Comparison TDDB Models

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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.



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:



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):



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



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−

:=



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



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



X T( )

expEb−

k

1

T

1

300−

⋅

1

expEb

k

1

T

1

300−

⋅

:=

300 320 340 360 380 4000

2

4

6

8

10

12

14

τ Tk ( )

X Tk ( )

Tk

A 108−

cm⋅≡ MV 106

volt⋅≡ mil 103−

in⋅≡

Documents

Comparison TDDB Models