Determination of thickness and optical constants of amorphous silicon films from transmittance data

Abstract

This work presents the application of a recently developed numerical method to determine the thickness and the optical constants of thin films using experimental transmittance data only. This method may be applied to films not displaying a fringe pattern and is shown to work for a−Si:H (hydrogenated amorphous silicon) layers as thin as 100 nm. The performance and limitations of the method are discussed on the basis of experiments performed on a series of six a−Si:H samples grown under identical conditions, but with thickness varying from 98 nm to 1.2 µm. 2Modern electronic devices, such as thin−film transistors, solar cells, active matrix displays and image sensors, possess thin semiconductor layers of hydrogenated amorphous silicon (a−Si:H). For most electronic applications, the optical properties and the thickness t of these films play an important role, in the sense that they govern the device performance. The quality of the as−deposited material can be monitored in production lines through the in−situ determination of its optical constants (refractive index n and extinction coefficient k) and the thickness homogeneity. For that aim, ellipsometry is the most appropriate tool [1] due to the fact that it is not influenced by the adopted substrate. Alternatively, for the ex−situ analysis of samples grown on top of transparent substrates like glass, the use of optical transmittance is the most attractive method because optical transmission is a very easy, accurate and non−destructive measure.The problem of estimating the thickness and the optical constants of thin films using transmission data only represents a very ill−conditioned inverse problem with many local A set of experimental data [λ i , T meas (λ i )], λ min ≤ λ i ≤ λ i+1 ≤ λ max , for i = 1,…,N, is given, and we want to estimate t, n(λ), and k(λ). The problem seems highly underdetermined. In fact, for known t and given λ, the following must hold [5]: T meas (λ) = T theor (λ, s(λ), t, n(λ)where T theor is the calculated transmission of the film+substrate [3] and s the refractive index of the transparent substrate. This equation has two unknowns n(λ) and k(λ) and, in general, its set of solutions (n,k) is a curve in the two−dimensional (n(λ), k(λ)) space. Therefore, the set of functions (n,k) satisfying T meas = T theor for a given t is infinite and, roughly speaking, is represented by a nonlinear manifold of dimension N in R 2n . However, physical constraints (PC) drastically reduce the range of variability of the unknowns n(λ), k(λ). The optimization process looks for a thickness that, subject to the physical input of the problem, minimizes the difference between the measured and the theoretical spectra, i. e.,(1)The minimization process starts sweeping a thickness range ∆t R divided into thickness steps ∆t S and proceeds decreasing ∆t R and ∆t S until the optimized thickness t opt is found. In the examples to follow, the starting ∆t R and ∆t S were 5 µm and 100 nm, respectively.As seen, the most important issue of the present method is the ret...

Determination of Optical Properties of Undoped Amorphous Selenium (a-Se) Films by Dielectric Modeling of Their Normal-Incidence Transmittance Spectra

“…Therefore, in principle, two independent optical measurements are necessary at each spectral wavelength to solve for the unknowns and but, in practice, the true geometric thickness of the film is not known in prior and is thus to be determined at the same time-that is, three unknowns to be retrieved from two independent optical measurements at the same spectral wavelength, which demands adequate model suppositions or an otherwise optical analytical approach (Tan, 2006;Tan et al, 2007;Tan et al, 2006;Vautier et al, 1988;Bettstelleret al, 1993;Mulama et al, 2014;Benkhedir, 2006;Solieman et al, 2014;Adachi & Kao1980;Kotkata & Abdel-Wahab, 1990;Kotkata et al, 2009;Navarrete et al, 1990;Solieman & Abu-Sehly, 2010;Nagels et al, 1995;Tichy et al, 1996;Nagels et al, 1997;Tauc, 1972;Tauc, 1979;Heavens, 1991;Azzam & Bashara, 1987;Ward, 1994;Dragoman & Dragoman, 2002;Stenzel, 2005;Joo et al, 1999;Dobrowolski et al, 1983;Klein et al, 1990;Kukinyi et al, 1996;Chiao et al, 1995;Case, 1983;Peng & Desu, 1994;Birgin et al, 1999;Mulato et al, 2000;Erarslan & Gungor, 2010;Ventura et al, 2005;Manifacier et al, 1976;Swanepoel, 1983;Minkov, 1990;El-Naggar et al, 2009;…”

“…Only when a global minimum solution for the curve-fitting problem is achieved, one obtains reliable and informative dielectric and optical parameters for the studied film. If the substrate in the {air/thin uniform film/thick substrate/air}-stacking is transparent ( ≅ 0) in the spectral region above the absorption edge of the film, the -formula given in Equation (4) reduces to a wieldy expression that forms the basis of the analytical envelope and PUMA methods (Birgin et al, 1999;Mulato et al, 2000;Erarslan & Gungor, 2010;Ventura et al, 2005;Manifacier et al, 1976;Swanepoel, 1983;Minkov, 1990;El-Naggar et al, 2009;Epstein et al, 1987;González-Leal et al, 1998;Tigau, 2006;Poelman & Smet, 2003;Swanepoel, 1984;Chambouleyron & Martínez, 2001;Truong & Tanemura, 2006;Kasap & Capper, 2006), the application of which do not necessitate in prior dispersion functions for the film's optical constants.…”

“…When a global optimum solution of the fitting problem is realized, the output parameters will be evocative and physically meaningful; otherwise, local multi solutions are obtained, and hence the output parameters must be identified with utmost care to reject vulnerable ones and select those yield reliable solution, a job that is virtually challenging unless it is assisted by related independent findings. The Pointwise Unconstrained Minimization Approach (PUMA) numerical method (Birgin et al, 1999;Mulato et al, 2000;Erarslan & Gungor, 2010;Ventura et al, 2005) and algebraic methods (Peng & Desu, 1994;Birgin et al, 1999), based on approximate equations derived from reasonable assumptions, were widely used to extract spectral dispersion of optical constants of a film and its geometric thickness from normal-incidence -and/or -spectra of its air-supported {thin film/thick substrate}-stack, both of which do not require in prior a dispersion functional model. An algebraic method in this category is the envelope method, developed and refined by a number of authors ( Manifacier et al, 1976;Swanepoel, 1983;Minkov, 1990;El-Naggar et al, 2009;Epstein et al, 1987;González-Leal et al, 1998;Tigau, 2006;Poelman & Smet, 2003), whose viability is based on the presence of many interference-fringe maxima and minima on transmittance (reflectance) spectra in transparent and weak absorption regions of the film, and cannot be applied to spectra displaying few interference fringes.…”

Preparation, structure and optical properties of transparent conducting gallium-doped zinc oxide thin films

“…From the measured transmittance spectra, the optical constants of the GZO thin films were determined using the method of optical spectrum fitting [46]. Fig.…”

Tópicos em otimização com restrições lineares

“…Comparamos os resultados com Algencan-G, Algencan-B, ve11 [47], Minos [41], Ipopt [54] e Lancelot B [18,19,20], outros conhecidos métodos para a resolução de (2.1). Apresentamos, também, os resultados da aplicação das versões dos métodos novosà resolução do problema de estimar a espessura e constantesópticas de filmes finos [5,6,16,39]. Finalmente, na Seção 2.8 apresentamos as conclusões obtidas.…”