Abstract
All materials tend to degrade, and will eventually fail, with time. For example, metals tend to creep and fatigue; dielectrics tend to trap charge and breakdown; paint tends to crack and peel; polymers tend to lose their elasticity and become more brittle, teeth tend to decay and fracture; etc. All devices (electrical, mechanical, electromechanical, biomechanical, bioelectrical, etc.) will tend to degrade with time and eventually fail. The rate of degradation and eventual time-to-failure will depend on the electrical, thermal, mechanical and chemical environments to which the device is exposed.
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Notes
- 1.
The divergence theorem states that: \(\int\limits_V {\mathop \nabla \limits^ \to } \bullet \mathop J\limits^ \to dV = \int\limits_A {\mathop J\limits^ \to } \bullet d\mathop A\limits^ \to\), where V is the volume of interest which is bounded by a surface of area A.
- 2.
The reaction-rate constant, in many cases, may not really be constant. It may, in general, be a function of time.
- 3.
We have inserted the identity: \(TF = \int\limits_0^{TF} {dt}\).
- 4.
This is generally true for electromigration-induced failure in conductors. Wider metal leads, at the same current density stress, tend to last longer. An apparent exception seems to exist in aluminum where very narrow metal leads can last longer than wider metal leads during electromigration testing. With aluminum, a bamboo-like grain-boundary microstructure can develop when the metal width and thickness are comparable to the Al grain size. Here, however, the amount of flux divergence is no longer constant, but is reduced by the bamboo grain structure thus causing the narrow metal leads to last longer than the wider leads.
- 5.
To properly use this equation, the temperature T must be expressed in Kelvin.
- 6.
This can be a very important issue if the stress also tends to serve as a significant source of self-heating, e.g., Joule heating can raise the temperature of the conductor when the current density stress is increased in a metal stripe during electromigration testing. This temperature rise (with the level of current-density stress) must be taken into account when determining the failure kinetics.
- 7.
Remember that one must convert the temperature from Centigrade to Kelvin. The conversion equation is T(K)= T(°C) + 273.
- 8.
Identity is used: \(\exp ( - \gamma \xi _{BD} )\exp (\gamma \xi _{BD} ) = 1\).
- 9.
In Problem 7, at the end of this chapter, it is shown that a stress dependent activation energy also develops if a Maclaurin Series expansion is used: \(\gamma (T) = a_o + (a_1 K_B )T\).
- 10.
The occurrence of a stress-dependent activation energy (for high level of stress) is discussed in detail in Chapter 10.
References
McPherson, J.: Stress Dependent Activation Energy, IEEE International Reliability Physics Symposium Proceedings, 12(1986).
McPherson, J. and E. Ogawa: Reliability Physics and Engineering. In: Handbook of Semiconductor Manufacturing Technology, 2 nd Edition, CRC Press 30-1(2008).
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McPherson, J. (2010). Time-To-Failure Modeling. In: Reliability Physics and Engineering. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-6348-2_4
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DOI: https://doi.org/10.1007/978-1-4419-6348-2_4
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