The Ballistic MOSFET - nanoHUB · nanoHUB.org online simulations and more Network for Computational Nanotechnology nanoscale MOSFETs Intel Technical J., Vol. 6, May 16, 2002. 130

nanoHUB.orgonline simulations and more

Network for Computational Nanotechnology

Mark LundstromPurdue University

Network for Computational Nanoechnology

Simple Theoryof the

Ballistic Nanotransistor



outline

I) Traditional MOS theoryII) A “bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary



nanoscale MOSFETs

Intel Technical J., Vol. 6, May 16, 2002.

130 nm technology (LG = 60 nm)

Low VT

I DS

(mA

/µm

)



MOSFET IV: low VDS

VG VD0

!

ID

=W Qix( )" x

(x) =W Qi0( )" x

(0)

!

ID =W Cox VGS "VT( )µeffE x

!

Ex

=VDS

L

!

Qix( ) = "C

oxVGS"V

T"V (x)( )

ID

VDS

VGS

!

ID =W

LµeffCox VGS "VT( )VDS



MOSFET IV: high VDS

VG VD0

!

ID

=W Qix( )" x

(x) =W Qi0( )" x

(0)

!

ID =W Cox VGS "VT( )µeffE x

!

V x( ) = VGS"V

T( )

VGS

ID

VDS

!

ID =W

LµeffCox VGS "VT( )

2

2GS T

x

V V

L

!"E



velocity saturation

electric field V/cm --->

velo

city

cm

/s --

->

107

104

!

" = µE!

" ="sat

41.5V25 10 V/cm

60nm

DSV

L= ! "



MOSFET IV: velocity saturation

VG VD0

!

ID

=W Qix( )" x

(x) =W Qi0( )" x

(0)

!

ID

=W CoxVGS"V

T( )# sat

( )D ox sat GS TI WC V V!= "

410

x>>E

0 0.4 0.8 1.2 1.4



MOSFET IV: velocity overshoot

Position along Channel (µm) Position along Channel (mm)

Frank, Laux, and Fischetti, IEDM Tech. Dig., p. 553, 1992

µ



the MOSFET as a BJT

S DG

ID

VDS

VGS

electron energy vs. position

VD≈ 0V

VD= VDD

E.O. Johnson, RCA Review, 34, 80, 1973



the MOSFET as a BJT

JC = qDn

WB

ni2

NA

eqVBE /kBT ! e

qVBC /kBT( ) = qDn

WB

ni2

NA

eqVBE /kBT 1! e

qVCE /kBT( )

VBE!"

S

VCE!V

SD

WB! L

Dn !kBT

qµeff

IDS!W t

inv

tinv !kBT q

E S

E S =qNA

(m !1)Cox

ni

NA

!

"#$

%&

2

= e'q2( B /kBT = e

'qVT /mkBT

!S=

Cox

Cox+ C

D

VG=VG

m

BJT Theory:

( ) ( ) ( )2

/ /1 1

GS T B DS Bq V V mk T qV k Tb

D eff ox

k TWI C m e e

L qµ

!" #" #== ! !$ %$ %

& ' & '

MOSFET Theory:

eqn. (3.36) on p. 128 of Fundamentals of Modern VLSI Devices,Yuan Taur and Tak Ning, Cambridge Univ. Press, 1998.



the MOSFET as a BJT

VGS

log 1

0 ID

S

( )/~

GS T Bq V V mk Te

!

( )~GS TV V!

above threshold:

( )i ox GS TQ C V V= !

/ 2~ S Bq k T

iQ e!

E.O. Johnson, RCA Review, 34, 80, 1973

( )~ lnS GS T

V V! "

( )/~ ~S B

k T

D GS TI e V V

!"



Outline

I) Traditional MOS theoryII) A “bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary



Gate

EF1 EF1-qVD( )SCF

D E U!

1

1

!"

=h

!

"2

=h

#2

a general view of nano-devices



“top of the barrier model”

ε(x)

EF1EF1-qVD

(0)CE

ener

gy

position

contact 1 contact 2

1 2! !

" #= = =

h h

xL/

‘device’

LDOS

!

L

FB

SCF C SU E q!= "



filling states from the left contact

1

1

!"

=h

( )SCF

D E U!

0

1

1

( )( ) N E NdN E

dt !

"=

0

1 1( ) ( ) ( )SCFN E D E U f E= !

EF1

Gate



filling states from the right contact

!

"2

=h

#2

µ2

0

2

2

( )( ) N E NdN E

dt !

"=

0

2 2( ) ( ) ( )SCFN E D E U f E= !

Gate

EF1-qVD( )SCF

D E U!



steady-state

0 0

1 2

1 2

( )0

N N N NdN E

dt ! !

" "= + =

[ ]1 1 2 2( ) ( ) ( ) ( )N D E f E D E f E dE= +!

( ) ( ) ( )2 1

1

1 2 1 2

SCF SCFD E D E U D E U

! "

! ! " "# $ = $

+ +

(ballistic)



steady-state current

( )00

21

1 2

D

N NN NI

! !

" ""= =

( )1 2

2( ) ( ) ( )D

qI M E f E f E dE

h= !"

( )1 2

1 2 1 2

( )( ) ( )

2

hD EM E D E

! !"

# # ! !$ =

+ +



NEGF theory

Non-equilibrium Green’s Function Approach (NEGF)

S. Datta, IEDM Tech. Dig., 2002

device

!

["1]

!

[H]

!

["2]

!

["S]

Gate

EF1 EF2



outline

I) Traditional MOS theoryII) A “Bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary



assumptionsen

ergy

positioncontact 1 contact 2

1 2

x

! !"

= =L

ε(x)

EF1 EF1-qVDLDOS

!

L

1) 2D, planar MOSFET

2) 1 subband occupied

3) parabolic E(k)( )

*

2( )

C

mD E W E E

!= " #

hL

1 2( ) ( ) ( ) 2D E D E D E= =

( )1

*2 2 /x E m

!" #

= =L L

*2( )

W m EM E

!=

h

FB

SCF C C SU E E q!" = #



procedureen

ergy

positioncontact 1 contact 2

1 2

x

! !"

= =L

ε(x)

EF1 EF1-qVDLDOS

!

L

1) assume a ψS(sets top of the barrier energy)

2) fill states

3) self-consistentelectrostatics

4) evaluate current

N N N+ !

= +

( )1 2

2( ) ( ) ( )D

qI M E f E f E dE

h= !"



filling states (ballistic)

N = D1(E) f

1(E) + D

2(E) f

2(E)[ ]! dE

[ ]*

1 22( ) ( )

2

mN W f E f E dE

!= +"

hL

[ ]20 1 0 2( ) ( )

2

D

F F

NN W ! != +L F F

*

2 2

B

D

m k TN

!=

h

!F1 = EF1 " EC

FB + q# S( ) kbT

!F2 = EF1 " qVD " EC

FB + q# S( ) kbT

k

E(k)

!

h2k2

2m*

EF1EF2

C

FB

SE q!"

N at the top of the barrierdepends on: VG (through ψS) VD (through EF2)



filled states in equilibrium

* 2( ) /( ) / / 2

( ) /

1( )

1F C BF B B

F B

E E k TE E k T m k T

E E k Tf E e e e

e

!""

"= # = $

+2 2 2 *( ) / 2

( , ) x y Bk k m k T

x yf k k e+

!h

f0 f (kx, ky)



filling states under bias

Increasing VDS

X (nm) --->-10 -5 0 5 10

ε1 vs. x for VGS = 0.5V

ε 1 (e

V) -

-->

f (kx, ky)



the ballistic current

[ ]1 2*

2( ) ( ) ( )D

qI M E f E f E dE

m= !"

*2( )

W m EM E

!=

h

[ ]21/ 2 1 1/ 2 2( ) ( )

2

DD T F F

qNI W! " "= #F F

*

2B

T

k T

m!

"=

D nI WQ !"

n

NQ

W=

L

alternatively:

1/ 2 2 1/ 2 1

0 2 0 1

1 ( ) / ( )

1 ( ) / ( )

F F

T

F F

! !" "

! !

# $%= & '

+( )%

F F

F F( )( )

1/ 2 1

*

0 1

2 FB

T

F

k T

m

!"

# !

$ %= & '& '

( )%

F

F



carrier velocity in a ballistic MOSFET

Increasing VDS

X (nm) --->

EC (e

V)

---> Increasing VDS

-10 -5 0 5 10




velocity saturation in a ballistic MOSFET

Increasing VDS

X (nm) --->-10 -5 0 5 10


!

"(0) # ˜ " T

ε 1 (e

V) -

-->

( )( )

1/ 2 1

*

0 1

2 FB

T

F

k T

m

!"

# !

$ %= & '& '

( )%

F

F“injection velocity”

VDS --->

υ(0)

υin

j (10

7 cm

/s)

--->



IV of a ballistic MOSFET

[ ]21/ 2 1 1/ 2 2( ) ( )

2

DD T F F

qNI W! " "= #F F

[ ]20 1 0 2( ) ( )

2

D

F F

NN W ! != +L F F

( )1 1

FB

F F C S bE E q k T! "= # +

( )2 1

FB

F F D C S bE qV E q k T! "= # # +

Key equations We must express ψSin terms of

VG (1D electrostatics)

or

VG and VD (2D electrostatics)



1D MOS electrostatics (above threshold)

[ ]21/ 2 1 1/ 2 2( ) ( )

2

DD T F F

qNI W! " "= #F F

[ ]20 1 0 2( ) ( )

2

D

F F

NN W ! != +L F F

Key equations

( )ox GS T

qNC V V

W! =

L

(1)

(2)

(3)

equations (1), (2), and (3) give…



1D MOS electrostatics (above threshold)

1/ 2 1

1/ 2 1

0 1

0 1

( / )1

( )( )

( / )1

( )

F DS B

FDS ox GS T T

F DS B

F

qV k T

I WC V VqV k T

!

!"

!

!

#$ %#& && &

= # '( )#& &+& &* +

%

F

F

F

F

( ) [ ]20 1 0 1( ) ( / )

2

Dox G T F F DS B

NC V V qV k T! !" = + "F F

for non-degenerate statistics:/

/

1( )

1

DS B

DS B

qV k T

DS ox GS T T qV k T

eI WC V V

e!

"

"

# $"= " % &

+' (



the ballistic MOSFET

1/ 2

1/ 2

0

0

( )1

( )( )

( )1

( )

F DS

F

DS ox GS T T

F DS

F

U

I WC V VU

!

!"

!

!

#$ %#& && &

= # ' (#& &+& &) *

%

F

F

F

F

VDS

!

IDS

(on) =W Cox

˜ " TVGS#V

T( )

quantum conductance

VG! V

T( )

"

IDS

!

" M2q2

h

K. Natori, JAP, 76, 4879, 1994.

idealelectrostatics



electrostatics (subthreshold and 2D)

CG

VS VD

CDCS

VG

!

Q = "qN

!S

( )SG SDS G D S

qNC CCV V V

C C C C

!!

" " " "

# $ # $ # $= + + %& ' & ' & '

( ) ( ) ( )



procedure

for a given VG, VD:

1) guess ψS2) fill states

3) compute improved ψS

4) iterate between (2) and (3)

5) compute current

6) select new VG, VD, and go to 1

( )SG SDS G D S

qNC CCV V V

C C C C

!!

" " " "

# $ # $ # $= + + %& ' & ' & '

( ) ( ) ( )

[ ]21/ 2 1 1/ 2 2( ) ( )

2

DD T F F

qNI W! " "= #F F

see FETToy at

www.nanohub.org



outline




the quantum capacitance

VG

insC

S QC C=

( )Gate G TQ C V V= !

ins Q

Gate

inc Q

C CC

C C=

+

( )( )2 *

2~

S

Q D F

S

qnC q D E m

!

" #= =

"S

!

if ,Q ins Gate insC C C C>> !

ID � Q!

inj



bandstructure effects in nano-MOSFETs

a

(001)

(100)

(010)

(111)

(110)

-tight binding model (sp3d5s*) (Boyken, Klimeck, et al.)

-bulk, UTB, nanowire MOSFETs

-Si, Ge, SiGe, GaAs, InAs, … (strained or unstrained) (heterostructure channels)

2 2

*( )

2

kE k

m=h

( ) : tabulatedE k

Top-of-the-barrier model

analytical numerical



scattering in nano-MOSFETs

measured ballistic

Chau et al, IEDM Technical Digest,2000, pp 45 -48

Intel 30nm bulk MOSFET

MOSFETs operate at ≈ 50% of their ballistic limit



relation to traditional MOSFET theory

!

ID =W

LµeffCox VGS "VT( )VDS

( )2

2D eff ox GS T

WI C V V

Lµ= !

low VDS

high VDS (long channel)

1/ 2 1

1/ 2 1

0 1

0 1

( / )1

( )( )

( / )1

( )

F DS B

FDS ox GS T T

F DS B

F

qV k T

I WC V VqV k T

!

!"

!

!

#$ %#& && &

= # '( )#& &+& &* +

%

F

F

F

F

ballistic MOSFET



relation to traditional MOSFET theory

( )( )21 2

1 2

( )2( ) ( )

2

DD

hD EqI f E f E dE

h ! !

" #= $% &

+' ()

ballistic transport:( )

1 2*

2 2 /x E m

! !" #

= = =L L

2

1 2

2 effD! != =

Ldiffusive transport:

see: “The Ballistic MOSFET,” unpublished notes by M.S. Lundstrom, 2005



Outline




summary

1) A ballistic, “top-of-the barrier” model for the MOSFETis easy to formulate.

2) The ballistic model provides new insights into thephysics of nanoscale MOSFETs.

3) Although not comprehensive, the top-of-the-barrierballistic model should prove useful in exploringnew materials and structures for ultimate CMOS.



references

the ballistic model:

scattering in nanotransistors:

Mark Lundstrom, “The Ballistic Nanotransistor,” unpublished notes, 2005.

Anisur Rahman, Jing Guo, Supriyo Datta, and Mark Lundstrom, “Theory ofBallistic Nanotransistors,” IEEE Trans. Electron. Dev., 50, 1853-1864, 2003.

Mark Lundstrom and Zhibin Ren, “Essential Physics of Carrier Transport inNanoscale MOSFETs,” IEEE Trans. Electron Dev., 49, pp. 133-141, 2002.

The Ballistic MOSFET - nanoHUB · nanoHUB.org online simulations and more Network for Computational Nanotechnology nanoscale MOSFETs Intel Technical J., Vol. 6, May 16, 2002. 130

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