1. What It Models

Quantity: The Arrhenius model gives the time to failure as $t_{f} = A e^{Δ H / (k_{B} T)}$ , so the ratio of lives at two temperatures, the acceleration factor, depends only on the activation energy $Δ H$ (or $E_{a}$ ) and the two absolute temperatures (NIST).
Physical basis: It describes mechanisms driven by thermally activated chemical reactions, diffusion or migration, with activation energies typically from about 0.3 to 1.5 $eV$ or more depending on mechanism and materials (NIST; Vigrass).
Eyring generalization: The Eyring model adds a temperature prefactor $T^{α}$ and terms for other stresses such as voltage, current or humidity, with optional temperature interactions; with $α = 0$ and no interactions it reduces to Arrhenius times separate stress factors (NIST).
Mechanism dependence: Each failure mechanism has its own activation energy, so one value applies only to one mechanism, and a population with several mechanisms has an apparent activation energy that changes with temperature (Vigrass; TI; computed in the table).
Scope: This page covers temperature acceleration. Humidity, voltage and cycling acceleration are on the next page of this folder, the conversion of test hours to FIT on the Failure Rate page, and the distributions on the Weibull and Lognormal pages.

2. Usage

Converting a test: Multiply the stress device hours by the acceleration factor between the junction temperatures in kelvin, including the self-heating at stress and at use (NIST; Vigrass).
Choosing the activation energy: Use the measured value of the mechanism when one exists, a mechanism value such as 0.3 to 0.5 $eV$ for oxide defects or 0.6 to 0.9 $eV$ for electromigration, and 0.7 $eV$ only for an unknown mechanism (Vigrass; TI).
Several mechanisms: Assign each failure its own activation energy and sum the rates, as in the Intersil example, rather than applying one factor to all failures (Vigrass).
Measuring it: Run at least two, better three, stress temperatures and fit $ln t_{f}$ against $1/ T$ ; the slope times $k_{B}$ is the activation energy (NIST; Lognormal page for the median).
Checking the sensitivity: Between 55 and 125 $^{\circ} C$ each 0.1 $eV$ of error changes the acceleration by 1.86 times (computed), so state the activation energy with every FIT value (Failure Rate page).
Adding stresses: Use an Eyring form when voltage, current or humidity matter, and keep interaction terms only when tests show them (NIST).

Arrhenius Acceleration

Name	Formula	Related Formulae	Variables/Constants
Arrhenius time to failure and acceleration factor	$t_{f} = A e^{Δ H / (k_{B} T)}, A F = exp [\frac{Δ H}{k _{B}} (\frac{1}{T _{1}} - \frac{1}{T _{2}})]$	$A F = exp [11605 Δ H (\frac{1}{T _{1, C} + 273.16} - \frac{1}{T _{2, C} + 273.16})]$	$t_{f} : t im e t o f ai l u r e$ $A : co n s t an t t ha t c an ce l s in A F$ $Δ H : a c t i v a t i o n e n er g y in eV$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K, N I S T)$ $T : ab so l u t e t e m p er a t u r e$ $A F : a cce l er a t i o n f a c t or o f t h e hi g h er t e m p er a t u r e T_{2} o v er t h e l o w er T_{1}$ $T_{1, C} an d T_{2, C} : t h e s am e in^{\circ} C (N I S T w r i t es 273.16), 11605 = 1/ k_{B} in K / eV; N I S T : Δ H t y p i c a l l y r an g es f r o m 0.3 or 0.4 u p t o 1.5 or hi g h er, d e p e n d in g o n m ec hani s m an d ma t er ia l s, A F g r o w s e x p o n e n t ia l l y w i t h Δ H, t h e m o d e l w or k s f or m ec hani s m s d r i v e n b y c h e mi c a l r e a c t i o n s, d i f f u s i o n or mi g r a t i o n, an d f r o m 25 t o 125^{\circ} C A F i s 133 f or 0.5 eV an d 17597 f or 1.0 eV$ $(co m p u t e d : 132.7 an d 17600 f r o m 25 t o 125^{\circ} C, ma t c hin g N I S T; f r o m 125 t o 55^{\circ} C, 6.5, 78, 501 an d 3240 f or 0.3, 0.7, 1.0 an d 1.3 eV)$
Junction temperature in the acceleration factor	$T_{J} = T_{A} + Δ T_{J} = T_{A} + θ_{J A} P_{D}$	$A F = exp [\frac{E _{a}}{k _{B}} (\frac{1}{T _{J, u se}} - \frac{1}{T _{J, s t r ess}})]$	$T_{J} : j u n c t i o n t e m p er a t u r e$ $T_{A} : ambi e n t t e m p er a t u r e$ $Δ T_{J} : se l f - h e a t in g$ $θ_{J A} : j u n c t i o n - t o - ambi e n t t h er ma l r es i s t an ce$ $P_{D} : d i ss i p a t e d p o w er$ $E_{a} : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T_{J, u se} an d T_{J, s t r ess} : ab so l u t e j u n c t i o n t e m p er a t u r es; V i g r a ss (I n t er s i l) : t e m p er a t u r es m u s t in c l u d e t h e j u n c t i o n r i se, t h e e x am pl e u ses 150^{\circ} C ambi e n t pl u s 20 K (443 K) f or s t r ess an d 55^{\circ} C pl u s 20 K (348 K) f or u se, w i t h k_{B} = 8.63 \times 1 0^{- 5} eV / K; t h er ma l m o d e l s a r e o n t h e T h er ma l S t r ess p a g e$ $(co m p u t e d : w i t h 8.63 \times 1 0^{- 5} eV / K, A F = 148.2 f or 0.7 eV an d 8.52 f or 0.3 eV b e tw ee n 348 an d 443 K, ma t c hin g t h e e x am pl e; w i t h 8.617 \times 1 0^{- 5} t h e 0.7 eV f a c t or f r o m 55 t o 150^{\circ} C i s 259.2 a g ain s t 257.1, a 0.8 p er ce n t d i f f er e n ce)$
Activation energies by failure mechanism	$E_{a} : 0.3 to 0.5 (oxide, bulk Si), 0.45 (corrosion), 0.5 to 0.7 (assembly) eV$	$E_{a} : 0.6 (Al EM), 0.9 (contact EM), 0.7 (mask), 1.0 (contamination), 1.3 (charge injection) eV$	$E_{a} : a c t i v a t i o n e n er g y; V i g r a ss (I n t er s i l, T ab l e 2) : o x i d e d e f ec t s an d b u l k s i l i co n d e f ec t s 0.3 t o 0.5 eV, cor r os i o n 0.45 eV, a sse mb l y d e f ec t s 0.5 t o 0.7 eV, e l ec t r o mi g r a t i o n 0.6 eV in A l l in es an d 0.9 eV a t co n t a c t s an d v ia s, ma s k an d p h o t or es i s t d e f ec t s 0.7 eV, co n t amina t i o n 1.0 eV, c ha r g e inj ec t i o n 1.3 eV, an d d e f a u l t s o f 0.7 eV f or an u nk n o w n m ec hani s m an d 1.0 eV w h e n n o f ai l u r es a r e r ecor d e d; T I (E 2 E n o t e) : e a c h f ai l u r e m o d e ha s i t s o w n E_{a}, an d a s in g l e 0.7 eV i s a r e a so nab l y a cc u r a t e a v er a g e f or d i o d e - t y p e se mi co n d u c t or s in l i e u o f e m p i r i c a l d a t a; T I c a l c u l a t or : 16.7 F I T a t 55^{\circ} C b eco m es 69.2 F I T a t 75^{\circ} C w i t h 0.7 eV$ $(co m p u t e d : A F = 4.146 f r o m 55 t o 75^{\circ} C a t 0.7 eV, so 16.7 F I T b eco m es 69.2 F I T, ma t c hin g T I)$
Sensitivity of the acceleration to the activation energy	$\frac{A F ( E _{a} + δ )}{A F ( E _{a} )} = exp [\frac{δ}{k _{B}} (\frac{1}{T _{1}} - \frac{1}{T _{2}})]$	$\frac{\partial l n A F}{\partial E _{a}} = \frac{1}{k _{B}} (\frac{1}{T _{1}} - \frac{1}{T _{2}})$	$δ : er r or o f t h e a c t i v a t i o n e n er g y$ $T_{1} an d T_{2} : ab so l u t e u se an d s t r ess t e m p er a t u r es$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K) (d er i v e d f r o m t h e A r r h e ni u s f or m, m y d er i v a t i o n)$ $(co m p u t e d : b e tw ee n 55 an d 125^{\circ} C an er r or o f 0.1 eV c han g es A F b y 1.86 t im es, w ha t e v er t h e E_{a})$

Eyring Model and Multiple Mechanisms

Name	Formula	Related Formulae	Variables/Constants
Eyring model with temperature and other stresses	$t_{f} = A T^{α} exp [\frac{Δ H}{k _{B} T} + (B + \frac{C}{T}) S_{1} + (D + \frac{E}{T}) S_{2}]$	$α = 0, C = E = 0 \Rightarrow A F = A F_{T} A F_{S_{1}} A F_{S_{2}}$	$t_{f} : t im e t o f ai l u r e$ $A : co n s t an t$ $T : ab so l u t e t e m p er a t u r e$ $α : e x p o n e n t o f t h e t e m p er a t u r e p r e f a c t or$ $Δ H : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $S_{1} an d S_{2} : n o n - t h er ma l s t r esses (v o l t a g e, l n v o l t a g e, c u r r e n t, h u mi d i t y)$ $B an d D : t h e i r coe f f i c i e n t s$ $C an d E : t e m p er a t u r e in t er a c t i o n coe f f i c i e n t s$ $A F_{T}, A F_{S 1}, A F_{S 2} : se p a r a t e a cce l er a t i o n f a c t or s; N I S T : α = 0 mak es t h e t e m p er a t u r e t er m t h e A r r h e ni u s o n e, m os t p r a c t i c a l m o d e l s d r o p t h e in t er a c t i o n t er m s so t ha t f a c t or s m u l t i pl y, t h e m o d e l han d l es man y s t r esses an d d e g r a d a t i o n d a t a, Δ H ha s p h y s i c a l m e anin g, b u t tw o s t r esses a l r e a d y n ee d f i v e p a r am e t er s, e a c h f u r t h er s t r ess a dd s tw o, an d t h e f u n c t i o na l f or m o f t h e o t h er s t r esses m u s t b e c h ose n f or e a c h m ec hani s m$ $(co m p u t e d : w i t h α = 1, t h e p r e f a c t or a l o n e c han g es A F b e tw ee n 55 an d 125^{\circ} C b y 398.15/328.15 = 1.21, s ma l l n e x t t o t h e e x p o n e n t ia l)$
Failure rate from several mechanisms	$λ = \frac{χ _{C L; 2 r + 2}^{2}}{2 r} \sum_{i} \frac{r _{i}}{D H \cdot A F _{i}}$	$D H = \sum_{j} N_{j} t_{j}, r = \sum_{i} r_{i}$	$λ : f ai l u r e r a t e a t u se$ $χ_{C L; 2 r + 2}^{2} : c hi - s q u a r e q u an t i l e a t co n f i d e n ce C L w i t h 2 r + 2 d e g r ees o f f r ee d o m$ $r_{i} : f ai l u r es o f m ec hani s m i$ $A F_{i} : i t s a cce l er a t i o n f a c t or$ $D H : d e v i ce h o u r s a t s t r ess$ $N_{j} : u ni t s o n t es t in in t er v a l j$ $t_{j} : i t s l e n g t h$ $r : t o t a l f ai l u r es (m y r eco n s t r u c t i o n o f t h e g a r b l e d E q . 2, w hi c h r e p r o d u ces b o t h p u b l i s h e d r es u l t s); V i g r a ss (I n t er s i l) : 600 p a r t s a t 150^{\circ} C f or 3000 h, o n e p h o t or es i s t f ai l u r e a t 2000 h (0.7 eV) an d o n e o x i d e f ai l u r e a t 3000 h (0.3 eV), 1.797 \times 1 0^{6} d e v i ce h o u r s, 218 F I T a t 95 p er ce n t an d 107 F I T a t 60 p er ce n t co n f i d e n ce (χ^{2} = 12.6 an d 6.2 f or 6 d e g r ees o f f r ee d o m), M T T F 4.59 \times 1 0^{6} an d 9.35 \times 1 0^{6} h$ $(co m p u t e d : 217 an d 107 F I T, ma t c hin g; t h e o x i d e f ai l u r e co n t r ib u t es 95 p er ce n t o f t h e r a t e b ec a u se i t s A F i s 17 t im es s ma l l er; tw o m ec hani s m s w i t h e q u a l r a t es a t 125^{\circ} C an d 0.3 an d 1.0 eV g i v e a co mbin e d A F o f o n l y 12.8 t o 55^{\circ} C, an a pp a r e n t E_{a} o f 0.41 eV)$

Important Questions

Ques: Should an unknown failure mechanism use 0.7 or 1.0 eV?

The sources use both, for different cases. Vigrass recommends 0.7 $eV$ for an unknown mechanism, and the TI note calls a single 0.7 $eV$ a reasonably accurate average for diode-type semiconductors when no empirical data exist. Vigrass also lists 1.0 $eV$ as the default when no failures are recorded. The higher value is not conservative: from 125 to 55 $^{\circ} C$ it gives an acceleration of 501 against 78 at 0.7 $eV$ (computed), so the same zero-failure test claims a 6.5 times lower FIT. For a reliability claim 0.7 $eV$ is therefore the more likely and the safer default, and 1.0 $eV$ should be used only when the dominant mechanism is known to have it (my reading). Oxide defects, at 0.3 to 0.5 $eV$ , accelerate far less than either. Reference: W. J. Vigrass, Intersil (Renesas), ‘Calculation of Semiconductor Failure Rates’ (year not stated; Eq. 2 garbled in the copy read, fetch-tool summary); Texas Instruments, activation-energy note shared on the TI E2E forum (Ea.pdf; title, author and year not in the copy read) (fetch-tool summary).

Ques: Does the Boltzmann constant of $8.617 \times 1 0^{- 5}$ or $8.63 \times 1 0^{- 5}$ eV/K matter?

Only slightly. NIST uses $8.617 \times 1 0^{- 5}$ $eV$ /K and writes the factor as 11605 times the activation energy, while the Intersil note by Vigrass uses $8.63 \times 1 0^{- 5}$ $eV$ /K. For 0.7 $eV$ between 55 and 150 $^{\circ} C$ the two values give acceleration factors of 259.2 and 257.1 (computed), a difference of 0.8 per cent, far below the uncertainty of the activation energy itself, which changes the factor by 1.86 times per 0.1 $eV$ between 55 and 125 $^{\circ} C$ (computed). The CODATA value is $8.617 \times 1 0^{- 5}$ $eV$ /K (background knowledge), so it is the correct one to use, but a published result computed with $8.63 \times 1 0^{- 5}$ does not need to be recalculated. The same applies to NIST’s 273.16 against 273.15 for the conversion from Celsius. Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.5.1 ‘Arrhenius’ (fetch-tool summary); W. J. Vigrass, Intersil (Renesas), ‘Calculation of Semiconductor Failure Rates’ (year not stated; Eq. 2 garbled in the copy read, fetch-tool summary).

Ques: Why must each failure mechanism get its own activation energy?

Because a single factor misweights the failures. In the Intersil example, 600 parts ran for 3000 $h$ at a 443 $K$ junction and two failed: a photoresist defect at 0.7 $eV$ with an acceleration of 148.2 to the 348 $K$ use junction, and an oxide defect at 0.3 $eV$ with only 8.52. Summing each failure with its own factor gives 218 FIT at 95 per cent and 107 FIT at 60 per cent confidence (computed, reproducing the paper), and the oxide failure contributes 95 per cent of the rate. Applying 0.7 $eV$ to both would have hidden this. More generally, two mechanisms with equal rates at 125 $^{\circ} C$ and 0.3 and 1.0 $eV$ give a combined acceleration of only 12.8 to 55 $^{\circ} C$ , an apparent 0.41 $eV$ , so the low-energy mechanism dominates at use. Reference: W. J. Vigrass, Intersil (Renesas), ‘Calculation of Semiconductor Failure Rates’ (year not stated; Eq. 2 garbled in the copy read, fetch-tool summary); Failure Rate and the Bathtub Curve page of this vault.

Ques: When should the Eyring model be used instead of Arrhenius?

When a stress other than temperature also accelerates the failure. The Eyring model writes the life as $A T^{α} exp [Δ H / (k_{B} T) + (B + C / T) S_{1} + ...]$ , so voltage, current or humidity enter as further exponential terms, and the $C / T$ terms describe interactions in which the effect of a stress depends on temperature. NIST notes that with $α = 0$ the temperature part is the Arrhenius one, that most practical models drop the interactions so that the separate factors multiply, and that the model can also describe degradation data. The cost is the number of parameters: two stresses already need five, and each further stress adds two, so the tests must cover enough combinations. The prefactor $T^{α}$ matters little, about 1.21 times between 55 and 125 $^{\circ} C$ for $α = 1$ (computed). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1.5.2 ‘Eyring’ (fetch-tool summary).

Ques: Why must the acceleration factor use the junction temperature rather than the ambient?

Because the failure happens in the device, and self-heating raises the junction above the ambient by an amount that differs between test and use. Vigrass states that temperatures must include the junction rise, and his example adds 20 $K$ to both the 150 $^{\circ} C$ stress ambient and the 55 $^{\circ} C$ use ambient, giving 443 and 348 $K$ . If the rise differs, for example a power device that dissipates more at stress, ignoring it misstates the acceleration exponentially, since the factor depends on $1/ T_{J}$ : at 0.7 $eV$ , the factor from 75 to 55 $^{\circ} C$ alone is 4.1 (computed from the TI example). The junction temperature follows from $T_{A} + θ_{J A} P_{D}$ , with the thermal resistance from the Thermal Stress page, and must be checked under the actual bias of the test. Reference: W. J. Vigrass, Intersil (Renesas), ‘Calculation of Semiconductor Failure Rates’ (year not stated; Eq. 2 garbled in the copy read, fetch-tool summary); Texas Instruments, ‘AFR FIT for lognormal distribution’ and ‘Temperature change FIT’ calculators, ti.com reliability pages (fetch-tool summary); Thermal Stress page of this vault.

Quartz 5

Explorer

Temperature Acceleration Models (Arrhenius and Eyring)

1. What It Models

2. Usage

Arrhenius Acceleration

Eyring Model and Multiple Mechanisms

Important Questions

Ques: Should an unknown failure mechanism use 0.7 or 1.0 eV?

Ques: Does the Boltzmann constant of $8.617 \times 1 0^{- 5}$ or $8.63 \times 1 0^{- 5}$ eV/K matter?

Ques: Why must each failure mechanism get its own activation energy?

Ques: When should the Eyring model be used instead of Arrhenius?

Ques: Why must the acceleration factor use the junction temperature rather than the ambient?

Graph View

Table of Contents

Backlinks

Quartz 5

Explorer

Temperature Acceleration Models (Arrhenius and Eyring)

1. What It Models

2. Usage

Arrhenius Acceleration

Eyring Model and Multiple Mechanisms

Important Questions

Ques: Should an unknown failure mechanism use 0.7 or 1.0 eV?

Ques: Does the Boltzmann constant of 8.617×10−5 or 8.63×10−5 eV/K matter?

Ques: Why must each failure mechanism get its own activation energy?

Ques: When should the Eyring model be used instead of Arrhenius?

Ques: Why must the acceleration factor use the junction temperature rather than the ambient?

Graph View

Table of Contents

Backlinks

Ques: Does the Boltzmann constant of $8.617 \times 1 0^{- 5}$ or $8.63 \times 1 0^{- 5}$ eV/K matter?