IEEE Semiconductor Interface Specialists Conference Tutorial, Nov. 30th, 2011
Understanding
the
Nanoscale MOSFET
Mark Lundstrom
Electrical and Computer Engineering
and
Network for Computational Nanotechnology
Birck Nanotechnology Center
Purdue University, West Lafayette, Indiana USA
1
nanoscale MOSFETs
S
G
D
source
drain
silicon
2
SiO2
gate
electrode
gate oxide
EOT ~ 1.1 nm
Lundstrom: SISC 2011
channel
~ 32 nm
Moore’s Law: 2011
3
http://en.wikipedia.org/wiki/Moore's_law
Lundstrom: SISC 2011
Moore’s Law: 2011
transistors per cpu chip
4
Lundstrom: SISC 2011
Moore’s Law: 2010
1980s
~2000
?
5
Lundstrom: SISC 2011
2010
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.
6
Lundstrom: SISC 2011
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10 nm
Summary
This work is licensed under a Creative Commons AttributionNonCommercial-ShareAlike 3.0 United States License.
http://creativecommons.org/licenses/by-nc-sa/3.0/us/
Lundstrom: SISC 2011
7
MOSFET IV characteristic
gate-voltage
controlled
current source
circuit
symbol
on-current
D
I DS
VGS
VDS
G
VGS
S
gate-voltage
controlled
resistor
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
8
Lundstrom: SISC 2011
MOSFET IV: low VDS
L
0
VG>VT
VD
VGS
()
(
Qi x ≈ −Cox VGS − VT
)
I D = W Qi ( x )υ x (x)
ID = W Cox (VGS − VT )µeffE x
V
E x = DS
L
gate-voltage
controlled
resistor
W
ID = µeff Cox (VGS − VT )VDS
L
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Lundstrom: SISC 2011
MOSFET IV: “pinch-off” at high VDS
0
VG
Qi ( x ) = −Cox (VGS − VT − V (x))
VD
V ( x ) = (VGS − VT )
Qi ( x ) ≈ 0
10
Lundstrom: SISC 2011
MOSFET IV: high VDS
0
VG
gate-voltage
controlled
current source
VD
V ( x ) = (VGS − VT )
Qi ( x ) = −Cox (VGS − VT − V (x))
VGS
ID = W Qi ( x )υ x (x) = W Qi (0)υ x (0)
I D = W Cox (VGS − VT ) µeffE x (0)
W
2
ID = µeff Cox (VGS − VT )
2L
V − VT
E x (0) ≈ GS
L
11
Lundstrom: SISC 2011
velocity saturation
velocity cm/s --->
VDS 1.0 V
≈
≈ 3 × 10 5 V/cm
L
30 nm
107
υ = υ sat
υ = µE
105
104
electric field V/cm --->
12
Lundstrom: SISC 2011
MOSFET IV: velocity saturation
0
VG
VD
E x >> 10 4
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
I D = W Qi ( x )υ x (x)
ID = W Cox (VGS − VT )υ sat
I D = W Cox υ sat (VGS − VT )
13
Lundstrom: SISC 2011
carrier transport nanoscale MOSFETs
Velocity (cm/s) 
quasi-ballistic
EC
( µm )
υ SAT
( µm )
D. Frank, S. Laux, and M. Fischetti, Int. Electron Dev. Mtg., Dec., 1992.
14
Lundstrom: SISC 2011
how transistors work
VDS = 1.0 V
2007 N-MOSFET
VGS
EC
electron energy
vs. position
electron energy
vs. position
VDS = 0.05 V
VGS
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
EC
E.O. Johnson, “The IGFET: A Bipolar Transistor in Disguise,”
RCA Review, 1973
15
Lundstrom: SISC 2011
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10 nm
Summary
Lundstrom: SISC 2011
16
MOS electrostatics: low drain voltage
VG
E
 low gate voltage
EC
EC = EC ( 0 ) − qψ s
EF
 high gate voltage
Qi (ψ S ) C/cm 2
n0 = N C e( EF − EC ) kB TL /cm 3
Lundstrom: SISC 2011
x
17
inversion charge vs. gate voltage
CG = Cox || CS
≤ Cox
Qi
Qi ≈ −CG (VGS − VT )
ε ox
Cox =
F cm 2
t ox
VT
threshold voltage
VG
Lundstrom: SISC 2011
18
MOS electrostatics: high drain voltage
top-of-the-barrier should be strongly
controlled by the gate voltage
Qinv ( 0 ) ≈ −CG (VGS − VT )
EC
E TOP
EF1
EF1 - qVD
E C (x)
x
Lundstrom: SISC 2011
19
2D MOS electrostatics
CΣ = CGB + CSB + C DB + C D
VG
EC
ψ S (0) =
CGB
CSB
VS
CGB
C
VG + DB VD
CΣ
CΣ
C DB
ψS
VD
CD
EC ( x )
0
x
Lundstrom: SISC 2011
20
geometric scaling length (double gate example)
Λ
Require L > Λ
TOX = 1nm
Off-state: VG = 0V, VD = 1V, Ioff = 0.1µA/µm (by H. Pal, Purdue)
Lundstrom: SISC 2011
21
nonplanar MOSFETs
“Transistors go Vertical,” IEEE Spectrum, Nov. 2007.
See also: “Integrated Nanoelectronics of the Future,” Robert Chau, Brian Doyle, Suman
Datta, Jack Kavalieros, and Kevin Zhang, Nature Materials, Vol. 6, 2007
Lundstrom: SISC 2011
22
MOSFETs are barrier controlled devices
2) region under strong\
1) “Well-tempered MOSFET”
Qi ( 0 ) ≈ −CG (VG − VT )
control of gate
E

EC ( x )
3) Additional increases in VDS
drop near the drain and
have a small effect on ID
x
M. Lundstrom, IEEE EDL, 18, 361, 1997.
A. Khakifirooz, O. M. Nayfeh, D. A. Antoniadis, IEEE TED, 56, pp. 1674-1680, 2009.
23
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional Approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10 nm
Summary
Lundstrom: SISC 2011
24
Landauer approach
B. J. van Wees, et al. Phys. Rev. Lett. 60, 848–851,1988.
W -->
2q 2
G=
T ( EF ) M ( EF )
h
Lundstrom: SISC 2011
25
molecular electronics
The Birth of Molecular Electronics, Scientific American, September 2001.
Lundstrom: SISC 2011
26
current flows when the Fermi-levels are different
gate
EF1
f1 ( E ) =
1
1 + e( E − EF1 ) kB TL
I=
27
EF 2
D(E)
τ1
τ2
f2 ( E ) =
2q
T ( E ) M ( E ) ( f1 − f2 ) dE
∫
h
Lundstrom: SISC 2011
1
1 + e( E − EF 2 ) kB TL
“top of the barrier model”
energy
U = EC − qψ S
EF1
LDOS
EC ( x )
“device”
EF 2
ε1 ( x )
contact 1 contact 2
position
28
Lundstrom: SISC 2011
ballistic MOSFET: linear region
I DS
VGS = VDD
VDS
near-equilibrium
f1 ≈ f2
I DS = GCH VDS
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linear region with MB statistics
GCH
2q 2
=
h
∞
⎛ ∂f0 ⎞
∫E T ( E ) M 2 D ( E ) ⎜⎝ − ∂E ⎟⎠ dE
C
(
Boltzmann statistics: f0 = e
EF −E ) kBTL
− ∂f0 ∂E = f0 kBTL
GCH
υT
= WnS
2k BTL q
nS ≈ Cox (VGS − VT ) (MOS electrostatics)
GCH = WCox
30
υT
VGS − VT )
(
2k BT q
✔
( )
M 2D E = gV W
(
2m* E − EC
π
( )
T E =1
( )
(
E− E
f0 E = 1 1+ e( F
k BTL
)
nS = N 2 D e( EF − EC ) kB TL
⎛ m * ⎞ ( EF − EC ) kB TL
= gV ⎜ 2 ⎟ e
⎝ π ⎠
υT = 2kBTL π m* = υ x+
Lundstrom: SISC 2011
)
)
ballistic MOSFET: linear region
ID
VGS = VDD
I DS = GCH VDS
VDS
near-equilibrium
I DS = GCH VDS
31
f1 ≈ f2
υT
I D = WCox
VGS − VT )VDS
(
2k BT q
Lundstrom: SISC 2011
relation to conventional expression
ballistic MOSFET
I DS
conventional MOSFET
υT
= WCox
(VGS − VT )VDS
2k BTL q
I DS =
W
υT L
Cox
(VGS − VT )VDS
L
2kBTL q
υ λ
Dn = T 0
2
I DS =
32
I DS
µn =
Dn
υT L
µB ≡
k
T
q
2 k B TL q
(B L )
W
Cox µ B (VGS − VT )VDS
L
Lundstrom: SISC 2011
W
= Cox µn (VGS − VT )VDS
L
µn → µ B
ballistic MOSFET: on-current
ID
VGS = VDD
f1 >> f2
VDS
I DS
33
2q
2q
=
T ( E ) M ( E ) ( f1 − f2 ) dE → I DS =
T ( E ) M ( E ) f1 dE
∫
∫
h
h
Lundstrom: SISC 2011
saturated region with MB statistics
I DS
2q 2
=
h
∞
∫ T ( E ) M ( E ) f ( E ) dE
2D
1
EC
Boltzmann statistics: f0 ≈ e− ( EC − EF ) kB TL
34
π
( )
T E =1
N 2 D ( EF − EC ) kB TL
e
2
⎛ m * ⎞ ( EF − EC ) kB TL
= gV ⎜
e
2⎟
⎝ 2π  ⎠
nS =
I DS = WqnSυT
I DS = WCoxυT (VGS − VT )
( )
M 2D E = gV W
(
2m* E − EC
✔
υT = 2kBTL π m* = υ x+
Lundstrom: SISC 2011
)
under low VDS
I+
E
EF1
I + = nS+υT
I−
+
I I
EF 2
I − = nS−υT
nS+  nS−
−
nS = nS+ + nS−
nS+  nS− ≈
x
35
Lundstrom: SISC 2011
nS
2
Cox
nS ≈
(VGS − VT )
q
under high VDS
E
I+
EF1

I − = nS−υT
I− ≈ 0
nS− ≈ 0
nS = nS+ + nS− ≈ nS+
EF 2
x
36
I + = nS+υT
Lundstrom: SISC 2011
Cox
nS ≈
(VGS − VT )
q
velocity vs. VDS
E
I+
I + = nS+υT
I−
EF1
I
υ = D
qnS

I
(
=
+
− I−
)
nS
nS = nS+ + nS−
EF 2
x
nS−
−qVDS
=
e
nS+
υ = υT
37
I − = nS−υT
k B TL
(1 − e
(1 + e
− qVDS kBT
− qVDS kBT
)
)
velocity vs. VDS
E
I+
I−
υ → υT
υ
EF1

EF 2
υ ∝ VDS
x
VDS
Velocity saturates in a ballistic
MOSFET but at the top of the
barrier, where E-field = 0.
38
Lundstrom: SISC 2011
velocity saturation in a ballistic MOSFET
υ = υT ≈ 1.2 × 10 7 cm/s
velocity
2007 N-MOSFET saturation
VDS = 1.0 V
VGS
υ = υ sat ≈ 1.0 × 10 7 cm/s
39
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
Lundstrom: SISC 2011
aside: relation to conventional expression
ballistic MOSFET
I ON = WCoxυT (VGS − VT )
conventional MOSFET
I DS = WCoxυ sat (VGS − VT )
υ sat → υT
40
Lundstrom: SISC 2011
the ballistic IV (Boltzmann statistics)
I DS (on) = W υT Cox (VGS − VT )
ID
ballistic
on-current
(
VG − VT
)
1
ballistic
channel resistance
VDS
I DS = GCH VDS
41
υT
=W Cox
(VGS − VT )VDS
( 2kBTL q )
Lundstrom: SISC 2011
K. Natori, JAP, 76, 4879, 1994.
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional Approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10 nm
Summary
Lundstrom: SISC 2011
42
comparison with experiment: Silicon
LG = 40 nm
I Dlin I ballistic ≈ 0.2
I ON I ballistic ≈ 0.6
•  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.
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.
43
Device characterization and simulation: Himadri Pal and Yang Liu, Purdue, 2010.
Jesus del Alamo group (MIT)
44
Lundstrom: SISC 2011
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional Approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10nm
Summary
Lundstrom: SISC 2011
45
transmission and carrier scattering
λ0
T=
λ0 + L
1
X
X
X
0 < T <1
R = 1− T
L
λ0 is the mean-free-path for backscattering
I DS → TI DS ?
46
Lundstrom: SISC 2011
the quasi-ballistic MOSFET
?
I DS
VDS
⎛
⎞
υT
I D = T ⎜ WCox (VGS − VT )
VDS
⎟
( 2kBTL q ) ⎠
⎝
47
Lundstrom: SISC 2011
→ Tlin ≈ 0.2
on current and transmission
+
I BALL
Qi ( 0 ) =
W υT
I + + (1 − T ) I +
Qi ( 0 ) =
W υT
+
I BALL
I =
(2 − T )
+
I ON
48
(1 − T ) I
+
EF1
⎛ T ⎞ +
=⎜
I
⎝ 2 − T ⎟⎠ BALL
Lundstrom: SISC 2011
I
+
I D = TI +
EF 2
the quasi-ballistic MOSFET
I DS
⎛ T ⎞
I D = W υT ⎜
Cox (VGS − VT )
⎟
⎝2−T ⎠
→ Tsat ≈ 0.7
Tsat > Tlin
why?
VDS
⎛
⎞
υT
I D = T ⎜ WCox (VGS − VT )
VDS
⎟
( 2kBTL q ) ⎠
⎝
49
Lundstrom: SISC 2011
→ Tlin ≈ 0.2
scattering under high VDS
λ0
Tlin =
λ0 + L
E
EF1
L→

Tsat
EF 2
x
λ0
=
λ0 + 
 << L
Tsat > Tlin
50
Lundstrom: SISC 2011
connection to traditional model (low VDS)
ID =
W
µnCox (VGS − VT )VDS
L
⎛
⎞
υT
I D = T ⎜ WCox (VGS − VT )
VDS
⎟
( 2kBTL q ) ⎠
⎝
W
ID =
µnCox (VGS − VT )VDS
L + λ0
ID =
λ0
T=
λ0 + L
W
µappCox (VGS − VT )VDS
L
1 µapp = 1 µn + 1 µ B
51
Lundstrom: SISC 2011
connection to traditional model (high VDS)
I D = WCoxυ sat (VGS − VT )
⎛ T ⎞
I D = WCox (VGS − VT )υT ⎜
⎝ 2 − T ⎟⎠
−1
⎡1
1 ⎤
ID = W ⎢ +
⎥ Cox (VGS − VT )
⎣ υT ( Dn  ) ⎦
how do we interpret this result?
52
Lundstrom: SISC 2011
λ0
T=
λ0 + 
the MOSFET as a BJT
I D = W υ (0) Cox (VGS − VT )
1
1
1
=
+
υ (0) υT Dn 
“base”
E
‘bottleneck’
“collector”

EC ( x )
x
53
Lundstrom: SISC 2011
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional Approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10 nm
Summary
Lundstrom: SISC 2011
54
limits to barrier control: quantum tunneling
55
4)
3)
2)
1)
Lundstrom:
SISC
2011 / Purdue
from M. Luisier,
ETH
Zurich
ultimate limits
Limits
ES
E(eV)
min
= ln(2) kB T
Lmin ≈ 
2m ES
τ min ≈  ES
min
min
(ΔpΔx = )
(ΔE Δt = )
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.
56
Lundstrom: SISC 2011
32 nm technology
32 nm node
Limits
ES
min
(ITRS 2009 ed.)
ES ≈ 2, 500 × ES
= ln(2) kB T
Lmin ≈
2mEmin =1.5nm(300K)
τ min ≈ ES
min
= 0.04 ps (300K)
nmax (at 100W/cm ) =1.5 B/cm
2
57
2
L ≈ 15 × Lmin
τ ≈ 17 × τ min
n ≈ 1.1 B/cm 2
(minimum size transistor – no parasitics)
min
Moore’s Law?
58
“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.”
Lundstrom: SISC 2011
outline
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional Approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10nm
Summary
Lundstrom: SISC 2011
59
physics of nanoscale MOSFETs
ID
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.
VDS
1) Transistor-like I-V characteristics are a result
of electrostatics.
60
Lundstrom: SISC 2011
limits of scaling
What happens below 10 nm?
61
Lundstrom: SISC 2011
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
3) “Electronics from the Bottom Up”
http://nanohub.org/topics/ElectronicsFromTheBottomUp
4) “Lessons from Nanoscience,”
http://nanohub.org/topics/LessonsfromNanoscience
62
Lundstrom: SISC 2011
nanoHUB-U
http://nanoHUB.org/u
63
Lundstrom: SISC 2011
Thank you!
1)
2)
3)
4)
5)
6)
7)
8)
Introduction
Traditional Approach
MOS electrostatics
The ballistic MOSFET
Comparison to experiments
Scattering in nano-MOSFETs
MOSFETs below 10nm
Summary
Lundstrom: SISC 2011
64