1. What It Models

Quantity: The lognormal distribution gives the fraction failed by time $t$ when the natural logarithm of the failure time is normally distributed, $F (t) = Φ [(ln t - ln t_{50}) / σ]$ , with the median $t_{50}$ as scale and the standard deviation $σ$ of $ln t$ as shape (NIST).
Origin: When degradation proceeds by many small multiplicative steps, the central limit theorem in the log domain gives a lognormal life; this fits corrosion, diffusion, migration, crack growth and electromigration, though it is not always correct for them (NIST).
Hazard: Unlike the Weibull, the lognormal hazard rises, reaches a maximum and then falls, so the distribution describes wear-out of a population with a spread of weaknesses rather than a single monotonic trend (background knowledge; computed in the table).
Threshold: A three-parameter form adds a waiting time before which no failure occurs; for copper electromigration it represents the time to form the smallest fatal void and improves the fit of early failures (NIST; Li et al.).
Relation to the Weibull: Small- $σ$ lognormals and large-shape Weibulls look alike, so empirical fits often cannot choose between them; the two differ mainly in the low tail (NIST; Weibull page).
Scope: This page covers the distribution, its estimation and its use in extrapolation. The Weibull alternative, the failure rate and FIT, and the temperature acceleration models have their own pages; electromigration physics is on the Metal and Interconnect page.

2. Usage

Estimating: For complete data, take the mean and standard deviation of $ln t$ as $ln t_{50}$ and $σ$ ; with censoring, plot $Φ^{- 1} (F)$ against $ln t$ or use maximum likelihood (Rovitto; NIST; Weibull page for plotting positions).
How many failures: At least two failures are needed for any estimate, about ten give reasonable estimates and fewer than five are often very inaccurate (NIST).
Extrapolating: Move the median from stress to use with the acceleration model, then to the tolerated fraction with $e^{z_{f} σ}$ , for example $z = - 3.719$ for 0.01 per cent (Rovitto; NIST).
Choosing the model: Use the lognormal for diffusion- and migration-driven wear-out such as electromigration, where it is the industry practice, and the Weibull for weakest-link breakdown; test the early-failure tail before relying on either, since a bent tail can make a two-parameter fit pessimistic (NIST; Li et al.).
Test planning: With a known $σ$ , the required use median follows from the acceptable fraction, and a lot test becomes an acceptance sampling plan (NIST).

Lognormal Functions and Estimation

Name	Formula	Related Formulae	Variables/Constants
Lognormal distribution and hazard	$F (t) = Φ (\frac{l n t - l n t _{50}}{σ}), f (t) = \frac{1}{σ t 2 π} exp [- \frac{( l n t - l n t _{50} ) ^{2}}{2 σ ^{2}}]$	$R (t) = 1 - F (t), h (t) = \frac{f ( t )}{R ( t )}$	$F (t) : c u m u l a t i v e f r a c t i o n f ai l e d b y t im e t$ $Φ : s t an d a r d n or ma l c u m u l a t i v e d i s t r ib u t i o n$ $t_{50} : m e d ian l i f e (sc a l e)$ $σ : s ha p e, t h e s t an d a r d d e v ia t i o n o f t h e na t u r a l l o g a r i t hm o f t h e f ai l u r e t im e, f (t) : d e n s i t y, R (t) : r e l iabi l i t y, h (t) : ha z a r d; N I S T : ln t i s n or ma l l y d i s t r ib u t e d w i t h m e an μ = ln t_{50} an d s t an d a r d d e v ia t i o n σ, a t h r ee - p a r am e t er f or m a dd s a w ai t in g t im e θ b e f or e w hi c h n o f ai l u r e occ u r s, s ma l l - σ l o g n or ma l s l oo k l ik e l a r g e - s ha p e W e ib u l l s, an d t h e m o d e l f o l l o w s f r o m m u l t i pl i c a t i v e d e g r a d a t i o n t h r o ug h t h e ce n t r a l l imi t t h eor e m in t h e l o g d o main, ma t c hin g cor r os i o n, d i f f u s i o n, mi g r a t i o n, cr a c k g r o w t h an d e l ec t r o mi g r a t i o n, t h o ug h i t i s n o t a l w a y s cor r ec t f or t h e m; N I S T e x am pl e : σ = 0.5, t_{50} = 20000 h, t = 5000 h g i v es f = 0.34175 \times 1 0^{- 5}, F = 0.002781 an d h = 0.3427 \times 1 0^{- 5} p er h o u r$ $(co m p u t e d : z = ln (0.25) /0.5 = - 2.773, F = 0.00278, f = 3.417 \times 1 0^{- 6} an d h = 3.427 \times 1 0^{- 6} p er h o u r, r e p r o d u c in g N I S T; t h e ha z a r d r i ses, p e ak s an d t h e n f a l l s (ba c k g r o u n d k n o w l e d g e), a t ab o u t 0.62 t_{50} f or σ = 1 an d 0.14 t_{50} f or σ = 1.5 (co m p u t e d n u m er i c a l l y))$
Mean, median and percentiles	$t_{p} = t_{50} e^{z_{p} σ}, \overset{ˉ}{t} = t_{50} e^{σ^{2} /2}$	$Var (t) = t_{50}^{2} e^{σ^{2}} (e^{σ^{2}} - 1)$	$t_{p} : t im e t o a f ai l e d f r a c t i o n p$ $z_{p} : s t an d a r d n or ma l q u an t i l e o f p (- 2.326 f or 1 p er ce n t, - 3.090 f or 0.1 p er ce n t, - 3.719 f or 0.01 p er ce n t, - 4.753 f or 1 pp m)$ $\overset{ˉ}{t} : m e an l i f e$ $V a r : v a r ian ce (N I S T)$ $(co m p u t e d : \overset{ˉ}{t} / t_{50} = 1.05, 1.13, 1.65 an d 5.1 f or σ = 0.3, 0.5, 1 an d 1.81; t_{p} / t_{50} f or 1 p er ce n t, 0.1 p er ce n t, 0.01 p er ce n t an d 1 pp m i s 0.50, 0.40, 0.33 an d 0.24 f or σ = 0.3, 0.31, 0.21, 0.16 an d 0.093 f or σ = 0.5 an d 0.098, 0.046, 0.024 an d 0.0086 f or σ = 1)$
Estimating the median and the shape	$ln \hat{t}_{50} = \frac{1}{S} \sum_{k = 1}^{S} ln T T F_{k}, \overset{σ}{^} = \frac{\sum _{k} ( l n T T F _{k} - l n t ^ _{50} ) ^{2}}{S - 1}$	$z = Φ^{- 1} (F_{j}) against ln t_{j} (probability plot, slope 1/ σ)$	$\hat{t}_{50} : es t ima t e d m e d ian$ $S : n u mb er o f f ai l u r es (co m pl e t e d a t a)$ $T T F_{k} : k - t h f ai l u r e t im e$ $\overset{σ}{^} : es t ima t e d s ha p e$ $F_{j} : pl o tt in g p os i t i o n o f t h e j - t h f ai l u r e, f or e x am pl e a m e d ian r ank (W e ib u l l p a g e)$ $Φ^{- 1} : in v er se s t an d a r d n or ma l; R o v i tt o : t h e m o m e n t s o f ln T T F g i v e M T T F (t h e m e d ian) an d σ f or e l ec t r o mi g r a t i o n t es t s; N I S T : n o es t ima t e i s p oss ib l e w i t h f e w er t han tw o f ai l u r es, a t l e a s t 10 e x a c t f ai l u r e t im es u s u a l l y g i v e r e a so nab l e es t ima t es, b e l o w 5 f ai l u r es t h e es t ima t es a r e o f t e n v er y ina cc u r a t e, an d r e a d o u t d a t a n ee d f ai l u r es in t h r ee or m or e in t er v a l s$ $(co m p u t e d : n o n e)$

Extrapolation and Test Planning

Name	Formula	Related Formulae	Variables/Constants
Lifetime extrapolation for electromigration	$t_{u se, f} = t_{50, s t r ess} (\frac{j _{s t r ess}}{j _{u se}})^{n} exp [\frac{E _{a}}{k _{B}} (\frac{1}{T _{u se}} - \frac{1}{T _{s t r ess}}) + z_{f} σ]$	$z_{f} = Φ^{- 1} (f_{ma x})$	$t_{u se, f} : u se - co n d i t i o n t im e t o t h e t o l er a t e d f ai l e d f r a c t i o n f_{ma x}$ $t_{50, s t r ess} : m e d ian l i f e in t h e t es t$ $j_{s t r ess} an d j_{u se} : c u r r e n t d e n s i t i es$ $n : c u r r e n t e x p o n e n t$ $E_{a} : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T_{u se} an d T_{s t r ess} : ab so l u t e t e m p er a t u r es$ $z_{f} : s t an d a r d n or ma l q u an t i l e o f f_{ma x}$ $σ : l o g n or ma l s ha p e (R o v i tt o, B l a c k - e q u a t i o n f or m; t h e e x cer pt r e a d p r in t s t h e t e m p er a t u r e t er m a s 1/ T_{s t r ess} - 1/ T_{o p er}, w hi c h w o u l d s h or t e n t h e l i f e a t a l o w er u se t e m p er a t u r e, so t h e p a g e u ses t h e s i g n t ha t l e n g t h e n s i t (m y r e a d in g)); R o v i tt o : f_{ma x} = 0.01 p er ce n t a s t h e e x am pl e, n i s t i e d t o t h e d o minan t f ai l u r e p ha se an d E_{a} t o t h e t r an s p or t p a t h, b o t h a r e t ak e n f r o m a cce l er a t e d t es t s an d a ss u m e d v a l i d a t u se, u se co n d i t i o n s a r e ab o u t r oo m t e m p er a t u r e an d b e l o w 5 mA / μ m^{2}, an d a l a r g er m e d ian an d a s ma l l er σ b o t h l e n g t h e n t h e l i f e; t h e m ec hani s m i s o n t h e M e t a l an d I n t er co nn ec t p a g e$ $(co m p u t e d : t_{50} = 100 h a t 300^{\circ} C, a c u r r e n t r a t i o o f 10 w i t h n = 2 an d E_{a} = 0.9 eV (a l l a ss u m e d) g i v e A F = 1.2 \times 1 0^{6} t o 105^{\circ} C, a u se m e d ian o f 13700 y e a r s an d 4500 y e a r s t o 0.01 p er ce n t a t σ = 0.3)$
Three-parameter lognormal with a threshold time	$F (t) = Φ (\frac{l n ( t - X _{0} ) - l n ( t _{50} - X _{0} )}{σ}) (t > X_{0})$	$F (t) = 0 (t \leq X_{0})$	$X_{0} : t h r es h o l d f ai l u r e t im e (N I S T c a l l s i t θ)$ $t_{50} : m e d ian$ $σ : s ha p e o f ln (t - X_{0}); L i e t a l . (C u in t er co nn ec t s) : t h e tw o - p a r am e t er l o g n or ma l f i t s w or se b ec a u se e a r l y f ai l u r es b e n d d o w n w a r d, i t s l a r g e σ g i v es u na cce pt ab l e u se - co n d i t i o n p r o j ec t i o n s, v ia an d l in e co n f i g u r a t i o n s w i t h o u t r e d u n d an cy ha v e w i d er d i s t r ib u t i o n s, t h e t h r ee - p a r am e t er f or m g r e a tl y im p r o v es t h e f i t in t h e e a r l y - f ai l u r e r e g i o n, an d X_{0} i s r e a d a s t h e t im e t o f or m t h e s ma l l es t v o i d t ha t c a u ses f ai l u r e, w i t h a l l f ai l u r es l a t er t han X_{0}$ $(co m p u t e d : n o n e; in t hi s f or m t h e s ha p e σ i s t ha t o f t - X_{0} (t h e p a r am e t er i z a t i o n i s m y r e a d in g, s in ce t h e ab s t r a c t g i v es n o n e))$
Lot acceptance with a known shape	$t_{50, u} = t_{0} exp [- σ Φ^{- 1} (p_{0})]$	$p_{a} = Φ (\frac{l n ( A F T / t _{50, u} )}{σ _{0}})$	$t_{50, u} : u se - co n d i t i o n m e d ian t ha t j u s t m ee t s t h e r e q u i r e m e n t$ $t_{0} : r e q u i r e d u se t im e (100000 h in N I S T^{'} s e x am pl e)$ $p_{0} : a cce pt ab l e f ai l e d f r a c t i o n a t t_{0}$ $Φ^{- 1} : in v er se s t an d a r d n or ma l$ $p_{a} : e x p ec t e d f ai l e d f r a c t i o n in a s t r ess t es t o f l e n g t h T$ $A F : a cce l er a t i o n f a c t or$ $σ_{0} : k n o w n s ha p e; N I S T : l o t r e l iabi l i t y v a r i es t h r o ug h t h e m e d ian f r o m l o t t o l o t, an d t h e p r o b l e m r e d u ces t o a l o t a cce pt an ce s am pl in g pl an$ $(co m p u t e d : p_{0} = 0.1 p er ce n t an d σ = 0.5 r e q u i r e t_{50, u} = 4.7 \times 1 0^{5} h, 4.7 t im es t h e 100 000 h t a r g e t)$

Important Questions

Ques: How much does sigma change a lognormal ppm life?

Very much, because the percentile is $t_{p} = t_{50} e^{z_{p} σ}$ and $z_{p}$ is large for small fractions, -4.753 for 1 ppm (computed). For $σ = 0.3$ the 1 ppm time is 0.24 of the median, for $σ = 0.5$ it is 0.093 and for $σ = 1$ only 0.0086, a factor of 28 between the narrow and the wide distribution at the same median (computed). The mean is affected in the opposite direction, $t_{50} e^{σ^{2} /2}$ , 1.05 times the median for $σ = 0.3$ but 1.65 times for $σ = 1$ , so a wide distribution can have a long mean life and a short ppm life at once. Rovitto states the practical consequence for electromigration: the life is lengthened by a larger median as well as by a smaller standard deviation. Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.6.4 ‘Lognormal’ (fetch-tool summary); M. Rovitto, ‘Electromigration Reliability Issue in Interconnects for Three-Dimensional Integration Technologies’, PhD thesis, TU Wien, Section 1.4.4 ‘Test Data Analysis and Lifetime Extrapolation’ (year not in the excerpt read) (fetch-tool summary).

Ques: Should failure data be fitted with a lognormal or a Weibull distribution?

The data often cannot decide. NIST notes that lognormals with small $σ$ look very similar to Weibulls with a large shape and that both are flexible enough that empirical fits rarely choose between them. The choice is therefore usually made from the mechanism: the lognormal follows from multiplicative degradation and is used for corrosion, diffusion, migration, crack growth and electromigration, while the Weibull follows from the weakest link and is used for dielectric breakdown and material strength (NIST; Weibull page). The two differ most in the low tail, where the Weibull usually predicts earlier failures for the same median and spread (background knowledge). When the mechanism is unclear, the more pessimistic tail is the safer choice for a reliability claim. Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.6.4 ‘Lognormal’ (fetch-tool summary); Weibull Distribution page of this vault.

Ques: How is a lognormal electromigration test turned into a use-condition life?

In two steps. First the median is moved from stress to use with Black’s equation, the current ratio to the power $n$ times an Arrhenius factor in $E_{a}$ ; then it is moved to the tolerated failed fraction with $e^{z_{f} σ}$ (Rovitto). As an example with assumed values, a median of 100 $h$ at 300 $^{\circ} C$ , a current ratio of 10, $n = 2$ and $E_{a} = 0.9$ $eV$ give an acceleration of $1.2 \times 1 0^{6}$ to 105 $^{\circ} C$ , a use median of 13 700 years and, with $σ = 0.3$ and 0.01 per cent tolerated, 4500 years (computed). Rovitto takes $n$ and $E_{a}$ from accelerated tests and assumes they remain valid at use, which is the main risk of the method. The excerpt read prints the sign of the temperature term reversed, so the sign was checked against its effect (my reading). Reference: M. Rovitto, ‘Electromigration Reliability Issue in Interconnects for Three-Dimensional Integration Technologies’, PhD thesis, TU Wien, Section 1.4.4 ‘Test Data Analysis and Lifetime Extrapolation’ (year not in the excerpt read) (fetch-tool summary); Metal and Interconnect page of this vault.

Ques: Why can a two-parameter lognormal be too pessimistic for copper interconnects?

Because the early failures of copper interconnects bend downward on a lognormal plot, and forcing a straight line through them increases $σ$ . Li et al. found that the two-parameter fit worsens for via and line configurations without redundancy, and that its large $σ$ gives use-condition projections that are unacceptable for chip reliability. A three-parameter lognormal with a threshold time $X_{0}$ greatly improved the fit in the early-failure region and gave acceptable projections. They interpret $X_{0}$ as the time needed to form the smallest void that causes failure, and all recorded failures came after it. The threshold has physical meaning here, since a void must grow to a critical size before the line can fail, so no failure is possible at very short times. Reference: B. Li, C. Christiansen, J. Gill, T. Sullivan, E. Yashchin and R. Filippi, ‘Threshold electromigration failure time and its statistics for Cu interconnects’, Journal of Applied Physics (14 December 2006), doi:10.1063/1.2397281 (abstract only); NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.6.4 ‘Lognormal’ (fetch-tool summary).

Ques: How many failures are needed to estimate a lognormal distribution?

NIST gives practical rules for lognormal and Weibull life tests. Since both models have two parameters, no estimate is possible without at least two failures; at least ten exact failure times usually give reasonable estimates, and below five failures the estimates are often very inaccurate. With readout data, failures in three or more readout intervals are needed. For planning, NIST suggests simulating test data from plausible parameter values to check whether a plan can meet its goal. When $σ$ is known from earlier lots, fewer failures suffice, because only the median must be estimated: to meet 0.1 per cent at 100 000 $h$ with $σ = 0.5$ , the use median must be at least $4.7 \times 1 0^{5}$ $h$ (computed), and a stress test then checks a lot against that value. Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.3.1.2 ‘Lognormal or Weibull tests’ (fetch-tool summary); NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.6.4 ‘Lognormal’ (fetch-tool summary).

Quartz 5

Explorer

Lognormal Distribution