

0,,

ECE 663

Exact Charge Model (Pao-Sah)– Long Channel MOSFET

http://www.nsti.org/Nanotech2006/WCM2006/WCM2006-BJie.pdf

ECE 663