8. Assessing Product Reliability
8.1. Introduction
8.1.5. What are some common acceleration models?

## Other models

Many useful 1, 2 and 3 stress models are simple Eyring models. Six are described This section will discuss several acceleration models whose successful use has been described in the literature.  The (Inverse) Power Rule for Voltage

This model, used for capacitors, has only voltage dependency and takes the form: $$t_f = AV^{-\beta} \,\, .$$ This is a very simplified Eyring model with $$\alpha, \Delta H,$$ and $$C$$ all 0, $$S = \mbox{ln} V$$, and $$\beta = -B$$.

The Exponential Voltage Model

In some cases, voltage dependence is modeled better with an exponential model: $$t_f = A e^{-BV} \,\, .$$ Two Temperature/Voltage Models

Temperature/Voltage models are common in the literature and take one of the two forms given below: $$t_f = A \cdot \mbox{exp} \left( \frac{\Delta H}{kT} \right) \cdot V^{-\beta}$$ or $$t_f = A \cdot \mbox{exp} \left( \frac{\Delta H}{kT} \right) \cdot e^{-BV} \,\, .$$

Again, these are just simplified two stress Eyring models with the appropriate choice of constants and functions of voltage.

The Electromigration Model

Electromigration is a semiconductor failure mechanism where open failures occur in metal thin film conductors due to the movement of ions toward the anode. This ionic movement is accelerated high temperatures and high current density. The (modified Eyring) model takes the form $$t_f = A \cdot J^{-n} \cdot \mbox{exp} \left( \frac{\Delta H}{kT} \right) \,\, ,$$ with $$J$$ denoting the current density. $$\Delta H$$ is typically between 0.5 and 1.2 electron volts, while an $$n$$ around 2 is common.

Three-Stress Models (Temperature, Voltage and Humidity)

Humidity plays an important role in many failure mechanisms that depend on corrosion or ionic movement. A common 3-stress model takes the form $$t_f = A \cdot \mbox{exp} \left( \frac{\Delta H}{kT} \right) \cdot V^{-\beta} \cdot R H^{-\gamma} \,\, .$$ Here $$RH$$ is percent relative humidity. Other obvious variations on this model would be to use an exponential voltage term and/or an exponential $$RH$$ term.

Even this simplified Eyring 3-stress model has 4 unknown parameters and an extensive experimental setup would be required to fit the model and calculate acceleration factors.

The Coffin-Manson Model is a useful non-Eyring model for crack growth or material fatigue The Coffin-Manson Mechanical Crack Growth Model

Models for mechanical failure, material fatigue or material deformation are not forms of the Eyring model. These models  typically have terms relating to cycles of stress or frequency of use or change in temperatures. A model of this type known as the (modified) Coffin-Manson model has been used successfully to model crack growth in solder and other metals due to repeated temperature cycling as equipment is turned on and off. This model takes the form $$N_f = A \cdot f^{-\alpha} \cdot \Delta T^{-\beta} \cdot G(T_{\mbox{max}}) \,\, ,$$ with

• $$N_f$$ = the number of cycles to fail
• $$f$$ = the cycling frequency
• $$\Delta T$$ = the temperature range during a cycle
and $$G(T_{\mbox{max}})$$ is an Arrhenius term evaluated at the maximum temperature reached in each cycle.

Typical values for the cycling frequency exponent $$\alpha$$ and the temperature range exponent $$\beta$$ are around -1/3 and 2, respectively (note that reducing the cycling frequency reduces the number of cycles to failure). The $$\Delta H$$ activation energy term in $$G(T_{\mbox{max}})$$ is around 1.25.