SiO₂ Thickness Measurement

Overview

When a Silicon wafer grows an oxide layer it picks up a visible color. This thin-film interference can be seen in more common things like soap bubbles. The color for Silicon Oxide however, encodes its thickness. I built a Python tool to decode it by measuring the reflectance spectrum using an OceanOptics HR2000+ and get back a SiO₂ thickness in nanometers.

This is used in Prof. Gokirmak's semiconductor fabrication lab to ensure proper characterization of wafers. Oxide thickness governs dielectric characteristics, anti-contamination, masking capability, and isolation potential. Being able to measure it with any physical contact on a nanometer-scale film is essentially impossible without damaging the structure. This is an optical, non destructive means of measuring the wafer with high accuracy.

The tool targets thin films in the 1–200 nm range, where the oxide is too thin to produce repeating interference fringes. That rules out the Fast Fourier Transform approach used for thicker films. Instead, thickness is extracted by fitting a Fresnel reflection model to the absolute reflectance measurements using bounded least-squares optimization.

How it works / the build

The Optical Fiber. The coupled fiber sits in a specialized 3D-printed mount that positions it in the center of a microscope eyepiece tube. A tungsten halogen source illuminates the sample and the HR2000+ captures the reflected spectrum to within 0.2 nm resolution.

Getting clean reflectance. Raw intensity from the spectrometer is contaminated by a wavelength-dependent coefficient S(λ) derived from the fiber coupling, lamp spectral shape, and ambient light. It can't be characterized directly, so instead you take a reference measurement off bare silicon with the exact same setup, and it divides out:

I_sample = S(λ) × R_sample
I_Si     = S(λ) × R_Si
I_sample / I_Si = R_sample / R_Si
(R_sample / R_Si) × R_Si = R_sample

Dark measurements. These are taken with a silicon wafer at a 45 degree angle under the light to reflect away all incident light and ideally expose sources of light leakage in. These are subtracted from both sample and reference first, like zeroing a scale before weighing. Skipping this step can lead to nanometers of error.

The Fresnel model. The SiO₂ can be treated as a thin film on a "semi-infinite" silicon substrate. Silicon's high absorption means light never reaches the back face, so only two interfaces matter: air–oxide and oxide–silicon. Summing the partial reflections with Fresnel equations gives the total amplitude reflectance coefficient r_total, which depends only on known wavelength λ and unknown oxide thickness d:

r_total = (r₁₂ + r₂₃ e^i2δ) / (1 + r₁₂r₂₃ e^i2δ)
δ = 2π n₂(λ) d / λ   // phase round-trip through the film
Power Reflectance = |r_total|²

Wavelength-dependent indices. SiO₂'s index uses the Sellmeier equation with well-documented coefficients. Silicon's complex index (real + imaginary, accounting for absorption) is trickier so I inserted 400 data points each for the real and imaginary components and fit a linear interpolation across the visible range. Its close enough for the application, and much simpler than a full dispersion model.

Optimization. The program runs the model at every wavelength in the measured spectrum and finds the value of d that minimizes error using scipy.optimize.minimize_scalar with method='bounded'. The bounds (currently 1–300 nm) prevent the optimizer from converging on a mirrored fringe cycle that would give the same reflectance curve at the wrong thickness.

Results & measurements

The strongest result is a wafer found in the lab labeled as ~100 nm with unknown fabrication history. Without dark subtraction: 96.70 nm, RMSE 0.0356. After dark subtraction: 90.36 nm, RMSE 0.0097. The fit is visibly cleaner with the vertical offset removed.

The noise is structural. Between the short integration time due to oversaturating the spectrometer, as well as the spectrometer possibly having a steep falloff range for wavelengths 450nm or less (according to Ocean Optics' website), it is possible I may need to truncate that data to promote accuracy.

The model currently returns no result outside the 1–300 nm bounds — it fails explicitly, flagging that it isnt prepared to handle thick films. The FFT coarse-ranging planned for the next version will relax this constraint.

Challenges & what's next

The fixed optimization bounds are the main limitation. SiO₂ reflectance curves repeat at fringe multiples, so the optimizer can converge confidently on the wrong alias if the bounds don't already bracket the true thickness. The solution is a coarse FFT pass first: use the frequency-domain fringe spacing to identify which ~100 nm window the thickness falls in, then let the Fresnel model refine to single-digit nanometers within that window.

The thin-film model also breaks for thicker oxides as once fringes become dense enough to resolve the FFT is actually the better primary tool anyway, and the two methods should hand off automatically. The current code handles neither case gracefully.

Longer term: the model is material-agnostic. With the right index tables it could measure oxide on silver, or nitride stacks, or anything with a well-characterized refractive index. That's prospectively interesting once the thin-SiO₂ case is fully validated.

Gallery