MODELING OF INTERCONNECT DIELECTRIC LIFETIME UNDER STRESS

CONDITIONS AND NEW EXTRAPOLATION METHODOLOGIES

FOR TIME-DEPENDENTDIELECTRIC BREAKDOWN

Gaddi S. Haase, Joe W. McPhersonTexas Instruments, Dallas, TX, USA

TEXAS INSTRUMENTS

4E.2

2

• The actual minimum line-to-line spacing is dropping under 60nm for the new technology nodes.

• The traditional methodology for BEOL low-k dielectric lifetime predictions (E-model + Weibull statistics) might seem overly conservative, and might restrict technology scaling.

• Do we have enough data to support more lenient models for extrapolating test data to predict product lifetime?

• Is there any other source of over-conservatism that can be readily removed?

Purpose

3

• Two critically important parts of reliability assessments.

• Physical models for dielectric degradation as a function of E-field.

• Statistical Modeling: – Is Weibull good for interconnect dielectrics?

– Actual line-to-line spacing distributions

– Lifetime simulations using within-DUT variations

– Lifetime simulations including DUT to DUT variations.

– Are we using the correct Weibull parameters?

– A couple methodologies for extracting the true shape parameter β

• Conclusions

Outline

4

Ch

arac

teri

stic

life

tim

e (u

sua

lly

on

a l

og

sca

le)

Stress

The two parts of reliability assessments

1.What is the functional dependence of the lifetime on the stress? •Can the lifetime be extrapolated to use conditions?

Example: pdf of the Weibull

distribution for differentβ

values

Time

Prob

abili

ty D

ensi

ty

2. What statistical function describes dielectric failure?•Extract lifetime at cumulative failure probability <0.1%, while testing small DUT samples

•Area scaling

5

1.E-01

1.E+01

1.E+03

1.E+05

1.E+07

1.E+09

1.E+11

1.E+13

1.E+15

1.E+17

1.E+19

1.E+21

1.E+23

1.E+25

0 1 2 3 4 5 6

Electric Field (MV/cm)

Lif

etim

e

E model

SQRT(E) model

1/E model

Power-law

Usag

e fieldL

ifet

ime

(Lo

g s

cale

)Accelerated test regime

Extrapolating interconnect dielectric accelerated test data

• The E-model is generally the most conservative.

• The E-model is backed by a plausible physical model.

• Other models either lack viable physical basis for BEOL-dielectrics, or have no experimental verification.

• Uncertainty in E during test, and limited test time make model exploration & verification difficult.

• Hence, at this point, there is no safe, verified alternative to the E-model.

6

( )[ ] ( )[ ] NN tFtF −=− 11

Area scaling requires that:

( )

−−=

β

αt

tF exp1

Weibull distribution

Is Weibull statistics viable for BEOL dielectric BD?

F(t) = cum. failure probability at ≤ t

N= area factor. Correct ONLY if the cap dielectric properties & thickness are homogeneous

A tw

o param

. solution

-3.5

-2.5

-1.5

-0.5

0.5

1.5

0 1 10 100 1000 10000 100000Time to Breakdown (s)

ln[-

ln(1

-F)]

40 V

45 V

50 V

Log. (45V)

Linearized Weibull Plot

β= ~0.9But trying to fit TDDB data to a Weibull distribution…

• The fit is often poor

• The slope, β, appears <1

7

The actual line-to-line spacing distribution in a 65nm technology comb-comb test structure

The observed spacing distributions, from many cross-sectional electron micrographs for each test structure, were fitted with asymmetric normal distributions (solid curves) for ease of use in the simulation.

M1

M2

0

0.2

0.4

0.6

0.8

1

60 70 80 90Spacing at the top of the lines (nm)

Cu

m.

Pro

bab

ilit

y

Center

Edge

M1

8

TDDB Simulation: Within DUT variation only• All DUTs in the simulated test have the same spacing distribution.• In this example: All DUTs have the “Center” spacing.• Divide the test structure into bins i with spacing Si

• Use the E-model to correlate spacing (Si) with Field (Ei) and with characteristic lifetime ( tbd,63%, i ) :

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

60 65 70 75 80 85 90

Line-to-Line spacing (nm)

Nor

mal

ized

bin

pro

babi

lity

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

Ass

ocia

ted

tbd(

63%

) (s

)

Spacing distribution(Normal)

The associatedtbd(63%) at 50 V

Line-to-line spacing (nm)

CENTER

Nor

amal

ized

bin

pro

babi

lity

tbd, 63%,i (s)

Acceleration factor γ=4.0 cm/MV

50V

⋅−⋅= V

sAt

iibd

γexp%,63,

9

• All DUTs in the simulated test have the same spacing distribution.• In this example: All DUTs have the “Center” spacing.• The probability to find spacing bin i is

The “real” Weibull β

was taken as 2.0

The “experimentally observed” β is <1.7 at 50 V stress

y = 1.69Ln(x) + 0.96

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.1 1.0 10.0

Lifetime (s)

Cu

m. P

rob

ab

ility

-8

-6

-4

-2

0

2

4

ln[-

ln(1

-F)]

F vs. Lifetime at 50 V

ln[-ln(1-F)] vs. lifetime at 50 V

Linearized Weibull Plot fit

Lifetime (s)

CENTER50 VC

um fa

ilure

pro

b.

Ln[-ln(1-F)]

isp

TDDB Simulation: Within DUT variation only

( ) ( )[ ]∏ −−=i

pbdibd

istFtF 11

• Each bin i, with a constant spacing Si, has its own Weibull-ditribution Fi(tbd) with the associated tbd,I,63% , but the same β.

10

• The wafer is divided to several zones (locations), each having a different spacing-distribution and a lifetime-distribution Fzone(tbd)

• The occurrence of a tested DUT in each zone is: Pzone

• The total observed cum. failure probability would be:

Adding the DUT-to-DUT (across wafer) variations:

Simulated TDDB with two distinct spacing-distributions in two wafer area-loactions which are equally represented in the tested site map.

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Time to breakdown (s)

ln[-

ln(1

-F)]

50 V stress45 V stress40 V stress35 V stress

Note: The lifetime of DUTs from different zones are independent.

Simulated, using only the “center” and “edge” zones with equal contribution (Pzone=0.5)

( ) ( )∑ ⋅=zone

bdzonezonebd tFPtF

11

Simulated apparent TDDB

Using DUTs from three spacing-distribution zones. Simulated TDDB with three distinct spacing-distributions in three wafer area-loactions which are equally represented in the tested site map.

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Time to breakdown (s)

ln[-

ln(1

-F)]

50 V stress45 V stress40 V stress35 V stress

However, the “true” β used in this

simulation was 2.0

The “apparent” β is only ~1

Simulated data

Note: The true β is dictated by the actual material variation

12

Simulated TDDB with three distinct spacing-distributions in three wafer area-loactions which are equally represented in the tested site map.

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

1.E-03 1.E-01 1.E+01 1.E+03 1.E+05 1.E+07 1.E+09 1.E+11 1.E+13

Time to breakdown (s)

ln[-

ln(1

-F)]

50 V stress45 V stress40 V stress35 V stress1.5 V stress

At low voltage, β appears

again as 2.0

Extending the simulation to operating voltage :

Simulated data

At operating voltages, the spacing variations do not affect the lifetime uncertainty as much as at test voltages !

The “true” β value can be used for product reliability assessment

13

-5-4-3-2-10123

0.001 0.01 0.1 1 10

Time to Breakdown (s)

ln[-

ln(1

-F)]

Area x104 x103x102 x10 x1• Area scaling tests Simulations using:

( ) ( )[ ]NN tFtF −−= 11

•Center zone sites only•Used “real” β=2.0

Simulated data

Experimental extraction of the correct β

50VCenter

β1

,%63

%63 Nt

t

N

=

“Observed” β is -1/slope ≈1.93

• Extract β from tbd,63 vs. area:Slope = -0.52

-6

-5

-4

-3

-2

-1

0

1 10 100 1000 10000

Area factor

ln(

t 63

%/s

ec

)

Center50V stress

=N

14

• Area scaling tests. Simulations using:

( ) ( )[ ]NN tFtF −−= 11

•All three zones•Used “real” β=2.0

Simulated data

50VAll three zones

slope = -0.51

-5

-4

-3

-2

-1

0

1

1 100 10000

Area factor

ln(

t 63

%/s

ec

)

All zones50V stress

“Observed” β is -1/slope ≈1.96

• Extract β from tbd,63 vs. area:

=N

-5

-3

-1

1

3

0.001 0.01 0.1 1 10

Time to Breakdown (s)ln

[-ln

(1-F

)]

Area x104 x103 x102 x10 x1

Experimental extraction of the correct β

Simulated data

15

Using V-ramp to breakdown for test structures with various line-length allows more sampling:

TDDB tests are slow. A faster way to evaluate β:

100mV/170ms

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

5.E-03

6.E-03

50 60 70 80 90

Line-line spacing (nm)

Pro

bab

ilit

y to

fi

nd

sp

acin

g

0

10

20

30

40

50

60

70

Vb

d,6

3%

testi

testiiibd V

t

VtsV +

=

,0

%63,%63,,

)(ln

γ

Vs

i

ie

t∆⋅−

−

∆= γτ

1,0

Every spacing bin Si

has its own Vbd distribution with a

characteristic Vbd,63%,i

A. Berman, 1981

0.001

0.002

0.003

0.004

0.005

0.006

16

100mV/170msAll three zones

-4

-3

-2

-1

0

1

35 40 45 50 55Vbd (Volts)

ln[-

ln(1

-F)]

Area X100 X10 X1

Simulated data

( )( )

βγ ⋅=%63

%,63,

%63,

exp

exp s

Nbd

bd NV

V

-7-6-5-4-3-2-10123

35 40 45 50 55Vbd (Volts)

ln[-

ln(1

-F)]

100mV/170ms

100mV/1770ms

All three zones 4.437V

4.201V

3.982V

( ) ( )[ ]

∆

∆−=

2

1

12%63

ln

%63,%63,

ττγ

RVRVs bdbd

Simulated data

Using V-ramp to breakdown at two ramp rates, si/γ can be extracted at F=63%, and used to derive β from Vbd,63% vs. area curves.

Evaluating the true β with V-ramp tests

17

Conclusions• Simulations of lifetime distributions were performed using:

– Actual line-to-line spacing distributions in a test structure– Actual across-wafer spacing variations– Weibull statistics for each spacing bin– E-model (the most conservative option)

• The resulting “observed” accelerated test lifetime does not appear Weibull distributed.

• A Weibull-plot fit for a typical sample size shows a substantially lower “observed” β (Weibull shape parameter).

• Simulations at low (usage) voltage show that only the true β matters! (which depends only on actual material variations)

• Using the true β buys us the extra lifetime margin that we need (until we have other validated physical degradation models).

• The true β can be approximated from line/area-scaled testing.

• A faster true-β extraction can be done using two-ramp rate V-ramp to breakdown on multiple length test structures.