physics of nanoscale transistors - an introduction to electronics from the bottom up

7/31/2019 Physics of Nanoscale Transistors - an Introduction to Electronics From the Bottom Up

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23rd Symposium on Microelectronics Technology and Devices: SB Micro 2008IEEE / EDS Mini Colloquium

September 1, 2008, Gramado, Brazil


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NCN

Mark Lundstrom

Network for Computational Nanotechnology

Birck Nanotechnology Center

Discovery Park, Purdue University

West Lafayette, Indiana USA

23rd Symposium on Microelectronics Technology and Devices: SB Micro 2008IEEE / EDS Mini Colloquium

September 1, 2008, Gramado, Brazil

Physics

ofNanoscale Transistors:

An Introduction to Electronics from the Bottom Up


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technology trends..

logL

1975

5 mMoores Law

5 nm

2005

50 nm

log # chip

10 3

10 9


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models for devices (conceptual and computational)

0.1 mm

10 m

1 m

0.1 m

10 nm

1 nm

0.1 nm

Macroscopicdimensions

Atomicdimensions

drift-diffusion

drift-diffusion + velocity saturation

Boltzmann for velocity overshoot

quasi-ballistic

quantum mechanical


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NCN

21st Century electronic devices

HfO2

10 nm SiO2

p++ Si

SD

Al

Gate

SWNTcarbon nanotube

electronics

CoFe (2.5)

Ru (0.85)

InsulatorCoFeB (3)

CoFeB (3)

MgO (0.85)

spin torque devices

nanowire PVnanowire

bio-sensors

molecular electronics

nanonets


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Electronics from the Bottom Up

1) Introduction

2) Generic model of a nanodevice

3) The ballistic MOSFET

4) Scattering in nano-MOSFETs

5) Discussion

6) Summary


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Gate

D(E U)

generic model

S. Datta, Quantum Transport: Atom to Transistor, Cambridge, 2005(Concepts of Quantum Transport nanohub.org)

21

EF1 EF2


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filling states from the left contact

1

d N(E)

dt =N1

0 (E) N1

N10 (E) = D(E U)f1(E)

Assumption:Each energy channel isindependent

includes spin

Gate

D(E U)EF1


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filling states from the right contact

2

d N(E)

dt=

N20(E) N

2

N20

(E) = D(E U)f2(E)

Gate

D(E U) EF2


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steady-state

d N(E)

dt=

N10 N

1+

N20 N

2= 0

1 1( )N10 1 1( )N+ 1 2( )N20 1 2( )N = 0

N E( )=1 1( )

1 1( )+ 1 2( )N1

0E( )+

1 2( )1 1( )+ 1 2( )

N20

E( )

N10

E( ) D E U( )f1 E( )

N20

E( ) D E U( )f2 E( )

1 = h 1

1 = h 2


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steady-state electron number, N(E)

N E( )= 1

1

+ 2

D E U( )f1 E( )+2

1

+ 2

D E U( )f2 E( )

N E( )= D1 E U( )f1 E( )+ D2 E U( )f2 E( )

D1 E U( )=1

1 + 2D E U( )

D2 E U( )=2

1 + 2D E U( )

DOS that can be filled by

contact 1

DOS that can be filled bycontact 2


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steady-state current, I

ID

ID

d N(E)

dt 1

=N1

0 (E) N E( )

1

d N(E)

dt 2

=N2

0(E) N

2

ID E( )= +qd N(E)

dt1

= qd N(E)

dt2

Gate

D(E U)f2 E( )f1 E( )

EF1 EF2 VDS


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result

I E( )= qh

12

1 + 2

D E U( ) f1 f2( )

ID = I E( )dE =2q

h

2

D E U( ) f1 f2( )dE

1 = 2 =

N = N E( )dE= D E U( )2 f1 E( )+ f2 E( )( ) dE


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final results

ID =2q

h D E U( ) f1 f2( )dE

1 = 2 = = h

N = D E U( ) f1 + f2( )dE

D E U( )= D E U( )2

density-of-states per spin


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determining ()

ID E( )=2q

h

D E U( ) f1 f2( )

N E( )= D E U( ) f1 E( )+ f2 E( )

energy channels are independent:

qN

ID

=h

=

if f1 >> f2(source injects, drain collects), then:

ID =Q

=

stored charge

transit time


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Nanoelectronics and the Meaning of Resistance

1) What and where is theresistance?

2) Microscopic model for electricalresistance

3) Spins and magnets

4) Energy conversion

5) Beyond the one-electronpicture

Electronics from the Bottom Up on nanoHUB.org

Supriyo Datta


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outline

1) Introduction



4) Scattering in nano-MOSFETs

5) Discussion

6) Summary


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electron energy vs. position

VDS = 0.05 VVGS

controlling current with energy barriers

E = qV

VDS = 1.0 V

VGS

E.O. Johnson, The Insulated-Gate Field EffectTransistor: A Bipolar Transistor in Disguise, RCAReview, 34, pp. 80-94, 1973.


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top of the barrier MOSFET model

energy

position

1 = 2 =

h

device

LDOS

L

contact 1 contact 2

U = EC = EC0 qS

EF1

EF2

EC x( )

EC 0( )

low drain bias:


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energy

position

contact 1 contact 2

1

= 2

=h

device


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electron density

U = EC(0)

Wx

y

contact 2contact 1

D =m*

2h2

Wl

N = D E U( ) f1 + f2( )dE

f1 E( )=1

1+ eEF1 EC (0)( ) kBT

f2 E( )=1

1+ eEF1 qVDS EC (0)( ) kBT


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electron density

N = D E U( ) f1 + f2( )dE

F1 EF1 EC(0)[ ] kB T

F2 = F1 qVDS kBT

N2D = m

*kBT h

2

nS 0( )=N

Wl=

N2D

2F0 F1( )+F0 F2( )


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current

U = EC(0)

Wx

y

contact 2contact 1

D =

m*

2h2 Wl

=h

=

h x

l

ID

=2q

h D f

1

f2

( )dE


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current (cont.)

U = EC(0)

Wx

y

contact 2contact 1

=h

=h x

l

x = cos = 2 E EC( ) m* cos

x = cos/2/2

d = 2

=

2 E EC( )m*

2


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current (cont.)

ID =2q

h D f1 f2( )dE

D E( )= hl

2 E EC( )m

*

2

m*

2h2Wl

D E( )= W2m* E EC( )

h= M E( )

ID =2q

hM E( ) f1 f2( )dE

M(E) =W

2( )


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current (final result)

ID =2q

hM E( ) f1 f2( )dE

ID = WqN2D

2T

F1/ 2 F1( )F1/ 2 F2( )

F1 EF1 EC(0)[ ] kB T

F2 = F1 qVDS kBTT = 2kBT m*

N2D = m

*kBT h

2


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re-cap

nS 0( )=N2D

2

F0 F1( )+F0 F2( )

(1)

(2)

Solve (2) for N2D, then insert in (1):

ID = WqN2D

2

T

F1/ 2 F1( )F1/ 2 F2( )


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I-V characteristic

ID = WqnS 0( )TF1/ 2 F1( )F1/ 2 F2( )F

0

F1

( )+F

0

F2

( )

qnS 0( ) Cox VGS VT( ) (simple, 1D MOS electrostaticsVGS> VT)

ID = WQI 0( ) T

F1/ 2 F1( )F0 F1( )

1F1/2 F2( ) F1/2 F1( )1 +F0 F2( ) F0 F1( )


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final result

ID = WCox VGS VT( )%T1-F1/ 2 F2( ) F1/2 F1( )1 +F0 F2( ) F10 F1( )

%T 2kBT

m*F1/ 2 F1( )F0 F1( )


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Boltzmann limit

ID = WCox VGS VT( )T 1- e qV

DS

/kB

T

1 + e qVDS /kBT

T =2kBT

m*

Fj F( ) eF


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on-current

IDS

VDS

VDSAT

kB

T q

VGS = VDD

ID = WCox %T VGS VT( )1-F1/ 2 F2( ) F1/2 F1( )1+F0 F2( ) F10 F1( )

%T 2kBTm*F1/2 F1( )F0 F1( )

(100) [110] Si (single subband)

ID WCox %T VDD VT( )


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velocity saturation in a ballistic MOSFET

VDS

VGS = VDD

ID = WCox %T VGS VT( )1-F1/2 F2( ) F1/ 2 F1( )1 +F0 F2( ) F10 F1( )

= WQI(0) 0( )

EF1 EC(0)[ ] q

IDS

VDS

(0)

T

ballistic injection

velocity


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channel resistance of a ballistic MOSFET

IDS

VDS

VDSAT

kB

T q

ID = WCox %T VGS VT( )1-F1/ 2 F2( ) F1/2 F1( )1+F0 F2( ) F10 F1( )

finite channelresistance

VGS = VDD

as T 0

GCH =1

RCH M EF1( )

2q2

h


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ION ballistic( ) = WQI (0 ) %T

T 1.8 107 cm/s

QI(0) q

is a nanoscale MOSFET really ballistic?

= Cinv VDD VT( )

0.8 1013

cm-2

ION W ballistic( ) 2 mA/m

Typical N-channel MOSFET:

ION 1 mA/m

(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)

about 10% of the ballistic limit.

about 50% of theballistic limit

li


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outline

1) Introduction



4) Scattering in nano-MOSFETs5) Discussion

6) Summary


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scattering in Si n-channel nano-MOSFETs

ID =

W

L

nCox VGS VT( )VDS ?


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energy

position

1

= 2

=h

device

LDOS

L

contact 1 contact 2

EF1

EF2

EC x( )

EC 0( )

low drain bias: =

2 E EC( )

m*

cos x + sin y( )

(ballistic)

ff


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diffusive current

U = EC(0)

L

W

x

ycontact 2contact 1

=h

=h x

L

ballistic

ID =2q

h

D f1 f2( )dE

=h

= ? diffusive

=L

2

2Dn

b b lli i d diff i


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between ballistic and diffusive

D =0

0 + L B D

=h

B + D=

1

1 + D B

h

B

ID = WCox VGS VT( )0

0 + L

%T

1-F1/ 2 F2( ) F1/ 2 F1( )1 +F0 F2( ) F10 F1( )

Dn (E) =x

20 =

2

2

0 T =0

0 + L

li i t


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linear-region current


0 + L

T1- e

qVDS /kBT

1 + eqVDS /kBT

Boltzmann statistics:

Low VDS:


0 + L

T

2 kBT q( )VDS

ID =W

L + 0nCox VGS VT( )VDS

tt i i Si h l MOSFET


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scattering in Si n-channel nano-MOSFETs

ID =W

LnCox VGS VT( )VDS ID =

W

L + 0nCox VGS VT( )VDS

on-current

where scattering matters (the most)


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where scattering matters (the most)

Increasing VDS

X(nm) --->

-10 -5 0 5 10

EC

(eV)

--->

X(nm) --->

-10 -5 0 5 10

EC

(eV)

--->

ECvs. xfor VGS= 0.5V

T =0

0 +LT =

0

0 + l

ECvs. xfor VGS= 0.5V


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outline

1) Introduction




6) Summary

Physics of Nanoscale Transistors


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Physics of Nanoscale Transistors

1) Review of MOSFET Fundamentals

2) Elementary Theory of the NanoscaleMOSFET

3) Theory of the Ballistic MOSFET

4) Scattering in Nanoscale MOSFETs

5) Application to State-of-the-ArtMOSFETs

6) Quantum Transport in NanoscaleMOSFETs

7) Connection to the Bottom UpApproach Mark Lundstrom


S/D quantum mechanical tunneling


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S/D quantum mechanical tunneling

from M. Luisier, ETH Zurich

4) 3)

2) 1)

generic model to NEGF


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Gate

D(E U)

2

generic model to NEGF

S. Datta, Quantum Transport: Atom to Transistor, Cambridge, 2005(Concepts of Quantum Transport nanohub.org)

1

EF1 EF2H[ ] S[ ]

1[ ]/ 1[ ] 2[ ]/ 2[ ]

randomness is the rule - not the exception!


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randomness is the rule - not the exception!

side view

top view

Random dopant fluctuations

nanonets

Percolation in Electronic Devices


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Percolation in Electronic Devices

1) Percolation in Electronic Devices

2) Thresholds, Islands, and Fractals

3) Nonlinear Electrical Conduction inPercolative Systems

4) Stick Percolation and Nanonet Electronics

5) 2D Nets in 3D World: Sensors, Solar Cells,

and Antennas

M. Ashraf Alam


outline


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outline

1) Introduction




6) Summary

summary


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y

1) The bottom-up view provides a simple, but rigorousapproach to nanoelectronics.

2) Its useful for familiar devices, like MOSFETs.

3) Its also a good starting point for new devices.

4) You can learn more on nanoHUB.org or by attending theannual Electronics from the Bottom Up summerschools at Purdue University.

nanoHUB.org


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g

Electronics from the Bottom Up

Questions & Answers


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Quest o s & s e s

physics of nanoscale transistors - an introduction to electronics from the bottom up

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