Ellipsometry Measurement

Invert ellipsometry data (Ψ, Δ) to the pseudo-dielectric function and pseudo-refractive index (two-phase ambient/substrate model), or to the thickness of a transparent film of known index on a known substrate.

Model: Exact

Ψ (degrees)Δ (degrees)Angle of Incidence (°)

Default: bulk Si at 633 nm and 70° (N = 3.882 − 0.019i in this page’s N = n − ik convention)

Pseudo-refractive index ⟨n⟩: 3.8820

Pseudo-extinction coeff ⟨k⟩: 0.0190

⟨ε₁⟩: 15.0699

⟨ε₂⟩: 0.1475

R_p at θ: 0.0242

R_s at θ: 0.6947

R_avg at θ: 0.3595

Film thickness (transparent film of known index)

Wavelength (nm)Film index n₁ (lossless)Substrate nSubstrate k (N = n − ik)

Default: thermal SiO₂ (n₁ = 1.457) on Si at 632.8 nm. The default Ψ, Δ are bare Si, so d ≈ 0.

Film thickness d: 0.00 nm + m·D_φ, m = 0, 1, 2, …

Thickness period D_φ: 284.16 nm

|X| (1 for a consistent fit): 1.0000

Ψ and Δ repeat every D_φ, so one measurement fixes d only modulo D_φ: a second angle or wavelength (or a rough idea of d) picks m. Where the white curve moves slowly with d (for SiO₂ on Si at 70°, around Ψ ≈ 34°, Δ ≈ 80°), small errors in Ψ and Δ move d the most; near its top (Ψ → 90°) d is pinned best.

Key Formulas

ρ = r_p/r_s = tan(Ψ)·e^iΔ

⟨ε⟩ = sin²θ [1 + tan²θ·((1−ρ)/(1+ρ))²] (pseudo-dielectric function)

⟨N⟩ = √⟨ε⟩ = ⟨n⟩ − i⟨k⟩, ⟨ε⟩ = ⟨ε₁⟩ − i⟨ε₂⟩ (Nebraska convention: Δ = 180° at normal incidence)

Exact for a bare substrate; with a film on top these are “pseudo” values, not material constants.

Film: R_ν = (r_01ν + r_12νX)/(1 + r_01νr_12νX), X = e^−i2β, β = 2πd·n₁cos φ₁/λ

ρ = R_p/R_s is a quadratic in X (Azzam & Bashara): take the root with |X| nearest 1; d = (−arg X)/(2π)·D_φ

D_φ = λ / (2√(n₁² − sin²θ))

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⟨n⟩ over the Ψ-Δ plane at the current angle of incidence. The white curve is the film model from d = 0 (bare substrate) to one period; the red × is the measurement.

Model and references

Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers

Two-phase inversion, exact for a bare isotropic substrate (else ⟨n⟩, ⟨k⟩ are pseudo-values). Film: exact three-phase inversion for one lossless film of known index, modulo the period D_φ; |X| ≠ 1 flags a misfit.

Azzam R. M. A., Bashara N. M. (1977). Ellipsometry and Polarized Light, ch. 4 (ambient-substrate and ambient-film-substrate systems). North-Holland.
Aspnes D. E. (1985). The accurate determination of optical properties by ellipsometry. In Palik E. D. (ed.), Handbook of Optical Constants of Solids, 89–112. Academic Press. doi:10.1016/b978-0-08-054721-3.50010-1

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