Time-to-Failure Modeling

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 (TF) will depend on the electrical, thermal, mechanical, and chemical environments to which the device is exposed.

Problems

• 1. If a constant flux divergence exists, and is given by:

  $$ \int \overrightarrow{J}\cdot \mathrm{d}\overrightarrow{A}=R=\frac{100,000\;\mathrm{Billion}\kern0.17em \mathrm{atoms}}{\mathrm{s}}, $$

  find the time required for 50 % of the atoms to flow out of 1 cm³ of aluminum. Hint:

  $$ {\displaystyle \begin{array}{c}{N}_{\mathrm{atoms}}=\frac{{\left(\mathrm{density}\right)}_{\mathrm{A}1}{\left(\mathrm{Vo}1\mathrm{ume}\right)}_{\mathrm{A}1}}{{\left(\mathrm{atomic}\kern0.17em \mathrm{weight}\right)}_{\mathrm{A}1}}=\frac{\left(2.7\mathrm{g}/{\mathrm{cm}}^3\right)\left(1{\mathrm{cm}}^3\right)}{\left(27.0\mathrm{g}\right)/\left(6.02\times {10}^{23}\mathrm{atoms}\right)}\\ {}=6.0\times {10}^{22}\;\mathrm{atoms}\end{array}} $$

  Answer: 9.5 years

• 2. If the reaction-rate constant k shows a monotonic time dependence, then Chap. 2 suggests that one can approximate the time dependence with:

  $$ k(t)={k}_0\left[1\pm {a}_0{t}^m\right], $$

  where the plus (+) sign is used for an increasing reaction rate constant and a minus (-) sign for a decreasing reaction rate constant. Using Eq. (5.3), show that the TF is given by the transcendental equation:

  $$ \mathrm{TF}=\frac{\ln \left[{N}_0/N\left(t=\mathrm{TF}\right)\right]}{k_0\left[1\pm {a}_0\frac{{\left(\mathrm{TF}\right)}^m}{m+1}\right]}. $$

• 3. EM testing of Cu produced the following table of TF results:

  Electromigration time-to-failure data
  	1 × 106 (A/cm2)	2 × 106 (A/cm2)	3 × 106 (A/cm2)
  280 °C	–	20.3 h	–
  300 °C	20 h	10 h	6.7 h
  320 °C	–	5 h	–

  • (a) Find the power-law exponent n for the current density.

  • (b) Find the activation energy Q for this failure mechanism.

  Answers: (a) n = 1 (b) Q = 1.0 eV

• 4. Corrosion testing of a metal produced the following table of TF results:

  Corrosion time-to-failure data

  Corrosion time-to-failure data
  	60 % RH	70 % RH	80 % RH
  25 °C	–	824 h	–
  50 °C	332 h	100 h	30 h
  75 °C	–	16.4 h	–

  • (a) Find the exponential-dependence parameter γ for the humidity.

  • (b) Find the activation energy for this failure mechanism.

  Answers: (a) γ = 0.12 (%RH)^-1 (b) Q = 0.7 eV

• 5. Testing for surface-inversion/mobile-ions in ICs produced the following TF results:

  Mobile-Ions time-to-failure data
  	3 V	6 V	9 V
  60 °C	–	67.70 h	–
  70 °C	40 h	20 h	13.30 h
  80 °C	–	6.33 h	–

  • (a) Find the power-law exponent n which describes the voltage dependence.

  • (b) Find the activation energy Q for this failure mechanism.

  Answers: (a) n = 1 (b) Q = 1.2 eV

• 6. Testing for channel hot-carriers in n-type MOSFETs produced the following TF results.

  Hot-Carrier injection time–to–failure data
  	5 μA/μm	15 μA/μm	25 μA/μm
  25 °C	–	5.65 h	–
  50 °C	324 h	12 h	2.60 h
  75 °C	–	22.90 h	–

  • (a) Find the power-law exponent n which describes the substrate current dependence.

  • (b) Find the activation energy Q for this failure mechanism.

  Answers: (a) n = 3 (b) Q = −0.25 eV

• 7. Using Eq. (5.13b) for TF, and assuming that the temperature dependence of γ can be expressed by the Maclaurin Series:

  $$ \gamma (T)\cong {a}_0+\left({a}_1{K}_B\right)T, $$

  show that a stress-dependent activation energy develops of the form:

  $$ {Q}_{\mathrm{eff}}=Q-{a}_1{\left({K}_BT\right)}^2\xi . $$