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Results concerning the optical characterization of two inhomogeneous polymer-like thin films deposited by the plasma enhanced chemical vapor deposition onto silicon single crystal substrates are presented. One of these films is deposited onto a smooth silicon surface while the latter film is deposited on a randomly rough silicon surface with a wide interval of spatial frequencies. A combination of variable-angle spectroscopic ellipsometry and spectroscopic reflectometry applied at near-normal incidence are utilized for characterizing both the films. An inhomogeneity of the films is described by the method based on multiple-beam interference of light and method replacing inhomogeneous thin films by multilayer systems. Homogeneous transition layers between the films and substrates are considered. The Campi–Coriasso dispersion model is used to express spectral dependencies of the optical constants of the polymer-like films and transition layers. A combination of the scalar diffraction theory and Rayleigh–Rice theory is used to include boundary roughness into formulae for the optical quantities of the rough polymer-like film. Within the optical characterization, the spectral dependencies of the optical constants at the upper and lower boundaries of both the polymer-like films are determined together with their thickness values and profiles of the optical constants. Roughness parameters are determined for the rough film. The values of the roughness parameters are confirmed by atomic force microscopy. Moreover, the optical constants and thicknesses of both the transition layers are determined. A discussion of the achieved results for both the polymer-like films and transition layers is performed.

In this study, a novel approach for characterizing the optical properties of inhomogeneous thin films is presented, with a particular focus on samples exhibiting absorption in some part of the measured spectral range. Conventional methods of measuring the samples only from the film side can be limited by incomplete information at the lower boundary of the film, leading to potentially unreliable results. To address this issue, depositing the thin films onto non-absorbing substrates to enable measurements from both sides of the sample is proposed. To demonstrate the efficacy of this approach, a combination of variable-angle spectroscopic ellipsometry and spectrophotometry at near-normal incidence was employed to optically characterize three inhomogeneous polymer-like thin films. The spectral dependencies of the optical constants were modeled using the Kramers–Kronig consistent model. It was found that it is necessary to consider thin, weakly absorbing transition layers between the films and the substrates. The obtained results show excellent agreement between the fits and the measured data, providing validation of the structural and dispersion models, as well as the overall characterization procedure. The proposed approach offers a method for optically characterizing a diverse range of inhomogeneous thin films, providing more reliable results when compared to traditional one-sided measurements.

An inhomogeneous polymer-like thin film was deposited by the plasma enhanced chemical vapor deposition onto silicon single-crystal substrate whose surface was roughened by anodic oxidation. The inhomogeneous thin film with randomly rough boundaries was created as a result. This sample was studied using the variable-angle spectroscopic ellipsometry and spectroscopic reflectometry. The structural model including the inhomogeneous thin film, transition layer, and identically rough boundaries was used to process the experimental data. The scalar diffraction theory was used to describe the influence of roughness. The influence of the scattered light registered by the spectrophotometer due to its finite acceptance angle was also taken into account. The thicknesses and optical constants of the inhomogeneous thin film and the transition layer were determined in the optical characterization together with the roughness parameters. The determined rms value of the heights of roughness was found to be in good agreement with values obtained using AFM. The results of the optical characterization of the studied inhomogeneous thin film with rough boundaries were also verified by comparing them with the results of the optical characterization of the inhomogeneous thin film prepared using the same deposition conditions but onto the substrate with a smooth surface. (C) 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

In this paper the overview of the most important approximate methods for the optical characterization of inhomogeneous thin films is presented. The following approximate methods are introduced: Wentzel–Kramers–Brillouin–Jeffreys approximation, method based on substituting inhomogeneous thin films by multilayer systems, method based on modifying recursive approach and method utilizing multiple-beam interference model. Principles and mathematical formulations of these methods are described. A comparison of these methods is carried out from the practical point of view, ie advantages and disadvantages of individual methods are discussed. Examples of the optical characterization of three inhomogeneous thin films consisting of non-stoichiometric silicon nitride are introduced in order to illustrate efficiency and practical meaning of the presented approximate methods.

Dispersion models are necessary for precise determination of the dielectric response of materials used in optical and microelectronics industry. Although the study of the dielectric response is often limited only to the dependence of the optical constants on frequency, it is also important to consider its dependence on other quantities characterizing the state of the system. One of the most important quantities determining the state of the condensed matter in equilibrium is temperature. Introducing temperature dependence into dispersion models is quite challenging. A physically correct model of dielectric response must respect three fundamental and one supplementary conditions imposed on the dielectric function. The three fundamental conditions are the time-reversal symmetry, Kramers-Kronig consistency and sum rule. These three fundamental conditions are valid for any material in any state. For systems in equilibrium there is also a supplementary dissipative condition. In this contribution it will be shown how these conditions can be applied in the construction of temperature dependent dispersion models. Practical results will be demonstrated on the temperature dependent dispersion model of crystalline silicon.

In this paper the complete optical characterization of an inhomogeneous polymer-like thin film of SiO x C y H z exhibiting a thickness non-uniformity and transition layer at the boundary between the silicon substrate and this film is performed using variable angle spectroscopic ellipsometry. The Campi-Coriasso dispersion model was utilized for describing the spectral dependencies of the optical constants of the SiO x C y H z thin film and transition layer. The multiple-beam interference model was used for expressing inhomogeneity of the SiO x C y H z thin film. The thickness non-uniformity of this film was taken into account by means of the averaging of the elements of the Mueller matrix performed using the thickness distribution for the wedge-shaped non-uniformity. The spectral dependencies of the optical constants of the SiO x C y H z thin film at the upper and lower boundaries together with the spectral dependencies of the optical constants of the transition layer were determined. Furthermore, the thickness values of the SiO x C y H z film and transition layer, profiles of the optical constants of the SiO x C y H z thin film and the rms value of local thicknesses corresponding to its thickness non-uniformity were determined. Thus, all the parameters characterizing this complicated film were determined without any auxiliary methods.

In this paper the exact approach of the Rayleigh–Rice theory enabling us to calculate optical quantities of multi-layer systems with boundaries exhibiting slight random roughness is presented. This approach is exact in the sense that it takes into account the propagation of perturbed electromagnetic fields (waves) among randomly rough boundaries including all cross-correlation and auto-correlation effects. The restriction to the second order of perturbation, which is the lowest order that gives nonzero corrections to coherent waves (obeying the Snell’s law), represents the only approximation used in our calculations. It is assumed that the layers and the substrates are formed by optically homogeneous and isotropic materials. The formulae obtained in the theoretical part are used to investigate the influence of layer thicknesses and roughness parameters on reflectances and associated ellipsometric parameters of the selected numerical examples of a three-layer system. The presented approach represents the generalization of the exact approach for single-layer systems and the improvement of the approximate approach for multi-layer systems published earlier. The exact approach of the RRT has a substantial importance for the optical characterization of multi-layer systems occurring in applied research and optics industry applications.

A new approach to calculation of optical quantities of inhomogeneous layers is presented. In this approach, the Richardson extrapolation is used to improve the accuracy of the method, in which the inhomogeneous layer is approximated by a stack of thin homogeneous layers. The results presented in this paper are based on the assumption that the media are isotropic and the inhomogeneity is along the axis normal to the boundary. The results obtained by the new method are compared with those obtained without the Richardson extrapolation. The Richardson extrapolation brings significant improvement in accuracy, especially if the number of approximating layers is large. Moreover, the method using the Richardson extrapolation proceeds in steps with an error estimate available in each step; thus, the calculation can be stopped when the desired accuracy is reached. The use of the method is illustrated by means of the optical characterization of strongly inhomogeneous film of non-stoichiometric silicon nitride.

Results of the optical characterization of randomly rough silicon surfaces covered with native oxide layers based on processing experimental data obtained by ellipsometry and reflectometry are presented. It is shown that the Rayleigh-Rice theory is suitable theoretical approach for characterizing micro-rough surfaces in contrast to effective medium approximation. Combination of the Rayleigh-Rice theory and scalar diffraction theory is efficient and reliable approach for characterizing rougher surfaces with the rms values of heights larger than 10 nm. Thickness of native oxide layers and roughness parameters, ie, the rms values of heights and autocorrelation lengths, are determined for micro-rough and rougher surfaces using the corresponding theoretical approaches.

In this paper epitaxial ZnSe thin films prepared by molecular beam epitaxy onto GaAs single crystal substrates exhibiting two defects, i.e. boundary roughness and thickness non-uniformity, are optically characterized using a combination of spectroscopic ellipsometry and near-normal spectroscopic reflectometry. The influence of boundary roughness is included into optical quantity formulae by the combination of the scalar diffraction theory and Rayleigh–Rice theory. Very thin overalyers modelled by rough thin films with identically rough boundaries are taken into account on the upper boundaries of the ZnSe thin films. Two approximations are used to express the local reflection coefficient of the rough ZnSe thin films covered with the overlayers within combination of both the theories. Thickness non-uniformity is incorporated by means of averaging the elements of the unnormalized Mueller matrices. The universal dispersion model of the optical constants of the ZnSe thin films based on parametrization of the joint density of electronic states is used. The spectral dependencies of the optical constants of the ZnSe thin films are determined within the wide spectral range (0.12–8.7 eV). Moreover, the mean thickness of the ZnSe thin films and thickness of overlayers are determined together with the other structural parameters characterizing the defects. The values of roughness parameters, determined by the optical method, are verified by a comparison with results achieved by atomic force microscopy. It is also shown that the approximations of the local reflection coefficient presented are usable for processing the experimental data.

Optical characterization of magnesium fluoride thin films is performed in a wide spectral range from far infrared to extreme ultraviolet (0.01-45 eV) utilizing the universal dispersion model. Two film defects, i.e. random roughness of the upper boundaries and defect transition layer at lower boundary are taken into account. An extension of universal dispersion model consisting in expressing the excitonic contributions as linear combinations of Gaussian and truncated Lorentzian terms is introduced. The spectral dependencies of the optical constants are presented in a graphical form and by the complete set of dispersion parameters that allows generating tabulated optical constants with required range and step using a simple utility in the newAD2 software package.

Hafnium oxide (HfO2) thin films could be potentially used as thermal diffusion barrier coatings for copper absorbers used in solar thermal collectors. Atomic layer deposition (ALD) was used to deposit thin, dense, non-columnar HfO2 films on copper sheets, which are established base materials for solar absorbers. The aim of this work is to find a simple and efficient method for quantitative characterization of thin hafnium oxide films deposited on imperfect rough copper sheets. The difference ellipsometry applied in infrared region was found to be a practical tool for non-destructive characterization of thin films deposited onto metal surfaces even when these surfaces are imperfect. The presented method enables us not only to identify the presence of hafnium oxide but also to perform quantitative characterization, i.e. to determine the thickness of the film. Moreover, a simple method, in which the thickness is determined from the height of the structure in the difference ellipsometric data, is presented. This method is demonstrated on HfO2 film with nominal thickness 60 nm deposited by ALD on copper cold-rolled sheet.

The modification of the effective medium approximation for randomly microrough surfaces covered by very thin overlayers based on inhomogeneous fictitious layers is formulated. The numerical analysis of this modification is performed using simulated ellipsometric data calculated using the Rayleigh–Rice theory. The system used to perform this numerical analysis consists of a randomly microrough silicon single crystal surface covered with a SiO2 overlayer. A comparison to the effective medium approximation based on homogeneous fictitious layers is carried out within this numerical analysis. For ellipsometry of the system mentioned above the possibilities and limitations of both the effective medium approximation approaches are discussed. The results obtained by means of the numerical analysis are confirmed by the ellipsometric characterization of two randomly microrough silicon single crystal substrates covered with native oxide overlayers. It is shown that the effective medium approximation approaches for this system exhibit strong deficiencies compared to the Rayleigh–Rice theory. The practical consequences implied by these results are presented. The results concerning the random microroughness are verified by means of measurements performed using atomic force microscopy.

Five dispersion models of interband electronic transitions in disordered solids based on the combination of the Tauc's law and the Lorentz model (or the Lorentz function) will be discussed. These models are the Campi–Coriasso, Jellison–Modine (Tauc–Lorentz), Ferlauto et al. (Cody–Lorentz) and we also propose two other models. The models that we propose can be considered to be modifications of the Jellison–Modine and Ferlauto et al. models. This modification consists of the fact that these models are based on the Lorentz function, not on the response function of the damped harmonic oscillator. The analytic expressions for the real part of the dielectric function of the Campi–Coriasso model and one of our models will be published. The differences between the presented models will be compared by means of fitting the optical constants (functions) calculated with one model by other models.

The study was devoted to optical characterization of non-stoichiometric silicon nitride films prepared by reactive magnetron sputtering in argon-nitrogen atmosphere onto cold (unheated) substrates. It was found that these films exhibit the combination of three defects: optical inhomogeneity (refractive index profile across the films), uniaxial anisotropy with the optical axis perpendicular to the boundaries and random roughness of the upper boundaries. The influence of the uniaxial anisotropy was included into the corresponding formulae of the optical quantities using the matrix formalism and the approximation of the inhomogeneous layer by a multilayer system consisting of large number thin homogeneous layers. The random roughness was described using the scalar diffraction theory. The processing of the experimental data was performed using the multi-sample modification of the least-squares method, in which experimental data of several samples differing in thickness were processed simultaneously. The dielectric response of the silicon nitride films was modeled using the modification of the universal dispersion model, which takes into account absorption processes corresponding to valence-to-conduction band electron transitions, excitonic effects and Urbach tail. The spectroscopic reflectometric and ellipsometric measurements were supplemented by measuring the uniformity of the samples using imaging spectroscopic reflectometry of the samples using imaging spectroscopic reflectometry. Je studována optická charakterizace nestechiometrických vrstev nitridu křemíku připravených reaktivním magnetronovým naprašováním na chladné (nezahřívané) podložky v argonové – dusíkově atmosféře. Bylo zjištěno, že tyto vrstvy vykazují kombinaci tří defektů: optické nehomogenity (profil indexu lomu napříč vrstvami), jednoosé anizotropie s optickou osou kolmou k rozhraním a náhodné drsnosti horních rozhraní. Vliv jednoosé anizotropie byl zahrnut do odpovídajících vzorců pro optické veličiny užitím maticového formalizmu a aproximací nehomogenní vrstvy mnohavrstevným systémem velkého počtu tenkých homogenních vrstev. Náhodná drsnost byla popsána v rámci skalární teorie difrakce. Zpracování experimentálních dat bylo provedeno užitím mnohavzorkové modifikace metody nejmenších čtverců, ve které byla experimentální data několika vzorků lišících se v tloušťce zpracovávána současně. Dielektrická odezva studovaných vrstev byla modelována pomocí modifikace univerzálního disperzního modelu, který bere v úvahu absorpční procesy odpovídající elektronovým přechodům mezi valenčním a vodivostním pásem, exitačním procesům a Urbachovu konci. Spektroskopická reflektometrická a elipsometrická měření byla doplněna měřením uniformity vzorků pomocí zobrazovací spektroskopické reflektometrie.

This review paper is devoted to optics of inhomogeneous thin films exhibiting defects consisting in transition layers, overlayers, thickness nonuniformity, boundary roughness and uniaxial anisotropy. The theoretical approaches enabling the inclusion of these defects into formulae expressing the optical quantities of these inhomogeneous thin films are summarized. These approaches are based on the recursive and matrix formalisms for the transition layers and overlayers, averaging of the elements of the Mueller matrix using local thickness distribution or polynomial formulation for the thickness nonuniformity, scalar diffraction theory and Rayleigh-Rice theory or their combination for boundary roughness and Yeh matrix formalism for uniaxial anisotropy. The theoretical results are illustrated using selected examples of the optical characterization of the inhomogeneous polymer-like thin films exhibiting the combination of the transition layers and thickness nonuniformity and inhomogeneous thin films of nonstoichiometric silicon nitride with the combination of boundary roughness and uniaxial anisotropy. This characterization is realized by variable angle spectroscopic ellipsometry and spectroscopic reflectometry. It is shown that using these optical techniques, the complete optical characterization of the mentioned thin films can be performed. Thus, it is presented that the values of all the parameters characterizing these films can be determined.

The effects of thickness non-uniformity on measured optical quantities must be often considered in the optical characterization. The effects of thickness non-uniformity can be taken into account by averaging the Mueller matrices over the distribution of local thicknesses within the measured area. The distribution of local thicknesses can be assumed in a certain form (e.g. the uniform distribution), or it can be derived on the basis of a model assuming a certain shape of thickness non-uniformity. The latter approach is especially useful for the variable-angle spectroscopic ellipsometry since it can take into account dependence on the incidence angle due to the changes in the size of the light spot. This paper presents results of the optical characterization of three polymer-like thin films highly non-uniform in thickness using variable-angle spectroscopic ellipsometry. The shapes of the thickness non-uniform films are determined on the basis of a model assuming local thicknesses given by quadratic polynomials in coordinates along the surfaces of the films. The studied areas on the films were also measured by the imaging spectroscopic reflectometry, which provides a more direct method to determine local thicknesses. The results achieved using the imaging spectroscopic reflectometry and variable-angle spectroscopic ellipsometry were then compared.

A least-squares data fitting procedure is developed for the analysis of measurements of thin films non-uniform in thickness by imaging spectroscopic reflectometry. It solves the problem of simultaneous least-squares fitting of film thicknesses in all image pixels together with shared dispersion model parameters. Since the huge number of mutually correlated fitting parameters prevents a straightforward application of the standard Levenberg-Marquardt algorithm, the presented procedure exploits the special structure of the specific least-squares problem. The free parameters are split into thicknesses and dispersion model parameters. Both groups of parameters are fitted alternately, utilising an unmodified Levenberg-Marquardt algorithm, correcting however the thicknesses during the dispersion model fitting step to preserve effective optical thicknesses. The behaviour of the algorithm is studied using experimental data of two highly non-uniform thin films of different materials, SiOxCyHz and CNx:H, and by numerical simulations using artificial data. It is found that the optical thickness correction enables the procedure to converge rapidly, permitting the analysis of large imaging spectroscopic reflectometry data sets with reasonable computational resources. © 2015 Elsevier B.V. All rights reserved.

In this chapter the theoretical aspects of ellipsometry and their applications in optics of layered systems are presented. The basic formulae of the theory of ellipsometric measurements are introduced. For this purpose the Jones and Stokes–Mueller matrix formalisms are used. By using these formalisms the individual types of ellipsometry and the most utilized ellipsometric techniques are briefly described. Furthermore, the matrix formalisms enabling us to derive the formulae for the optical quantities of optically isotropic and anisotropic layered systems are described as well. Applications of the matrix formalisms in practice are illustrated by means of three examples.

In this chapter the influence of the main defects on the optical characterization of thin films is described. These defects are random roughness of boundaries, thickness non-uniformity, optical inhomogeneity corresponding to refractive index profiles, overlayers and transition layers. The theoretical approaches and the formulae for the corresponding optical quantities of the thin films exhibiting these defects are presented. The attention is concentrated on the ellipsometric parameters and reflectance of these thin films belonging to the specular reflection. The selected numerical examples illustrating the influence of the defects are introduced. Several experimental examples of the optical characterization of the thin films with the defects are also shown. The discussion of both the numerical and experimental results is carried out too.

A common approach to non-uniformity is to assume that the local thicknesses inside the light spot are distributed according to a certain distribution, such as the uniform distribution or the Wigner semicircle distribution. A model considered in this work uses a different approach in which the local thicknesses are given by a polynomial in the coordinates x and y along the surface of the film. An approach using the Gaussian quadrature is very efficient for including the influence of the non-uniformity on the measured ellipsometric quantities. However, the nodes and weights for the Gaussian quadrature must be calculated numerically if the non-uniformity is parameterized by the second or higher degree polynomial. A method for calculating these nodes and weights which is both efficient and numerically stable is presented. The presented method with a model using a second-degree polynomial is demonstrated on the sample of highly non-uniform polymer-like thin film characterized using variable-angle spectroscopic ellipsometry. The results are compared with those obtained using a model assuming the Wigner semicircle distribution.

Different types of broadening of the dielectric response are studied with respect to the preservation of the Thomas-Reiche-Kuhn sum rule. It is found that only the broadening of the dielectric function and transition strength function conserve this sum rule, whereas the broadening of the transition probability function (joint density of states) increases or decreases the sum. The effect of different kinds of broadening is demonstrated for interband and intraband direct electronic transitions using simplified rectangular models. It is shown that the broadening of the dielectric function is more suitable for interband transitions while broadening of the transition strength function is more suitable for intraband transitions. (C) 2013 Elsevier B.V. All rights reserved.

The distribution of the total transition strength, i.e. the right hand side of the integral form of Thomas-Reiche-Kuhn sum rule, into individual absorption processes is described for crystalline silicon containing interstitial oxygen. Utilization of the sum rule allows the construction of a dispersion model covering all elementary excitations from phonon absorption to core electron excitations. The dependence of transition strength of individual electronic and phonon contributions on temperature and oxygen content is described. (C) 2014 Elsevier B.V. All rights reserved.

Standard variable-angle spectroscopic ellipsometry, mapping spectroscopic ellipsometry with microspot and imaging spectroscopic reflectometry are applied to optical characterisation of a thin SiOxCyHz film considerably non-uniform in thickness and which is also suspected of non-uniformity also in the optical constants. It is shown that using the combination of these three optical methods, enables us to determine the spectral dependencies of the optical constants of the film together with parameters characterising the shape of thickness non-uniformity and fine map of local thickness. The mapping spectroscopic ellipsometry with microspot enables deciding whether the film is non-uniform in optical constants. For the thin film studied it is found that the non-uniformity in optical constants is under experimental accuracy. The consistency of results obtained using individual techniques is checked and the advantages and disadvantages of the techniques are discussed. (C) 2013 Elsevier B.V. All rights reserved.

Expressions for ellipsometric quantities and reflectance presented are based on a heuristic combination of the Rayleigh-Rice theory and the scalar diffraction theory. The latter takes into account the local slopes and shadowing. The second-order Rayleigh-Rice theory is used to express the local electric field on the rough surface and the obtained expressions are then used in the scalar diffraction theory instead of the expressions corresponding to a smooth surface. A numeric method of evaluation of the quadruple integral resulting from this combination of the two theories is developed, utilising a Gauss-like quadrature. The efficiency of the formulae is illustrated by optical characterisation of rough silicon and gallium arsenide surfaces created using anodic and thermal oxidation, respectively, and covered with native oxide layers. The optical characterisation employs variable-angle spectroscopic ellipsometry and spectroscopic reflectometry. (C) 2014 Elsevier B.V. All rights reserved.

A multiple-beam interference model is used to derive approximate formulae for the reflection and transmission coefficients of inhomogeneous thin films exhibiting large gradients of refractive index profiles. It is shown that these formulae are constituted by series containing the Wentzel-Kramers-Brillouin-Jeffreys term and correction terms with increasing order corresponding to number of considered internal reflections inside the films. A numerical analysis enabling us to show the influence of a degree of inhomogeneity on spectral dependencies of reflectance and ellipsometric parameters of inhomogeneous films is performed. Advantages and disadvantages of our approach compared with other approximate approaches are discussed. The optical characterization of a selected non-stoichiometric silicon nitride film prepared by reactive magnetron sputtering onto silicon single crystal substrate is performed for illustration of using our formulae in practice.

This paper presents the results of the optical characterization of inhomogeneous thin films of polymer-like SiO x C y H z and non-stoichiometric silicon nitride SiN x. An efficient method combining variable angle spectroscopic ellipsometry and spectroscopic reflectometry applied at the near-normal incidence based on the multiple-beam interference model is utilized for this optical characterization. The multiple-beam interference model allows us to quickly evaluate the values of ellipsometric parameters and reflectance of the inhomogeneous thin films, which exhibit general profiles of their optical constants. The spectral dependencies of the optical constants of the inhomogeneous SiO x C y H z and SiN x thin films are determined using the Campi–Coriasso dispersion model. The profiles of the optical constants of these films can also be determined. Furthermore, the transition layers at the lower boundaries of the characterized films are also taken into account. Spectral dependencies of the optical constants of these transition layers are also determined using the Campi–Coriasso dispersion model.This paper presents the results of the optical characterization of inhomogeneous thin films of polymer-like SiO x C y H z and non-stoichiometric silicon nitride SiN x. An efficient method combining variable angle spectroscopic ellipsometry and spectroscopic reflectometry applied at the near-normal incidence based on the multiple-beam interference model is utilized for this optical characterization. The multiple-beam interference model allows us to quickly evaluate the values of ellipsometric parameters and reflectance of the inhomogeneous thin films, which exhibit general profiles of their optical constants. The spectral dependencies of the optical constants of the inhomogeneous SiO x C y H z and SiN x thin films are determined using the Campi–Coriasso dispersion model. The profiles of the optical constants of these films can also be determined. Furthermore, the transition layers at the lower boundaries of the characterized films are also taken into account. Spectral dependencies o...

The optical properties of a slightly boron doped float-zone crystalline silicon wafer are studied using ellipsometry and spectrophotometry in a wide spectral range from far IR to vacuum UV. One side of the wafer was cleaned in an argon plasma, which influenced the optical properties of silicon near the surface. The dielectric response of silicon was modeled using a simplified universal dispersion model which is constructed on the basis of parameterization of the joint density of states describing both the electronic and phonon excitations. Several variants of models describing phonon absorption and interband transitions are discussed. It was possible to accurately determine the optical constants of bulk silicon and the optical constants near the perturbed surface over a wide spectral range. These optical constants agree well with those found in other works. From the optical measurements, it was also possible to determine the thickness of the wafer and the static value of resistivity, and the determined values agreed with nominal values specified for the wafer.

A combined method of spectroscopic ellipsometry and spectroscopic reflectometry is employed for the optical characterization of randomly rough surfaces of silicon single crystal covered with very thin native oxide layers within spectral range 197-1240 nm (1.0-6.3 eV). It is shown that random roughness of the samples of this system exhibits a wide interval of spatial frequencies so that the Rayleigh–Rice theory, scalar diffraction theory for coherent light and scalar diffraction theory with including light scattering must be used to achieve good fits of experimental data. The ellipsometric data are compatible with the Rayleigh–Rice theory within the entire spectral range while the reflectometric data must be processed separately within two sub-ranges using two different theoretical approaches. In one of the sub-ranges the combination of Rayleigh–Rice theory and scalar diffraction theory for coherent light is utilized for obtaining good fits of the corresponding experimental data of reflectance. Within the latter sub-range the scalar diffraction theory including the influence of light scattering on reflectance enables us to achieve good fits. Further, it is shown that using this combined method the values of basic roughness parameters, i.e. the rms values of heights, autocorrelation lengths and rms values of slopes of roughness irregularities, can be determined together with thickness values of the native oxide layers. A comparison of the results achieved optically with those determined by atomic force microscopy is performed. The assessment of efficiency of spectroscopic ellipsometry and spectroscopic reflectometry in characterizing random roughness with the wide interval of spatial frequencies is also presented.

Amorphous hydrogenated silicon (a-Si:H) films deposited on glass and crystalline silicon substrates are analyzed using a multi-sample method combining ellipsometry and spectrophotometry in a spectral range of 0.046–8.9 eV, utilizing an analytical dispersion model based on parametrization of joint density of states and application of sum rule. This model includes all absorption processes from phonon absorption to core electron excitations. It is shown that if films deposited on both substrates are characterized together it is possible to study both phonon absorption and weak absorption processes below the band gap, i.e. the Urbach tail and absorption on localized states.

Optical data obtained for a-Si:H films by ellipsometry and spectrophotometry in the wide photon energy range 0.046–8.9 eV are fitted using the analytical dispersion models based on the application of the sum rule. The models include all absorption processes ranging from phonon absorption in IR region to core electron excitations in X-ray region. They take into account the existence of extended and localized states of valence electrons and distinguish transitions to conduction band and higher energy electron states. It is demonstrated that a combination of optical measurements over the wide range, combined with reasonable assumptions about the optical response in regions where no experimental data are available can lead to dispersion models enabling to determine the mass density of the film. Comparing the density of states determined by tight-binding method with that obtained from optical data, it is shown that an excitonic effect is significant in a-Si:H and causes a redistribution of transition probability from higher energies to the broad peak centered at 3.5 eV. Moreover, it is suggested how to apply the sum rule in the commercial ellipsometric software implementing the Tauc–Lorentz model.

In this paper, we present the temperature dependent dispersion model of float zone crystalline silicon. The theoretical background for valence electronic excitations is introduced in the theoretical part of this paper. This model is based on application of sum rules and parametrization of transition strength functions corresponding to the individual elementary phonon and electronic excitations. The parameters of the model are determined by fitting ellipsometric and spectrophotometric experimental data in the spectral range from far infrared (70 cm-1) to extreme ultraviolet (40 eV). The ellipsometric data were measured in the temperature range 5–700 K. The excitations of the valence electrons to the conduction band are divided into the indirect and direct electronic transitions. The indirect transitions are modeled by truncated Lorentzian terms, whereas the direct transitions are modeled using Gaussian broadened piecewise smooth functions representing 3D and 2D van Hove singularities modified by excitonic effects. Since the experimental data up to high energies (40 eV) are available, we are able to determine the value of the effective number of valence electrons. The Tauc–Lorentz dispersion model is used for modeling high energy electron excitations. Two slightly different values of the effective number of valence electrons are obtained for the Jellison–Modine (4.51) and Campi–Coriasso (4.37) parametrization. Our goal is to obtain the model of dielectric response of crystalline silicon which depends only on photon energy, temperature and small number of material parameters, e.g. the concentration of substituted carbon and interstitial oxygen. The model presented in this paper is accurate enough to replace tabulated values of c-Si optical constants used in the optical characterization of thin films diposited on silicon substrates. The spectral dependencies of the optical constants obtained in our work are compared to results obtained by other authors.

Variable-angle spectroscopic ellipsometry is employed for the optical characterization of non-stoichiometric silicon nitride thin films exhibiting inhomogeneity formed by refractive index and extinction index changes through the film thickness. For all the film samples, the best fit of the experimental data is achieved if, in addition to the inhomogeneity, an overlayer or roughness of the upper boundary is included. However, distinguishing of these two defects is found not to be possible. The influence of working gas ratio, deposition temperature and on/off time on the film properties is studied. The refractive index and extinction coefficient is found to increase with increasing working gas ratio and less significantly with decreasing deposition temperature. It is also found that the inhomogeneity increases with decreasing deposition temperature, and the deposition rate of the films decreases with increasing working gas ratio. The influence of the on/off time on the film properties is practically unimportant.

Wire grid polarizers (WGPs), periodic nano-optical meta-surfaces, are convenient polarizing elements for many optical applications. However, they are still inadequate in the deep ultraviolet spectral range. We show that to achieve high performance ultraviolet WGPs a material with large absolute value of the complex permittivity and extinction coefficient at the wavelength of interest has to be utilized. This requirement is compared to refractive index models considering intraband and interband absorption processes. We elucidate why the extinction ratio of metallic WGPs intrinsically humble in the deep ultraviolet, whereas wide bandgap semiconductors are superior material candidates in this spectral range. To demonstrate this, we present the design, fabrication and optical characterization of a titanium dioxide WGP. At a wavelength of 193 nm an unprecedented extinction ratio of 384 and a transmittance of 10 % is achieved., 21 pages, Advanced Optical Materials 2016

Vacuum evaporated SiO2 thin films are very important in a design and manufacturing of optical devices produced in optics industry. In this contribution a reliable and precise optical characterization of such SiO2 thin films is performed using the combined method of spectrophotometry at normal incidence and variable-angle spectroscopic ellipsometry applied over spectral range from far IR to extreme UV (0.01-45 eV). This method uses the Universal Dispersion Model based on parametrization of the joint density of states and structural model comprising film defects such as nanometric boundary roughness, inhomogeneity and area non-uniformity. The optical characterization over the wide spectral range provides not only the spectral dependencies of the optical constants of the films within the wide range but, more significantly, it enables their correct and precise determination within the spectral range of interest, i.e. the range of their transparency. Furthermore, measurements in the ranges of film absorption, i. e. phonon excitations in IR and electron excitations in UV, reveal information about the material structure. The results of the optical characterization of the SiO2 thin films prepared on silicon single crystal substrates under various technological conditions are presented in detail for two selected samples. Beside film thicknesses and values of dispersion parameters and spectral dependencies of the optical constants of the SiO2 films, the characterization also enables quantification of film defects and their parameters are presented as well. The results concerning the optical constants of SiO2 films are compared with silica optical constants determined in our earlier studies.

article i nfo Hafnia films prepared onto silicon wafers at three substrate temperatures of 40, 160 and 280 °C are optically characterized utilizing the multi-sample method. The characterization uses the combination of variable angle spectroscopic ellipsometry and spectroscopic reflectometry within the spectral region 1.24-6.5 eV (190- 1000 nm). The structural model of the HfO2 films includes boundary nanometric roughness, thickness non- uniformity and refractive index profile. Spectral dependences of the film optical constants are expressed using a recently developed parametrized joint density of states model describing the dielectric response of both interband transitions and excitations of localized states below the band gap. It is shown that the observed weak absorption below the band gap does not correspond to the Urbach tail.

An efficient and numerically stable method for calculating the optical quantities of multi-layer systems with slightly rough boundaries using the second order Rayleigh–Rice theory is developed. It is assumed that the mean planes of the boundaries are parallel and all the media forming the system are nonmagnetic, isotropic and homogeneous. The perturbation series is formulated using the four-dimensional formalism inspired by the Yeh matrix formalism, but the final result is written using the two-dimensional formalism which is more efficient for the numerical calculations. The final formulae, which are expressed using an arbitrary power spectral density function (PSDF), include the mixing between the p and s polarizations occurring for anisotropic roughness. Although in the general case the calculation of optical quantities requires evaluation of double integrals, it is shown that for the PSDF given by the isotropic Gaussian function some integrals can be calculated analytically and only single integrals have to be evaluated numerically. The random roughness of boundaries is a defect that occurs frequently in practice, and it must be taken into account in the optical characterization and synthesis of thin film systems exhibiting this defect. The presented method is suitable for these purposes, since both of the mentioned applications require methods that are very fast.

The combination of spectrophotometry and variable-angle spectroscopic ellipsometry is used for the optical characterization of a system consisting of transparent polymer film on a transparent polycarbonate substrate. In the region of sample transparency a relatively large depolarization is observed in ellipsometry. This can be explained by the presence of anisotropy in the film or substrate. The possibility of distinguishing between the depolarization enhanced by film and substrate anisotropy is discussed and tested using several models. It is shown that it is not possible to distinguish between these effects.

Optical measurements of transmittance in the far infrared region performed on crystalline silicon wafers exhibit partially coherent interference effects appropriate for the determination of thicknesses of the wafers. The knowledge of accurate spectral and temperature dependencies of the optical constants of crystalline silicon in this spectral region is crucial for the determination of its thickness and vice versa. The recently published temperature dependent dispersion model of crystalline silicon is suitable for this purpose. Because the linear thermal expansion of crystalline silicon is known, the temperatures of the wafers can be determined with high precision from the evolution of the interference patterns at elevated temperatures.Optical measurements of transmittance in the far infrared region performed on crystalline silicon wafers exhibit partially coherent interference effects appropriate for the determination of thicknesses of the wafers. The knowledge of accurate spectral and temperature dependencies of the optical constants of crystalline silicon in this spectral region is crucial for the determination of its thickness and vice versa. The recently published temperature dependent dispersion model of crystalline silicon is suitable for this purpose. Because the linear thermal expansion of crystalline silicon is known, the temperatures of the wafers can be determined with high precision from the evolution of the interference patterns at elevated temperatures.

Complete optical characterization of diamond-like carbon (DLC) films non-uniform in thickness is performed using spectroscopic imaging reflectometry (SIR). It is shown that by using this technique it is possible to determine the area distribution (area map) of the local thickness of these films with arbitrary shape of this thickness non-uniformity. Furthermore, it is shown that in principle it is possible to determine the distributions of the refractive index and extinction coefficient of these films simultaneously with the thickness distribution, if a suitable dispersion model of these optical constants is chosen. In this paper the dispersion model of the optical constants of the DLC films based on parameterization of density of electronic states (DOS) is used. The values of the material parameters of this dispersion model are determined too. It is shown that the DLC films studied do not exhibit the area non-uniformity in material parameters and optical constants. The method presented can be used to characterize the non-uniform films consisting of other materials.

Optical characterization of diamond-like carbon (DLC) films non-uniform in thickness is performed using spectroscopic phase-modulated ellipsometry. This characterization is based on new formulas for the associated ellipsometric parameters of thin films exhibiting a wedge-shaped thickness non-uniformity. These formulas express the associated ellipsometric parameters by means of the density of distribution of local film thickness. The spectral dependences of the optical constants of these non-uniform DLC films are expressed using the dispersion model based on parametrization of density of electronic states. It is shown that this model of the thickness non-uniformity provides a relatively good agreement between the experimental and theoretical data, indicating that the results of the optical characterization of the non-uniform DLC films are close to the correct results. Moreover, it is shown that the model of uniform thin films is unsuitable for the optical characterization of the non-uniform DLC films studied.

Complete optical characterization of SiOx films non-uniform in thickness is performed using imaging spectroscopic reflectometry. It is shown that by using this technique it is possible to determine the area distribution of the local thickness (area map) of these films with arbitrary shape of this thickness non-uniformity. Furthermore, it is shown that the SiOx films studied do not exhibit the area non-uniformity in dispersion (material) parameters and optical constants. This is possible because imaging spectroscopic reflectometry enables us to determine the area distributions of local thickness and local refractive index simultaneously in an independent way under the assumption that a suitable dispersion model of the refractive index of the films is used. In this paper the dispersion model corresponding to the Cauchy's formula is used. On the basis of this dispersion model the spectral dependence of the refractive index of the SiOx films is determined. The method presented can be used to characterize the non-uniform films consisting of other non-absorbing materials. [DOI: 10.1380/ejssnt.2009.409]

The variable angle spectroscopic ellipsometry and spectroscopic reflectometry is used for the optical characterization of the as-deposited and annealed Ge2Sb2Te5 (GST) films. The dispersion model of the GST films is based on the parameterization of the density of electronic states. As a consequence of annealing, roughness of the upper boundaries of the GST films originates. This annealing also causes the evident redistribution of the density of electronic states from the higher energies to lower energies. The roughening and redistribution of the electrons are explained by means of the phase change of the GST films from amorphous to polycrystalline structure. This phase change is accompanied by changing the conductivity of the films that was proved by introducing the terms of the free carriers in the dispersion model of the GST films. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Experimental ellipsometric and spectroscopic data of a randomly rough silicon substrate are treated using several theoretical approaches. It is shown that the effective medium approximation and scalar diffraction theory are unsuitable approaches for treating the experimental data of rough surfaces containing spatial frequencies comparable with the wavelength of incident light. Using the Rayleigh-Rice theory one can obtain an excellent fit of the experimental data and very close agreement between the values of the statistical parameters determined using the optical method and atomic force microscopy.

Two optical methods for the optical characterization of the statistically rough surfaces exhibiting wide intervals of spatial frequencies are presented. These methods employ the combination of variable angle spectroscopic ellipsometry and near-normal spectroscopic reflectometry. The first method is based on combining the scalar diffraction theory and effective medium theory while the second method combines the scalar diffraction theory with Rayleigh-Rice theory. Both the methods are applied to the optical characterization of statistically rough GaAs surfaces prepared by thermal oxidation. It is shown that both the methods can be utilized for characterization of these surfaces in a reasonable way, however, the latter is more suitable for this purpose. The results of the optical characterization of the selected rough GaAs surface are supported by those obtained using atomic force microscopy.

The formulae enabling us to calculate the values of the ellipsometric parameters of randomly rough surfaces and very thin films with randomly rough boundaries with respect to the influence of the shadowing between the irregularities of roughness are presented. These formulae are derived within the scalar diffraction theory of light. The first formula allows the calculation of the ellipsometric parameters using a numerical method, while the second formula is the approximative one but it expresses the ellipsometric parameters in a closed form. The numerical analysis of both formulae is performed for several examples of randomly rough surfaces and thin films. Moreover, the comparison of the results achieved using both formulae respecting the shadowing and an earlier formula not including this effect is performed. The experimental data of two samples of the randomly rough silicon surfaces covered with very thin surface layers are interpreted using all the formulae mentioned. Using this experimental study, the correctness of both formulae taking into account the shadowing, is confirmed.