section1p2
TRANSCRIPT
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EE231 Vivek Subramanian Slide 1-1
A quick review of MOS Capacitors
EE231 Vivek Subramanian Slide 1-2
MOS Capacitors
What happens when thework function is different?
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EE231 Vivek Subramanian Slide 1-3
MOS Band Diagram Different Work function
EE231 Vivek Subramanian Slide 1-4
E0 : Vacuum levelE0Ef: Work functionE0Ec : Electron affinitySi/SiO2 energy barrier
sMfbV =
SiO2=0.95 eV
9 eV
Ec,Ef
Ev
Ec
Ev
Ef
3.1 eV qs= Si + (EcEf)
qMSi
E0
3.1 eV
Vfb
N+ -poly-Si P-body
4.8 eV
=4.05eV
Ec
Ev
SiO2
Flat Band Condition
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EE231 Vivek Subramanian Slide 1-5
voltage
acrossthe oxide
voltage across the substrate,
i.e. the band bending in the
substrate, also called surface potential
fbg VV
SOxfbg VVV ++=
VOx
Vg
S
EfEf
What if
substrate charge
Ox
SOx
CQV = accinvdepS QQQQ ++=
Non-Flat-band conditions
EE231 Vivek Subramanian Slide 1-6
oxsfbg VVV ++=
s is negligible
3.1eV
Ec ,Ef
Ev
E0
E
c
E
f
Ev
M O S
qVg
Vox
qs
Surface Accumulation
kTEE
vvfeNp
/)( =
( )fbgOx
OxOxS
SOxfbg
Ox
SOx
VVC
VCQ
VVV
CQV
=
=
++=
=
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EE231 Vivek Subramanian Slide 1-7
Can solve for (Vg) and VOx(Vg)
SOxfbg VVV ++=
Ox
SSa
Ox
depa
Ox
dep
Ox
SOx
C
qN
C
XqN
C
Q
C
QV
2 =
+=
=
=
S
Ec,Ef
Ev
Ec
EfEv
M O S
qVg
depletion
region
qs
Wdep
qVox
----
Depletion
EE231 Vivek Subramanian Slide 1-8
aS Npn == 0
( )
Vn
N
q
kT
EEq
V.
EEEE
i
a
bulkfiB
BS
bulkVfSfC
4.0ln
1
802
)()(
=
=
=
=
VtAt
( )
Ox
BaS
C
Nq
BfbT
OxSfbBSgT
VV
VVVV
222
2
++=
++=== ( )
( )
319
/
/
10
=
=
cm
eNp
eNn
kTEE
V
kTEE
C
Vf
fC
C=D
BA =
(Alternative definition: S,th = B + 0.45V)
0.15V
Threshold (of Inversion)
Ec,Ef
M O S
Ev
Ef
Ei
Ec
A
B
C= q
Ev
D
qVg
=qVt
st
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EE231 Vivek Subramanian Slide 1-9
BSBS 2~2
Ox
dep
BfbT
Ox
invT
Ox
inv
Ox
BaS
Bfb
Ox
invdep
Bfbg
C
QVV
C
QV
C
Q
C
NqV
C
QQVV
+=
=++=
++=
2
222
2
( TgOxinv VVCQ =
Ox
S
inv
BS
V
Q
Q
large
large
large
2
EC
EF
Inversion
EE231 Vivek Subramanian Slide 1-10
s
2B
Vf b Vt
Vg
accumulation depletion inversion
Wdep
Wdmax
Vfb Vt
Vg
accumulation depletion inversion
(s)1/2
Wdmax
= (2s2/qa)1/2
Review : Basic MOS Capacitor Theory
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EE231 Vivek Subramanian Slide 1-11
Qdep
= qNaW
dep
0
Vfb
Vt
Vg
accumulation depletion inversion
qNaWdep
Qinv
Vfb
Vt
Vg
accumulation depletion inversion
slope =Cox
(a)
(b)
Qacc
Vfb
Vt
Vg
accumulation depletion inversion
(c)
qNaW
dmax
slope =Cox
Qs
0
Vfb
Vt
Vg
accumulationregime
depletionregime
inversionregime
Qinv
slope= Cox
invdepaccs QQQQ ++=
Review : Basic MOS Capacitor Theory
VgVT
C
Vfb
COx
small-signal capacitance:
g
S
dV
dQC =
EE231 Vivek Subramanian Slide 1-12
Advanced MOSCAP Physics and Technology
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EE231 Vivek Subramanian Slide 1-13
Na
( ) ( ) ( )
kT
q
epepxp xkTxq
==
0/
0
( ) ( ) kTxqenxn /0=
x
(x)
(0) = S
SiSiO2
EV
Na, p-type
0
( ) 0=
a
i
Nn
2
Eq. 3-9
EF
So were we too simplistic?
Problem with previous analysis Assumes that there are no free carriers in the depletion region
(depletion approximation)
Obviously, this is not true (else, how could we have inversioncharge?)
EE231 Vivek Subramanian Slide 1-14
A more general and accurate MOSCAP analysis
If we dont assume complete absence of carriers in thedepletion region, we have:
Now, we can proceed by noting that:
( ) ( )
( ) ( )( )daxxS
ad
SS
NNenepq
NnNpqx
dx
d
+=
+==
+
00
2
2
2
2
2
2
1
=dx
d
d
d
dx
d
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EE231 Vivek Subramanian Slide 1-15
Calculation of E(x)
Therefore, we have:
Which is E(x)
( )
( )
[ ]
( )2/1
00
00
0 00
2
00
2
2
2
)1()1(2
)1()1(2
2
2
1
++=
+
=
+
=
+
=
=
enepq
dx
d
enepq
dNNenepq
dx
d
NNenepq
dx
d
d
d
dx
d
S
S
da
S
da
S
EE231 Vivek Subramanian Slide 1-16
Calculation of QS
Now, we can solve this equation at x=0 to find the peakfield. By Gauss law, we can therefore find the totalcharge in the silicon
This equation is valid in all regions of operation, since wehavent made any region-specific assumptions so far
( ) ( )[ ] 2/10 112
LawsGauss'by
++=
==
=
=
SoSS
S
S
S
SSS
SS
S
enepq
dx
d
dx
d
EQ
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EE231 Vivek Subramanian Slide 1-17
Calculation of inversion and depletion charge
From the previous analysis, we can also calculate the various charge
components, namely the inversion layer charge and the depletion layercharge (and, of course, the accumulation charge in accumulation).
For example, in inversion, we find:
Since we know E(x) and n(x), we can solve for the total inversion layercharge.
Similarly, we can also solve for QB, which consists of ionizedacceptors and holes.
==c c
surf
x
dxd
dnqdxxnq
0
I/
)()(Q
EE231 Vivek Subramanian Slide 1-18
Graphical analysis of charge distributions The results of the previous analysis are plotted below:
Accumulation Depletion Onset of Inversion
Specific conclusions:
1. The inversion layer charge is extremely close to the surface
2. There are very few mobile carriers in the depletion region
i.e., our simplistic analysis in the previous section is actually reasonablyaccurate.
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EE231 Vivek Subramanian Slide 1-19
Extraction of C-V characteristics
From our equation for QS, we can also determine more accurate C-V
characteristics that predicted by the previous analysis.
( ) ( )[ ] 2/10 112
++= SoSS
SSS enep
qQ
( ) ( )
( ) ( )[ ]
( ) 0
2/1
0
0
2)(12
11
11
2
nNNnp
NNnpq
enep
enepq
d
dQC
daSSS
daSSS
SoS
oS
S
SS
SS
SS
++
+=
++
++==
C
V gV
-- --
-- -
- --
--
E s
accdepinvS QQQQ ++=
EE231 Vivek Subramanian Slide 1-20
Extraction of C at VT
We can use this equation to determine specific capacitance values
At VG = VT, we have:
Where
This value is different from the value predicted by the simple modelby a factor if 2, since the latter does not include the inversion chargepresent at VT
( ) 02)(12
nNNnp
NNnpqC
daSSS
daSSSS
++
+=
maxd
Sdep
XC
=
OxC2//2
2
2 maxd
S
aSB
S
B
aS
aB
aSS
XqN
Nq
N
NqC
====
a
BSd
qNX
22max =
S
invinv
d
dQC
=
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EE231 Vivek Subramanian Slide 1-21
Extraction of C at VFB
Similarly, at VG = VFB, we find that:
Again, we find that the simplistic model is somewhat inaccurate,since it doesnt include the wiggle around flat-band, which affectsthe charge in the silicon. This effect is small, of course.
S
COx
Cinv Cdep Cacc
EV
x=0
s
( )
a
S
SfbS
qN
VC =lengthDebye
kT/2qofbending
bandforXdep
q
kT=
1
EE231 Vivek Subramanian Slide 1-22
C Ox
C s
C
0
S
C
0Vg
CS(S)
S2B
Cdepacc
S
LD
max
2
d
S
X
max
max
d
S
inv
d
Sdep
XC
XC
=
( )
Ox
SSSfbg
C
QVV
+=
m.equilibriuatarepandni.e.
,ppeg.,,inchanges
torespondpn,i.e.,,Q,Q,Q:LF
0S
depaccinv
= e
MOSCAP LFCV Characteristics
We can plot the variation in CS in the various regions to find theMOSCAP LFCV characteristics:
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EE231 Vivek Subramanian Slide 1-23
MOSCAP LFCV Characteristics Graphically:
CS
S
C Ox
M
S
V g
C Ox
C s
s
S
s
2 Bacc
0
S
SS
d
dQC
=
S
Cg
QS
Substrate cap.
=> Vg
EE231 Vivek Subramanian Slide 1-24
MOSCAP HFCV Characteristics We can perform a similar analysis for HFCV:
S
Vg
C
C
S
COx
CinvCdep Cacc
COx
Cs
( )
( ) ( )
( ) ( )[ ]enough)quicklyrespondnotdoeslayerinversionthe(since0nLet
11
11
2
C
HF?forC
0
2/1
00
00S
S
++
++=
SS
S
S
SS
SS
enep
enepq
S
CS(HF)
HF or DDmax
~d
S
X
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EE231 Vivek Subramanian Slide 1-25
LF: n in equilibrium with s(Vg)
tVg
C
HF
n
t
LF
( ) ( )gacacgbiasbias VnVnn +:
HF:
n:nac=0
n
t
HF
Ox
SBfbg
C
QVV ++= 2
n
t
Deep Depletion:
n:nac=0
t
S
B2
C
Vg
VgLinear ramp
+ ac signal
LF
DD
I.
II
III
nbias (Vgbias)s2 B
Xd Xdmax
Ox
S
Bfbg C
Q
VV ++= 2
s>2 B Xd > Xdmax
HF
DD
nbias=0
Summary of MOSCAP CV Characteristics
EE231 Vivek Subramanian Slide 1-26
The Charge Sheet Model
Problems with the simple model
Inaccurate in depletion
Inaccurate in accumulation
Inaccurate in weak inversion (2B > S > B) Problems with the general model
Requires numerical solution for Q I
The charge sheet model provides a reasonable tradeoffbetween the two. It isnt as accurate in depletion or
accumulation, but these regions arent as important forMOSFET operation
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EE231 Vivek Subramanian Slide 1-27
Main Assumptions
Mobile charge exists beyond the onset of weak inversion
(i.e., QI > 0 forS > B unlike the simple model, which assumes that the inversion chargeis zero forS < 2B
Mobile charge is present in a negligibly thin layer similar to the simple model
Depletion region has a sharp boundary similar to the simple model
The surface potential is not clamped past threshold
unlike the simple model, which assumes that S is clamped at 2Bfor all values of VGpast threshold
EE231 Vivek Subramanian Slide 1-28
Derivation of QS
As in the general model, we have:
To simplify, assume we only care about QS in weak and stronginversion (i.e., S > B). Then:
( )
( )2/1
00
002
2
)1()1(2
++=
+
=
enepq
dx
d
NNenepq
dx
d
S
da
S
( ) ( )[ ]
2/1
0 11
2
++=
SoS
S
S
SS
enep
q
Q
||2
||2
2
depinv
a
iS
SaS QQe
N
nqNQ S +=
+=
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EE231 Vivek Subramanian Slide 1-29
Then, we can determine Qdep as in the simple model, and subtract tofind Qinv
Note that Qinv(VG) still requires an iterative solution
In strong inversion, since the exponent dominates, we can simplify to:
This exponential dependence implies that large changes in Qinv result
from small changes in S, which means that S is essentially clamped,as was assumed in the simple model.
( )
Ox
SSSfbg
SSa
a
iS
Sainv
SSadep
C
QVV
qNeN
nqNQ
qNQ
S
+=
+=
=
22
2
2
2
Derivation of Qdep and Qinv
a
iSinv
N
enqQ
S
22
=
EE231 Vivek Subramanian Slide 1-30
In weak inversion, Qinv < QB. This allows us to simplify Qinv:
This equation clearly incorporates the effect of subthreshold current inMOSFETs, unlike the simple VT equations studied previously (forexample, in EE130)
Vg
1 decade ofincrease
for 60/ mV in Vg
typically 80 mV
log Qinv
ss eN
nNqe
N
nqNQ
a
i
S
aS
aS
iSaSinv
BSB
2
2
2
2
2
112
2
+=
>>
( )
Ox
SSSfbg
C
QVV
+=
S
1 decade ofincrease
for 60 mV in s
log Qinv
kTq Se /
Simplification under weak inversion
From first 2terms of taylor
series
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EE231 Vivek Subramanian Slide 1-31
Oxide Charges
In general, these charges all
modify the threshold voltagebased on their charge centroid
In addition, they may altermobility due to coulombicscattering
=0
2 00
)(1
x
ox
SiO
T dxxxV
( )
Ox
SSfb
Ox
T
Ox
itSSMSg
CQV
dxxx
CQQV
Ox
+=++=
0
EE231 Vivek Subramanian Slide 1-32
Mobile Ions People observed odd shifts in C-Vs
Reason: Mobile charge was moving towards /away from interface, changing charge centroid
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EE231 Vivek Subramanian Slide 1-33
Interface traps
Traps cause sloppy C-V and alsogreatly degrade mobility in channel
EE231 Vivek Subramanian
Telegraphic noise in Id of a smallMOSFET is the signature of asignal interface trap.
When a single trap changes fromempty to filled Ninv=-1.
Id Ninv
ninvWL
Id Ninv + Ninv = - + Ninv
+
=
invd
d
NI
I 1
0 : donor type
Noise due to InterfaceTraps
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EE231 Vivek Subramanian
Physical ToxElectrical Toxe
Electrical ToxRaised by Inversion
Change Centroid and Gate Depletion
EE231 Vivek Subramanian
= ginv CdVQ
)1(60Ox
dep
C
CmvS +=
Npoly
CpolyCOx
Cdep
XdepImpact on IV
increase the effective Tox by ~5 -20
decrease of Vg by ~0.2V,
also has impact on CV and S
3/
1111
dpolyox
ox
s
dpoly
ox
ox
polyox
WT
WT
CCC
+=
+=
+=
Polysilicon gate depletion
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EE231 Vivek Subramanian
How to Reduce Gate-Depletion Effect
Metal gate - process integration issues
Or increase active doping concentration in the gate:
In-situ or POCl3 doped poly-Si gate Not suitable for dual-gate CMOS technology
Higher dosage for ion-implanted poly-Si gate Cost, damage and boron penetration issues
or higher activation temperature
S/D diffusion and boron penetration
Poly-Si1-xGex-gate technology
EE231 Vivek Subramanian
Sec 3.3 presents the classical analysis based on Poissons equationand Fermi-Dirac function (or Boltzmanns relations)
Quantum confinement in the potential well at the Si/SiO2 interfacecreates discrete subbands of energy levels.
Ref. Stern, self-consistent , Phy, Rev. B. vol. 5, p.4891, 1972
40
E0~60mV, dependingon Nsub and Qinv
SiO2
x
Ec 0,1,2j4
3
24
23/2
=
+ jm
hqE
x
Sj
Assume only ground subband is populated
+
invdepx
Siinv
QQqm
hX
3
116
9
2
2
( ) Simmx 100for9.0 0
Quantum effect (in the inversion layer)
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EE231 Vivek Subramanian
-Effect on VTS has to be larger than 2Bby, say 60mV depending on Nsub.
mVC
CVieV
Ox
dep
Sgt 100~,1,
+=
Empirical model: Rios, A Physical Compact MOSFET Model ., IEDMP.937, 1995
-Effect on CV
-Effect on IVSimilar to CV, but there is a subtle difference between AC chargecentroid and DC charge centroid.
inv
Siinv
XC
Quantum effect (in the inversion layer)
EE231 Vivek Subramanian
-4 -3 -2 -1 0 1 2 3 4
2x10-7
4x10-7
6x10-7
8x10-7
1x10-6
Measured Data
Cox, Tox=30A
Classical
QM+PD
Tox=30A
Nsub=5.2x1017cm-3
Npoly=4.5x1019cm-3
Capacitance(F/cm
2)
Gate Voltage(V)
Real MOSCAP CV Characteristics