The Simultaneous Determination of Thickness and Complex Refractive Index of a Thin Film on a Substrate
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Date
2007-08-07
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Abstract
The problem of the simultaneous determination on a wavelength-by-wavelength basis of the complex refractive index ñ = n + iκ and thickness d of an isotropic thin film deposited on an isotropic substrate has never been solved analytically over the last 35 years. Here I find the analytic solution of the three-phase model in the limit d/λ << 1 on a wavelength-by-wavelength (λ-by-λ) basis from polarimetric data Δρ/ρ and ΔR⁄R, where ρ and R are the complex reflectance ratio and either the p- or s-polarized power reflectance, respectively. My relatively simple quadratic equation eliminates several disadvantages of previous numerical-analysis approaches, including intrinsic instabilities and a need for accurate estimates of starting parameters for convergence, which become particularly acute as d⁄λ→0. I also investigate the source of the numerical difficulties, the correlations in both parameters and data, with an approach based on the Jacobian of the exact equations. These correlations prevent a simple quadrature addition of uncertainties in Δρ⁄ρ and ΔR⁄R in the calculation of uncertainties in the parameters. I also show that a simple mathematical transformation of the data can improve the accuracy of the first-order equations. The results will be useful directly for optically analyzing films with d ≤ 1 nm, and for providing starting values for more accurate numerical calculations for thicker films. Results are demonstrated for the cyclic physisorption and desorption of a layer of H₂O on oxidized GaAs, and as numerical estimates of performance specifications for application to several materials combinations important to semiconductor technology.
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Keywords
polarimetry, thin film, analytic solution, ellipsometry, three-phase model
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Degree
PhD
Discipline
Electrical Engineering
