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 (8.6e5 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/Tuse1/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 linetoline short circuits (due to the hillocks).
Electromigration is accelerated by
temperature and current density, and is modeled as follows:
tf =
CJ^{n}e^{(Ea/kT)
}
AF = tf_{use }/ tf_{test}
AF = (J_{test}/J_{use})^{n
}e^{(Ea/k)
(1/Tuse1/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
= tf_{use }/ tf_{test}
and
tf
= C(RH)^{3}e^{(0.9/kT)}.
With no applied
voltage:
AF = (RH_{test}/RH_{use})^{3
}e^{(0.9/k)
(1/Tuse1/Ttest)
}
With voltage V
applied:
AF =
(V) (RH_{test}/RH_{use})^{3
}e^{(0.9/k)
(1/Tuse1/Ttest)
}
where:
C =
a constant
RHuse, RHtest = relative humidity during use
and under test, respectively
Tuse,
Ttest = temperature during use and under test, respectively
Timedependent Dielectric Breakdown (TDDB)
Timedependent Dielectric Breakdown, or TDDB, is the destruction of
dielectric layers occurring over time.
R = A_{1}e^{(Ea/kT+CV)}
AF =
tf_{use }/ tf_{test}_{
}
= R_{test}
/R_{use}
AF
= e^{([Ea/k]
[1/Ttest1/Tuse] + C [VtestVuse])}
where:
A_{1},
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 buildup that can
shift transistor threshold voltages. This mechanism is
accelerated by low temperatures.
AF = tf_{use }/ tf_{test }
AF
= e^{([Ea/k]
[1/Tuse1/Ttest] + C [VtestVuse])}
^{
}
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, CuSn intermetallic growths towards
the leadfinish surface, etc. are modeled as follows.
^{
}
tf
= Ae^{(Ea/kT) }
AF = tf_{use }/ tf_{test }
AF
=
e^{(Ea/k)
(1/Tuse1/Ttest)}
^{
}
where:
A = constant
Ea = 1 eV for AuAl bonds
Ea = 0.50.75 eV for Snbased leadfinish
TCinduced Package Cracking
The
occurrence
of fracture anywhere in the package after it has undergone several
temperature cycles has also been modeled.
Since
the zerostress 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 propertydependent)
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
See also:
Reliability Engineering;
Failure
Analysis;
Process
Qualification;
Package
Failures;
Die
Failures
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