A Confidence Interval Calculations

For conventional products where hundreds of failures are seen during a single exposure, a user 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 as a result, have 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 an 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 designed 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 test) (11). 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 approximately 95% of the cases.

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 x² 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, x² 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. As a result, 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. Let’s also assume that we have a number of devices with 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. As more events are observed, the statistics are improved so that uncertainty in the actual device performance is reduced.