SNAK009A April 2022 – February 2024 ADC128S102-SEP

7 Confidence Interval Calculations

For conventional products where hundreds of failures can occur during a single exposure, the average failure rate of devices can be determined by being tested in a heavy-ion beam as a function of fluence with a high degree of certainty and reasonably tight standard deviation, and thus obtain 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, using 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, a failure rate is expected 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) and the DUT is monitored for failures. This process 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 is therefore a fixed fluence test (5)). Calculating a confidence interval specifically provides a range of values that is likely to contain the parameter of interest (the actual number of failures per 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 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, start with the standard reliability determination of the lower-bound (minimum) mean-time-to-failure for fixed-time testing (an exponential distribution is assumed) in Equation 2:

Equation 2.

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 to this equation for the purposes of this test, invert the inequality and substitute F (fluence) in the place of T, as shown in Equation 3:

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)

Equation 3.

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

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 4:

Equation 4.

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. 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). When 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 difference makes sense when considering that as more events are observed the statistics are improved such that uncertainty in the actual device performance is reduced.

Table 7-1 Experimental Example Calculation of MFTF and σ Using a 95% Confidence Interval⁽¹⁾

Degrees-of-Freedom (d)	2(d + 1)	χ² at 95%	Calculated Cross Section (cm²)
Degrees-of-Freedom (d)	2(d + 1)	χ² at 95%	Upper-Bound at 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. When the number of observed events increases, the confidence interval approaches the mean.