Alpha Law Directional SEU Tool

1. What this tool does

Conventional heavy ion rate prediction tilts the beam, converts the tilt to an effective LET with the cosine law, fits a Weibull to the result, and integrates. The cosine law assumes the sensitive volume is a thin sheet. For a FinFET, and for any element whose sensitive volume is taller than it is wide, that assumption fails, and it fails differently along the two die axes. This tool fits Edmonds' alpha law instead, which replaces the cosine with a two parameter shape function, recovers the true normal incidence curve, tells you how far and in which direction the cosine law was wrong for your part, and computes the on orbit rate by direct integration of the directionally averaged cross section, with bootstrap confidence intervals and a set of quality checks.

2. Entering data

• One row per exposure: ion label, LET, tilt, rotation, counts, fluence. Or paste a CSV with a header; column order does not matter. If you only have cross sections, use columns sigma and err instead of counts and fluence.
• LET is the true LET of the ion at the die. Never enter an effective LET; the tool computes its own angular transformation and an effective LET would apply the cosine law twice.
• Fluence convention. Edmonds defines the directional cross section with fluence measured perpendicular to the beam. If your facility reported device plane fluence, select that convention and the tool converts. It also fits the other convention silently and warns you if that one fits much better, which usually means the data were already converted once.
• Bits under test. Counts are divided by fluence and by this number to give a per bit cross section. Enter 1 if the data are already per bit.
• Systematic floor. A fractional error added in quadrature to the Poisson error of every point. With thousands of counts the Poisson error is a percent, and beam runs do not repeat to a percent. The default of ten percent stands in for dosimetry, LET and run to run scatter; without it the fit is controlled by whichever exposure was longest. Lower it if you have repeat runs that justify it.
• Normal incidence curve. The four parameter Weibull needs three or more distinct LET values and ideally reaches saturation. With sparse LET coverage choose the two parameter exponential.
• Hold out. Tick a row to exclude it from the fit. The tool predicts it and reports the pull. This is how to test the model against a measurement it has not seen.

3. Reading the results

• Collapse scan. A heat map of the Patterson and Edmonds scatter score over A and B, with no assumption about the shape of the curve. A sharp minimum means the data pin both parameters. A valley along the B axis means only one rotation was measured and B is not determined; treat B as zero for a conservative rate, or set B = A if the part is believed to have no azimuthal dependence.
• Model ladder. Cosine, isotropic, axisymmetric and full alpha law fitted to the same data, with reduced chi square and delta AIC. A delta AIC above ten in favour of the alpha law is decisive. The sum rule row fits independent exponents inside and outside the normal incidence function; if they differ, no single sensitive volume model describes the data and no rate from any such model should be trusted.
• Collapse plot. Every exposure transformed to σ / α against L / α, with the fitted normal incidence curve. If the law holds, everything lies on the line. Look at this before the numbers.
• Tilt plot. Data and model against tilt for each ion at the two rotations. The dotted curve is the cosine law fitted to the same data. Where solid and dotted diverge is where a conventional analysis is wrong.
• Residuals. Per exposure pulls. Anything beyond three is highlighted.
• Quality. Plain language warnings: too few rotations or LET values, low counts, a reduced chi square that the floor does not explain, disagreement between the scan and the fit, extrapolation of the normal incidence curve beyond the data, and which regime the fitted A and B place the part in.

4. Rates and the new Weibulls

• Environment. Five CREME96 integral omnidirectional spectra are built in, all behind 100 mil aluminium: solar minimum galactic cosmic rays, ISS, and the peak 5 minute, worst day and worst week solar particle events. For any other environment paste a two column CREME96 integral LET spectrum, LET in MeV cm2/mg and flux above that LET in per m2 s sr.
• Rate. Edmonds' direct integration: the directionally averaged cross section on a 45 by 45 angular grid, folded with the integral flux by summation by parts on a 400 point log LET grid. Reported per bit per day, then per device per day and per year using the bits per device you enter. The cosine law rate is from the cosine model fitted to the same data, and their ratio is the angular correction factor.
• Interval. The 68 and 95 percent intervals come from re-integrating the bootstrap replicates. They cover fit statistics only, not the environment model or part to part variation.
• Normal incidence Weibull. The part's physical curve with the angular distortion removed. Do not feed it to a code that applies its own cosine law; that puts the distortion back.
• Directional average Weibull. A Weibull fitted to the isotropic average of the directional cross section, for use in any tool that computes rate as the integral of a cross section against the omnidirectional differential flux with no angular machinery of its own. The rms log residual tells you how faithful the Weibull is; the exact table is downloadable and is preferable where a tool can take it.

5. Validating the method on a new device

The alpha law with A or B above one fits FinFET angular data well; it has not yet been shown to predict a flight rate. The tool is built to close that gap one measurement at a time. Fit on the runs you have, read the next measurements panel, which ranks unmeasured directions by how sharply they separate the alpha and cosine laws among runs that can reach a hundred counts in a practical fluence, go and measure the top one or two, then paste them in with hold out ticked. If the pulls are inside about two, the prediction held. Do that on a few parts and the case for the method makes itself. The alternative, a flight experiment in a cosmic ray dominated orbit with enough bits to count, is the definitive test but is rarely available.

6. Caveats

• LET and direction are assumed to describe the ion completely. Low energy proton direct ionisation, nuclear reactions at angle and same LET, different energy effects are outside the model.
• A single ellipsoid is fitted. Fin plus sub fin volumes with different aspect ratios, or paired transistors quenching each other, will show as collapse scatter and a poor reduced chi square rather than being resolved.
• Per bit data only. Event cross sections with clustered multiple cell upsets have their own angular dependence and need not obey the sum rule.
• Range at grazing angles can fake an angular effect. Stay below about 80 degrees and choose long range ions.
• The example data set is one part, one lot, one bias. The A and B it yields are not a property of the technology node.

7. References

Back to the tool