What the alpha law is, and whose it is. The physics, the plotting format and the rate integration on this page are the work of L. D. Edmonds at JPL (IEEE Transactions on Nuclear Science 49(3), 1522, 2002), with the automated parameter search due to J. D. Patterson and Edmonds. This tool is an implementation and an extension of the parameter range, not a new model. Edmonds showed that if a device's charge collection efficiency falls off on concentric ellipsoids around the sensitive node, the directional cross section obeys σ(L, θ, φ) = α · σN(L / α), α² = (A² cos²φ + B² sin²φ) sin²θ + cos²θ where σN is the normal incidence curve, θ is tilt, φ is rotation, and A, B are the lateral to vertical aspect ratios of the ellipsoid along the two rotation axes. A = B = 0 is the cosine law exactly. A = B = 1 is isotropic. The planar literature has always kept A, B below 1. Nothing in the derivation requires that: A or B above 1 describes a volume taller than it is wide, which is a fin, and it predicts a cross section that falls with tilt, which is what bulk FinFETs measurably do.
What the tool does. It first scans A and B with no assumption about the shape of the curve, looking for the pair that collapses every measurement onto one line. It then fits a ladder of models, cosine law through full alpha law, with a chosen normal incidence form, reports which the data prefer and by how much, bootstraps the parameters, integrates the directionally averaged cross section against a CREME96 environment exactly as Edmonds' DIAL_E code does, and hands back a Weibull for the normal incidence curve and another for the isotropic average, so the result can be used elsewhere.
One row per exposure. LET is the true LET of the ion at the die, never an effective LET, and fluence is measured in a plane perpendicular to the beam unless you tell the tool otherwise. Counts and fluence let the tool compute Poisson errors itself; if you only have cross sections, paste those with an error column instead. Tick hold out on a row to exclude it from the fit and test the prediction against it.
| ion | LET | tilt deg | rot deg | counts | fluence /cm2 | hold out |
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The rate is the integral of the directionally averaged cross section against the omnidirectional differential flux, evaluated with Edmonds' summation by parts on a fine log spaced LET grid. The cosine law rate uses the cosine model fitted to the same data, which is what a conventional analysis would have produced. Their ratio is the angular correction factor for this part.
There are two ways to find out whether this works for a device like a FinFET. Fly it in a cosmic ray dominated environment and compare. Or fit the model, predict the cross section at angles you have not measured, then go and measure them. The second is cheaper and the tool supports it directly: hold out rows and it predicts them; with nothing held out it ranks the unmeasured directions by how strongly they would separate the alpha law from the cosine law, restricted to runs that could reach a hundred counts in a practical fluence.
This section exists because the method asks for something most heavy ion campaigns do not collect, and the tool is only as good as the campaign behind it.
The cosine law says that tilting the beam by θ is the same as raising the LET by 1 / cosθ, because the ion's path through a thin sheet grows by that factor. That is a statement about geometry, and it is right for a sheet. A fin is a wall a few tens of nanometres wide and taller than it is wide. Tilt the beam across the wall and the track exits through the sidewall into the isolation oxide; the chord through silicon gets shorter, not longer, and the deposited charge falls. Tilt it along the wall and the chord grows more or less as the sheet picture says. So the same part obeys something close to the cosine law in one rotation and something close to its inverse in the other. Vanderbilt measurements at the 14 and 16 nm node established this experimentally, with normal incidence the worst case for across fin tilting at low LET. The alpha law captures both behaviours with two numbers, A along one rotation axis and B along the other, and the sign of A minus 1 tells you which regime you are in.
Normal incidence first. At least three ions spanning from below threshold to as high an LET as the facility can deliver with adequate range, because the normal incidence curve is what everything else is referred to. Five or more is much better. Saturation must be reached or the Weibull saturation is an extrapolation, and the tool will say so.
Then tilt, at two rotations. Tilt at 45 and 60 degrees, and ideally 30, at two rotations 90 degrees apart. If you know where the fins run, align the rotations with them, since those are the axes on which A and B are defined. If you do not, use the die edges and add a 45 degree rotation if beam time allows; the ellipsoidal form predicts a specific interpolation between the two principal axes, and the third rotation tests it. With only one rotation the data cannot determine B at all, and the tool will tell you to treat B as zero, the conservative choice, or to set it equal to A.
Use true LET and beam normal fluence. Facilities habitually report an effective LET and a device plane fluence for tilted runs. Both bake the cosine law into the raw data, which is the assumption under test. Enter the LET of the ion at the die and the fluence in the plane perpendicular to the beam. If you only have device plane numbers, select that convention and the tool converts, but check that the facility did not already convert once. The tool fits both conventions internally; if the one you selected fits much worse, it says so.
Per bit counts, not per event. The alpha law is a statement about a single cell. It applies to bit cross sections, what Edmonds calls U type, and it need not apply to event cross sections where multiple cell upsets have been clustered, because clustering has an angular dependence of its own. Fit bit data, handle multiple cell statistics separately. Fit all ones and all zeros patterns separately too, since they can have different sensitive nodes with different aspect ratios.
Mind the range at grazing angles. The ion must cross the overlayers and the full sensitive depth after projection by cosθ. A short range ion at 60 or 70 degrees can produce an apparent angular effect that is a range artefact. Nsengiyumva and co-workers flag overlayer range effects above about 80 degrees; stay below that, and choose the longest range ion available at each LET.
Count. Aim for a hundred or more upsets per exposure. The tool's next measurement recommender only proposes runs that can reach a hundred counts inside 1e8 ions per square centimetre for exactly this reason. And put a realistic systematic floor on the errors. With thousands of counts the Poisson error is a percent, and no two beam runs agree to a percent; the default floor of ten percent reflects dosimetry, LET and run to run scatter, and without it the fit is dominated by whichever exposure happened to be longest.
Look at the collapse plot first. If the alpha law holds for your part, every exposure, at every tilt and rotation, lies on one line. Scatter about the line is what the model cannot explain, and it should look like counting noise. Systematic structure, for example fine tilt runs that wobble above and below the line, is the signature of multiple fin crossings that a single ellipsoid cannot follow, and the reduced chi square will say so. The scan heat map shows whether A and B are pinned: a sharp minimum is good; a valley along the B axis means you have one rotation and B is unconstrained.
Then the model ladder. The cosine law is the null hypothesis, isotropic and axisymmetric are the intermediate cases, and the full alpha law is the alternative. A delta AIC above ten in favour of the alpha law is decisive. If it is not decisive, your part may not need the correction, which is a perfectly good result. The sum rule row is a consistency check: if the fitted exponents p and q differ materially, no single sensitive volume model of any kind fits the data, and neither this tool nor an RPP calculation will give a reliable rate.
Finally the rate ratio. A ratio below one means the cosine law analysis was conservative for this part, which is common for fins tilted across their width; a ratio above one means it was optimistic. The bootstrap interval on the rate is the statistical uncertainty of the fit only. It does not include the environment model, part to part variation, or the same LET, different energy effects that no LET based model can represent.
The normal incidence Weibull is the part's physical curve. Do not give it to CREME96 or any code that applies a cosine law internally, because it will put the angular distortion back in. The directional average Weibull is the isotropic average of the directional cross section; a tool that computes rate as the integral of a cross section against the omnidirectional differential flux, with no angular integration of its own, can use it directly. The exact table is available as a download and is preferable to the Weibull approximation where a tool can accept it.
The honest position is that the alpha law with A or B above one has been shown to fit FinFET angular data well, and has not yet been shown to predict a flight rate. Two things would change that. One is a flight experiment in a cosmic ray dominated orbit with enough bits to count. The other is cheaper: fit the model on the runs you have, let the tool tell you which unmeasured direction would most sharply distinguish it from the cosine law, go and measure that direction, and see whether the point lands on the prediction. Do that a few times on a few parts and the case makes itself. The hold out feature is there so that the prediction is written down before the measurement is made.
It assumes LET and direction describe the ion completely. That excludes low energy proton direct ionisation, nuclear reactions at angle, and the same LET, different energy effects that appear in highly scaled parts; for those, Edmonds' energy based extension or a Monte Carlo code is needed. It fits a single ellipsoid, so a device whose fin and sub fin volumes feed one node with different aspect ratios, or whose paired transistors quench each other's charge, is beyond it; the collapse scatter and the sum rule diagnostic will show the strain but will not resolve it. It does not handle multiple cell upset statistics. And it is a fit to one part in one lot at one bias, with everything that implies.
The directional cross section is counts divided by beam normal fluence, per bit. The alpha law is applied with the ellipsoidal shape function of Edmonds' equation 5, with A and B free on the interval 0 to 10 rather than 0 to 1. Any positive shape function with α(0, φ) = 1 satisfies Edmonds' direction independence sum rule, his equation 7, because substituting u = L / α in the integral of (1 / L) dσ / dL removes the direction dependence; the ellipsoid is the two parameter member of that family. The normal incidence curve is either a four parameter Weibull, σsat (1 − exp(−((L − L0) / w)s)), or the two parameter exponential used in the original paper and by Kobayashi and Ikuta, σsat exp(−L1/e / L).
Following Patterson and Edmonds, for each (A, B) on a grid the exposures are transformed to (L / α, log(σ / α)), sorted by the first coordinate, and scored by the sum of absolute steps in the second divided by its total rise. A perfectly monotone collapse scores one. This is model free in the sense that it commits to no form for the curve, and it is guaranteed to find the global minimum on the grid, which is why it is used to seed and to sanity check the parametric fit.
Weighted least squares on the directional cross section, with the error on each point the Poisson error from its counts combined in quadrature with a fractional systematic floor, minimised by a bounded Nelder Mead simplex on logistic transformed parameters with a dozen jittered restarts and a polishing pass. The same code runs in Python in the development harness and in the browser, and the two agree to all displayed digits on the example. Model comparison uses AIC. The sum rule diagnostic refits with independent exponents on α inside and outside the normal incidence function.
A parametric bootstrap: for each replicate the expected counts at every exposure are taken from the fitted model, Poisson resampled, multiplied by a lognormal factor with the systematic floor as its width, and refitted from the base solution. Parameter intervals and the rate interval are percentiles over replicates. The interval is statistical only.
The directional average cross section is Edmonds' equation 12, evaluated by his 45 by 45 midpoint rule over one quadrant of solid angle using the symmetry of the ellipsoid. The rate is his equation 13, the summation by parts of the integral flux against differences of the average cross section, with the CREME96 integral omnidirectional flux converted from per square metre per second per steradian to per square centimetre per day by the factor 108.57 and interpolated in log log onto a four hundred point grid, which removes the conservatism Edmonds notes for coarse LET grids. The integrator was checked against the analytic limits: an isotropic device returns the normal incidence curve exactly, and a saturated cosine law device returns half its saturation cross section.
The worked example is a per bit SEU data set on an AMD Versal device taken at the Berkeley 88-Inch Cyclotron in November 2023: seven ions at normal incidence from LET 1.7 to 63, then tilts of 45 and 60 degrees at two rotations 90 degrees apart, about 4.8e8 bits under test. At LET 3.3 and 60 degrees the two rotations differ by a factor of nearly seventy. The fit puts A near 2.3 on one rotation axis, which is the inverse cosine regime of a tall volume, and B near 0.26 on the other, which is close to the cosine law, and the ratio of the alpha law rate to the cosine law rate in a solar minimum cosmic ray environment is about one third. The normal incidence Weibull the tool recovers agrees with an independent hand analysis of the same spreadsheet to within a few percent on saturation, width and shape.
Related tools: SEE Rate Assessment, SEL Threshold Predictor, Analog SET Guideline, Standards and Data Sources. Corrections and additional angular data sets are welcome through the community forum.
