Alpha Law Directional SEU Tool

Preliminary analysis aid. Check the result against your own analysis and, where needed, a test on the intended part. Full disclaimer.

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 normal incidence curve the law implies, shows how far and in which direction the data depart from the cosine law, 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.

Status: for FinFETs this is one possible approach, not a validated one. What is known is that the integral RPP (IRPP) model is insufficient for them. The alpha law fits FinFET angular data, but its rates have not yet been validated (section 7 describes how that could be done). Until a better model exists, this is what we have: treat the rate as the best available estimate, with the cosine law rate beside it, not as a settled answer.

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.
• Zero rows. A row with sigma 0 (or counts 0) is a direction with no response. The alpha law and HICUP fits skip it, because it carries no counting error; the dual-node fit under the report's weighting uses it, since the directions where a two node cell shows nothing are half of what defines its alignment window. Keep them.
• Examples. The list beside the run button loads the single node worked example (an AMD Versal per bit SEU set) or one of the four dual-node data sets of JPL Publication 11-6, each with the settings the report used, so that its tables can be reproduced before your own data go in. The Versal set is also downloadable as a CSV file from the exposures table caption.
• Calculated cells. When a row has counts and fluence, the sigma and error cells are filled in with the per bit cross section and its Poisson error (counts over fluence times bits, root counts over fluence times bits, in the fluence convention you entered) and tinted to show they are calculated; the systematic floor is added in the fit, not here. Typing into a tinted cell makes it an entered value; clearing both cells returns them to calculated.
• Export. Download the table as CSV, or copy it tab separated for a spreadsheet. The file carries every column with the calculated cross sections filled in and a sigma_source column that says which were calculated and which were entered, so pasting it back reproduces the table exactly.

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 favor 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, the cosine law does not describe the data.
• 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.
• Is the minimum sharp. The same panel reports how many of the thirteen simplex starts ended at the same chi square (the test for local minima), and draws the profile chi square along A and along B, each held at a series of values with everything else refitted. Where the profile rises by 2.30 and 5.99 above the minimum are the joint 68 and 95 percent intervals, scaled by chi square per degree of freedom when that exceeds one. A flat profile means the data do not pin that parameter; the scan's contours at 5, 10 and 25 percent above its minimum show the same thing from the model free side. On the worked example A is sharp and B is flat below about 0.3, which is real: along that axis the exposures are close to the cosine law and the rate moves by about ten percent over the flat range.

4. Rates and the new Weibulls

• Environment. Fourteen CREME96 integral omnidirectional spectra are built in (the site's environment set, run on 11 and 12 September 2026 at solar minimum with the quiet magnetosphere unless the label says otherwise, all behind 100 mil aluminum): solar minimum galactic cosmic rays at GEO, ISS quiet and stormy, LEO 400 km 28.5 degrees, the 550 km 53 degree constellation orbit, sun synchronous 600 km, polar 1200 km, MEO 20,200 km, GTO, the lunar and Mars surfaces, 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 algorithm of his DIAL_E program: the directionally averaged cross section on a 45 by 45 angular grid, folded with the integral flux by summation by parts on a 2000 point log LET grid (DIAL_E sums on the file's own LET points, which is conservative when they are coarse). 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.
• Curve for the rate. Either the fitted function (Weibull or exponential) with its fitted A and B, or Edmonds' DIAL_T path: a monotone table through the collapsed points with A and B from the model free collapse scan, no functional form anywhere. The panel always shows the rate from the other choice next to it. Prefer the table when the normal incidence points do not look like the function, as in the worked example; prefer the function when the points are few or noisy and the monotone table would follow their scatter.
• Cutoff LET. DIAL_E's LCUT input: the normal incidence cross section is taken as zero below it. Leave it at 0 to use the fitted threshold (the Weibull's L0, or L1/e over 30 for the exponential form, as DIAL_E does). Set it to the lowest LET over alpha your data reach to see how much of the rate the fitted function takes from below the data; the caption reports that share in any case.
• 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 curve. The part's curve with the angular distortion removed, in the fitting function you chose. The Weibull form has no physical content; where the data stop short of saturation its saturation and width are extrapolations and the curve will not look like a textbook Weibull, as in the worked example. An integral RPP code applies the angular dependence of the box it is given, so hand this curve to CREME96 only with a box that carries the fitted A and B: Edmonds' equation 15 box, printed in the rate panel, or the HICUP box fitted alongside.
• 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. The HICUP comparison

The same exposures are fitted with HICUP, the rectangular parallelepiped model of Connell, Sexton and Prinja (Sandia, 1995): a box l by w by h with a funnel length, the exact chord length distribution through it for every direction, and a Weibull distributed critical energy. It runs with the analysis unless switched off.

• Settings. Aspect ratio free with the funnel held at zero is the default and needs two rotations, otherwise the aspect ratio is held at one. Freeing the funnel needs three or more tilts, because the funnel and the depth are separated only by how the effective LET grows with tilt. A square cell with no funnel is the classic RPP of the rate codes.
• Parameters. A (top area), aspect ratio, depth h, funnel, and the normal incidence Weibull L0, WL and b, with bootstrap intervals from a shorter replicate set than the alpha law's (at most 40) because each refit is slower. The box dimensions and the critical charge at E0 are derived. Read the box as a parameterization of the angular data, not as a picture of the device: Edmonds showed the RPP depth is a fitting parameter.
• Comparison table. Cosine law, alpha law and HICUP side by side with chi square, delta AIC and the rate in the selected environment. A delta AIC under about four is a dead heat, and the two rates then bracket the model uncertainty of the angular treatment, which neither bootstrap interval contains.
• Which to use. HICUP assumes a Weibull normal incidence curve, which is what makes it a conventional least squares fit and also what limits it. The alpha law with the Weibull option is the same kind of fit and gives nearly the same rate; the alpha law's own advantage is the table curve path, where no function is assumed at all. Prefer that path when the normal incidence points do not look like a Weibull; use HICUP for the comparison and for the hand off to RPP based codes.
• Scaling plot. Every tilted point corrected to normal incidence by HICUP's scaling laws (effective LET from the longest chord and the funnel, effective cross section from the projected area and the removal term). If the model holds the points sit on the Weibull; the small open markers show where the inverse cosine law would have put them.
• Tilt plot. Both models against tilt for each ion at the two rotations; where they separate, a measurement there would tell them apart.
• Rate. The I-HICUP rate is the angle integrated RPP cross section folded with the same environment by the same summation by parts as the alpha law rate, so the ratio of the two is a like for like comparison. The angle average saturates at half the box surface and starts below L0, at L0 (h + Sf) / (diagonal + Sf).
• Sensitivity. Percent change of the I-HICUP rate for plus and minus 10 percent in each of A, aspect ratio, h, Sf, E0, b and W, the others held (the paper's equation 22). E0 usually dominates.
• CREME96 hand off. X, Y, Z, funnel, sigma sat, L0, W, S and the critical charge, in the form the integral RPP codes take. The I-HICUP rate is the closed form of that calculation, so the code should reproduce it for the same environment. Do not combine these with the alpha law's normal incidence curve.

6. The dual-node model

A DICE latch, or any cell hardened so that no single node can flip it, upsets only when two particular nodes collect their critical charge from the same ion. Such data violate the rule every single node model obeys (the integral of sigma over LET squared is the same in every direction), so neither the alpha law nor HICUP can fit them at any parameter values. Edmonds' dual-node model (JPL Publication 11-6, 2011) keeps the alpha law for each node, turns the two upset regions into circles in a transformed plane and takes the cross section to be the lens the circles share; their center distance carries the node separation T, a vertical offset h and the beam direction. The fit runs alongside the others when it is switched on.

• When to use it. The sum rule row of the model ladder is the fingerprint: fitted exponents p and q that differ by more than about half say the data violate the invariant integral and no single node model will fit. The reading panel repeats the verdict once the dual-node fit has run, and warns the other way when the data show no such violation, because the model can always be fitted and a fit is not evidence of the mechanism.
• Settings. Symmetric nodes (one node curve for both, h = 0, six parameters) or two node curves with h and the rotation offset free (nine). T in micrometers is held during the fit; give it from the layout, or enter a list to sweep and the smallest objective is kept. The report found that floating T produces a field of local minima, which is why it is never free here. The objective is the alpha law's counting error chi square, or the report's weighted least squares (1 / L cubed alone as in its Section VIII, or with the cross section factor of its Section IX), which counts the zero rows. An additive floor can be freed for data with a residual response outside the alignment window, and the node thresholds can be held at zero for single LET data.
• Parameters. A and B (shared by both nodes), the two node curves L0, S0 and the common exponent P, h, the rotation offset that puts the laboratory rotation zero on the node line, and T. Node 1 is the softer node; the labels are interchangeable. The last row gives the normal incidence center distance, sqrt(A / B) T, against the two radii at the largest LET, which says whether the cell has any normal incidence response or only the grazing spike. Intervals are from a shorter bootstrap than the alpha law's (40 replicates, 20 above 400 rows).
• Comparison table. Cosine law, alpha law, HICUP and dual-node on the same nonzero rows with chi square, delta AIC and the rate; the zero rows, which only the dual-node model can honor, are not in the chi square. When the report's weighting was used the chi square is evaluated afterwards with the counting errors, so the curves are compared fairly even though the objectives differ.
• Maps. A polar map of the model over the hemisphere at a chosen LET (tilt outward, rotation around, color by log cross section) with the measured points on the same scale, the zero rows as crosses and the node line as a white spoke; and the rotation sweep at the best populated tilt and the tilt sweep at the best populated rotation, data against model. These are the plots that show whether the alignment window is in the right place.
• Rate. The directional average is evaluated on the report's 45 by 180 grid (its Appendix F) and folded with the environment by the same summation by parts as the other rates. The report's own spreadsheet takes the average at the upper edge of each LET bin, which is about twelve percent conservative; the page reports the converged integral, so expect its numbers for the report's data sets to sit ten to fifteen percent below the report's.
• Hand off. The report's Table 4 recipe: a Weibull fitted to the directional average over LET 1 to 30 and a cube of side equal to the square root of its limiting cross section, for a code with its own RPP machinery. The report found the cube about twenty percent below its direct sum; quote the direct rate.
• Preloaded data. The report's four data sets (early SRAM6 all-1's, 106 rows; later SRAM6 all-1's, 976 rows, and all-0's, 576 rows; SRAM11 at one LET, 43 rows) load with the report's settings. The early set reproduces its Table 2 within ten percent in A, B and P; the two large sets take a minute or two to fit.
• The campaign. Rotation sweeps at fixed tilt and LET through the node line, tilt sweeps along it, and an LET sweep along it at the tilt where the window is widest, at tilts from about 55 degrees to as near grazing as the overlayers allow; record the directions with no response; take T from the layout; fit all ones and all zeros separately. Details are in the campaign section of the tool page.

7. 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.

8. Caveats

• LET and direction are assumed to describe the ion completely. Low energy proton direct ionization, nuclear reactions at angle and same LET, different energy effects are outside every model here.
• The single node models fit one ellipsoid or one box. Fin plus sub fin volumes with different aspect ratios will show as collapse scatter and a poor reduced chi square rather than being resolved. The dual-node model covers two nodes that must both collect, one pair at a time, through the geometric overlap of their collection regions; several vulnerable pairs, or coupling through the well potential, are beyond it, and it has not been tested on a FinFET cell.
• 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.

9. References

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