SPRADI9 June 2024 AM623 , AM625

B Appendix – The Mathematic Basis for EM Reliability Estimates

Electro-migration failures are generally recognized to follow the Log-normal statistical reliability distribution with acceleration factors following Black’s Law. The parameters are such that the instantaneous failure rate increases with time, reflective of the latest stage of the bathtub curve. It is beneficial to first describe Black’s Law in basic form, then proceed to the Lognormal CDF. The fundamental CDF equation for EM Lognormal failures is in Equation 4.

Equation 4.

t_{f a i l u r e} = A J^{- n} e^{- \frac{E_{a}}{k T}}

J is the current density (effectively average current density in practice) through the wire or via.
n is the current density exponent depend on the metallization used. Generally, n = 1is applied for Copper (Cu) and n =2 for Aluminum. In the case of the Sitara products, both Cu and Al metals are used, but the limiting metallization components are typically Cu.
A is a fitting constant that divides away when taking ratios of time to failures at different stress or use conditions, which is the definition of an acceleration factor (AF).
k is Boltzmann’s constant, 8.617x10^-5 eV/°K. T in this case is temperature in Kelvin.

Equation 5.

F (t) = Φ [σ^{- 1} \ln ([\frac{1}{{s^{- n} t}_{50 - r e f}}] \sum_{i = 1}^{N} [\frac{t_{i}}{{(\frac{V_{i}}{V_{r e f}})}^{- n} {(\frac{f_{i}}{f_{r e f}})}^{- n} e^{- \frac{E_{a}}{n k} (\frac{1}{T_{r e f}} - \frac{1}{T_{i}})}}])]

In Equation 5,

Ф is the Standard Normal CDF.
t_50-ref is the Median Time to Failure at a design reference condition.
σ is the Standard Deviation of the natural logarithims of times to failure, which depends on the technology node, but 0.2 to 0.5 is typical (which is constant with wear-out).
s is a ratio of average current density for the specific wire or via over the allowed maximum current density limit. The maximum limit corresponds to a ceiling for allowed reliability for a single component. As s trends lower (i.e. average reducing current density) effect is to scale up the effective t₅₀ versus the reference condition. Higher t₅₀implies improved reliability.
V and f refer to voltage and frequency respectively.

The terms summed from i=1 to N (behind Σ) require further explanation. E_a and n are are the Black’s Law parameters, previously described. The numerator term t₁ refers to time at an application use condition tier in the mission profile. For example, t₁ can represent 20000 POH at 95°C Tj, at some operating some application voltage and frequency (an Operating Performance Point, or OPP, within data sheet specifications).

Another case:

Equation 6.

t_{2}

can be 50000 POH at 95°C Tj, and this can be at the same or a different OPP allowed within TI data sheet compared to (T_ref, V_ref, f_ref). The times for each tier must be scaled, either up or down, relative to the equivalent time at the reference condition. Finally, these scaled times must be summed for the entire mission profile (all appliation use tiers).

We have described the (Electro-migration) reliability of a single wire or via (component) up to now. How, then, is the total reliability for the SoC calculated? Mathematically, this is relatively straightforward.

Use the identity for the Reliability function:

Equation 7.

R = 1 - F

where F is the CDF, described previously.

If there are N total components, then the total Reliability function the overall product of each individual component's Reliability function:

Equation 8.

R_{t o t} = \prod_{i = 1}^{N} R_{i}

Finally, the total CDF is 1 minus the total Reliability:

Equation 9.

F_{t o t} = 1 - R_{t o t}

The Standard Normal CDF also has a convenient mathematical identity that can be used simply translation of F to R or vice versa. That identity is:

Equation 10.

Φ (z) = 1 - Φ (- z)

Equation 11.

F = Φ (z)

then

Equation 12.

R = Φ (- z)

In Equation 4, the sign of z can be changed from positive to negative by taking the reciprocal of the argument of the natural logarithm.