1. What It Models

Quantity: These models give the time or the number of cycles to failure as a function of stresses other than steady temperature: relative humidity (Peck), voltage (inverse power and exponential laws), current density (electromigration), several stresses together, and temperature cycling (Coffin-Manson and Norris-Landzberg) (NIST; Peck; Vasudevan and Fan).
Form: Most are simplified Eyring models in which each stress contributes its own factor, multiplied by the Arrhenius temperature factor; interaction terms are dropped unless tests show them (NIST; Temperature Acceleration page).
Humidity: Peck’s model, $t_{f} \propto R H^{n} e^{E_{a} / (k_{B} T)}$ with $n \approx - 2.7$ and $E_{a} \approx 0.8$ $eV$ , was fitted to median lives of epoxy-encapsulated parts and is the basis of HAST and 85/85 correlation (Peck; Teverovsky).
Cycling: The Coffin-Manson family describes fatigue crack growth in solder and metals under repeated temperature swings, with the range, the frequency and the maximum temperature as stresses (NIST; Vasudevan and Fan).
Limits: Each model holds only within the range of its data and for one mechanism: Peck’s model for corrosion-type failures of epoxy packages below their glass transition, voltage models within the voltage range tested, and cycling constants for a given solder and package (Peck; Teverovsky; Stathis; Vasudevan and Fan).
Scope: This page covers the non-thermal acceleration laws. The Arrhenius and Eyring forms are on the Temperature Acceleration page, the mechanisms on the Environmental and Chemical, Mechanical and Package Stress and Metal and Interconnect pages, and oxide voltage models in detail on the CMOS Logic Transistors page.

2. Usage

Humidity tests: Convert HAST or 85/85 hours to use with Peck’s factor, and state the temperature, RH, $n$ and $E_{a}$ used; check that the test temperature stays below the glass transition of the encapsulant (Peck; Teverovsky).
Voltage tests: Fit both the power law and the exponential to data at several voltages and use the more conservative extrapolation, since the two can differ by orders of magnitude at use voltage (NIST; computed in the table).
Electromigration: Combine the current-density exponent with the Arrhenius term, and then the lognormal percentile (NIST; Lognormal page).
Thermal cycling: Use Norris-Landzberg constants that match the solder, the package and the strain level, and verify the factor with at least two test profiles (Vasudevan and Fan).
Combining stresses: Multiply the separate factors only if the mechanism is the same at all stress levels and no interaction is known (NIST).

Humidity Acceleration

Name	Formula	Related Formulae	Variables/Constants
Peck temperature-humidity model	$t_{f} = A R H^{n} e^{E_{a} / (k_{B} T)}, A F = (\frac{R H _{s}}{R H _{u}})^{- n} exp [\frac{E _{a}}{k _{B}} (\frac{1}{T _{u}} - \frac{1}{T _{s}})]$	$n = - 2.66, E_{a} = 0.79 eV (n = - 2.5, 0.81 eV; n = - 3.0, 0.77 eV)$	$t_{f} : m e d ian t im e t o f ai l u r e$ $A : co n s t an t$ $R H : r e l a t i v e h u mi d i t y in p er ce n t$ $n : h u mi d i t y e x p o n e n t (n e g a t i v e)$ $E_{a} : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e t e m p er a t u r e$ $A F : a cce l er a t i o n f a c t or o f s t r ess s o v er u se u; P ec k (61 m e d ian l i v es o f e p o x y p a c k a g es, u p t o a t l e a s t 140^{\circ} C an d 100 p er ce n t R H) : cor r e l a t i o n 0.986 t o 0.987 f or t h e t h r ee co mbina t i o n s, 0.81 eV f or d a t a f r o m 1979 o n, in se n s i t i v e t o n f r o m - 2.5 t o - 3.0, l o g n or ma l s p r e a d o f r es u l t s 0.4 t o 0.5, 90 p er ce n t in t er v a l o f t h e e x t r a p o l a t e d m e d ian ab o u t a f a c t or o f 1.8, v a l i d d o w n t o ab o u t 30 p er ce n t R H (l i v es b e l o w ma y b e l o n g er), t h e g l a ss t r an s i t i o n o f t h e e p o x y l imi t s t h e t es t t e m p er a t u r e (a 145^{\circ} C T_{g} p r o d u c t f e l l a l m os t t e n t im es b e l o w t h e m o d e l a t 150^{\circ} C), r es u l t s ab o v e ab o u t 150^{\circ} C a r e u n cer t ain, an d u p t o ab o u t 10 p er ce n t f r e ak f ai l u r es d i s t or t m e d ian s; P ec k^{'} s r a t i os : a l m os t 20 f or 130/85 a g ain s t 85/85, o v er 4900 f or 30^{\circ} C /20 p er ce n t an d 1877 f or 40/20 a g ain s t 85/85, 1000 h a t 85/85 e q u a l t o ab o u t 48, 28 an d 20 h a t 121, 135 an d 140^{\circ} C an d 100 p er ce n t R H; T e v er o v s k y : b es t f i t 0.8 eV an d n = 2.7, r an g es 0.7 t o 1.1 eV an d 1 t o 5$ $(co m p u t e d w i t h n = - 2.66 an d 0.79 eV : 17.4 f or 130/85 (18.7 w i t h 0.81 eV), 4880 f or 30/20 an d 1858 f or 40/20; 1000 h a t 85/85 e q u a l s 28.2 an d 21.5 h a t 135 an d 140^{\circ} C an d 100 p er ce n t R H, ma t c hin g P ec k, b u t 62.6 h a t 121^{\circ} C a g ain s t P ec k^{'} s 48 h)$
HAST and its equivalence to 85/85	$t_{85/85} = A F_{P ec k} t_{H A S T}$	$13 0^{\circ} C /85% : 96 h, 11 0^{\circ} C /85% : 264 h$	$t_{85/85} : e q u i v a l e n t t im e a t 85^{\circ} C an d 85 p er ce n t R H$ $A F_{P ec k} : P ec k a cce l er a t i o n f a c t or$ $t_{H A S T} : hi g h l y a cce l er a t e d s t r ess t es t t im e; T e v er o v s k y (N A S A) : J E S D 22 - A 110 an d A 118 C o n d i t i o n A i s 130^{\circ} C /85 p er ce n t f or 96 h an d C o n d i t i o n B 110^{\circ} C /85 p er ce n t f or 264 h, o t h er s p ec i f i c a t i o n s u se 150, 250 or 500 h, a 250 h H A S T i s s t a t e d a s e q u i v a l e n t t o 1000 h a t 85/85, 11 o f 25 l in e a r p a r t t y p es f ai l e d 250 h a t 130/85, 130^{\circ} C e x cee d s t h e o p er a t in g r an g e o f man y p a r t s, m o l d in g co m p o u n d s c an l ose u p t o ab o u t 30^{\circ} C o f T_{g}, cr ee p an d l ose a d h es i o n, s w e l l in g a f t er H A S T w a s i r r e v er s ib l e, m o i s t u r e - a ss i s t e d h o t - c a r r i er d e g r a d a t i o n ma y ha v e a n e g a t i v e a c t i v a t i o n e n er g y, an d cor r os i o n i s n o l o n g er a p r im e co n cer n; m ec hani s m s o f m o i s t u r e a r e o n t h e E n v i r o nm e n t a l an d C h e mi c a l p a g e$ $(co m p u t e d w i t h P ec k^{'} s 0.79 eV an d - 2.66 : 96 h a t 130/85 e q u a l s 1670 h an d 264 h a t 110/85 e q u a l s 1400 h a t 85/85, an d 1000 h a t 85/85 e q u a l s 57 h a t 130/85, so t h e s t a t e d 250 h f or 1000 h im pl i es an A F o f o n l y 4)$

Voltage and Current Acceleration

Name	Formula	Related Formulae	Variables/Constants
Voltage acceleration: power law and exponential	$t_{f} = A V^{- β} or t_{f} = A e^{- B V}$	$t_{f} = A e^{Δ H / (k_{B} T)} V^{- β}, t_{f} = A e^{Δ H / (k_{B} T)} e^{- B V}$	$t_{f} : t im e t o f ai l u r e$ $A : co n s t an t$ $V : v o l t a g e$ $β : v o l t a g e e x p o n e n t$ $B : e x p o n e n t ia l v o l t a g e coe f f i c i e n t$ $Δ H : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e t e m p er a t u r e; N I S T : t h e in v er se p o w er r u l e i s a v er y s im pl i f i e d E y r in g m o d e l u se d f or c a p a c i t or s, t h e e x p o n e n t ia l m o d e l f i t s so m e c a ses b e tt er, an d t h e co mbin e d f or m s a r e tw o - s t r ess E y r in g m o d e l s; S a l v a d or e t a l . : t h e d i e l ec t r i c b r e ak d o w n sc a l e f o l l o w s τ = τ_{0} e^{- γ (V - I R_{S})} (E - m o d e l); S t a t hi s : t h e t im e t o b r e ak d o w n o f u l t r a t hin o x i d es d oes n o t o b ey an y s im pl e l a w s u c h a s e x p o n e n t ia l d e p e n d e n ce o n E, 1/ E or V_{g}, an d t h e d e f ec t g e n er a t i o n p r o babi l i t y ha s a s i g m o i d a l in f l ec t i o n b e tw ee n 2 an d 3 V; o x i d e v o l t a g e m o d e l s a r e o n t h e C M O S L o g i c T r an s i s t or s p a g e$ $(co m p u t e d : i f l i v es a t 6 an d 5 V d i f f er b y t e n t im es, β = 12.6 or B = 2.30 p er v o l t; e x t r a p o l a t e d t o 1.5 V f r o m 5 V t h e p o w er l a w g i v es 4.0 \times 1 0^{6} t im es an d t h e e x p o n e n t ia l 3.2 \times 1 0^{3} t im es l o n g er l i f e, 1300 t im es a p a r t)$
Current-density acceleration of electromigration	$t_{f} = A J^{- n} e^{Δ H / (k_{B} T)}$	$A F = (\frac{J _{s}}{J _{u}})^{n} exp [\frac{Δ H}{k _{B}} (\frac{1}{T _{u}} - \frac{1}{T _{s}})]$	$t_{f} : t im e t o f ai l u r e$ $A : co n s t an t$ $J : c u r r e n t d e n s i t y$ $n : c u r r e n t e x p o n e n t$ $Δ H : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e t e m p er a t u r e$ $s an d u : s t r ess an d u se; N I S T : a m o d i f i e d E y r in g m o d e l f or o p e n f ai l u r es o f t hin - f i l m co n d u c t or s b y i o n m o v e m e n t t o w a r d t h e an o d e, w i t h Δ H t y p i c a l l y 0.5 t o 1.2 eV an d n a r o u n d 2 co mm o n; t h e l o g n or ma l e x t r a p o l a t i o n i s o n t h e L o g n or ma l D i s t r ib u t i o n p a g e an d t h e p h y s i cs o n t h e M e t a l an d I n t er co nn ec t p a g e$ $(co m p u t e d : w i t h n = 2, a t e n t im es l o w er c u r r e n t d e n s i t y g ain s a f a c t or o f 100)$
Three-stress model of temperature, voltage and humidity	$t_{f} = A e^{Δ H / (k_{B} T)} V^{- β} R H^{- γ}$	$A F = A F_{T} A F_{V} A F_{R H} (no interaction terms)$	$t_{f} : t im e t o f ai l u r e$ $A : co n s t an t$ $Δ H : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e t e m p er a t u r e$ $V : v o l t a g e$ $β : v o l t a g e e x p o n e n t$ $R H : r e l a t i v e h u mi d i t y in p er ce n t$ $γ : h u mi d i t y e x p o n e n t$ $A F_{T}, A F_{V}, A F_{R H} : se p a r a t e f a c t or s; N I S T : e x p o n e n t ia l v o l t a g e or R H t er m s a r e o b v i o u s v a r ia t i o n s, an d t h e m o d e l ha s f o u r u nk n o w n p a r am e t er s t ha t n ee d an e x t e n s i v e e x p er im e n t a l se t u p; P ec k^{'} s m o d e l i s t h e c a se w i t h o u t t h e v o l t a g e t er m$ $(co m p u t e d : n o n e)$

Thermal Cycling Acceleration

Name	Formula	Related Formulae	Variables/Constants
Coffin-Manson model for thermal cycling	$N_{f} = A f^{- α} Δ T^{- β} G (T_{ma x})$	$G (T_{ma x}) = e^{Δ H / (k_{B} T_{ma x})}$	$N_{f} : cy c l es t o f ai l u r e$ $A : co n s t an t$ $f : cy c l in g f r e q u e n cy$ $α : f r e q u e n cy e x p o n e n t$ $Δ T : t e m p er a t u r e r an g e o f a cy c l e$ $β : r an g e e x p o n e n t, G (T_{ma x}) : A r r h e ni u s t er m a t t h e ma x im u m t e m p er a t u r e T_{ma x} (ab so l u t e)$ $Δ H : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K); N I S T : u se d f or cr a c k g r o w t h in so l d er an d o t h er m e t a l s, α a r o u n d - 1/3, β a r o u n d 2 an d Δ H a r o u n d 1.25, an d l o w er cy c l in g f r e q u e n cy r e d u ces t h e cy c l es t o f ai l u r e; p a c k a g e m ec hani cs a r e o n t h e M ec hani c a l an d P a c k a g e S t r ess p a g e$ $(co m p u t e d : w i t h α = - 1/3 an d β = 2, ha l v in g Δ T m u l t i pl i es N_{f} b y 4 an d s l o w in g t h e cy c l e t e n t im es d i v i d es i t b y 2.15)$
Norris-Landzberg acceleration for solder joints	$A F = (\frac{Δ T _{t}}{Δ T _{f}})^{n} (\frac{f _{f}}{f _{t}})^{m} exp [\frac{E _{a}}{k _{B}} (\frac{1}{T _{ma x, f}} - \frac{1}{T _{ma x, t}})]$	$SnPb: n = 1.9, m = \frac{1}{3}, E_{a} / k_{B} = 1414 K; SAC (Pan): 2.65, 0.136, 2185 K$	$A F : a cce l er a t i o n f a c t or o f t h e t es t t o v er t h e f i e l d f$ $Δ T : t e m p er a t u r e r an g e$ $f : cy c l in g f r e q u e n cy$ $T_{ma x} : ab so l u t e ma x im u m t e m p er a t u r e$ $n : r an g e e x p o n e n t$ $m : f r e q u e n cy e x p o n e n t$ $E_{a} : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K); V a s u d e v an an d F an : or i g ina l S n P b co n s t an t s 1/3, 1.9 an d 1414 K, P an e t a l . f or S A C 0.136, 2.65 an d 2185 K, t h e i r p r o p ose d S A C co n s t an t s 0.33, 1.9 an d 1414 K w i t h a m e an er r or o f 5.8 p er ce n t o v er 17 d a t a se t s a g ain s t 60.66 p er ce n t f or P an e t a l ., m e a s u r e d t es t A F s o f 1.2 t o 3.7 b e tw ee n t h er ma l cy c l in g p r o f i l es, an d C - M coe f f i c i e n t s o f 2.1 t o 2.2 f or l o w - s t r ain co m p o n e n t s$ $(co m p u t e d f or - 40 t o 125^{\circ} C a t 24 cy c l es p er d a y a g ain s t 0 t o 60^{\circ} C a t o n e p er d a y (a ss u m e d) : A F = 10.2 w i t h t h e S n P b or p r o p ose d S A C co n s t an t s an d 80.9 w i t h P a n^{'} s; E_{a} / k_{B} = 1414 an d 2185 K a r e 0.122 an d 0.188 eV)$

Important Questions

Ques: Is 1000 hours at 85/85 equal to 48 or 63 hours at 121 degrees Celsius and 100 per cent RH?

Peck states about 48 $h$ , but his own model gives 62.6 $h$ with $n = - 2.66$ and 0.79 $eV$ , and 62.9 $h$ with $- 3.0$ and 0.77 $eV$ (computed). The same calculation reproduces his values at 135 and 140 $^{\circ} C$ , 28.2 and 21.5 $h$ against 28 and 20 $h$ , and his factor of almost 20 for 130/85 (17.4 computed), so the 121 $^{\circ} C$ value is the outlier. The likely reason is that the 121 $^{\circ} C$ figure came from a different calculation or measured data in the paper, which is OCR-garbled in the copy read (my reading). The model value, about 63 $h$ , is therefore the more consistent one, and it is also the more conservative test time, since a test of only 48 $h$ would apply about a quarter less stress time. Reference: D. S. Peck, ‘Comprehensive Model for Humidity Testing Correlation’, Proc. IEEE International Reliability Physics Symposium (IRPS) 1986, 44-50 (Portland State University copy; equation partly garbled, fetch-tool summary).

Ques: Is a 96 hour HAST equivalent to 1000 hours at 85/85?

By Peck’s model it is more severe. With $n = - 2.66$ and 0.79 $eV$ , 96 $h$ at 130 $^{\circ} C$ and 85 per cent RH equals about 1670 $h$ at 85/85, and the 264 $h$ of the 110 $^{\circ} C$ condition equals about 1400 $h$ (computed). Teverovsky’s NASA slides, however, state that a 250 $h$ HAST is equivalent to 1000 $h$ at 85/85, which implies an acceleration of only 4, against 17 to 19 from the model. The difference reflects caution rather than a different model: at 130 $^{\circ} C$ molding compounds can lose up to about 30 $^{\circ} C$ of glass transition, creep and swell irreversibly, so HAST can create failures that never occur at 85/85, and other mechanisms may not follow Peck at all. Peck’s factor is the more likely physical value, the NASA equivalence a conservative rule. Reference: D. S. Peck, ‘Comprehensive Model for Humidity Testing Correlation’, Proc. IEEE International Reliability Physics Symposium (IRPS) 1986, 44-50 (Portland State University copy; equation partly garbled, fetch-tool summary); A. Teverovsky (QSS Group, NASA Goddard), ‘Moisture Diffusion in Molding Compounds and Quality Assurance of PEMs for Space Applications’, MRQW workshop, December 2002 (NASA NEPP slides, fetch-tool summary).

Ques: Should voltage acceleration be extrapolated with a power law or an exponential?

The choice can change the answer by orders of magnitude, and the data at high voltage rarely decide it. NIST gives both, the inverse power rule $t_{f} \propto V^{- β}$ , used for capacitors, and the exponential $e^{- B V}$ , which fits some cases better. If lives at 6 and 5 $V$ differ by ten times, the power law has $β = 12.6$ and the exponential $B = 2.30$ per volt, and extrapolated from 5 to 1.5 $V$ they predict $4.0 \times 1 0^{6}$ and $3.2 \times 1 0^{3}$ times longer life, 1300 times apart (computed). For thin oxides the problem is worse: Stathis states that the time to breakdown obeys no simple law in field, inverse field or gate voltage. The safe practice is to test as close to use as practical and to take the shorter extrapolated life (my reading). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.5.3 ‘Other models’ (fetch-tool summary); J. H. Stathis, IBM Research, ‘Physical and Predictive Models of Ultra Thin Oxide Reliability in CMOS Devices and Circuits’ (abstract only; venue and year not given); E. Salvador, R. Rodriguez and E. Miranda, ‘SPICE simulation of the time-dependent clustering model for dielectric breakdown’, Solid-State Electronics 215, 108895 (2024) (fetch-tool summary).

Ques: Which Norris-Landzberg constants apply to lead-free SAC solder?

Vasudevan and Fan compared two sets. Pan et al. proposed $n = 2.65$ , $m = 0.136$ and $E_{a} / k_{B} = 2185$ $K$ for SAC, while Vasudevan and Fan found that 1.9, 0.33 and 1414 $K$ , almost the original SnPb values, fit 17 data sets with a mean error of 5.8 per cent against 60.66 per cent for Pan’s constants. The difference matters: for a test from -40 to 125 $^{\circ} C$ at 24 cycles per day against a field cycle of 0 to 60 $^{\circ} C$ once a day, the acceleration is 10.2 with the SnPb-like constants and 80.9 with Pan’s (computed, assumed profiles). The constants of Vasudevan and Fan are therefore the more likely ones for typical SAC assemblies, and Pan’s would overstate the field life about eight times. Low-strain components may need exponents near 2.1 to 2.2. Reference: V. Vasudevan and X. Fan, ‘An Acceleration Model for Lead-Free (SAC) Solder Joint Reliability under Thermal Cycling’, Proc. Electronic Components and Technology Conference (ECTC) 2008, 139-145 (equations garbled in the copy read, fetch-tool summary); Mechanical and Package Stress page of this vault.

Ques: Does the maximum-temperature term of cycling fatigue have an activation energy of 0.12 or 1.25 eV?

The sources differ greatly. Norris-Landzberg writes the term as $exp (1414/ T_{ma x})$ , which is 0.122 $eV$ , and Pan’s SAC constant of 2185 $K$ is 0.188 $eV$ (computed), while NIST gives about 1.25 for the activation energy of $G (T_{ma x})$ in its Coffin-Manson model without units. Between a 60 and a 125 $^{\circ} C$ maximum, 1414 $K$ gives a factor of 2.0 but 1.25 $eV$ would give about 1220 (computed), which is not consistent with the measured test factors of 1.2 to 3.7 that Vasudevan and Fan list between cycling profiles. The 0.12 $eV$ form is therefore the more likely one for solder fatigue, and the NIST value may refer to another quantity or unit (my reading). The difference shows why constants must be taken from data for the same solder. Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.5.3 ‘Other models’ (fetch-tool summary); V. Vasudevan and X. Fan, ‘An Acceleration Model for Lead-Free (SAC) Solder Joint Reliability under Thermal Cycling’, Proc. Electronic Components and Technology Conference (ECTC) 2008, 139-145 (equations garbled in the copy read, fetch-tool summary).

Ques: When does Peck’s humidity model stop being valid?

Peck fitted 61 median lives of epoxy packages up to at least 140 $^{\circ} C$ and 100 per cent RH and lists the limits himself. The glass transition of the epoxy limits the test temperature: one product with a 145 $^{\circ} C$ transition failed almost ten times sooner than the model at 150 $^{\circ} C$ , and results above about 150 $^{\circ} C$ are uncertain. Below about 30 per cent RH the model may underestimate life. Phenolic and silicone packages matched but were not fitted, so other encapsulants need their own data. Teverovsky adds that the model mainly describes corrosion, which is no longer the main concern, that moisture-assisted hot-carrier degradation may have a negative activation energy, and that molding compounds swell irreversibly after HAST. The extrapolated median carries an uncertainty of about 1.8 times at 90 per cent. Reference: D. S. Peck, ‘Comprehensive Model for Humidity Testing Correlation’, Proc. IEEE International Reliability Physics Symposium (IRPS) 1986, 44-50 (Portland State University copy; equation partly garbled, fetch-tool summary); A. Teverovsky (QSS Group, NASA Goddard), ‘Moisture Diffusion in Molding Compounds and Quality Assurance of PEMs for Space Applications’, MRQW workshop, December 2002 (NASA NEPP slides, fetch-tool summary); Environmental and Chemical page of this vault.

Quartz 5

Explorer

Humidity, Voltage and Cycling Acceleration Models