Lundstrom SISC_11.30.11v2

IEEE Semiconductor Interface Specialists Conference Tutorial, Nov. 30th, 2011

Understanding the

Nanoscale MOSFET

Mark Lundstrom

1

Electrical and Computer Engineering and

Network for Computational Nanotechnology Birck Nanotechnology Center

Purdue University, West Lafayette, Indiana USA

2

nanoscale MOSFETs

source drain

SiO

2

silicon

channel ~ 32 nm

gate oxide EOT ~ 1.1 nm

gate electrode

S G D

Lundstrom: SISC 2011

3

Moore’s Law: 2011


http://en.wikipedia.org/wiki/Moore's_law

4

Moore’s Law: 2011

transistors per cpu chip


5

Moore’s Law: 2010


~2000

1980s

? 2010

6

objectives

1)  Present a simple, physical picture of the nanoscale MOSFET (to complement, not supplement simulations).

2)  Discuss ballistic limits, velocity saturation, and quantum limits in nanotransistors.

3)  Compare to experimental results for Si and III-V FETs.

4)  Discuss scattering in nano-MOSFETs.

5)  Consider ultimate limits.


7

outline


1) Introduction 2) Traditional approach 3) MOS electrostatics 4) The ballistic MOSFET 5) Comparison to experiments 6) Scattering in nano-MOSFETs 7) MOSFETs below 10 nm 8) Summary

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

8

MOSFET IV characteristic

VGS

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

S

D

G

circuit symbol

VDS

VGS

IDS

gate-voltage controlled resistor


gate-voltage controlled

current source on-current

9

MOSFET IV: low VDS

VG>VT VD 0

ID =WQi x( )υx (x)

�

ID =W Cox VGS −VT( )µeffE x

Qi x( ) ≈ −Cox VGS −VT( )

�

ID = WL

µeffCox VGS −VT( )VDS

VGS

gate-voltage controlled resistor


�

E x = VDS

L

L

10

MOSFET IV: “pinch-off” at high VDS

VG VD 0

�

Qi x( ) = −Cox VGS −VT −V (x)( )

�

V x( ) = VGS −VT( )

Qi x( ) ≈ 0


11

MOSFET IV: high VDS

VG VD 0

ID =W Cox VGS −VT( )µeffE x (0)

�

ID = WL

µeffCox VGS −VT( )22

E x (0) ≈

VGS −VTL

�

Qi x( ) = −Cox VGS −VT −V (x)( )�

V x( ) = VGS −VT( )

VGS

gate-voltage controlled

current source

�

ID =W Qi x( )υ x (x) =W Qi 0( )υ x (0)


12

velocity saturation

electric field V/cm --->

velo

city

cm

/s --

->

107

104

�

υ = µE�

υ = υ sat

VDSL

≈ 1.0V30nm

≈ 3×105 V/cm

105


13

MOSFET IV: velocity saturation

VG VD 0

ID =WQi x( )υx (x)

�

ID =W Cox VGS −VT( )υ satID =WCoxυsat VGS −VT( )

E x >> 104



µm( )

Velo

city

(cm

/s)

14

carrier transport nanoscale MOSFETs

υSAT


D. Frank, S. Laux, and M. Fischetti, Int. Electron Dev. Mtg., Dec., 1992.

µm( )

EC

quasi-ballistic

15

how transistors work

2007 N-MOSFET

electron energy vs. position

VDS = 0.05 VVGS

EC


VDS = 1.0 V

VGS

electron energy vs. position

EC

E.O. Johnson, “The IGFET: A Bipolar Transistor in Disguise,” RCA Review, 1973


16

outline


1) Introduction 2) Traditional approach 3) MOS electrostatics 4) The ballistic MOSFET 5) Comparison to experiments 6) Scattering in nano-MOSFETs 7) MOSFETs below 10 nm 8) Summary

17

MOS electrostatics: low drain voltage


x

E

EF

EC

VG

Qi ψ S( )C/cm2

low gate voltage

high gate voltage

n0 = NC eEF −EC( ) kBTL /cm3

EC = EC 0( ) − qψ s

18

inversion charge vs. gate voltage


VG

Qi

Qi ≈ −CG VGS −VT( )

VTthreshold voltage

CG = Cox ||CS

≤ Cox

Cox =εoxtox

F cm2

19

MOS electrostatics: high drain voltage


E TOP

EC

EF1 EF1 - qVD

E C (x)

x

top-of-the-barrier should be strongly controlled by the gate voltage

Qinv 0( ) ≈ −CG VGS −VT( )

20

2D MOS electrostatics


ψ SCDBCSB

CGB

EC x( )CD

EC

x

CΣ = CGB + CSB + CDB + CD

ψ S 0( ) = CGB

CΣ

VG +CDB

CΣ

VD

VG

VS VD

0

21

geometric scaling length (double gate example)


TOX = 1nm

Off-state: VG = 0V, VD = 1V, Ioff = 0.1µA/µm (by H. Pal, Purdue)

Λ

Require L > Λ

22

nonplanar MOSFETs


See also: “Integrated Nanoelectronics of the Future,” Robert Chau, Brian Doyle, Suman Datta, Jack Kavalieros, and Kevin Zhang, Nature Materials, Vol. 6, 2007

“Transistors go Vertical,” IEEE Spectrum, Nov. 2007.

MOSFETs are barrier controlled devices

EC x( )

x

E

3) Additional increases in VDS drop near the drain and have a small effect on ID

A. Khakifirooz, O. M. Nayfeh, D. A. Antoniadis, IEEE TED, 56, pp. 1674-1680, 2009.

2) region under strong\

control of gate Qi 0( ) ≈ −CG VG −VT( )1) “Well-tempered MOSFET”

M. Lundstrom, IEEE EDL, 18, 361, 1997. 23

24

outline


1) Introduction 2) Traditional Approach 3) MOS electrostatics 4) The ballistic MOSFET 5) Comparison to experiments 6) Scattering in nano-MOSFETs 7) MOSFETs below 10 nm 8) Summary

25

Landauer approach


W --> B. J. van Wees, et al. Phys. Rev. Lett. 60, 848–851,1988.

G =2q2

hT EF( )M EF( )

26

molecular electronics


The Birth of Molecular Electronics, Scientific American, September 2001.

27

current flows when the Fermi-levels are different

D(E)

τ 2

EF1 EF2

τ1f1 E( ) = 11+ e E−EF1( ) kBTL

f2 E( ) = 11+ e E−EF 2( ) kBTL

gate

I = 2qh

T E( )M E( ) f1 − f2( )dE∫


28

“top of the barrier model” en

ergy

position

contact 1 contact 2

“device”

LDOS

U = EC − qψ S

EF1

EF2

ε1 x( )EC x( )


29

ballistic MOSFET: linear region

VDS

IDS VGS =VDD


near-equilibrium f1 ≈ f2IDS = GCHVDS

30

linear region with MB statistics


GCH = 2q2

hT E( )M 2D E( ) − ∂f0

∂E⎛⎝⎜

⎞⎠⎟ dE

EC

∞

∫ M2D E( ) = gVW

2m* E − EC( )π

T E( ) = 1

nS = N2DeEF −EC( ) kBTL

= gVm*

π2⎛⎝⎜

⎞⎠⎟e EF −EC( ) kBTL

GCH =WnSυT

2kBTL q

GCH =WCoxυT

2kBT qVGS −VT( )

nS ≈ Cox VGS −VT( ) (MOS electrostatics)

✔

f0 E( ) = 1 1+ e E−EF kBTL( )( )

υT = 2kBTL πm* = υx+

Boltzmann statistics: −∂f0 ∂E = f0 kBTL

f0 = eEF −E( ) kBTL

31

ballistic MOSFET: linear region

VDS

ID VGS =VDD


near-equilibrium f1 ≈ f2IDS = GCHVDS

IDS = GCHVDS

ID =WCoxυT

2kBT qVGS −VT( )VDS

32

relation to conventional expression


IDS =WCoxυT

2kBTL qVGS −VT( )VDS

ballistic MOSFET conventional MOSFET

IDS =WLCoxµn VGS −VT( )VDS

IDS =WLCox

υT L2kBTL q

VGS −VT( )VDS

Dn =υTλ02

µn =Dn

kBTL q( )

IDS =WLCoxµB VGS −VT( )VDS

µB ≡υT L

2 kBTL q

µn → µB

33

ballistic MOSFET: on-current

VDS

IDVGS =VDD


f1 >> f2

IDS =2qh

T E( )M E( ) f1 − f2( )dE∫ → IDS =2qh

T E( )M E( ) f1 dE∫

34

saturated region with MB statistics


IDS =2q2

hT E( )M 2D E( ) f1 E( )dE

EC

∞

∫

M2D E( ) = gVW

2m* E − EC( )π

T E( ) = 1

Boltzmann statistics: f0 ≈ e− EC −EF( ) kBTL

nS =N2D

2e EF −EC( ) kBTL

= gVm*

2π2⎛⎝⎜

⎞⎠⎟e EF −EC( ) kBTL

IDS =WqnSυT

υT = 2kBTL πm* = υx+

IDS =WCoxυT VGS −VT( ) ✔

35

under low VDS

x

E

EF1 EF2

I +I + = nS

+υT


I −I − = nS

−υT

I + I − nS+ nS

−

nS = nS+ + nS

−

nS+ nS

− ≈ nS2

nS ≈Cox

qVGS −VT( )

36

under high VDS

x

E

EF1

EF2

I +


I + = nS+υT I − = nS

−υT

I − ≈ 0 nS− ≈ 0

nS = nS+ + nS

− ≈ nS+

nS ≈Cox

qVGS −VT( )

37

velocity vs. VDS

x

E

EF1

EF2

I + I −

υ =IDqnS

=I + − I −( )nS

I + = nS+υT I − = nS

−υT

nS = nS+ + nS

−

nS−

nS+ = e−qVDS kBTL

υ = υT

1− e−qVDS kBT( )1+ e−qVDS kBT( )

38

velocity vs. VDS

x

E

EF1

EF2

I + I −

υ ∝VDS

υ →υT

Velocity saturates in a ballistic MOSFET but at the top of the barrier, where E-field = 0.


υ

VDS

39

velocity saturation in a ballistic MOSFET

VDS = 1.0 V

VGS


2007 N-MOSFET velocity

saturation

υ = υsat ≈ 1.0 ×107 cm/s


υ = υT ≈1.2 ×107 cm/s

40

aside: relation to conventional expression


ION =WCoxυT VGS −VT( )

ballistic MOSFET conventional MOSFET

IDS =WCoxυsat VGS −VT( )

υsat →υT

41

the ballistic IV (Boltzmann statistics)

VG −VT( )1

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

IDS (on) =W υTCox VGS −VT( )

IDS = GCHVDS =W CoxυT

2kBTL q( ) VGS −VT( )VDS

VDS

ID

ballistic channel resistance

ballistic on-current


42

outline



43

comparison with experiment: Silicon

A. Majumdar, Z. B. Ren, S. J. Koester, and W. Haensch, "Undoped-Body Extremely Thin SOI MOSFETs With Back Gates," IEEE Transactions on Electron Devices, 56, pp. 2270-2276, 2009.

Device characterization and simulation: Himadri Pal and Yang Liu, Purdue, 2010.

LG = 40 nm

ION Iballistic ≈ 0.6

IDlin Iballistic ≈ 0.2

•  Si MOSFETs deliver > one-half of the ballistic on-current. (Similar for the past 15 years.)

•  MOSFETs operate closer to the ballistic limit under high VDS.

44

Jesus del Alamo group (MIT)


45

outline


1) Introduction 2) Traditional Approach 3) MOS electrostatics 4) The ballistic MOSFET 5) Comparison to experiments 6) Scattering in nano-MOSFETs 7) MOSFETs below 10nm 8) Summary

46

transmission and carrier scattering

T =λ0

λ0 + L

0 < T <1

R = 1− T

1

L

X X X

λ0 is the mean-free-path for backscattering

IDS → TIDS ?Lundstrom: SISC 2011

?

ID = T WCox VGS −VT( ) υT

2kBTL q( )⎛

⎝⎜⎞

⎠⎟VDS

47

the quasi-ballistic MOSFET

VDS

IDS

→ Tlin ≈ 0.2


48

on current and transmission


ION =T2 − T

⎛⎝⎜

⎞⎠⎟IBALL+

EF1 EF2

ID = TI +I +1− T( ) I +Qi 0( ) = I + + 1−T( ) I +WυT

Qi 0( ) = IBALL+

WυT

I + =IBALL+

2 − T( )

ID =WυTT2 − T

⎛⎝⎜

⎞⎠⎟Cox VGS −VT( )


2kBTL q( )⎛

⎝⎜⎞

⎠⎟VDS

49

the quasi-ballistic MOSFET

VDS

IDS→ Tsat ≈ 0.7

→ Tlin ≈ 0.2

Tsat > Tlin why?


50

scattering under high VDS

x

E

EF1

EF2

Tsat =λ0

λ0 +

<< L

Tlin =λ0

λ0 + L

L→

Tsat > Tlin


51

connection to traditional model (low VDS)

ID =WLµnCox VGS −VT( )VDS

ID =W

L + λ0µnCox VGS −VT( )VDS


2kBTL q( )⎛

⎝⎜⎞

⎠⎟VDS


T =λ0

λ0 + L

ID = WLµappCox VGS −VT( )VDS

1 µapp = 1 µn +1 µB

52

connection to traditional model (high VDS)

how do we interpret this result?

ID =WCox VGS −VT( )υTT2 − T

⎛⎝⎜

⎞⎠⎟

ID =W1υT

+1

Dn ( )⎡

⎣⎢

⎤

⎦⎥

−1

Cox VGS −VT( )

ID =WCoxυsat VGS −VT( )


T =λ0

λ0 +

53

the MOSFET as a BJT

ID =W υ(0) Cox VGS −VT( )

EC x( )x

E

1υ(0)

=1υT

+1

Dn

‘bottleneck’ “collector”

“base”


54

outline



55

limits to barrier control: quantum tunneling

from M. Luisier, ETH Zurich / Purdue

4) 3)

2) 1)


56

ultimate limits


V. V. Zhirnov, R.K. Cavin III, J.A. Hutchby, and G.I. Bourianoff, “Limits to Binary Logic Switch Scaling—A Gedanken Model,” Proc. IEEE, 91, 1934, 2003.

�

ES min = ln(2) kBT

Lmin ≈ 2mES min

τmin ≈ ES min

Limits

�

ΔpΔx = ( )

�

ΔEΔt = ( )

E(eV)

57

32 nm technology

�

ES min = ln(2) kBT

�

Lmin ≈ 2mEmin =1.5nm(300K)

�

τmin ≈ ES min = 0.04 ps (300K)

Limits

�

nmax (at 100W/cm2) =1.5 B/cm2

32 nm node (ITRS 2009 ed.)

ES ≈ 2,500 × ES min

L ≈ 15 × Lmin

τ ≈ 17 × τmin

n≈ 1.1 B/cm2

(minimum size transistor – no parasitics)

58

Moore’s Law?

“Moore’s Law is about lowering cost per function….progress continues at a breathtaking pace, but transistor scaling is approaching limit. When that limit is reached, things must change, but that does not mean that Moore’s Law has to end.”


59

outline



60

physics of nanoscale MOSFETs

VDS

ID

1) Transistor-like I-V characteristics are a result of electrostatics.

2) The channel resistance has a lower limit - no matter how high the mobility or how short the channel.

3) The on-current is controlled by the ballistic injection velocity - not the high-field, bulk saturation velocity.

4) Channel velocity saturates near the source, not at the drain end.


61

limits of scaling


What happens below 10 nm?

62

for more information


1) “Physics of Nanoscale MOSFETs,” a series of eight lectures on the subject presented at the 2008 NCN@Purdue Summer School by Mark Lundstrom, 2008.

http://nanohub.org/resources/5306

2) “Electronic Transport in Semiconductors,” Lectures 1-8, by Mark Lundstrom, 2011. http://nanohub.org/resources/11872

4) “Lessons from Nanoscience,” http://nanohub.org/topics/LessonsfromNanoscience

3) “Electronics from the Bottom Up” http://nanohub.org/topics/ElectronicsFromTheBottomUp

nanoHUB-U

63 Lundstrom: SISC 2011

http://nanoHUB.org/u

64

Thank you!



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