cmos device scaling past, present, and future · at the si/oxide interface, the oxide region is...
TRANSCRIPT
2013/9/4 1
Yuan Taur
Electrical & Computer Engineering
University of California, San Diego
CMOS Device Scaling—
Past, Present, and Future
2013/9/4 2
Outline
How short can a MOSFET be?
Early Short-Channel Models (1970s-1980s)
Charge sharing model.
Polynomial potential models.
Solving 2D Poisson’s equation analytically (1980s-1990s)
In silicon only.
In silicon and in SiO2.
Two-region, three-region scale length models (1990s-2000s)
Bulk CMOS.
Double-gate MOSFET, Nanowire MOSFET.
Projecting scaling limits.
The part on bulk MOSFET follows “Review and Critique of Analytic Models of MOSFET Short-Channel
Effects in Subthreshold,” Qian Xie, Jun Xu, and Yuan Taur, IEEE Transaction on Electron Devices, 2012.
9/4/2013 3
History of transistors and VLSI
9/4/2013 4
MOSFET Schematic
From “Guide to State-of-the-Art Electron
Devices,” IEEE Press/Wiley, 2013.
Can be turned on
with no conduction
current in silicon.
9/4/2013 5
MOSFET Scaling
Currents scale as 1/; Capacitances scale as 1/.
Gate delay scales as 1/.
Power per circuit scales as 1/ .2
Dennard et al., 1974
But no specific relation
of Lg to other device
parameters.
2013/9/4 6
If L ,
and , delay CV/I . But
I I CW
L
V V
mds dsat eff ox
g t
( )2
2
C WLCg ox2
3
When the channel length is
too short, the drain begins to
gain control of the barrier,
thus losing transistor action. source
gate-controlled
barrier
drain
Why Short Channel? How short?
Short-channel threshold voltage roll-off.
Drain Induced Barrier Lowering (DIBL).
9/4/2013 7
Short-Channel Vt Roll-off
DIBL
Degradation of subthreshold
slope.
9/4/2013 8
2D potential contours
(Same gate voltage)
Long channel Short channel
9/4/2013 9
Charge Sharing Model
n+ DrainWdm
L
tox
Gate
L
n+ Source
p-type Substrate
+ + + ++
+
+++++
+
+ + + +
Gate
oxide
xj
(Yau, SSE 1974)
1
2112
j
dmj
ox
dmaBfbt
x
W
L
x
C
WqNVV
• First to point out the importance of max. gate depletion depth Wdm.
• But, rather arbitrary partition of charge with no tox dependence.
• Difficult to deal with high drain bias.
Fixed
depletion
charge
Fixed
depletion
charge
9/4/2013 10
Polynomial Potential Models
sourcen+ drainn+
Gate
p-type substrate
x
y0 L
Wdm
tox
Na
Vgs
Vds
Vbs
(Toyabi and Asai, TED 1979)
a
si
Nq
yx
2
2
2
2
3
3
2
210 )()()()(),( xyaxyaxyayayx
),0(),0( yx
tVyVVox
sioxoxfbgs
bsdm VyW ),( 0),(
yW
xdm
Assume the solution to 2D
Poisson’s eq. in subthreshold
is of the form
Boundary condition at the Si-oxide interface: Continuity of vertical displacement,
Boundary conditions at the bottom of depletion region: Constant potential, zero field,
9/4/2013 11
Polynomial Potential Models
sourcen+ drainn+
Gate
p-type substrate
x
y0 L
Wdm
tox
Na
Vgs
Vds
Vbs
Using the 3 BC’s and the Poisson’s
eq., one obtains an ordinary
differential eq. for the surface
potential, s(y) = a0(y),
where
With the source and drain boundary conditions, s(0) = bi and s(L) = bi + Vds,
s has a minimum value (maximum energy barrier for electrons),
0
2
02
2
)/2( Aldy
dsa
s
dm
oxsidmox
oxsia W
tW
tl
32
20
1
/)(2
1
/2
100 )()()( AeAVeAy aa lLy
dsbi
ly
bis
1
/
11min0))((2 AeAVA alL
dsbibi
9/4/2013 12
Polynomial Potential Models
The L-dependent term of min describes SCE.
For SCE to be negligible (< 100 mV), Lmin ~ 3l0a.
l0a is called the “scale length”.
dm
oxsidmox
oxsia W
tW
tl
32
20
1
/
11min0))((2 AeAVA alL
dsbibi
• The first introduction of exponential SCE and “scale length” concept.
However, l0a is clearly incorrect because it is independent of tox in the limit
of thick tox.
• The polynomial (x,y) satisfies the 2D Poisson’s eq. only at x = 0, not the
region below the interface.
• The boundary condition at x = 0 is for long-channel MOSFETs which
implicitly assumes 2/x2 = 0 in the oxide, i.e., 1D Poisson’s eq.,
inconsistent with the 2D fields in Si.
9/4/2013 13
Other Polynomial Potential Models
• Yan et al., TED 1992:
Assumed a parabolic potential with constant potential at the bottom
boundary (field 0).
But same problem as Toyabe and Asai, l0b is independent of tox in the limit of
thick tox.
• Liu et al. TED 1993 (later BSIM3):
Assumed zero field at the bottom boundary with no restriction on potential
at x = Wdm.
Disqualifies as a predictive scale length as it all hinges on the fitting
parameter h.
dm
oxsidmox
oxsib W
tW
tl
20
ox
dmoxsic
Wtl
h
20
9/4/2013 14
Solving 2D Poisson’s eq. in silicon
• Ratnakumar and Meindl, JSSC 1982
Top: (mostly constant)
Bottom: (zero field)
Left:
Right:
)(),0( yy s
0),(
yW
xd
bix )0,(
dsbi VLx ),(
0
2
02
)12(sinh
2
)()12(sinh
2
)12(sinh
2
)12(sin
1),(n d
n
d
n
d
d
d
sW
ynB
W
yLnA
W
Ln
W
xn
W
xyx
a
si
Nq
yx
2
2
2
2
Scale length: l0d = 4Wdm/.
Key problem: Top boundary condition is conductor-like.
Long
channel
term
Satisfies 2D Poisson’s eq. at every (x, y).
9/4/2013 15
Solving 2D Poisson’s eq. in silicon
• Poole and Kwong, EDL 1984
Changed the top condition to:
Continuation of vertical displacement at
Si-SiO2 interface.
a
si
Nq
yx
2
2
2
2
Eigenvalue eq. from the top boundary condition:
where SCE ~ exp(L/l0e) with l0e = 2/g0 (longest).
Scale length, thin oxide: l0e = 4Wdm/, thick oxide:
),0(),0( yx
tVyVVox
sioxoxfbgs
ox
dmoxsie
Wtl
20
noxsi
oxdn
tW
g
g )tan(
0
2
0 sinh)(sinhsinh
)(cos1),(
n
nnnn
n
dn
d
s yDyLCL
Wx
W
xyx gg
g
g
9/4/2013 16
Two Issues: Top and Bottom BC
• A popular top boundary condition:
Continuation of vertical displacement at
Si-SiO2 interface.
But, it implicitly assumes constant vertical field in the oxide, i.e., 2/x2 = 0.
Lateral field will not be continuous at the Si-SiO2 interface if 2D in Si but 1D
in oxide.
Must solve 2D eq., 2/x2 + 2/y2 = 0, in both Si and SiO2.
),0(),0(
yxt
yVVsi
ox
fbgs
ox
9/4/2013 17
Two Issues: Top and Bottom BC
• Bottom boundary condition:
Zero field or
constant potential?
• 1D depletion approximation,
uniform doping:
both zero field and constant potential.
• But for 2D, must make a choice. If zero
field, then potential is not constrained. If
constant potential, then field may not be 0.
• Must choose constant potential because
field may not be continuous at x = Wdm, but
potential must be continuous everywhere.
0),(
yW
xdm
bsdm VyW ),(
Extreme retrograde
doped channel:
9/4/2013 18
Gate
DrainSourceH G L y
A
B C D E
F
-toxn+poly
Nan+ n+
xSubstrate
0
Wd
Boundary conditions:
Top: GH,
Left: AB,
Right: EF,
Bottom: CD.
( , ) 3t y V Vox g fb
( , )x bi0
( , )x L Vbi ds
( , )W yd 0
To eliminate the boundary condition
at the Si/oxide interface, the oxide
region is replaced by an equivalent
Si region (si/ox)tox 3tox thick.
In AFGH
(oxide),
In ABEF
(silicon),
2
2
2
2 0x y
2
2
2
2x y
qNa
si
One region scale length
Solving 2D Poisson’s eq. in silicon and oxide
Nguyen and Plummer, IEDM 1981.
In subthreshold,
9/4/2013 19
General approach to a 2-D boundary value problem
Let:
v(x,y) is a solution to the inhomogeneous equation (with Na) and
satisfies the top boundary condition.
uL, uR, uB are solutions to the homogeneous equation such that
(x,y) satisfies the other B.C.’s.
( , ) ( , ) ( , ) ( , ) ( , )x y v x y u x y u x y u x yL R B
Gate
DrainSourceH G L y
A
B C D E
F
-toxn+poly
Nan+ n+
xSubstrate
0
Wd
Superposition
uL = 0 on top, bottom, right; uR = 0 on top, bottom, left; etc.
9/4/2013 20
1 3
)3(sin
3sinh
3sinh
),( n oxd
ox
oxd
oxd
nRtW
txn
tW
Ln
tW
yn
cyxu
Gate
DrainSourceH G L y
A
B C D E
F
-toxn+poly
Nan+ n+
xSubstrate
0
Wd
For satisfying the
Boundary conditions:
Note that for u=sin(kx),
d2u/dx2=-k2u;
And that for u=sinh(ky),
d2u/dy2=k2u.
1 3
)3(sin
3sinh
3
)(sinh
),( n oxd
ox
oxd
oxd
nLtW
txn
tW
Ln
tW
yLn
byxu
1
sin)3(
sinh
)3(sinh
),( n oxd
ox
nBL
yn
L
tWn
L
txn
dyxu
uL = 0 on top, bottom, right; uR = 0 on top, bottom, left; etc.
y = 0 y = L
9/4/2013 21
Approximate 2-D Solution of Potential
Neglect uB and higher order terms in uL, uR:
oxd
ox
oxd
oxdoxd
d
stW
tx
tW
L
tW
yc
tW
yLb
W
xyx
3
)3(sin
3sinh
3sinh
3
)(sinh
1),(
112
oxd
oxtW
L
sctW
tecby oxd
3
)3(sin2),0(
3
2/
11
source
gate-controlled
barrier
drain
For x = 0, apply sinh z ez/2 and u + v 2(uv)1/2, the surface potential has a
minimum at y = yc:
SCE ~ exp(L/l0g),
Scale length:
Lmin 3l0g (Note: b1 bi, c1 bi + Vds)
ox
ox
sidmg tWl
20
Long channel
9/4/2013 22
One region scale length model
The scale length is given by the sum of Wdm and (si/ox)tox.
Valid for tox << Wdm so vertical field dominates in oxide.
Must reduce both Wdm (by higher doping) and tox to scale to shorter
gate lengths.
tox cannot scale much below 1 nm because of tunneling. The industry
went to high- insulators.
However, high- insulators can be physically thick. The one-region
model based on vertical fields is no longer correct.
Both the lateral and the vertical fields must be taken into
consideration two-region scale length model.
It is not only i/ti (vertical field), but also ti (lateral field).
ox
ox
sidmg tWl
20
9/4/2013 23
Two-Region Scale Length
Source Drain
Gate
Body
ti
Wd
L
si
i
1
2
D. Frank et al., EDL 10/98
In the one-region model,
the eigenvalues are:
For two regions, assume
the eigenvalues are kn.
for ti ≤ x ≤ 0,
for 0 ≤ x ≤ Wdm.
oxd
ntW
nk
3
1
111 )(sin
sinh
sinh)(sinh),(
n
in
n
nnnn txkLk
ykcyLkbyxu
1
222 )(sin
sinh
sinh)(sinh),(
n
dmn
n
nnnn WxkLk
ykcyLkbyxu
Note that u1 = 0 at the top (x = ti) while u2 = 0 at the bottom (x = Wdm).
9/4/2013 24
Two-Region Scale Length
Source Drain
Gate
Body
ti
Wd
L
si
i
1
2
D. Frank et al., EDL 10/98
),0(),0( 21 yuyu
),0(),0( 21 yx
uy
x
usii
At the interface (x = 0),
match potential,
and normal displacement,
(Tangential field matched
if potential matched.)
0tan1
tan1
dmn
si
in
i
Wktk
Every term must match since the BCs apply for any y. For nontrivial solutions, obtain
eigenvalue eq.:
(same eq. from cn1, cn2.)
1
2
1
1 )sin()(sinh)sin)(sinh n
dmnnn
n
innn WkyLkbtkyLkb
1
2
1
1 )cos()(sinh)cos)(sinh n
dmnnsinn
n
inninn WkkyLkbtkkyLkb
9/4/2013 25
Example: Two-Region
0tan1
tan1
dmn
si
in
i
Wktk
For si =11.70, i =7.80, ti =5.0 nm, and Wdm =10.0 nm,
k1 = 0.20 nm-1, k2 = 0.43 nm-1,
k3 = 0.63 nm-1,
Each SCE term exp(knL/2).
k1 term dominates.
Define l0i = 2/k1 so SCE exp(L/l0i)
and Lmin 3l0i.
Note that
5.31~2exp
2exp
min2
min1
Lk
Lk
02
tan12
tan1
00
i
dm
sii
i
i l
W
l
t
k1 k2 k3
kn
An infinite number of discrete,
irregularly spaced eigenvalues kn.
9/4/2013 26
Two-Region Scale Length
Lowest eigenvalue:
Note that:
l0i (2/)Wdm or (2/)ti,
whichever is longer.
If ti=Wdm, l0i=(4/)Wdm =
(4/)ti, regardless of i, si.
If i=si, l0i=(2/)[Wdm+ti],
the total height of box.
If ti<<Wdm, l0i (2/)[Wdm
+ (si/i)ti] (dotted line),
back to the one-region l0g.
02
tan12
tan1
00
i
dm
sii
i
i l
W
l
t
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Normalized Si Depletion Depth,
No
rmali
ze
d G
ate
In
su
lato
r T
hic
kn
es
s,
10
31
/ =i si
1/3 (SiO )2
W /d
t /i
(2/)(Wdm/l0i)
(2/)(ti/l0i)
In general, requires ti < ( /4)l0i or Lmin/4, even if i >>si.
9/4/2013 27
Graphical Method
Previous example, si =11.70, i =7.80, ti =5.0 nm, and Wdm =10.0 nm.
Draw a straight line
with slope ti/Wdm
(=1/2) through (0, 0).
Interpolate the i/si
(=2/3) value between
the curves.
Read off (2/)(Wdm/l0i)
(=0.63).
Obtain l0i = 10 nm.
Lmin 3l0i = 30 nm.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Normalized Si Depletion Depth,
No
rmalized
Gate
In
su
lato
r T
hic
kn
ess,
10
31
/ =i si
1/3 (SiO )2
(2/)(ti/l0i)
(2/)(Wdm/l0i)
9/4/2013 28
Verification by 2-D simulation
0 10 20 30 40 5010
-3
10-2
10-1
100
s, m
in (
V)
L (nm)
Vds = 0.05V
Vds = 1V
Lmin
~ 3l0
s,min s,min(short ch.)
s,min(long channel)
exp(-L/l0) for low/high Vds.
Pre-exponential factor1.7V,
Lmin 3l0 so s,min < 0.1 V.
source
gate-controlled
barrier
drain
(For L 2l0, s,min below exp(-L/l0) because exp(-L/l1) term is negative.)
(l0 = 8.6 nm)
9/4/2013 29
Comparison of 9 scale length models
0 5 10 15 200
5
10
15
20
Points: 2-D simulation
Lines: models l0a
- l0i
tox
(nm)
Sca
le l
ength
(n
m)
a
f
b
i
d
gc
eh
0 10 20 30 40 50 60 70 800
5
10
15
20
25
EOT = 1nm
Circles: 2-D Simulation
Lines: Models l0a
- l0i
i
h
gf
e
d
c
b
a
Sca
le l
en
gth
(n
m)
Dielectric constant of gate insulator
l0a, l0b : Polynomial potential model.
l0c: BSIM3.
l0d, l0e, l0h: Zero field BC at bottom.
l0g = (2/)[Wdm + (si/i)ti] (one-region).
l0i : Two region scale length model.
For a fixed EOT (ox/i)ti,
only the two-region l0i model
depicts the correct
dependence on i (due to
lateral field in oxide.)
Wdm=10 nm
ti
2013/9/4 30
Bulk, SOI and DG-MOSFET
S D
G
S D
G
BOX
S D
G
G
Bulk Fully-depleted
SOI Double-Gate
Bulk MOSFETs scale by decreasing depletion width with higher body doping. (High Vt, tunneling, poor sub-Vt slope, dopant fluctuation.)
FD-SOI and double-gate MOSFETs scale by silicon film thickness without doping.
SCE in Thin Body (FD) SOI MOSFETs
Bulk MOSFET
Wdm=10nm
tox=1nm
SOI MOSFET
tsi=10nm
tox=1nm
SOI: Field penetration into BOX (200nm).
No analytic scale length.
Scale length: Bulk vs. SOI
Bulk MOSFET
Wdm=10nm
tox=1nm
SOI MOSFET
tsi=10nm
tox=1nm
No analytic scale length.
Lmin 3.5l0i 28nm (50mV Vt)
, l0i = 8 nm
Lmin 58nm (50mV Vt)
02
tan12
tan1
00
i
dm
sii
i
i l
W
l
t
Charge and potential in short-channel SOI
For Vbs=0, inversion charge is higher at back surface in a short device.
Large substrate reverse bias restore the channel to the front surface.
sisi t
kTq
i
tkTq
ii ddxd
endxenQ
0
/
0
/
/
In subthreshold
Lmin hits a floor until very large reverse bias.
Lmin of SOI vs. reverse substrate bias
?
Xie et al.,
TED 2013.
Very Large Substrate Reverse Bias:
Backside Accumulation
SOI becomes bulk-like
with Wdm = tsi.
Body effect > 60 mV/dec.
Backside accumulation starts at
si
BOXg
BOX
ox
si
si
gg
subt
t
q
Et
qt
E
q
EV
31
min
But, high field issues.
9/4/2013 36
Double-Gate MOSFETs: Three-region scale length model
Three-region scale length model:
1
111
)(2sin
2sinh
)(2sinh
),( n n
n
n
nLl
tx
l
L
l
yL
byxu
1
22
2sin
2sinh
)(2sinh
),( n n
n
n
nLl
x
l
L
l
yL
byxu
1
3233
)(2sin
2sinh
)(2sinh
),( n n
n
n
nLl
ttx
l
L
l
yL
byxu
l
t
l
t
l
t
l
t
l
t
l
t 3
3
2
2
1
1
321
31
2 2tan
12tan
12tan
12tan
2tan
2tan
Eliminate from the two boundary conditions:
Eigenvalue eq.
Bottom conductor: Either gate or substrate.
9/4/2013 37
Scale Length of Double-Gate MOSFETs
Let 1 = 3 = i, t1 = t3 = ti, and 2 = si, t2 = tsi :
l
t
l
t
l
t
l
t si
si
i
i
sii
i
si 2tan
12tan
22tan
2tan 2
2
012
cot2
tan22
tan2
2
i
si
l
t
l
t
l
t sii
i
sii
lt
lt
l
t
l
t
l
t
si
sisisii
i
si
/2sin
1/2cos1
2cot
2cot
2tan 2
si
isii
l
t
l
t
tan
2tan
si
isii
l
t
l
t
cot
2tanTwo solutions: and
The longest l comes from the first eq. The second eq. requires
one of the angles to exceed /2.
9/4/2013 38
Scale Length of Double-Gate MOSFETs
For i = si, l = (2/tsi + 2ti), tsi +
2ti is the physical height between
the two gates.
DG MOSFET scales better than
bulk because tsi can be < Wdm with
no doping.
In any case, l (2/tsi and l
(4/ti regardless of i/si.
si
isii
l
t
l
t
tan
2tan
(2/)(tsi/l)
(2/)(ti/l)
Lmin 3l max(2tsi, 4ti)
Nanowire MOSFETs
]/)(2[
]/)(2[1
)/2(
)/2(
)/2(
)/2(
0
0
0
0
0
0
ltRJ
ltRY
lRJ
lRY
lRJ
lRY
i
i
i
si
i
si
0)()(1
yy
2-D Poisson’s eq. in
cylindrical coordinates:
Scale length l:
Nanowire MOSFETs
If si =i, l = 0.83(R +ti )
If i >>si (very high-),
Lmin=the larger of 2.5R , 3.5ti .
(tox=ti)
[DG: l = 0.64tsi +2ti)]
l (nanowire0.65l (DG)
if tsi 2R (why?)
Scaling limits?
(2/)(R/l)
(2/)(ti/l)
So where is the limit?
Limited by quantum
confinement:
DG
E0=h2/(8mtsi2)
Nanowire
E0=a2h2/(8m2R2)
For same confinement,
tsi (/a)R 1.3R
How thin can silicon be?
a2.405 is the 1st zero of J0(x).]
So where is the limit?
DG: l = 0.64tsi +2ti) tsi,min= 2.1 nm, Lmin 6.0 nm.
Nanowire: l = 0.83(R +ti )
Rmin=1.6 nm, Lmin 5.2 nm.
Assume si = i, ti = 0.5 nm,
InGaAs nanowire can only be scaled to Lmin ~ 10 nm because of
the stronger QM confinement from small electron effective mass.
Conclusion
Many evolutions after the first introduction of
“scale length” in 1979, we now have a good
handle on analytically assessing short-channel
effect in MOSFETs.
The “scale length” concept has been generalized
to high- dielectrics, double-gate MOSFETs, and
nanowire MOSFETs.
Combining scale length with quantum mechanical
considerations allows the projection of scaling
limits.
List of References
1. L. D. Yau, "A simple theory to predict the threshold voltage of short- channel IGFET’s," Solid-State
Electronics, vol. 17 , pp. 1059-1063, Oct. 1974.
2. T. Toyabe and S. Asai, "Analytical models of threshold voltage and breakdown voltage of short-channel
MOSFET's derived from two-dimensional analysis," IEEE Trans. Electron Devices, vol.26, no.4, pp. 453-
461, Apr. 1979.
3. R. H. Yan, A. Ourmazd, and K. F. Lee, "Scaling the Si MOSFET: from bulk to SOI to bulk," IEEE Trans.
Electron Devices, vol.39, no.7, pp.1704-1710, Jul. 1992.
4. Z.-H. Liu, C. Hu, J.-H. Huang, T.-Y. Chan, M.-C. Jeng, P. K. Ko, and Y.C. Cheng, "Threshold voltage model
for deep-submicrometer MOSFETs," IEEE Trans. Electron Devices, vol.40, no.1, pp.86-95, Jan. 1993.
5. K. Ratnakumar, and J. Meindl, "Short-channel MOST threshold voltage model," IEEE J. Solid-State
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