# [Beowulf] looking for a reference on failure rates

Don Holmgren djholm at fnal.gov
Mon Mar 7 14:59:56 EST 2005

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On Mon, 7 Mar 2005, Joe Landman wrote:

> Hi folks:
>
>    I am looking for a reference which describes failure rates of modern
> computer components as a function of temperature.  The usual rule of
> thumb is that every 10 degrees above a certain value doubles the failure
> rate (or decreases lifetime).  I would like to look at this analysis and
> refer to it for something I am working on.
>
>    Thanks
>
> Joe
>
>

Joe -

Take a look at this Test and Measurement World article for starters:

http://www.reed-electronics.com/tmworld/article/CA187523.html

The rule of thumb that you mention comes from using an Arrhenius model
to describe the relationship between temperature and failure rates.
Arrhenius first published this equation (now named after him) in 1889

k(T)  =  A  exp ( -Ea / RT)

to explain the variation of reaction rates with temperature of several
elementary chemical reactions.   Here, k is the reaction rate, A is a
constant, Ea is the activation energy for the reaction, R is the ideal
gas constant, and T is the temperature in Kelvin.  It turns out that
many semiconductor degradation mechanisms - electromigration, corrosion,
defect growth, etc. - fit this relationship well.  Note that you'll
usually see Boltzmann's constant (another 'k') instead of 'R' in the
semiconductor reliability literature.  Chemists use R and express Ea in
units of kJoule/mole, physicists and engineers tend to use k and express
Ea in electron volts.

In the reliability literature, you'll often see the Arrhenius model
written in term of time to failure, which is proportional to the inverse
of the reaction rate.  At two different temperatures T1 and T2, the
times to failure would be given by

t1 = A exp (Ea / kT1)               # k = Boltzmann's constant
t2 = A exp (Ea / kT2)

and so the ratio of lifetimes is given by

t1/t2 = exp [ Ea/k  * (1/T1 - 1/T2) ]

If t1 is room temperature (~ 298K), an activation energy of about
0.54 eV would give a doubling in failure rate at a 10 degree C higher
temperature.

There's a handy chemist's page at

http://antoine.frostburg.edu/chem/senese/101/kinetics/faq/temperature-and-reaction-rate.shtml

that will let you plug in 3 of the 4 variables (T1, T2, Ea, reaction
rate ratio) and it will give you the third.

I've got a number of semiconductor reliability texts with tables of Ea
versus failure mechanism - I can post the references if you request,
though they're a bit dated (15 years old).  Ea varies widely in these
tables from about 0.3 eV to as high as 2.0 eV.  There are even some
negative Ea's, corresponding to failure mechanisms that decelerate with
increasing temperature.  The "factor of 2 with every 10 degrees" is only
a very rough rule of thumb.

Don Holmgren

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