 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 tfuse / tftest .

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 = tfuse / tftest

AF = (Jtest/Juse)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

Corrosion

Corrosion is metal degradation due to chemical or electrolytic reactions in the presence of moisture, contaminants, and bias.

Corrosion rate is a function of temperature (T), relative humidity (RH), and bias (V).

Let AF = tfuse / tftest

and

tf   = C(RH)-3e(0.9/kT).

With no applied voltage:

AF = (RHtest/RHuse)3 e(0.9/k) (1/Tuse-1/Ttest)

With voltage V applied:

AF = (V) (RHtest/RHuse)3 e(0.9/k) (1/Tuse-1/Ttest)

where:

C = a constant

RHuse, RHtest = relative humidity during use and under test, respectively

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

Time-dependent Dielectric Breakdown (TDDB)

Time-dependent Dielectric Breakdown, or TDDB, is the destruction of dielectric layers occurring over time.

R = A1e(-Ea/kT+CV)

AF = tfuse / tftest  = Rtest /Ruse

AF =  e([-Ea/k] [1/Ttest-1/Tuse] + C [Vtest-Vuse])

where:   A1,  C = constants

Ea = 0.8 - 0.9 eV

Vuse, Vtest = voltage applied during use and under test, respectively

Hot Carrier Effects

Hot carrier effects is a phenomenon involving the injection of highly energetic carriers into the gate oxide layer and the silicon substrate, resulting in volume charge build-up that can shift transistor threshold voltages.  This mechanism is accelerated by low temperatures.

AF = tfuse / tftest

AF = e([Ea/k] [1/Tuse-1/Ttest] + C [Vtest-Vuse])

where:

V = voltage accelerating the carriers

Ea = -0.2 eV to -0.06 eV

C = constant

Bond/Solderability Failures

Bond/solderability failures related to intermetallic growths, e.g., ball lifting due to Kirkendall voids, Cu-Sn intermetallic growths towards the leadfinish surface, etc. are modeled as follows.

tf   = Ae(Ea/kT)

AF = tfuse / tftest

AF = e(Ea/k) (1/Tuse-1/Ttest)

where:

A = constant

Ea = 1 eV for Au-Al bonds

Ea = 0.5-0.75 eV for Sn-based leadfinish

TC-induced Package Cracking

The occurrence of fracture anywhere in the package after it has undergone several temperature cycles has also been modeled. Since the zero-stress condition of the package is at a high temperature (around 175 deg C) , the low temperature (cooling) cycle has the main effect on this mechanism.

AF = (DTaccel/DTuse)m

where:

DTaccel = Tmin(accel) - Tneutral

DTuse = Tmin(use) - Tneutral

Tneutral = zero stress temperature (approx. 175 deg C)

m = 20 (fracture property-dependent)

Fatigue Failures

Fatigue failures are failures due to application of cyclical stresses.

AF = (DTaccel/DTuse)n

Nf = C(DT)-n

where:

Nf = cycles to failure

DT = temperature difference

n = temperature difference factor