Reliability Models for Failure Mechanisms  

Failure Mechanism Reliability Modeling, or reliability modeling, or acceleration modeling, or simply modeling, is the mathematical representation of a failure mechanism in terms of a set of algebraic or differential equations from the perspective of its reliability implications. The term failure mechanism refers to the actual physical phenomenon behind a failure occurrence. Modeling is a means of determining and understanding the different variables or factors that bring out and accelerate a failure mechanism. 

Being able to model a mechanism and quantify how it is affected by various environmental factors will allow a reliability engineer to develop appropriate reliability tests for estimating field failure rates and predicting when failures will begin to occur. Modeling is often expressed in the form of time to failure, or tf, or the acceleration factor, AF.

The Arrhenius Equation

Everything in this universe will decay or degrade with time, and the Second Law of Thermodynamics is there to make sure of this. Destruction or degradation of matter is generally due to atomic or molecular changes accelerated by external factors, one of which is temperature. The response dependence of degradation or failure mechanisms on temperature is given by the Arrhenius equation:

R = Ae^(-Ea/kT)

where    R=reaction rate, A=constant, Ea=activation energy,

k= Boltzmann’s constant (8.6e-5 eV/K), T=absolute temperature              

For any given reaction obeying the Arrhenius equation, R1t1=R2t2=constant, where R is the reaction rate and t is the elapsed reaction time. To illustrate this, consider a reaction process that occurs at a high temperature T1 and low temperature T2.  Since temperature increases the reaction rate, then R1 is faster than R2, or R1 > R2 .  However, the reaction process also takes a shorter duration at T1, or  t1 < t2, such that  R1t1=R2t2 =constant.

Now, let tf=time to failure, then Rtf =constant, or tf=C1/R.

Thus, tf = C1/(Ae^(-Ea/kT)) = (C)(e^(Ea/kT)).

Let the acceleration factor AF be the ratio tf_use / tf_test .

Thus, AF=[(C)(e^(Ea/kTuse)) / (C)(e^(Ea/kTtest))]= e^{(Ea/k) (1/Tuse-1/Ttest)}

Estimating Ea and tf using Arrhenius Plots

Recall that tf = (C)(e^(Ea/kT)).  Then,  ln(tf) =  lnC + Ea/kT.

Thus, the plot of ln(tf) vs. 1/T yields a straight line whose slope

corresponds to Ea/k.

Electromigration 

Electromigration is the movement of metal atoms of a metal line in the direction of the current flow through that metal line. This mechanism is similar to pebbles in a stream, which are picked up and transported by the water in the direction of the water currents.  As such, during electromigration, metal atoms are removed from the starting end of the metal line and accumulates at the other end, forming voids at the entrance and hillocks at the exit of the metal line.  Thus, electromigration can result in open circuits (due to the voids) or line-to-line short circuits (due to the hillocks).  

Electromigration is accelerated by temperature and current density, and is modeled as follows:

tf = CJ^-ne^(Ea/kT)                                                                           

AF = tf_use / tf_test

AF = (J_test/J_use)ⁿ e^{(Ea/k) (1/Tuse-1/Ttest)}       

where:    

C = a constant based on metal line properties                      

n = integer constant from 1 to 7                                                  

Tuse, Ttest = temperature during use and under test, respectively

Juse, Jtest = current density during use and under test, respectively

Ea = 0.5 - 0.7 eV for pure Al