Inverse RCWA Metrology
completeProblem
Summer 2025 at KLA: recover trench geometry from broadband reflectance of patterned GAAFET stacks.
Observable: Reflectance from 300–2500 nm, polarization-resolved
Goal: Extract trench depth and layer thicknesses from spectral data
Challenge: Multiple geometries can produce nearly identical spectra
This was my first real encounter with inverse problems in industry — where “good enough to control the process” matters more than “perfectly understood.”
What worked
Trench depth was reliably recovered across 20+ stack and geometry variants. Depth creates specific interference patterns in the spectral shape that don’t correlate strongly with other parameters when you restrict to 0° polarization and apply fabrication process bounds. Validation was stable across multi-start optimization, consistent within ±5%, and matched CD-SEM measurements on calibration samples.
Result: 47% reduction in inline measurement time-to-solution by focusing on the one parameter we could actually resolve.
What didn’t work
Secondary parameters — sidewall angle and several layer thicknesses — remained degenerate. Changing sidewall angle by 5° and adjusting a nearby thickness by 10 nm produced nearly identical spectra. The χ² residual difference was sub-1% across these degenerate solutions, which is below the noise floor of the instrument.
Adding more wavelength points didn’t help — same degeneracy, more data. Using both polarizations made it worse by introducing additional parameter correlations. Tighter process bounds helped marginally.
The core issue: you cannot optimize your way out of a fundamentally ill-posed problem. More data points don’t fix degeneracy if the physics couples the parameters.
What the degeneracy actually looks like
The sensing matrix for this problem has a specific structure: depth-sensitive spectral features are well-separated from thickness-sensitive features in wavelength space, which is why depth is recoverable. Secondary parameters share spectral sensitivity regions — their Jacobian columns are nearly parallel, producing high covariance in the posterior and multiple local minima with similar fit quality.
Multi-start optimization from 20+ random initializations converged to geometrically different solutions with χ² < 0.03 in each case. That’s the signature of structural non-identifiability, not optimizer failure.
The more interesting observation is that the degeneracy is not polarization-symmetric. Under 0° polarization, a sidewall angle perturbation produces a peak position shift localized to the 400–500 nm window — the signature my depth assignment script used. Under 90°, the difference between degenerate solutions shifts to amplitude changes concentrated below 400 nm, in a region that doesn’t overlap with the 0° signal. The two polarizations are sensitive to different aspects of the same geometric perturbation, but neither alone resolves the sidewall/thickness trade-off within the ±5° process control window relevant to TSMC-tier fabrication.
The tool below illustrates this mechanism. The spectra are physically motivated — Fabry-Perot interference with polarization-dependent modulation — but are not direct RCWA output. Treat them as a model of the mechanism, not a reproduction of the measured data.
The bottom panel is the key one. The difference spectra for 0° and 90° occupy non-overlapping spectral regions. This is why using both polarizations didn’t resolve the degeneracy — the additional information sits outside the region where the existing model was fitting well, so it introduced new parameter correlations rather than constraining the existing ones.
What I’d do differently
Angle-resolved measurements break some of these degeneracies by adding diversity in the sensitivity structure. Multiple azimuthal orientations similarly. Better sensitivity analysis upfront — before building the full pipeline — would have identified which parameters were non-identifiable earlier and avoided wasted effort on secondary parameter extraction.
In a production environment, “good enough to control depth” was the right answer. Perfect geometric reconstruction wasn’t necessary and wasn’t achievable with this measurement configuration.
What I learned: Inverse problems fail quietly. Good fits don’t mean unique solutions. Identifiability matters more than optimization skill.
Status: Completed Summer 2025. Pipeline deployed for trench depth monitoring. Secondary parameter extraction acknowledged as future work requiring additional measurement diversity.
Constraint analysis: /constraints/inverse-rcwa
Methods: RCWA, Identifiability, Dispersion models