SLOK017 June 2024 TLV1H103-SEP
PRODUCTION DATA
For conventional products where hundreds of failures are seen during a single exposure, one can determine the average failure rate of parts being tested in a heavy-ion beam as a function of fluence with high degree of certainty and reasonably tight standard deviation, and thus have a good deal of confidence that the calculated cross-section is accurate.
With radiation hardened parts however, determining the cross-section becomes more difficult since often few, or even, no failures are observed during an entire exposure. Determining the cross-section using an average failure rate with standard deviation is no longer a viable option, and the common practice of assuming a single error occurred at the conclusion of a null-result can end up in a greatly underestimated cross-section.
In cases where observed failures are rare or non-existent, the use of confidence intervals and the chi-squared distribution is indicated. The Chi-Squared distribution is particularly well-suited for the determination of a reliability level when the failures occur at a constant rate. In the case of SEE testing, where the ion events are random in time and position within the irradiation area, one expects a failure rate that is independent of time (presuming that parametric shifts induced by the total ionizing dose do not affect the failure rate), and thus the use of chi-squared statistical techniques is valid (since events are rare an exponential or Poisson distribution is usually used).
In a typical SEE experiment, the device-under-test (DUT) is exposed to a known, fixed fluence (ions/cm2) while the DUT is monitored for failures. This is analogous to fixed-time reliability testing and, more specifically, time-terminated testing, where the reliability test is terminated after a fixed amount of time whether or not a failure has occurred (in the case of SEE tests fluence is substituted for time and hence it is a fixed fluence testEquation 2). Calculating a confidence interval specifically provides a range of values which is likely to contain the parameter of interest (the actual number of failures/fluence). Confidence intervals are constructed at a specific confidence level. For example, a 95% confidence level implies that if a given number of units were sampled numerous times and a confidence interval estimated for each test, the resulting set of confidence intervals would bracket the true population parameter in about 95% of the cases.
In order to estimate the cross-section from a null-result (no fails observed for a given fluence) with a confidence interval, we start with the standard reliability determination of lower-bound (minimum) mean-time-to-failure for fixed-time testing (an exponential distribution is assumed):
Where MTTF is the minimum (lower-bound) mean-time-to-failure, n is the number of units tested (presuming each unit is tested under identical conditions) and T, is the test time, and χ2 is the chi-square distribution evaluated at 100(1 – α / 2) confidence level and where d is the degrees-of-freedom (the number of failures observed). With slight modification for our purposes we invert the inequality and substitute F (fluence) in the place of T:
Where now MFTF is mean-fluence-to-failure and F is the test fluence, and as before, χ2 is the chi-square distribution evaluated at 100(1 – α / 2) confidence and where d is the degrees-of-freedom (the number of failures observed). The inverse relation between MTTF and failure rate is mirrored with the MFTF. Thus the upper-bound cross-section is obtained by inverting the MFTF:
Assume that all tests are terminated at a total fluence of 106 ions/cm2. Also assume there are a number of devices with very different performances that are tested under identical conditions. Assume a 95% confidence level (σ = 0.05). Note that as d increases from 0 events to 100 events the actual confidence interval becomes smaller, indicating that the range of values of the true value of the population parameter (in this case the cross-section) is approaching the mean value + 1 standard deviation. This makes sense when one considers that as more events are observed the statistics are improved such that uncertainty in the actual device performance is reduced.
Degrees-of-Freedom (d) | 2(d + 1) | χ2 @ 95% | Calculated Cross Section (cm2) | ||
---|---|---|---|---|---|
Upper-Bound @ 95% Confidence | Mean | Average + Standard Deviation | |||
0 | 2 | 7.38 | 3.69E–06 | 0.00E+00 | 0.00E+00 |
1 | 4 | 11.14 | 5.57E–06 | 1.00E–06 | 2.00E–06 |
2 | 6 | 14.45 | 7.22E–06 | 2.00E–06 | 3.41E–06 |
3 | 8 | 17.53 | 8.77E–06 | 3.00E–06 | 4.73E–06 |
4 | 10 | 20.48 | 1.02E–05 | 4.00E–06 | 6.00E–06 |
5 | 12 | 23.34 | 1.17E–05 | 5.00E–06 | 7.24E–06 |
10 | 22 | 36.78 | 1.84E–05 | 1.00E–05 | 1.32E–05 |
50 | 102 | 131.84 | 6.59E–05 | 5.00E–05 | 5.71E–05 |
100 | 202 | 243.25 | 1.22E–04 | 1.00E–04 | 1.10E–04 |