Apendix : sidewall widget math
The widget is built on one core geometric relationship between sidewall angle, feature height, and the CD error introduced into a top-down SEM measuremen
The geometry
When a feature has a perfectly vertical sidewall (θ = 90°), the top width and the bottom width are identical. When the sidewall tilts by an angle δ away from vertical (so θ = 90° − δ), the base of the feature is wider than the top.
W_top
┌───────────┐ ← SEM scans here, reads W_top
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└───────────────────┘ ← true CD at base = W_bot
W_bot
The overhang on each side is the horizontal distance gained by travelling down the full feature height h at angle δ from vertical:
\[\text{overhang} = h \cdot \tan\delta = h \cdot \tan(90° - \theta)\]Since there are two sides, the total extra width at the base is:
\[W_{bot} - W_{top} = 2h\tan(90° - \theta)\]The CD error
A top-down SEM cannot see the base of the feature — it measures the top width W_top only. If the true critical dimension of interest is W_bot, the CD error is:
\[\text{CD error} = W_{bot} - W_{top} = 2h\tan(90° - \theta)\]In the interactive model, h = 40 nm is fixed. At θ = 85° (δ = 5°):
\[\text{CD error} = 2 \times 40 \times \tan(5°) = 80 \times 0.0875 \approx 7.0 \text{ nm}\]As a percentage of a 20 nm CD feature:
\[\text{CD error (%)} = \frac{2h\tan(90°-\theta)}{CD} \times 100 = \frac{7.0}{20} \times 100 = 35\%\]The simulated SEM intensity profile
The profile in the right panel models what a secondary electron detector sees as the beam scans across the feature. Three contributions are summed:
\[I(x) = I_{\text{bulk}}(x) + I_{\text{top edges}}(x) + I_{\text{base edges}}(x)\]$I_{\text{bulk}}$ is a flat elevated signal over the feature. Metal produces more backscattered electrons than the silicon substrate, so the feature appears brighter overall.
$I_{\text{top edges}}$ is a Gaussian spike at each top corner, arising from the edge enhancement effect — a sharp corner produces a burst of secondary electrons as the beam passes over it:
\[I_{\text{edge}}(x) = A \cdot \exp\!\left(-\frac{(x - x_{\text{edge}})^2}{2\sigma^2}\right)\]$I_{\text{base edges}}$ is a smaller Gaussian at the projected base corners, which becomes visible when the sidewall tilts significantly because the angled wall intercepts part of the incident beam.
Why the top markers are fixed
The red dashed lines marking W_top do not move regardless of sidewall angle. This is the key point — a standard top-down SEM always reports the top width, so it always reads 20 nm no matter how sloped the walls are. It is the green W_bot markers that slide outward as the angle decreases, showing the true base width growing while the SEM measurement stays fixed. The gap between the two is the CD error.
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