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DEFECT RELATED BREAKDOWN MODEL (Oxide thinning = ∆Xox): Xeffective = tox - ∆Xox .
VoxPos Vgate 0.2−:= VoxNeg Vgate 1.3−:=
τo 1011−
sec⋅:= B 240 106
⋅volt
cm⋅:= Go 320 10
6⋅
volt
cm⋅:= tbdLee V( ) τo exp
Go Xeff ⋅
V volt⋅
⋅:=
JFN V( ) 3 108⋅ exp B Xeff ⋅V volt⋅
− ⋅ amp
cm2
⋅:= JFN 7.9( ) tbdLee 7.9( )⋅ 8.943 coul
cm2
= Qbd 10 coul
cm2
⋅:=
c 0.7:= slope70
c−
ln3 10
1−⋅
4 105
⋅
:=
TWK Approx for Defect Density, D(∆X)
tox = 107A, A = 500 x 500 umD ∆Xox( ) 4 10
5⋅ exp
∆Xoxc
−
slope
⋅:=
a1 13.1:= a2 6.3:=
b1 0.26−:= b2 0.175−:= Df ∆Xox( ) a1 exp b1 ∆Xox⋅( )⋅ a2 exp b2 ∆Xox⋅( )⋅+( ) 104
⋅:=Note: Article says b2 = -0.11. Added 10^4 above. This is a bi-Poisson Defect Distribution. See Yugami 1994
For a fixed Vox, the distribution D(∆X) of ∆X can be
extracted from the tbd distribution, i.e. ∆X =∆X(tbd).
∆X tbd( ) tox
Vox
Go
lntbd sec⋅
τo
⋅−:=
COMPARISON AND EVOLUTION OF GATE OXIDE TBD MODELS
http://www.leapcad.com/Other_Tech/Comparison_TDDB_Models.MCD
PHYSICAL CONSTANTS: k 8.62 105−
⋅ ev⋅:= ev 1.6 1019−
⋅ joule⋅≡ q 1.6 1019−
⋅ coul⋅:=
DEVICE DATA: thick SiO2 600 A⋅:= Xeff 79 A⋅:= tox 107 A⋅:= Vgate 1:=
VARIABLES: Tk 300 400..:= V 5 60..:= ∆Xox 1 70..:= Vox 11.2 volt⋅:=
1985, K. Yamabe, "Time dependent dielectric breakdown ... SiO2 films", SC 20, pg. 343:
TBD has a Thickness dependent Field Acceleration Factor, β =β =β =β =d(log(tbd)/dEox
β Tox( ) 4.2 log Tox( )⋅ 6.95−( )cm
MV⋅:= β 200( ) 2.714
cm
MV=
The time to fail for 200A oxide is shortened by 10^ -2.7 with an increase of the stress field by 1MV/cm.
1988, Qp Model, J. Lee, "Modeling and Char of Gate Oxide Relibility", ED 35, pg. 2268:
INTRINSIC BREAKDOWN MODEL, Critical Hole Fluence, Qp:
Oxide lifetime is is the time required for the hole fluence Qp, to reach a critical value.
Qp ~ J α t, where J is the FN current ~ e(-B/Eox) and α is the hole generation coefficient
~ e(-H/Eox). Then Qp ~ e(-G/Eox) t, where G = B + H.
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The Temperature Variation is modeled the same as R. Moazzami, which follows.
1 .105
1 .104
1 .1030.01 0.1 1 10 100
1
10
100
1 .103
1 .104
1 .105
1 .106
1/ Fail Rate vs time
Fig. 11a. Time (sec)
1 / F a i l R a t e ( % p
e r s e c )
λ tbdi 4,( )
λ tbdi 0.25,( )
λ tbdi 0.01,( )
tbdi
1 .1051 .10
41 .10
30.01 0.1 1 10
0.01
0.1
1
10
100Cumulative % Fail vs time
Fig. 10a. Time (sec)
C u m
% F
a i l u r e
CumFail tbdi 4,( )
CumFail tbdi 0.25,( )
CumFail tbdi 0.01,( )
tbdi
Area: 4, 0.25, 0.01 mm^2, Xox = 107A, Vgate = 11V
0 10 20 30 40 50 60 700.1
1
10
100
1 .103
1 .104
1 .105
1 .106
Defect Density as a function of delta X
Fig 9. Oxide Thinning (A) for 107A Gate
A r e a D e n s i t y ( 1 / c m ^ 2 )
D ∆Xox( )
Df ∆Xox( )
∆Xox
6 8 10 12 14 16 181 .10
41 .10
30.01
0.1
1
10
100
1 .103
1 .104
1 .105
1 .106
Time to Failure (sec)
Fig. 1. 79A Gate
tbdLee V( )
sec
V
λ tbd Area,( )
Area−Vox
tbd Go⋅⋅
1Area mm
2⋅
cm2
Df ∆X tbd( ) A1−
⋅⋅+
a1 b1⋅ exp b1∆X tbd( )
A⋅
⋅ a2 b2⋅ exp b2
∆X tbd( )
A⋅
⋅+
⋅
104
A⋅:=
Note: Added - sign to Area
Assume the defect distribution is
clustered, i.e. gamma distribution
with cluster factor, s = 0.6.
CumFail tbd Area,( ) 100 11
1Area mm2⋅
cm2
D ∆X tbd( ) A1−
⋅⋅ s⋅+
1
s
−
⋅:=
s 0.6:=tbdi
tbmin e
ir
n⋅
⋅:=i 1 n..:=r lntbmax
tbmin
:=n 200:=tbmax 1000:=tbmin 105−
:=
PREDICTING RELIABILITY FROM GAMMA DEFECT DISTRIBUTION:
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30 35 40 45 50 55 600.5
0.55
0.6
0.65
0.7
0.75
0.8Reduction of Activation Energy vs. Field
Fig. 3.
Q V( )
V
tbd.McP V T,( ) Ao e
Q V( ) ev⋅
k T⋅⋅:=Time to 50% Failure, TF:Q V( )
Ho 7.2 q⋅ A⋅ V⋅volt
thick SiO2
⋅−
ev:=
Ao 1 sec⋅:=Ho 1.15 ev⋅:=Enthalpy of Activation for Si-Si Breakage: Ho
Warranty 2000 hr = 7.2M sec. For 0.1ppm Failures,
need factor of 0.5/0.0000001 ==> t50 of 1.4 E14 sec.
1997, J. W. McPhearson, "Field enhanced Si-Si bond breakage", AP Phys Let, Aug, p.1101 Low Field TBD data reveals that TBD had field dependence ~ E and not 1/E.
1 10 100 1 .10310
100Poisson Cum % Fail vs time for D(Xeff)
Scale Time
C u m
% F
a i l u r e
CF tbdi 0.2,( )
tbdi
RELIABILITY PROJECTION, CUMULATIVE FAILURE PROBABILITY, CF:
Assume that defects are distributed randomly across the wafer. Poisson Distribution.
CF tbd Area,( ) 100 1 expArea mm2⋅
cm2
− D ∆X tbd( ) A1−
⋅⋅
−
⋅:=∆X tbd( ) Xeff Vox
Goln
tbd sec⋅
τo
⋅−:=
tbdMz V T,( ) τ T( ) τo⋅ expG T( ) Xeff ⋅
V volt⋅
⋅:=τ T( ) exp
Eb−
k
1
T
1
300−
⋅
:=G T( ) Go 1
δ
k
1
T
1
300−
⋅+
⋅:=
Go 300 106
⋅volt
cm⋅:=Xeff 100 A⋅:=Eb 0.28 ev⋅:=δ 0.0167 ev⋅:=τo 10
11−sec⋅:=
Vox 8 volt⋅:=Temp Variation on Qbd/Qp Model:
.
1990, R. Moazzami, "Projecting Gate Oxide Rel", ED 37, pg 1643):
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0.8 0 .9 1 1.1 1.2 1.3 1.4 1 .5 1.61
10
100
1 .103
1 .104
1 .105
1 .106
Median Time to Failure x Eox^2 vs. 1/Eox
Fig. 4 1/Eox (10^-7 cm/V)
tbdShi Vi InvE( ) 423,( )Vi InvE( )
tox
2
⋅
hrMV
cm
2
⋅
InvE
CF h MTF,( ) 1 exph
η MTF( )
mp
−
−:=
η MTF( )MTF
ln 2( )
1
mp
:=
hi
hmin e
ir
n⋅
⋅:=r ln
hmax
hmin
:=i 1 n..:=n 200:=hmax 2 104
⋅:=hmin 1:=
Vi InvE( ) tox InvE 107−
⋅cm
volt⋅
1−
⋅ volt1−
⋅:=InvE 0.8 0.82, 1.54..:=Plot Inverse of
E = 1/Eox:
mn 3:=tbdShi V T,( ) τos exp EaPlus
k T⋅ ⋅ exp Gsp
tox
V volt⋅⋅
⋅ V volt⋅tox
2−
⋅ SArea
1
mp−
⋅:=
Gsn 125:=
EaMinus 0.56 ev⋅:=
Area 0.09 mm2
⋅:=S 0.01 mm2
⋅:=τos 1 hr⋅volt
cm
2
⋅:=tox 110 A⋅:=Gsp 150MV
cm⋅:=EaPlus 0.63 ev⋅:=
mp 4:=
1993, N. Shiono, "A lifetime projection method using series ...TDDB...", IRPS, p. 1!/E overestimates low field lifetime. log(MTF Eox2) vs. !/Eox fits low fields.
Use full FN, JFN = A. Eox2 exp(-B/Eox), then MTF ~ exp(G/Eox)/Eox
2. Note τos Differences
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MODEL FOR TDDB STATISTICS: The Weibull is the extreme distribution for minimum value. TDDB
breakdown is min value. Therefore the Weibull is a more suitable statistical distribution model than log
normal. DMOS is composed of many cells, the failure of any one causes device failure. This is a series
model. Normalized Cumulative Failure CF(t) = 1- exp(-(t/ η)m), η = ηi/(η1/m) η is the characteristic
lifetime, m is the dispersion of lifetime, and n is the number of elements which is proportional to Area. Then
MTF~ Area1/m Let mp = β and η = α.
0.1 1 10 100 1 .103
1 .104
1 .105
1
10
100TDDB Failure Distribution (Cum %)
Fig. 2b Aging Time (hours)
C u m u l a t i v e F a i l u e s ( % )
CumFail tbd 4,( )
100 CF hi 2,( )⋅
100 CF hi 20,( )⋅
100 CF hi 50,( )⋅
100 CF hi 5 102
⋅,( )⋅
100 CF hi 104
,( )⋅
hi
η i 1:= ηη i
n
1
mp
:=
LLCF h MTF,( ) ln ln 1 CF h MTF,( )−( )1−
:=
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FAILURE RATE PROJECTIONS:Shiono's definition of acceptable failure rate Spec (FIT)
Failure Rate, λ(t): λ t MTF,( )mp
η MTF( )mp
tmp 1−
⋅:=
FailSpec 0.001:=
τoss 1.9 1012−
⋅ hr⋅MV
cm
2
⋅:= tbdShiE E T,( ) τoss expEaPlus
k T⋅
⋅ exp Gspcm
E MV⋅⋅
⋅ EMV
cm⋅
2−
⋅S
Area
1
mp−
⋅:=
λE t E,( )mp
ηtbdShiE E 358,( )
hr
mptmp 1−
⋅:=
t25yr 25 365⋅ 24⋅:= E 3 3.1, 6..:=
Ta = 85C, Area = 10 mm2
3 3.5 4 4.5 5 5.5 61 .10
12
1 .1011
1 .1010
1 .109
1 .108
1 .107
1 .106
1 .105
1 .104
1 .10
3
0.01
0.1Projected Failure Rate @25 Years
Fig 11. Eox (MV/cm)
F I T s
λE t25yr E,( )
109−
FailSpec
E
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300 320 340 360 380 4001 .10
6
1 .105
1 .104
1 .103
0.01
0.1
1
10
100
1 .103
1 .104
Median Time to Failure vs. Temp
Temperature Dependence
tbdMz 40 Tk ,( )
yr
tbdShi 40 Tk ,( )
yr
tbdMcP 40 Tk ,( )
yr
TFIRF 40 Tk ,( )
TenYears
WarrantyMiles
yr
Tk
10 15 20 25 30 35 40 45 50 55 600.01
0.1
1
10
100
1 .103
1 .104
Median Time to Failure @150C
Field Dependence
tbdMz V 423,( )
yr
tbdShi V 423,( )
yr
tbdMcP V 423,( )
yr
TFIRF V 423,( )
TenYears
UseFullLife
yr
WarrantyMiles
yr
V
tbdMcP V T,( ) Ao e
Q V( ) ev⋅
k T⋅⋅ 0.02⋅:=tbdMz V T,( ) τ T( ) τo⋅ exp
G T( ) thick SiO2⋅
V volt⋅
⋅:=
tbdShi V T,( ) τos expEaPlus
k T⋅ ⋅ exp Gsp
thick SiO2
V volt⋅⋅
⋅
V volt⋅
thick SiO2
2−
⋅S
M7
1
mp−
⋅:=
M7 1300 mil2
⋅:=
TFIRF V T,( ) 7 1028
⋅ e
V− volt⋅
Gaccel thick SiO2⋅⋅
sec
yr τ T( )⋅⋅:=Gaccel 0.112
MV
cm⋅:=
Projected LOG NORMAL Plot
for Median Time to Fail vs.Vgate
from Power DMOS Data Sheet.
Gate oxide thickness unknown.
UseFullLife 1.141yr=UseFullLifeUseFullLifeMiles
20 mi⋅ hr1−
⋅
:=UseFullLifeMiles 2 105
⋅ mi⋅:=
WarrantyMiles 0.228yr=WarrantyMiles 50000mi hr⋅
25 mi⋅
⋅:=TenYears 10:=
AF T( ) exp0.4 ev⋅
k
1
T
1
423−
⋅
:=
COMPARISON OF E AND 1/E MODELS