Dear Dr. Palakkandy,

Are you trying to perform point-by-point fitting? If so, the point-by-point method should be used with great caution. First of all, the correct initial model must be used for such direct data inversion, i.e., all other parameters, except the n&k values for the material under study, need to be known with high accuracy, otherwise those modeling errors can lead to unphysical dispersion or discontinuities in the optical constants. Moreover, existing measurement noise will be directly translated into extracted n&k values. In addition, the method does not enforce Kramers-Kronig (K-K) consistency between the real and imaginary part of the obtained optical constants (or, dielectric function) and, therefore, can result in unphysical shape of the (n,k)-curve in the considered spectral range.

Many analytical physics-based and Kramers-Kronig consistent expressions have been developed which describe various types of materials – amorphous and crystalline semiconductors and dielectrics, metals, organic films, optical metamaterials, etc. One of the first oscillator models, widely used for the optical modeling of metals and other conductive materials (for instance, indium tin oxide (ITO)), is simple phenomenological Drude–Lorentz (DL) oscillator model [for instance, see R.W. Collins, A.S. Ferlauto, Optical physics of materials, in: H.G. Tompkins, E.A. Irene (Eds.), Handbook of Ellipsometry, William Andrew Publishing/ Noyes, Norwich NY, 2005, p. 93]. Another well-known optical function parameterization has been developed by Forouhi and Bloomer for amorphous semiconductors and dielectrics [A.R. Forouhi, I. Bloomer, Optical dispersion relations for amorphous semiconductors and amorphous dielectrics, Phys. Rev. B 34 (1986) 7018] and later for crystalline materials [A.R. Forouhi, I. Bloomer, Optical properties of crystalline semiconductors and dielectrics, Phys. Rev. B 38 (1988) 1865]. Although the FB model fits satisfactory certain materials, in many cases it provides reasonable fit only in very limited spectral range or even yields unphysical results. Because of that some modifications of the original FB model have been derived by McGahan et al. [W.A. McGahan, T. Makovicka, J. Hale, J.A. Woollam, Modified Forouhi and Bloomer dispersion model for the optical constants of amorphous hydrogenated carbon thin films, Thin Solid Films 253 (1994) 57], Liu et al. [Y. Liu, G. Xu, C. Song, W. Weng, P. Du, G. Han, Modification on Forouhi and Bloomer model for the optical properties of amorphous silicon thin films, Thin Solid Films 515 (2007) 3910] and Laidani et al. [N. Laidani, R. Bartali, G. Gottardi, M. Anderle, P. Cheyssac, Optical absorption parameters of amorphous carbon films from Forouhi–Bloomer and Tauc–Lorentz models: a comparative study, J. Phys.: Condens. Matter 20 (2008) 015216]. More realistic Tauc-Lorentz (TL) and Cody-Lorentz (CL) parameterizations for the optical functions of amorphous materials have been proposed by Jellison and Modine [G.E. Jellison, Jr., F.A. Modine, Parameterization of the optical functions of amorphous materials in the interband region, Appl. Phys. Lett. 69 (1996) 371–373; Erratum: “Parameterization of the optical functions of amorphous materials in the interband region” [Appl. Phys. Lett. 69, 371 (1996)] idid. 69 (1996) 2137] and Ferlauto et al. [A.S. Ferlauto, G.M. Ferreira, J.M. Pearce, C.R. Wronski, R.W. Collins, X. Deng, G. Ganguly Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics, J. Appl. Phys. 92 (2002) 2424].

Those analytical models describes the imaginary part of the complex dielectric function and the real part of the dielectric function is obtained by the Kramers–Kronig integration, - therefore, those models are K-K consistent and can be used to fit the experimental data.

An alternative approach to express the dielectric function of the thin film is to use a parameterization by polynomial spline functions, which does not required any assumptions about the film properties and interaction of light with the material. Recently, Johs and Hale suggested a Kramers-Kronig consistent B-spline representation for the dielectric functions which has been successfully utilized to multiple applications [B. Johs, J.S. Hale, Dielectric function representation by B-splines, phys. status solidi (a) 205 (2008) 715]

Thanks Brain. Will try to get the book and go through it.

A little bit more advanced and modern texts on the theory of ellipsometry:

Handbook of Ellipsometry, Edited by H. G. Tompkins and E. A. Irene, 2005

Online version: http://www.sciencedirect.com/science/book/9780815514992#ancPT2

Hiroyuki Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, 2007

Online version: http://onlinelibrary.wiley.com/book/10.1002/9780470060193

Dr Likhachev, I have tried my hand at writing a code for fitting models to my ellipsometry data but am having problem in how to deal with the KK Integral relating the real epsilon to the imaginary epsilon. Can you give some help?

Dear Dr. Palakkandy,

Are you trying to perform point-by-point fitting? If so, the point-by-point method should be used with great caution. First of all, the correct initial model must be used for such direct data inversion, i.e., all other parameters, except the n&k values for the material under study, need to be known with high accuracy, otherwise those modeling errors can lead to unphysical dispersion or discontinuities in the optical constants. Moreover, existing measurement noise will be directly translated into extracted n&k values. In addition, the method does not enforce Kramers-Kronig (K-K) consistency between the real and imaginary part of the obtained optical constants (or, dielectric function) and, therefore, can result in unphysical shape of the (n,k)-curve in the considered spectral range.

Many analytical physics-based and Kramers-Kronig consistent expressions have been developed which describe various types of materials – amorphous and crystalline semiconductors and dielectrics, metals, organic films, optical metamaterials, etc. One of the first oscillator models, widely used for the optical modeling of metals and other conductive materials (for instance, indium tin oxide (ITO)), is simple phenomenological Drude–Lorentz (DL) oscillator model [for instance, see R.W. Collins, A.S. Ferlauto, Optical physics of materials, in: H.G. Tompkins, E.A. Irene (Eds.), Handbook of Ellipsometry, William Andrew Publishing/ Noyes, Norwich NY, 2005, p. 93]. Another well-known optical function parameterization has been developed by Forouhi and Bloomer for amorphous semiconductors and dielectrics [A.R. Forouhi, I. Bloomer, Optical dispersion relations for amorphous semiconductors and amorphous dielectrics, Phys. Rev. B 34 (1986) 7018] and later for crystalline materials [A.R. Forouhi, I. Bloomer, Optical properties of crystalline semiconductors and dielectrics, Phys. Rev. B 38 (1988) 1865]. Although the FB model fits satisfactory certain materials, in many cases it provides reasonable fit only in very limited spectral range or even yields unphysical results. Because of that some modifications of the original FB model have been derived by McGahan et al. [W.A. McGahan, T. Makovicka, J. Hale, J.A. Woollam, Modified Forouhi and Bloomer dispersion model for the optical constants of amorphous hydrogenated carbon thin films, Thin Solid Films 253 (1994) 57], Liu et al. [Y. Liu, G. Xu, C. Song, W. Weng, P. Du, G. Han, Modification on Forouhi and Bloomer model for the optical properties of amorphous silicon thin films, Thin Solid Films 515 (2007) 3910] and Laidani et al. [N. Laidani, R. Bartali, G. Gottardi, M. Anderle, P. Cheyssac, Optical absorption parameters of amorphous carbon films from Forouhi–Bloomer and Tauc–Lorentz models: a comparative study, J. Phys.: Condens. Matter 20 (2008) 015216]. More realistic Tauc-Lorentz (TL) and Cody-Lorentz (CL) parameterizations for the optical functions of amorphous materials have been proposed by Jellison and Modine [G.E. Jellison, Jr., F.A. Modine, Parameterization of the optical functions of amorphous materials in the interband region, Appl. Phys. Lett. 69 (1996) 371–373; Erratum: “Parameterization of the optical functions of amorphous materials in the interband region” [Appl. Phys. Lett. 69, 371 (1996)] idid. 69 (1996) 2137] and Ferlauto et al. [A.S. Ferlauto, G.M. Ferreira, J.M. Pearce, C.R. Wronski, R.W. Collins, X. Deng, G. Ganguly Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics, J. Appl. Phys. 92 (2002) 2424].

Those analytical models describes the imaginary part of the complex dielectric function and the real part of the dielectric function is obtained by the Kramers–Kronig integration, - therefore, those models are K-K consistent and can be used to fit the experimental data.

An alternative approach to express the dielectric function of the thin film is to use a parameterization by polynomial spline functions, which does not required any assumptions about the film properties and interaction of light with the material. Recently, Johs and Hale suggested a Kramers-Kronig consistent B-spline representation for the dielectric functions which has been successfully utilized to multiple applications [B. Johs, J.S. Hale, Dielectric function representation by B-splines, phys. status solidi (a) 205 (2008) 715]

Thanks Sir. Infact to be very honest ellipsometry is a new topic I have started and like you said, trying point-to-point fitting is dangeorus. But if you look at the various models being used and the sources of contribution to refractive index etc and also the number of variable parameters, all in all ending as a polynomial fitting (with over 200 data points in principle you can fit polynomial of order 200), one is going to get a fit!!! How worthy of physics it is ofcourse another matter (faster processors dont sweat to compute).

I shall look into the references you cited and see if I can use any of it.

Data interpolation by splines doesn't requre the polinomials of very large degree, in fact. I specially referenced to the paper by Johs and Hale on so-called B-spline interpolation [B. Johs, J.S. Hale, Dielectric function representation by B-splines, phys. status solidi (a) 205 (2008) 715].

These are more references on applications of such method to real ellipsometric studies:

J.W. Weber, T.A.R. Hansen, M.C.M. van de Sanden, R. Engeln, B-spline parametrization of the dielectric function applied to spectroscopic ellipsometry on amorphous carbon, J. Appl. Phys. 106 (2009) 123503;

J.W. Weber, V.E. Calado, M.C.M. van de Sanden, Optical constants of graphene measured by spectroscopic ellipsometry, Appl. Phys. Lett. 97 (2010) 091904;

S.G. Choi, J. Zúñiga-Pérez, V. Muñoz-Sanjosé, A.G. Norman, C.L. Perkins, D.H. Levi, Complex dielectric function and refractive index spectra of epitaxial CdO thin film grown on r-plane sapphire from 0.74 to 6.45 eV, J. Vac. Sci. Technol. B 28 (2010) 1120;

H.T. Beyene, J.W. Weber, M.A. Verheijen, M.C.M. van de Sanden, M. Creatore, Real time in situ spectroscopic ellipsometry of the growth and plasmonic properties of Au nanoparticles on SiO2, Nano Res. 5 (2012) 513;

S.G. Choi, H.Y. Zhao, C. Persson, C.L. Perkins, A.L. Donohue, B. To, A.G. Norman, J. Li, I.L. Repins, Dielectric function spectra and critical-point energies of Cu2ZnSnSe4 from 0.5 to 9.0 eV, J. Appl. Phys. 111 (2012) 033506;

A. Jimenez, D. Lepage, J. Beauvais, J.J. Dubowski, Study of surface morphology and refractive index of dielectric and metallic films used for the fabrication of monolithically integrated surface plasmon resonance biosensing devices, Microelectron. Eng. 93 (2012) 91.

Good luck!