Modeling Low-k Dielectric Breakdown to Determine Lifetime Requirements

Modeling Low-k DielectricBreakdown to DetermineLifetime RequirementsMuhammad Bashir and Linda Milor

Georgia Institute of Technology

�HISTORICALLY, THE MAJOR CAUSE of interconnect

wearout has been electromigration. Recently, be-

cause of the introduction of new materials (copper,

low-k dielectrics), the increase in the number of inter-

connect layers with smaller geometries and higher

current densities, and the concomitant increase in

on-chip temperatures, low-k dielectric breakdown

and stress migration have emerged as new sources

of interconnect wearout. This article focuses on mod-

eling one of these new wearout mechanisms: low-k

dielectric breakdown.

In the past, the lifetime of the dielectric between

interconnect lines has not been a concern, because

of the dielectric’s thickness. However, in recent

years copper and low-k interconnect systems have

become vulnerable to breakdown because of the

lower breakdown field strengths of porous low-k

materials, the susceptibility of low-k materials to

mechanical damage by chemical mechanical pol-

ishing (CMP), and the high susceptibility of low-k

materials to copper drift. These problems are com-

pounded because the supply voltage is not scaled

as aggressively as feature size, which results in

exponentially escalating electric fields among inter-

connects in each technology generation. Porosity

degrades the electrical and structural properties

of copper and low-k systems further, because of,

for example, the absorption of chemi-

cals through pores that have an open

connection to the surface.

Back-end dielectrics differ from thin

gate oxides in several ways. First, unlike

the gate oxide, the back-end dielectric

undergoes many process steps that

can potentially damage the interfaces,

which can become trap sites and assist in conduc-

tion. Second, the quality of the back-end dielectric,

which is deposited rather than thermally grown, is

far poorer, resulting in higher defect densities. Third,

back-end geometries in chips have a wide variety of

geometries that might impact chip lifetime. The ap-

propriate features to extract from a chip and the fail-

ure rates to measure from test structures are not

known. As a result, time-dependent dielectric break-

down (TDDB) of damascene structures must be

assessed as a system consisting of a dielectric, diffu-

sion barrier, cap layer (SiC, for example), and copper

interconnect. Figure 1a shows an example of this

system.

In this article, we explore how interconnect geom-

etry impacts failure rates and describe our failure-rate

models for determining the time-to-fail at small per-

centiles (the tails of the distribution) while account-

ing for the observed curvature in Weibull plots. We

use the data we obtained from back-end dielectric

breakdown lifetime measurements directly to deter-

mine the impact of line width variation. We also dis-

cuss how we use this information to determine

equivalent lifetime requirements that consider both

the impact of die-to-die line width variation and the

traditional failure rate distribution, modeled by Wei-

bull statistics.

Design for Reliability at 32 nm and Beyond

Editor’s note:

Low-k dielectric breakdown and stress migration have emerged as new

sources of wearout for on-chip interconnect. This article analyzes statistical

data from a 45-nm test chip and constructs a methodology to determine the

lifetime of low-k materials under process variations.

��Yu Cao, Arizona State University

0740-7475/09/$26.00 �c 2009 IEEE Copublished by the IEEE CS and the IEEE CASS IEEE Design & Test of Computers18

Back-end dielectric reliabilityBack-end dielectric reliability is measured with

comb test structures, as illustrated in Figure 1b. The

comb structures create a lateral stress across the dielec-

tric between the comb fingers, which are separated by

the minimum-space design rule. A voltage difference is

applied to the comb, creating a lateral electric field

through the intralayer dielectric. The current between

the comb fingers is monitored, and breakdown is

observed when the current exceeds a fixed threshold.

In collecting data for a sample of comb structures,

we order the data from shortest to longest breakdown

time. We assign each time point a probability point, P,

by partitioning the probability scale equally. Figure 2

shows an example.

The data is fit by a distribution, either the Weibull

or the lognormal distribution, in order to enable

extrapolations to lifetimes at low percentiles. Let’s

consider the Weibull distribution, as an example.

When constructing a model by fitting a Weibull distri-

bution to a dataset, two parameters are extracted: the

characteristic lifetime (62.5% probability point) Z and

the shape parameter b. The intercept of the x-axis is Z,

and the slope of the curve is b. The resulting data is

then scaled to use conditions and to the vulnerable

area corresponding to the chip.

When considering layout geometries, we note that

field enhancement occurs at the tips of the comb

structures. Prior work has indicated that failure sites

correspond with locations where there is field en-

hancement.2,3 To address this concern, we have

designed a set of test structures that can separate

the impact of field enhancement at the comb tips.

Analysis of data from these test structures has

required developing a methodology to determine fail-

ure rates for structures containing multiple feature

geometries��that is, parallel lines and tips.

The result of data analysis is the construction of an

accurate model to determine failure rates at low per-

centiles. As Figure 2 shows, the data points do not fall

on a straight line, as expected for Weibull statistics.

Chen et al. noted that line width variation can be

as large as �30%.4 This variation distorts the Weibull

curves used to determine a structure’s lifetime. As a

result, direct extraction of parameters Z and bwould be inaccurate.

Die-to-die line width variation can be eliminated

during Weibull parameter extraction through calibra-

tion of lifetime measurements based on capacitance

measurements, which can be used to compute the

mean distance between the lines of the comb struc-

ture.4 However, because of the structure’s complexity,

capacitance is also affected by variation in the low-k

dielectric constant as a function of the stack’s compo-

sition, and variation in the dielectric constant is con-

sequently confounded with variation in line space.

Hence, capacitance measurements overestimate

line width variation.

Impact of field enhancementTo determine the impact of field enhancement, we

conducted several experiments. Here, we discuss

(a) (b)

BarrierTEOS

SiC

Cu Cu

SiONPSG

SiC

Dielectric Dielectric

Figure 1. Cross-section of an example copper and low-k

interconnect system, including the barrier layer and cap

layer (a), and comb test structure used to measure back-end

dielectric breakdown (b).

–6

–5

–4

–3

–2

–1

0

1

2

3

4 6 8 10 12

ln(time) [a.u.]

ln(–

ln(1

–P))

1X 3X 4.5X 9X

Figure 2. Example Weibull plot of ln(2ln(1 2 P)) vs.

ln(time-to-failure(i)) for comb test structures with four

areas: 13, 33, 4.53, and 93. (Source: Bashir and Milor.1)

19November/December 2009

previous data acquired from some of our earlier

work, which indicated the potential impact of field

enhancement at the tips of combs on low-k dielectric

lifetimes.

Role of field enhancement in breakdown

It is generally assumed that the vulnerable area in

the comb test structure is solely a function of the area

and distance between the comb fingers. However, the

electric field between the comb fingers is nonuni-

form, and there is significant field enhancement at

the comb tips.

To analyze back-end geometries, a set of circuits

were synthesized using standard placement and rout-

ing tools from which we extracted the most frequent

patterns in the back-end geometries. Besides parallel

lines, several patterns were frequently found,2 includ-

ing terminating lines and lines with bends.

For each of these structures, a 3D finite element

was constructed to determine the electric field distri-

bution. The field was found to be highly nonuniform,

with peak fields from some geometries exceeding

that of the parallel line structure by a factor of 2

to 3.2 High fields could also be found at the top

(cap layer) and bottom interfaces for all structures

because of the sharp edges.2 These high fields at

the top, cap layer interface are especially problematic

because this interface is formed by CMP. This is a low-

quality interface, which contains many dangling

bonds, facilitating copper ion drift or the formation

of a percolation path.

Note that the test structure in Figure 1b has two

types of geometry: parallel lines with a minimum dis-

tance between them, and tips. Field enhancement

can potentially occur at the tips. Our experimental

results for 0.18-micron technology, involving stressing

a set of industrial comb structures made with copper

and low-k materials, indicated the potential role of

field enhancement, because even though the test

structures have long parallel lines with minimum

space, most failure sites coincided with peaks in elec-

tric field at the tips in the structure. Examples, avail-

able elsewhere,2 are similar to results achieved by

Chen et al.,3 in which failure sites also coincided

with electric field enhancement.

Test structures and data collection

In the work we report on here, we designed test

structures to isolate the failure rates of parallel lines

and tips. To isolate the rates, pairs of test structures

are needed to extract each target feature. For extrac-

tion of area and tips, one of the test structure pairs

holds area constant and varies tips, whereas another

pair holds the number of tips constant and varies

area. The smallest test structure with unit area and

tips is labeled as (1�, 1�).

Other test structures have area and tips that are a

multiple of the unit test structure. A test structure

with M times more area and N times more tips is la-

beled as (M�, N�). We used a matrix of test struc-

tures that include the following area and tip pairs:

(1�, 1�), (3�, 1�), (9�, 1�), (3�, 3�), (4.5�, 9�),

and (9�, 9�).

Each die contained two copies of the (1�, 1�)

structure and single copies of the other test structures.

Wafers contained 206 dies. In this study, 30 dies were

randomly selected for testing among the 206 dies. All

tests were performed on a single wafer.

Data analysis methodology

Suppose we have two independent failure mecha-

nisms, each associated with a defect density, �1(t)

and �2(t), which are functions of time. Each of

these defect densities is associated with feature

areas, A1 and A2. In the case of tips, the feature

area would be the number of tips. Using the Poisson

model, the probability of survival for a structure con-

taining feature i is R(t) ¼ exp(��i(t)Ai). If the failure

mechanisms are independent, then the joint probabil-

ity of failure, P(t) ¼ 1�R(t), is

PðtÞ ¼ 1� expð�l1ðtÞA1Þexpð�l2ðtÞA2Þ (1)

The standard statistical function to character-

ize reliability failure distributions is the Weibull

distribution:

PðtÞ ¼ 1� expð�ðt=hbÞÞ (2)

To characterize a failure rate distribution, a set of

time-to-failures is measured from a test structure. We

order them from smallest to largest and assign proba-

bility points to them by partitioning the probability

scale into sets of equal probabilities. Then, we can

plot ln(t) versus ln(�ln(1�P(t))), because Equation 2

can be rewritten as

lnðtÞ ¼ lnðhÞ þ ðl=bÞlnð�lnð1� PðtÞÞÞ (3)

The Weibull distribution parameters are extracted

by finding the slope b and the ln(t)�axis intercept

at the 62.5% probability point, ln(Z).


20 IEEE Design & Test of Computers

If we have multiple failure mechanisms, then by

rearranging Equation 1 we have

lnð�lnð1� PðtÞÞÞ ¼X

i

liðtÞAi

!(4)

Hence, we can see that the Weibull distribution

plot also gives an indication of the probability distri-

bution function for the number of defects at break-

down, �iAi.

Suppose two test structures contain all the same

features, except that in one structure there is 3� of

a target feature (area) versus 1� in the other struc-

ture. The difference between these two structures is

2� area. The number of defects in the 2� area is

l2�ðtÞA2� ¼ l3�ðtÞA3� � l1�ðtÞA1� (5)

We can, therefore, extract values for �2�(t)A2�

using the Weibull distribution plots for the 3� and

1� test structures. Using Equation 4, at any time t*,

we have

lnð�lnð1� P2�ðt�ÞÞÞ ¼ lnðlnð1� P1�ðt�ÞÞ� lnð1� P3�ðt�ÞÞÞ (6)

where the values for P1�(t*) and P3�(t*) are known

from measured data from the 1� and 3� test

structures. From Equation 6, we solve for P2�(t*).

Impact of field enhancement at tips

The test structures with 1� tips were used to ex-

tract the failure rate due to parallel lines with the min-

imum distance between the lines. The analysis

methodology subtracted the impact of 1� tips from

the measured results for these test structures, in accor-

dance with Equation 6.

The measurement data related to area indicated a

strong impact of area. We synthesized data sets for 2�of the target feature (area) and no tips using 1� and

3� test structures, for 6� area and no tips using 9�and 3� test structures, and for 8� area and no tips

using 9� and 1� test structures.

The data sets were merged so that they all corre-

sponded to 1� area. This was done with a transforma-

tion of the probability scale before extracting the

model,5,6 by plotting ln(�ln(1� P(t))/N) versus

ln(t), where N ¼ 2, N ¼ 6, and N ¼ 8, for the 2�,

6�, and 8� area data sets, respectively. This transfor-

mation of the probability scale is based on the Pois-

son model, which can show that if a test structure

has an area that is N times larger, the corresponding

plot for the larger area structure is the following:

lnðtÞ ¼ ln hþ 1

bln � 1

Nlnð1� PðtÞÞ

� �(7)

When the data sets are merged, the resulting distri-

butions fall on a straight line, as Figure 3 shows. We

denote the extracted model with the Weibull charac-

teristic lifetime Z* and the Weibull shape parameter

b*, to get the equation for 1� area:

lnðtÞ ¼ lnðh�Þ þ ð1=b�Þlnð�lnð1�PðtÞÞÞ (8)

The data on tips is inconclusive. A comparison of

the 3� area structures (which vary the number of

tips) indicated an impact of tips, whereas a compari-

son of the 9� area structures (which also vary the

number of tips) showed no impact of tips. The com-

plete data set must be considered to determine the

impact of tips.

We use the model in Equation 8 to synthesize data

sets that correspond to 9� area with varying numbers

of tips. To do this, we add defect densities corre-

sponding to the appropriate area correction. In

other words, for the 1� model, we add defect den-

sities corresponding to 8� area. If Z* and b* are

the parameters corresponding to the 1� area

model, and if P(ti) is the probability point corre-

sponding to breakdown time ti, then the probability

–6

–5

–4

–3

–2

–1

0

1

2

4 6 8 10 12ln(time) [a.u.]

ln(–

ln(1

–P))

2X 6X 8X Model

Figure 3. Merged data set for 23, 63, and 83 area, with the

probability scale modified to correspond to 13 area. The plots

are for In(Sili(t)Ai), but labeled as ln(2ln(12P)), because they

are equivalent, according to Equation 4. (Source: Bashir and

Milor.1)


point P0(ti), corresponding to 9� area at breakdown

time ti, is

lnð�lnð1� P 0ðtiÞÞÞ ¼ lnð�lnð1� PðtiÞÞþ 8ðexpðb�ðlnðtiÞ � lnðh�ÞÞÞÞÞ (9)

The number of defects at failure for 8� area is

8(exp(b*(ln(ti) � ln(Z*)))). Similarly, for 4.5� area,

we add defects corresponding to 4.5� area. The syn-

thesized Weibull curves obtained after converting all

data sets to 9� area show no impact of tips. Variation

among data sets appears to be random.

Failure rate modelingHere, we discuss our results in modeling the fail-

ure rate in the presence of die-to-die line width varia-

tion. We first extracted the Weibull shape parameter

via area scaling, and then we extracted the impact

of die-to-die line width variation via the slope of the

Weibull curve.

Extracting the Weibull shape parameter

In this study, we’ve used test structures with four

areas: 1�, 3�, 4.5�, and 9�, implemented with

45-nm technology. For this set of test structures,

the distance between the lines of the comb,

which determines the applied electric field through

the dielectric, is fixed. In addition, we’ve also used

a test structure with a smaller distance between the

lines with 1� area.

Figure 2 shows the data for the four different areas.

Unlike what is assumed with the Weibull distribution

model, the failure rate distributions aren’t linear.

Through simulations, we observed that random varia-

tion in line width creates curvature in the failure rate

distributions. It also degrades the measured Weibull

shape parameter b, although the characteristic life-

time (x-intercept) is less affected. Modeling required

that we determine the Weibull shape parameter

in the absence of line width variation.

The standard approach to determine the charac-

teristic lifetime Z and the Weibull shape parameter

b is to transform the probability scales of all of the

data sets by plotting ln(�ln(1�P(t))/N) versus

ln(t), where N is the area ratio, and then to fit a

straight line to the data by regression, as Figure 4a

shows. However, curvature could lead to erroneous

extraction of the Weibull parameters. The extracted

Weibull parameters are used to find the lifetime at

a low probability point, such as the 0.0001 point.

This is well beyond the data points in the data

set, since ln(�ln(1� 0.0001))¼�9.21, as Figure 4a

shows.

Errors in the extraction of slope can create sig-

nificant errors in the estimated lifetime at low prob-

abilities. Consider a data set with a true Weibull

(a)

(b)

–7

–6

–5

–4

–3

–2

–1

0

1

2

3

4 5 6 7 8 9 10 1211ln(time) [a.u.]

ln(–

ln(1

–P))

1X 3X 4.5X

9X Model

Upper bound

Lower bound

–15

–13

–11

–9

–7

–5

–3

–1

–2 0 2 4 6ln(time) [a.u.]

ln(–

ln(1

–P))

P = 0.0001Ideal case1X 3X4.5X 9X

Model

Figure 4. Model for 13 area for the merged data sets,

combined with 90% confidence bounds (a); and simulated

Weibull distributions with and without die-to-die line width

variation, merged in accordance with Equation 7 and modeled

with a straight line (b). The ‘‘ideal case’’ corresponds to the

distribution without die-to-die line width variation.



distribution. If we introduce variation in the die-to-

die line width with a standard deviation of 10%

and merge the data sets via Equation 7, we achieve

the result shown in Figure 4b, together with the

extracted model. Die-to-die line width variation cre-

ates at least an order of magnitude of error in the

expected lifetime at the 0.0001 probability point.

Instead, as in the research reported by Chen et al.,4

we use the area-scaling formula to extract the Weibull

shape parameter. The relationship between character-

istic lifetimes for structures with different areas is

given by

lnðhN�Þ ¼ lnðh1�Þ þ ð1=bÞlnðA1�=AN�Þ (10)

where ZN� and Z1� are characteristic lifetimes for the

N� and 1� area test structures, and AN�/A1� ¼ N is

the area ratio between the structures.

To extract the characteristic lifetime, we fit a qua-

dratic model to each data set to find the x-intercept.

Fitting a regression line to the characteristic lifetime

data points determines the Weibull shape parameter

b, with less degradation due to die-to-die line width

variation. However, the extracted models do not

match well with the original data, because of die-to-

die line width variation.

Extracting die-to-die line width variation

The impact of random die-to-die line width varia-

tion is to degrade the slope of the Weibull curves.

We can therefore determine die-to-die line width vari-

ation by extracting the difference in slope between band the data.

The slope depends on die-to-die variation and

whether we assume an E model7,8 or affiffiffiEp

model,9-11

where E is the electric field between the lines. The elec-

tric field is a function of line spacing. To determine the

impact of line spacing on electric field, we use the data

set with a different distance between the lines. With the

E model,

lnðhÞ ¼ a1 �b1

s(11)

where s is the distance between the lines, and a1

and b1 are arbitrary fitting constants. With theffiffiffiEp

model,

lnðhÞ ¼ a2 �b2ffiffiffi

sp (12)

where a2 and b2 are arbitrary fitting constants. With

two values for s, we determine values of the

constants. If we were to plot the relationship

between the characteristic lifetime and distance be-

tween the lines, these models would be indistinguish-

able for the range of data available from the two test

structures.

In this analysis, we’ve assumed normally distrib-

uted die-to-die line width variation and optimized

the fit between the data and the model, given the

value of b as we determined earlier, to find the stan-

dard deviation of die-to-die variation. Figure 5a shows

the results for theffiffiffiEp

model. Note that the models re-

semble the data sets. The result for the E model is

similar.

Figure 5b shows the confidence bounds for the 1�model. As we can see, the confidence bounds are

similar in size to those in Figure 4a. The variation is

dominated by accuracy in extraction of the Weibull

parameters with Equation 10. We also found that

the best fit model was essentially independent of

using either the E orffiffiffiEp

models.

In fitting the models in Equations 11 and 12, we

must consider any print bias, because the drawn

distance between the lines in the layout might not

match the printed distance exactly. Print bias is

the difference between the distance between the

lines in the layout and those in silicon. The manu-

facturer selects it to minimize interconnect delays.

For our test structures, we didn’t know the exact

values for print bias. So, we considered several val-

ues of print bias and found that for each value of

bias the optimum fit resulted in the same value

for the standard deviation for the distance between

the lines.

Lifetime and probability pointsThe traditional approach to determine lifetime for

a test structure (after area scaling and projection to

use conditions) for a fixed probability point, such

as 0.0001, involves analyzing lifetime at a single

value of distance between the lines. For example,

for the probability of 0.0001, which corresponds to

�9.21 on the Weibull scale, we simply find the life-

time that corresponds to this probability point. This

method involves solving a 1D problem and is illus-

trated in Figure 4b.

The introduction of random die-to-die line width

variation impacts determination of the expected life-

time at the target probability point. Random die-to-

die line width variation introduces a second dimen-

sion into the problem, because there are now two


simultaneous mechanisms that can degrade life-

time. Specifically, if we consider variation in line

width, the characteristic lifetime varies in accor-

dance with Equations 11 and 12. Figure 6a illustrates

probability versus distance between the lines for

several fixed values of lifetime. We can see that

the probability of having a lifetime worse than a

fixed value increases drastically as the distance be-

tween the lines decreases.

When the distance between lines varies, the prob-

ability of having a lifetime worse than a specified

value involves integrating the probabilities in

Figure 6a over the distribution of values of distance

between the lines. For a normal distribution of dis-

tance, the integral is a function of the variation’s stan-

dard deviation. Figure 6b shows the impact of

variation on lifetime. Clearly, when line width varia-

tion approaches a standard deviation of 10%, life-

times improved by more than an order of

magnitude are required to achieve the same probabil-

ity of failure. In addition, the results are almost insen-

sitive to the choice of model, E orffiffiffiEp

at test

conditions; however, the choice of model does im-

pact the projection to use conditions. Note that the

expected lifetime depends on two parameters: print

bias and random die-to-die variation in distance.

Extracted die-to-die variation in distance between

the lines was found to be insensitive to bias. However,

lifetime projections at small percentiles are sensitive

to print bias. Hence, it’s important to verify print

bias with additional data prior to making lifetime

projections.

THIS RESEARCH has looked at some interconnect geo-

metries to determine their potential impact on failure

rates. Although no significant impact was found to

have resulted from field enhancement at the comb

tips, future work will look into a larger set of geomet-

ric features, such as line width, vias, and bends, to im-

prove our understanding of the role of interconnect

geometry on time-to-failure.

Failure rates vary both as a function of Weibull sta-

tistics and as a function of die-to-die line width varia-

tion. Determination of whether or not a process

satisfies a lifetime requirement should take into ac-

count both factors. As variation becomes large, the

lifetimes that achieve the same target probability of

failure are orders of magnitude lower than without

die-to-die line width variation. �

AcknowledgmentsWe thank the Semiconductor Research Corpora-

tion for financial support, under task 1376.001, and


(a)

(b)

–6

–5

–4

–3

–2

–1

0

1

2

3

4 6 8 10 12ln(time) [a.u.]

ln(–

ln(1

–P))

1X 3X4.5X 9X1X Model3X Model4.5X Model9X Model

–7

–6

–5

–4

–3

–2

–1

0

1

2

3

4 5 6 7 8 9 10 11 12

ln(time) [a.u.]

ln(–

ln(1

–P))

Data Model

Upper bound

Lower bound

Figure 5. Models for the data sets using the area-scaling

formula and adding die-to-die variation with theffiffiffiffiEp

model (a),

and the extracted model for the 13 data set and the 90%

confidence bounds (b). (Source: Bashir and Milor.1)


AMD for providing the wafers used to collect the data

in this study. We also thank Changsoo Hong and Soh-

rab Aftabjahani for designing and laying out the test

structures.

�References1. Reprinted from Microelectronics Reliability, vol. 49,

nos. 9-11, M. Bashir and L. Milor, ‘‘A Methodology to

Extract Failure Rates for Low-k Dielectric Breakdown

with Multiple Geometries and in the Presence of Die-to-

Die Linewidth Variation,’’ pp. 1096-1102, �c 2009, with

permission from Elsevier.

2. C. Hong, L. Milor, and M.Z. Lin, ‘‘Analysis of the Layout

Impact on Electric Fields in Interconnect Structures Using

Finite Element Method,’’ Microelectronics and Reliability,

vol. 44, Sept.�Nov. 2004, pp. 1867-1871.

3. F. Chen et al., ‘‘Reliability Characterization of BEOL Ver-

tical Natural Capacitor Using Copper and Low-k SiCOH

Dielectric for 65nm RF and Mixed-Signal Applications,’’

Proc. IEEE Int’l Reliability Physics Symp., IEEE Press,

2006, pp. 490-495.

4. F. Chen et al., ‘‘The Effect of Metal Area and Line Spac-

ing on TDDB Characteristics of 45nm Low-k SiCOH

Dielectrics,’’ Proc. IEEE Int’l Reliability Physics Symp.,

IEEE Press, 2007, pp. 382-389.

5. E.Y. Wu and R.P. Vollertsen, ‘‘On the Weibull Shape

Factor of Intrinsic Breakdown of Dielectric Films and

Its Accurate Experimental Determination��Part 1:

Theory, Methodology, Experimental Techniques,’’

IEEE Trans. Electron Devices, vol. 49, no. 12, 2002,

pp. 2131-2140.

6. E.Y. Wu et al., ‘‘Challenges for Accurate Reliability

Projections in the Ultra-thin Oxide Regime,’’ Proc. IEEE

Int’l Reliability Physics Symp., IEEE Press, 1999,

pp. 57-65.

7. E.T. Ogawa et al., ‘‘Leakage, Breakdown, and

TDDB Characteristics of Porous Low-k Silica-

Based Interconnect Dielectrics,’’ Proc. IEEE

Int’l Reliability Physics Symp., IEEE Press, 2003,

pp. 166-171.

8. G.S. Haase, E.T. Ogawa, and J.W. McPherson, ‘‘Reli-

ability Analysis Method for Low-k Interconnect Dielectrics

Breakdown in Integrated Circuits,’’ J. Applied Physics,

vol. 98, no. 3, 2005, article 34503.

9. F. Chen et al., ‘‘A Comprehensive Study of Low-k

SiCOH TDDB Phenomena and Its Reliability Lifetime

Model Development,’’ Proc. IEEE Int’l Reliability Physics

Symp., IEEE Press, 2006, pp. 46-53.

10. F. Chen et al., ‘‘Investigation of CVD SiCOH Low-k

Time-Dependent Dielectric Breakdown at 65nm Node

Technology,’’ Proc. IEEE Int’l Reliability Physics Symp.,

IEEE Press, 2005, pp. 501-507.

11. N. Suzumura et al., ‘‘A New TDDB Degradation Model

Based on Cu Ion Drift in Cu Interconnect Dielectrics,’’

Proc. IEEE Int’l Reliability Physics Symp., IEEE Press,

2006, pp. 484-489.

–15

–13

–11

–9

–7

–5

–3

180 200 220 240 260 280 300 320 340 360

Distance(a)

ln(–

ln(1

–P))

Lifetime 1 Lifetime 2 Lifetime 3

(b)

ln(time) [a.u.]

ln(–

ln(1

–P))

–16

–14

–12

–10

–8

–6

–4

–3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2

0 0.050.1 0.150.2 0.25

Limit

Figure 6. Probabilities of having a lifetime not as bad as a

fixed value (labeled as lifetimes 1 to 3) as a function of

distance between the lines for theffiffiffiffiEp

model, where the fixed

lifetime requirement is longest for lifetime 1 and shortest for

lifetime 3 (a). Probability of failure as a function of lifetime for

theffiffiffiffiEp

model, for various values of the percentage of variation

of line width. Line width variation has been assumed to be

normally distributed (b). (Source of Figure 6b: Bashir and Milor.1)


Muhammad Bashir is a doctoral candidate in elec-

trical and computer engineering at the Georgia Insti-

tute of Technology. His research interests include

modeling yield and reliability of semiconductors. He

has an MS in electrical and computer engineering

from the Georgia Institute of Technology.

Linda Milor is an associate professor of electrical

and computer engineering at the Georgia Institute of

Technology. Her research interests include yield and

reliability modeling, testing, and design-for-testability

of analog and digital circuits. She has a PhD in electri-

cal engineering from the University of California,

Berkeley.

�Direct questions and comments about this article to

Linda Milor, School of Electrical and Computer

Engineering, Georgia Institute of Technology, Atlanta,

GA 30332; [email protected].



Modeling Low-k Dielectric Breakdown to Determine Lifetime Requirements

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