The Simultaneous Determination of Thickness and Complex Refractive Index of a Thin Film on a Substrate

polarimetry;thin film;analytic solution;ellipsometry;three-phase model

Kim, In Kyo ; Gerald J. Iafrate, Committee Member,David E. Aspnes, Committee Co-Chair,John F. Muth, Committee Chair,Salah M. Bedair, Committee Member,Kim, In Kyo ; Gerald J. Iafrate ; Committee Member ; David E. Aspnes ; Committee Co-Chair ; John F. Muth ; Committee Chair ; Salah M. Bedair ; Committee Member

University:North Carolina State University

关键词: polarimetry;    thin film;    analytic solution;    ellipsometry;    three-phase model;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/3001/etd.pdf?sequence=1&isAllowed=y

美国|英语

来源: null

【 摘 要 】

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.