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Transcript of The Ballistic MOSFET - nanoHUB · nanoHUB.org online simulations and more Network for Computational...
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Mark LundstromPurdue University
Network for Computational Nanoechnology
Simple Theoryof the
Ballistic Nanotransistor
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outline
I) Traditional MOS theoryII) A “bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary
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nanoscale MOSFETs
Intel Technical J., Vol. 6, May 16, 2002.
130 nm technology (LG = 60 nm)
Low VT
I DS
(mA
/µm
)
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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
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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
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velocity saturation
electric field V/cm --->
velo
city
cm
/s --
->
107
104
!
" = µE!
" ="sat
41.5V25 10 V/cm
60nm
DSV
L= ! "
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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
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MOSFET IV: velocity overshoot
Position along Channel (µm) Position along Channel (mm)
Frank, Laux, and Fischetti, IEDM Tech. Dig., p. 553, 1992
µ
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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
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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.
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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
!"
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Outline
I) Traditional MOS theoryII) A “bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary
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Gate
EF1 EF1-qVD( )SCF
D E U!
1
1
!"
=h
!
"2
=h
#2
a general view of nano-devices
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“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!= "
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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
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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!
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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)
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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
! !"
# # ! !$ =
+ +
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NEGF theory
Non-equilibrium Green’s Function Approach (NEGF)
S. Datta, IEDM Tech. Dig., 2002
device
!
["1]
!
[H]
!
["2]
!
["S]
Gate
EF1 EF2
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outline
I) Traditional MOS theoryII) A “Bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary
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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!" = #
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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= !"
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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)
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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)
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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)
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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
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carrier velocity in a ballistic MOSFET
Increasing VDS
X (nm) --->
EC (e
V)
---> Increasing VDS
-10 -5 0 5 10
ε1 vs. x for VGS = 0.5V
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velocity saturation in a ballistic MOSFET
Increasing VDS
X (nm) --->-10 -5 0 5 10
ε1 vs. x for VGS = 0.5V
!
"(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)
--->
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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)
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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…
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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!
"
"
# $"= " % &
+' (
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the ballistic MOSFET
1/ 2
1/ 2
0
0
( )1
( )( )
( )1
( )
F DS
F
DS ox GS T T
F DS
F
U
I WC V VU
!
!"
!
!
#$ %#& && &
= # ' (#& &+& &) *
%
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
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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
!!
" " " "
# $ # $ # $= + + %& ' & ' & '
( ) ( ) ( )
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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
!!
" " " "
# $ # $ # $= + + %& ' & ' & '
( ) ( ) ( )
[ ]21/ 2 1 1/ 2 2( ) ( )
2
DD T F F
qNI W! " "= #F F
see FETToy at
www.nanohub.org
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outline
I) Traditional MOS theoryII) A “Bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary
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the quantum capacitance
VG
insC
S QC C=
( )Gate G TQ C V V= !
ins Q
Gate
inc Q
C CC
C C=
+
( )( )2 *
2~
S
Q D F
S
qnC q D E m
!
" #= =
"S
!
if ,Q ins Gate insC C C C>> !
ID � Q!
inj
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bandstructure effects in nano-MOSFETs
a
(001)
(100)
(010)
(111)
(110)
-tight binding model (sp3d5s*) (Boyken, Klimeck, et al.)
-bulk, UTB, nanowire MOSFETs
-Si, Ge, SiGe, GaAs, InAs, … (strained or unstrained) (heterostructure channels)
2 2
*( )
2
kE k
m=h
( ) : tabulatedE k
Top-of-the-barrier model
analytical numerical
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scattering in nano-MOSFETs
measured ballistic
Chau et al, IEDM Technical Digest,2000, pp 45 -48
Intel 30nm bulk MOSFET
MOSFETs operate at ≈ 50% of their ballistic limit
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relation to traditional MOSFET theory
!
ID =W
LµeffCox VGS "VT( )VDS
( )2
2D eff ox GS T
WI C V V
Lµ= !
low VDS
high VDS (long channel)
1/ 2 1
1/ 2 1
0 1
0 1
( / )1
( )( )
( / )1
( )
F DS B
FDS ox GS T T
F DS B
F
qV k T
I WC V VqV k T
!
!"
!
!
#$ %#& && &
= # '( )#& &+& &* +
%
F
F
F
F
ballistic MOSFET
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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
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Outline
I) Traditional MOS theoryII) A “Bottom-up” approachIII) The ballistic nanotransistorIV) DiscussionV) Summary
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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.
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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.