SNOAA62B February 2023 – October 2024 LMP7704-SP

A Confidence Interval Calculations

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 is difficult because often few or 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 an excellent choice 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 (because 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 being a fixed fluence test). 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 brackets the true population parameter in about 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) in Equation 3:

Equation 3.

M T T F = \frac{2 n T}{χ_{2 (d + 1); 100 (1 - \frac{α}{2})}^{2}}

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)
T is the test time
χ² is the chi-square distribution evaluated at 100(1 – α / 2) confidence level
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 as shown in Equation 4:

Equation 4.

M F T F = \frac{2 n F}{χ_{2 (d + 1); 100 (1 - \frac{α}{2})}^{2}}

where

MFTF is mean-fluence-to-failure
F is the test fluence
Χ² is the chi-square distribution evaluated at 100(1 – α / 2) confidence
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 as shown in Equation 5:

Equation 5.

σ = \frac{χ_{2 (d + 1); 100 (1 - \frac{α}{2})}^{2}}{2 n F}

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.

Table A-1 Experimental Example Calculation of MFTF and σ Using a 95% Confidence Interval⁽¹⁾

Degrees-of-Freedom (d)	2(d + 1)	χ² @ 95%	Calculated Cross-Section (cm²)
Degrees-of-Freedom (d)	2(d + 1)	χ² @ 95%	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

(1) Using a 95% confidence interval for several different observed results (d = 0, 1, 2,…100 observed events during fixed-fluence tests) assuming 10⁶ ions/cm² for each test. Note that as the number of observed events increases the confidence interval approaches the mean.