Determination of Optical and Structural Parameters of Thin Films with Differently Rough Boundaries

Abstract

The optical characterization of non-absorbing, homogeneous, isotropic polymer-like thin films with correlated, differently rough boundaries is essential in optimizing their performance in various applications. A central aim of this study is to derive the general formulae necessary for the characterization of such films. The applicability of this theory is illustrated through the characterization of a polymer-like thin film deposited by plasma-enhanced chemical vapor deposition onto a silicon substrate with a randomly rough surface, focusing on the analysis of its rough boundaries over a wide range of spatial frequencies. The method is based on processing experimental data obtained using variable-angle spectroscopic ellipsometry and spectroscopic reflectometry. The transition layer is considered at the lower boundary of the polymer-like thin film. The spectral dependencies of the optical constants of the polymer-like thin film and the transition layer are determined using the Campi–Coriasso dispersion model. The reflectance data are processed using a combination of Rayleigh–Rice theory and scalar diffraction theory in the near-infrared and visible spectral ranges, while scalar diffraction theory is used for the processing of reflectance data within the ultraviolet range. Rayleigh–Rice theory alone is sufficient for the processing of the ellipsometric data across the entire spectral range. We accurately determine the thicknesses of the polymer-like thin film and the transition layer, as well as the roughness parameters of both boundaries, with the root mean square (rms) values cross-validated using atomic force microscopy. Notably, the rms values derived from optical measurements and atomic force microscopy show excellent agreement. These findings confirm the reliability of the optical method for the detailed characterization of thin films with differently rough boundaries, supporting the applicability of the proposed method in high-precision film analysis.

1. Introduction

Thin films are frequently encountered in many branches of fundamental and applied research. The optical properties of these films have been investigated extensively by means of their optical characterization. Considerable attention has been devoted to homogeneous thin films especially. Homogeneous thin films exhibit the same optical constants across their entire volumes. Many methods have been published for the optical characterization of these films so far. Some of these methods are based on processing photometric data consisting of reflectance and transmittance data. Such methods were utilized, for example, in [1,2,3,4]. Other methods are based on processing ellipsometric data [5,6,7]. The most efficient methods for the optical characterization of homogeneous thin films utilize the simultaneous processing of photometric and ellipsometric data [8,9]. The optical characterization of inhomogeneous thin films is more complicated compared to that of homogeneous thin films because these films have continuously varying optical constants along the axes perpendicular to their boundaries. Nevertheless, a certain number of papers have been devoted to the optical characterization of these films. The photometric and ellipsometric methods mentioned above are also used for the optical characterization of inhomogeneous thin films. For example, photometric methods, as published in papers [10,11], were utilized to determine the thickness values and profiles of the refractive index. Of course, it is necessary to point out that the photometric quantity (e.g., reflectance) must be measured at the oblique incidence of light within these methods. The ellipsometric methods are mostly used to characterize inhomogeneous thin films (see, e.g., [12,13,14,15,16]). The combined method of the spectroscopic ellipsometry and spectroscopic reflectometry was applied in [17].

Many thin films exhibit various defects. The parameters characterizing these defects must be determined together with the optical parameters within the optical characterization of homogeneous and inhomogeneous thin films. The defects encountered most often are thickness non-uniformity, boundary roughness and transition layers.

Considerable attention has been devoted to studies of surfaces and thin films with random roughness. Several theoretical approaches have been developed, enabling the processing of experimental data obtained by various experimental arrangements. The choice of the theoretical approach to processing the experimental data depends on the properties and structure of the roughness. If random roughness exhibits lateral dimensions larger than the wavelength, scalar diffraction theory (SDT) can be used. This theory is thus utilized for locally smooth roughness. For single thin films exhibiting locally smooth roughness in the boundaries, SDT has been utilized for optical characterization, as in papers [18,19,20,21]. SDT was also used to study multilayer systems with rough boundaries (see, e.g., [22,23,24]). If the heights of the roughness irregularities are substantially smaller than the wavelengths and the slopes of roughness are sufficiently small, one can utilize the Rayleigh–Rice theory (RRT) of the second order to describe the interaction of light with homogeneous single films with rough boundaries (see, e.g., [25,26,27,28]). Of course, the RRT is also utilized to derive formulae describing the interaction of light with multilayer systems and their applications in optical characterization (see, e.g., [29,30]).

A very small amount of attention has been devoted to the optical characterization of inhomogeneous thin films with randomly rough boundaries so far. Only a few papers dealing with the optical characterization of such films have been published (see [31,32,33]). In [31], the optical characterization of thin films originating from the thermal oxidation of the surfaces of a GaAs single crystal was performed. Variable-angle spectroscopic ellipsometry (VASE) and spectroscopic reflectometry (SR) were utilized for this purpose. The spectral dependencies of the optical constants, their profiles, the values of the mean thickness and the roughness parameters were determined for these oxide films (the roughness parameters corresponded to the rms values of the height and autocorrelation length). The RRT was utilized in this work. In [32], the results of the optical characterization of inhomogeneous polymer-like thin films with randomly rough boundaries were published. It was assumed that the roughness of the boundaries was identical, i.e., the model of a so-called identical film was used. The roughness only exhibited low spatial frequencies and therefore SDT was used to process the experimental data corresponding to VASE and SR. The optical characterization of inhomogeneous polymer-like thin films with roughness corresponding to the identical model of the boundaries was also performed using VASE and SR in [33]. However, the boundaries of the films exhibited wide intervals in their spatial frequencies, i.e., these boundaries exhibited low, moderate and high spatial frequencies, in contrast to the thin films characterized in [32]. Thus, the combination of the SDT and RRT had to be utilized to process the experimental data. Although the number of the searched parameters was larger than for the films characterized in [32], successful characterization was again carried out.

Note that, in some studies, the optical characterization of homogeneous and inhomogeneous thin films exhibiting several defects is performed. Inhomogeneous non-stoichiometric silicon nitride films represent such films (see [34]). These silicon nitride films exhibited two defects consisting of uniaxial anisotropy with the optical axis perpendicular to the boundaries and the slight random roughness of the upper boundaries. The roughness corresponded to low spatial frequencies and therefore SDT was employed to process the experimental data measured within VASE and SR. The spectral dependencies and profiles of the optical constants belonging to ordinary and extraordinary waves were simultaneously determined, together with the values of the mean thickness and roughness parameters for these films (see [34]).

The approach whereby the roughness of the boundaries is included by means of fictitious layers with optical constants expressed using effective medium approximation (EMA) is often utilized [35,36,37]. Another instance where the boundary roughness is accounted for by EMA is in the field of surface plasmon resonance (SPR) biosensors, where it is well known that the surface roughness of the metallic plasmonic layer significantly influences the sensor’s sensitivity [38,39,40,41,42]. Yet another area where boundary roughness has a critical impact and is modeled by EMA is the field of solar cells. Currently, graphene oxide layers play a significant role in improving solar cells’ efficiency; however, their surface roughness has proven to be a challenge [43,44,45]. This approach is, however, very limited compared to the RRT and SDT. It does not give such accurate predictions and it is limited only to roughness with very small heights. Moreover, it is not possible to relate the parameters of the fictitious layers to the roughness parameters, such as the rms values of the heights or the autocorrelation length, in a simple and reliable way.

The thickness non-uniformity of thin films also represents an important defect occurring in practice. The influence of this defect on the optical quantities of the thin films is different from the influence of the boundary roughness, as implied in [46,47,48,49,50,51,52]. Thus, these two defects can be distinguished when characterizing thin films.

The method presented in this paper enables us to perform the optical characterization of homogeneous thin films exhibiting different random roughness at the upper and lower boundaries. It can be considered as the generalization of the method used in [33], which assumed identical roughness at the film boundaries. It is assumed that the roughness of the boundaries exhibits spatial frequencies with wide intervals. If the boundaries of the characterized films have differently rough boundaries, the number of the sought roughness parameters is larger compared to films with identical roughness. As a consequence of this fact, the correlation among the searched optical and roughness parameters is increased, which complicates the optical characterization process. Moreover, the formulae for the optical quantities of films with differently rough boundaries are much more complicated than those for identical thin films. Despite this reality, it is shown that, under certain circumstances, it is possible to perform the successful optical characterization of films exhibiting differently rough boundaries by means of a combination of VASE and SR. The application of this method is carried out using the optical characterization of polymer-like thin films deposited onto silicon single-crystal substrates.

2. Sample Preparation

Smooth silicon wafers were anodically oxidized at a constant voltage following the procedure outlined by Manara [53]. The resulting anodic oxide thin films were dissolved in a mixture of water and hydrofluoric acid, creating randomly rough silicon crystal surfaces.

The rough silicon substrates were pre-treated with argon discharge for 5 min prior to film deposition in order to enhance the film adhesion.

Polymer-like thin films (${\mathrm{SiO}}_{\mathrm{x}}$${\mathrm{C}}_{\mathrm{y}}$${\mathrm{H}}_{\mathrm{z}}$) were deposited onto the rough silicon surfaces using plasma-enhanced chemical vapor deposition (PECVD). This process preserved the random roughness at both boundaries of the deposited films. A parallel-plate PECVD reactor with capacitively coupled glow discharge was used, operating at a frequency of 13.56 MHz. The reaction chamber consisted of a glass cylinder with stainless-steel flanges and graphite parallel electrodes. The supplied RF power was 50 W, and it was applied to the lower electrode. The polymer-like films were deposited from a mixture of methane and a hexamethyldisiloxane (${\mathrm{C}}_6$${\mathrm{H}}_{18}$${\mathrm{Si}}_2\mathrm{O}$, HMDSO) precursor. Details of the HMDSO-based polymer-like thin films can be found in [54,55].

3. Experimental Arrangements

The ellipsometric data were measured by a Horiba Jobin Yvon UVISEL phase-modulated ellipsometer for four angles of incidence ($60°$, $65°$, $70°$ and $75°$) of light within the spectral range of 0.6–6.5 eV (190–2066 nm). The experimental data consisted of the associated ellipsometric parameters $I_{\mathrm{s}}$, $I_{\mathrm{c}}$ and $I_{\mathrm{n}}$, which are related to $\mathrm{\Psi }$ and $\Delta$ as follows:

$$\begin{array}{cccccc}\hfill I_{\mathrm{s}}& =Psin2\mathrm{\Psi }sin\Delta ,\hfill & \hfill I_{\mathrm{c}}& =Psin2\mathrm{\Psi }cos\Delta ,\hfill & \hfill I_{\mathrm{n}}& =Pcos2\mathrm{\Psi },\hfill \end{array}$$

where P represents the degree of polarization. Note that these associated ellipsometric parameters are independent components of the normalized Mueller matrix of the optically isotropic system (see, e.g., [56,57,58]):

$$M=R\left(\begin{array}{cccc}1& -I_{\mathrm{n}}& 0& 0\\ -I_{\mathrm{n}}& 1& 0& 0\\ 0& 0& I_{\mathrm{c}}& I_{\mathrm{s}}\\ 0& 0& -I_{\mathrm{s}}& I_{\mathrm{c}}\end{array}\right),$$

(1)

where $R=(R_{\mathrm{p}}+R_{\mathrm{s}})/2$ is the average of the reflectance.

The reflectance at the near-normal incidence of $6°$ was measured using a Perkin Elmer Lambda 1050 spectrophotometer in the spectral range of 1.44–6.5 eV (190–860 nm). The Bruker Dimension Icon atomic force microscope was used to measure the topography of the upper boundaries of the films, as well as the topography of the substrate surfaces prior to the deposition.

4. Theory

4.1. Structural Model

In order to create a correct structural model of the polymer-like thin films characterized, first, it was necessary to study the randomly rough silicon surfaces using atomic force microscopy (AFM) before depositing these films. After film deposition, AFM scans of the upper boundaries were performed. For a selected sample of a polymer-like thin film, the AFM scan of the rough silicon surface is shown in Figure 1a. These scans were processed to obtain histograms of the heights using the Gwyddion software and fitted by the Gaussian distribution functions. The rms value for the heights of the rough silicon substrate, found from the fit of the heights distribution by the Gaussian function, was ${\sigma }_L^{AFM}=$ 18.19 ± 0.06 nm. This rough silicon corresponded to the lower boundary of the polymer-like film. In Figure 1b, the AFM scan of the upper boundary of this film is presented. The rms value of the height determined from the fit of the height distribution by the Gaussian function was ${\sigma }_U^{AFM}=$ 16.45 ± 0.03 nm. From Figure 1, it is evident that the topographies of the rough silicon substrate and upper boundary are close each other. However, despite this reality, it is evident that the rms value of the height of the upper boundary ${\sigma }_U^{AFM}$ is smaller than that ${\sigma }_L^{AFM}$ of the lower boundary. This can be explained by the certain smoothing of the upper boundary during the growth of the polymer-like thin film. The small difference between the rms values ${\sigma }_L^{AFM}$ and ${\sigma }_U^{AFM}$ allows us to assume that the smoothing of the roughness with high and moderate spatial frequencies is realized, while the roughness at low frequencies is not influenced by this smoothing during the growth of the film. This indicates that the roughness exhibiting low spatial frequencies is identical for both boundaries. From Figure 2 it is evident that the Gaussian distributions describe the height distributions very well, which indicates that the roughness of both boundaries is generated by the Gaussian process. It was found that the rms values ${\sigma }_L^{AFM}$ and ${\sigma }_U^{AFM}$ were the same across the whole illuminated area within the experimental accuracy.

From the above, it is clear that the structural model of the rough polymer-like thin film can be specified using the following assumptions.

• The roughness of both boundaries of the films is generated by a stationary stochastic Gaussian process. This roughness is homogeneous and isotropic.
• The roughness of both boundaries has wide intervals regarding the spatial frequencies, i.e., it exhibits low, moderate and high spatial frequencies.
• The roughness corresponding to the low spatial frequencies is identical for both boundaries.
• The roughness of both boundaries corresponding to moderate and high spatial frequencies is statistically correlated. The rms values of the heights are different in the upper and lower boundaries.
• The slopes of the roughness of both boundaries corresponding to the low spatial frequencies are very small and therefore their influence on the optical quantities can be neglected.
• The transition layers occur between the substrates and thin films, i.e., they occur at the lower boundaries of the films.
• The transition layers are formed by identical thin films (ITF) in which the roughness of both boundaries is identical. The roughness of both boundaries of the transition layers is the same as the roughness of the lower boundaries of the thin films, i.e., as the roughness of the silicon surfaces.
• The thin films, transition layers and substrates consist of isotropic and homogeneous materials from the optical point of view.

The transition layers must be considered in the structural model because of the pre-treatment of the rough silicon substrate in argon discharge prior to depositing the polymer-like thin films. The schematic diagram of the structural model is shown in Figure 3.

4.2. Dispersion Model

The Campi–Coriasso dispersion model [59] can be used to express the spectral dependencies of the optical constants of the polymer-like thin films within the spectral range of interest. The imaginary part of the dielectric function of these thin films is expressed as follows:

$${\epsilon }_{\mathrm{i}}\left(E\right)={\displaystyle \frac{2N_{\mathrm{vc}}}{\pi E}}{\displaystyle \frac{B{\left(E-E_g\right)}^2\mathrm{\Theta }\left(E-E_g\right)}{{\left[{\left(E_c-E_g\right)}^2-{\left(E-E_g\right)}^2\right]}^2+B^2{\left(E-E_g\right)}^2}},$$

(2)

where E denotes the photon energy, $N_{\mathrm{vc}}$ is the strength of the interband electronic transitions, $\mathrm{\Theta }$ represents the Heaviside function, $E_g$ denotes the band gap energy and $E_c$ and B are parameters ($E_c>E_g$). The real part of the dielectric function is calculated using the Kramers–Kroning relation [60].

The spectral dependencies of the optical constants of the transition layers have been again modeled by the Campi–Coriasso dispersion model. However, to correctly describe the interband transitions in these layers, it is necessary to consider the sum of three Campi–Coriasso terms instead of one, as utilized for the polymer-like thin films.

4.3. Scalar Diffraction Theory

4.3.1. Coherent Reflectance Corresponding to SDT

For the normal incidence of light, the reflection coefficient $r_0$ of the thin films with perfectly smooth boundaries is given as follows:

$${\widehat{r}}_0=\sum _{m=0}^{\infty }{\widehat{c}}_m,$$

(3)

where the individual terms correspond to the expansion in the number of internal reflections inside the film and are given as

$$\begin{array}{cccccc}\hfill {\widehat{c}}_m& ={\widehat{t}}_U{\widehat{t}\prime }_U{\left[{\widehat{r}\prime }_U\right]}^{m-1}{\left[{\widehat{\overline{r}}}_L\right]}^m{\mathrm{e}}^{\mathrm{i}m\widehat{x}},\mathrm{for}m>0,\hfill & \hfill \hspace{1em}{\widehat{c}}_0& ={\widehat{r}}_U,\hfill & \hfill \hspace{1em}{\widehat{r}\prime }_U& =-{\widehat{r}}_U,\hfill \end{array}$$

(4)

where ${\widehat{r}}_U$, ${\widehat{r}\prime }_U$ and ${\widehat{\overline{r}}}_L$ represent the reflection coefficients, while ${\widehat{t}}_U$ and ${\widehat{t}\prime }_U$ represent the transmission coefficients. The subscripts U and L are used to distinguish the upper and lower boundaries of the films. The prime is used for coefficients representing the incidence of light from the bottom side. The quantity $\widehat{x}$ determines the phase shift and extinction of light inside the film. The structural model also includes the transition layer, whose influence is included in the expression for the reflection coefficient ${\widehat{\overline{r}}}_L$ for the lower boundary of the film. These quantities are given as follows:

$$\begin{array}{ccccccccccc}\hfill {\widehat{r}}_U& ={\displaystyle \frac{n_0-{\widehat{n}}_1}{n_0+{\widehat{n}}_1}},\hfill & \hfill & \hfill {\widehat{\overline{r}}}_L& ={\displaystyle \frac{{\widehat{r}}_T+{\widehat{r}}_S{\mathrm{e}}^{\mathrm{i}{\widehat{x}}_T}}{1+{\widehat{r}}_T{\widehat{r}}_S{\mathrm{e}}^{\mathrm{i}{\widehat{x}}_T}}},\hfill & \hfill & \hfill {\widehat{r}}_T& ={\displaystyle \frac{{\widehat{n}}_1-{\widehat{n}}_T}{{\widehat{n}}_1+{\widehat{n}}_T}},\hfill & \hfill & \hfill {\widehat{r}}_S& ={\displaystyle \frac{{\widehat{n}}_T-{\widehat{n}}_S}{{\widehat{n}}_T+{\widehat{n}}_S}},\hfill \\ \hfill {\widehat{t}}_U& ={\displaystyle \frac{2n_0}{n_0+{\widehat{n}}_1}},\hfill & \hfill & \hfill {\widehat{t}\prime }_U& ={\displaystyle \frac{2{\widehat{n}}_1}{n_0+{\widehat{n}}_1}},\hfill & \hfill & \hfill \widehat{x}& ={\displaystyle \frac{4\pi }{\lambda }}{\widehat{n}}_1{d}_1,\hfill & \hfill & \hfill {\widehat{x}}_T& ={\displaystyle \frac{4\pi }{\lambda }}{\widehat{n}}_T{d}_T,\hfill \end{array}$$

(5)

where the symbols $\lambda$, $n_0$, ${\widehat{n}}_1$, ${\widehat{n}}_T$ and ${\widehat{n}}_S$ denote the wavelength and refractive indices of the ambient and polymer-like thin films, the transition layers and the substrates, respectively. Note that, apart from the refractive index $n_0$ of the ambient, which is real, all of the refractive indices can be complex, i.e., they can also include absorption in the media. The thicknesses of the films and transition layers are denoted by the symbols $d_1$ and $d_T$, respectively.

The influence of the boundary roughness of the thin films on their optical quantities can be expressed by means of the SDT. The starting point of this theory is the Helmholtz–Kirchhoff (HK) integral, given as follows (see, e.g., [18,61,62,63,64,65,66]):

$$\widehat{E}\left(P\right)={\displaystyle \frac{1}{4\pi }}{\int \int }_{S_1}\left({\widehat{E}}_{loc}{\displaystyle \frac{\partial \widehat{\mathrm{\Psi }}}{\partial n}}-\widehat{\mathrm{\Psi }}{\displaystyle \frac{\partial {\widehat{E}}_{loc}}{\partial n}}\right)\mathrm{d}{S}_1,$$

(6)

where $\widehat{E}\left(P\right)$ denotes the electric field in the observing point P, ${\widehat{E}}_{loc}$ is the local electric field on the upper boundaries of the films and function $\widehat{\mathrm{\Psi }}=exp\left(\mathrm{i}k\rho \right)/\rho$ with $k=2\pi n_0/\lambda$. The symbol $\rho$ denotes the distance between the observing point P and a given point on the upper boundary $S_1$.

If the conditions of the Fraunhofer diffraction are fulfilled, then, after simple mathematical operations, it is shown that Equation (6) can be rewritten in the following way [18]:

$$\widehat{E}\left(P\right)=-\mathrm{i}{\displaystyle \frac{n_0{\mathrm{e}}^{\mathrm{i}k{R}_1}}{\lambda R_1}}\widehat{A}{\int \int }_S{\widehat{\overline{r}}}_0{\mathrm{e}}^{-\mathrm{i}{v}_z{\eta }_1}{\mathrm{e}}^{\mathrm{i}(v_{x}x+v_{y}y)}\mathrm{d}x\mathrm{d}y,$$

(7)

where $\widehat{A}$ is the amplitude of incident light, $R_1$ denotes the distance of point P from the origin in the mean plane of the upper boundary, S is the illuminated area in the mean plane of the upper boundary and quantities $v_x$, $v_y$ and $v_z$ are defined as

$$\begin{array}{cccccc}\hfill v_x& =-{\displaystyle \frac{2\pi }{\lambda }}{n}_0{\theta }_{2}cos{\theta }_3,\hfill & \hfill v_y& =-{\displaystyle \frac{2\pi }{\lambda }}{n}_0{\theta }_{2}sin{\theta }_3,\hfill & \hfill v_z& =-{\displaystyle \frac{4\pi }{\lambda }}{n}_0,\hfill \end{array}$$

(8)

where ${\theta }_2$ and ${\theta }_3$ are the polar and azimuthal angles determining the position of the observing point P. The symbol ${\widehat{\overline{r}}}_0$ denotes the reflection coefficient of the rough film at the point $(x,y)$. This reflection coefficient is given by the reflection coefficient of the film with the smooth boundaries having the thickness $d_1$ substituted by the local thickness $d_{1,loc}$. This local thickness changes from point to point due to the roughness of the boundaries. It is given as

$$d_{1,loc}=d_1+{\eta }_1-{\eta }_2,$$

(9)

where ${\eta }_1$ and ${\eta }_2$ denote the heights of the roughness of the upper boundary and lower boundary, respectively, and $d_1$ corresponds to the mean thickness. The local reflection coefficient of the rough thin films is then expressed by the product of ${\widehat{\overline{r}}}_0$ and ${\mathrm{e}}^{-\mathrm{i}{v}_z{\eta }_1}$, i.e.,

$${\widehat{r}}_{loc}={\widehat{\overline{r}}}_0{\mathrm{e}}^{-\mathrm{i}{v}_z{\eta }_1}=\sum _{m=0}^{\infty }{\widehat{c}}_m{\mathrm{e}}^{\mathrm{i}{\widehat{B}}_m{\eta }_1+\mathrm{i}{\widehat{M}}_m{\eta }_2},$$

(10)

where ${\widehat{B}}_m$ and ${\widehat{M}}_m$ are given as

$$\begin{array}{cccc}\hfill {\widehat{B}}_m& =-{\displaystyle \frac{4\pi }{\lambda }}\left(n_0-m{\widehat{n}}_1\right),\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}{\widehat{M}}_m& =-{\displaystyle \frac{4\pi }{\lambda }}m{\widehat{n}}_1.\hfill \end{array}$$

(11)

The statistical mean value of the electric field $\langle \widehat{E}\left(P\right)\rangle$ is expressed as follows:

$$\langle \widehat{E}\left(P\right)\rangle =\left(-\mathrm{i}{\displaystyle \frac{n_0{\mathrm{e}}^{\mathrm{i}k{R}_1}}{\lambda R_1}}{\int \int }_S{\mathrm{e}}^{\mathrm{i}(v_{x}x+v_{y}y)}\mathrm{d}x\mathrm{d}y\right)\langle {\widehat{r}}_{loc}\rangle ,$$

(12)

where the symbol $\langle ·\rangle$ denotes the statistical mean value of a quantity written inside these brackets.

The mean value of the local reflection coefficient $\langle {\widehat{r}}_{loc}\rangle$ is as follows:

$$\widehat{r}=\langle {\widehat{r}}_{loc}\rangle ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\sum _{m=0}^{\infty }{\widehat{c}}_m{\mathrm{e}}^{\mathrm{i}{\widehat{B}}_m{\eta }_1+\mathrm{i}{\widehat{M}}_m{\eta }_2}w({\eta }_1,{\eta }_2)\mathrm{d}{\eta }_1\mathrm{d}{\eta }_2=\sum _{m=0}^{\infty }{\widehat{c}}_m\chi ({\widehat{B}}_m,{\widehat{M}}_m),$$

(13)

where $w({\eta }_1,{\eta }_2)$ represents the two-dimensional Gaussian distribution of ${\eta }_1$ and ${\eta }_2$, i.e.,

$$w({\eta }_1,{\eta }_2)={\displaystyle \frac{1}{2\pi {\sigma }_U{\sigma }_L\sqrt{1-Q_{UL}^2}}}exp\left\{-{\displaystyle \frac{1}{2(1-Q_{UL}^2)}}\left[{\displaystyle \frac{{\eta }_1^2}{{\sigma }_U^2}}-2Q_{UL}{\displaystyle \frac{{\eta }_1{\eta }_2}{{\sigma }_U{\sigma }_L}}+{\displaystyle \frac{{\eta }_2^2}{{\sigma }_L^2}}\right]\right\},$$

(14)

where ${\sigma }_U$ and ${\sigma }_L$ are the rms values of the heights of the upper and lower boundaries, respectively. The symbol $Q_{UL}$ denotes the cross-correlation coefficient of the upper and lower boundaries. One then obtains, for $\langle {\widehat{r}}_{loc}\rangle$, that the characteristic function is given as

$$\chi ({\widehat{B}}_m,{\widehat{M}}_m)=exp\left[-{\displaystyle \frac{1}{2}}\left({\widehat{B}}_m^2{\sigma }_U^2+2Q_{UL}{\widehat{B}}_m{\widehat{M}}_m{\sigma }_U{\sigma }_L+{\widehat{M}}_m^2{\sigma }_L^2\right)\right].$$

(15)

The last step of this derivation consists of the simple integration of the product $\langle \widehat{E}\left(p\right)\rangle \langle \widehat{E}{\left(p\right)}^{\ast }\rangle$ over the solid angle corresponding to the cone in which coherent light is concentrated. After this integration, it is obtained that the coherent reflectance of the film with the correlated differently rough boundaries $R_{\mathrm{c}}$ is given as follows:

$$R_{\mathrm{c}}={\left|\widehat{r}\right|}^2.$$

(16)

The ITFs represent the special case of the thin films with rough boundaries whose coherent reflectance is expressed by Equations (15) and (16) with $Q_{UL}=1$ and ${\sigma }_U={\sigma }_L=\sigma$. The coherent reflectance of these ITFs is thus expressed as follows:

$$R_{\mathrm{c}}=R_{0}exp\left({\displaystyle \frac{-16{\pi }^2{n}_0^2{\sigma }^2}{{\lambda }^2}}\right),$$

(17)

where $R_0$ denotes the reflectance of the corresponding thin film with perfectly smooth boundaries and $\sigma$ is the rms value of the heights of both boundaries.

4.3.2. Incoherent Reflectance Corresponding to SDT

The normal incidence of light on the mean planes of the rough boundaries will again be considered. If the roughness of the thin film boundaries and the acceptance angles of spectrophotometer detectors are sufficiently large, then scattered light is detected while measuring the reflectance. The total (measured) reflectance $R_{\mathrm{T}}=R_{\mathrm{c}}+R_{\mathrm{i}}$ includes the coherent reflectance as well as the incoherent reflectance $R_{\mathrm{i}}$ representing the scattered light. The general formula for the incoherent reflectance $R_{\mathrm{i}}$ of the thin films with different roughness at the boundaries derived using the SDT is given as follows:

$$\begin{array}{c}{R}_{\mathrm{i}}=\sum _{m=0}^{\infty }\sum _{l=0}^{\infty }{\widehat{c}}_m{\widehat{c}}_l^{\ast }\left[\chi ({\widehat{B}}_m,{\widehat{M}}_m)\right]{\left[\chi ({\widehat{B}}_l,{\widehat{M}}_l)\right]}^{\ast }\underset{j+s+n\ne 0}{\sum _{j=0}^{\infty }\sum _{s=0}^{\infty }\sum _{n=0}^{\infty }}{\displaystyle \frac{{\left[{\sigma }_U^2{\widehat{B}}_m{\widehat{B}}_l^{\ast }\right]}^j}{j!}}\times \hfill \\ \hspace{1em}\times {\displaystyle \frac{{\left[{\sigma }_L^2{\widehat{M}}_m{\widehat{M}}_l^{\ast }\right]}^s}{s!}}{\displaystyle \frac{{\left[Q_{UL}{\sigma }_U{\sigma }_L({\widehat{B}}_m{\widehat{M}}_l^{\ast }+{\widehat{M}}_m{\widehat{B}}_l^{\ast })\right]}^n}{n!}}\left[1-exp\left(-{\displaystyle \frac{{\pi }^2{n}_0^2{\tau }^2(j,s,n){\alpha }_0^2}{{\lambda }^2}}\right)\right],\hfill \end{array}$$

(18)

where

$${\displaystyle \frac{1}{{\tau }^2(j,s,n)}}={\displaystyle \frac{j}{{\tau }_U^2}}+{\displaystyle \frac{s}{{\tau }_L^2}}+{\displaystyle \frac{n}{{\tau }_{UL}^2}}.$$

(19)

${\tau }_U$ and ${\tau }_L$ are the autocorrelation lengths corresponding to the upper boundary and lower boundary, respectively, and ${\tau }_{UL}$ is the cross-correlation length. The symbol ${\alpha }_0$ denotes the acceptance angle of the detector. The foregoing Formula (18) is explained in detail in Appendix A of this paper. It should be noted that incoherent reflectance includes only contributions from light scattering to small polar angles, which is determined mostly by the roughness with low spatial frequencies (see, e.g., [67]). It also should be emphasized that the formulae for the incoherent reflectance with differently rough boundaries represent a new, so far unpublished, result.

4.3.3. Rayleigh–Rice Theory

As mentioned above, the RRT of the second order can be utilized to express the optical quantities of thin films with randomly rough boundaries if the two following assumptions are fulfilled: the rms values of the heights are much smaller than the wavelengths of incident light and the rms values of the slopes are sufficiently small. Within the RRT, the reflection coefficients ${\widehat{r}}_q$ (where $q=p,s$) of the rough thin films for the p- and s- polarizations at the oblique incidence are expressed as follows (see, e.g., [68,69]):

$${\widehat{r}}_q={\widehat{r}}_q^0+\Delta {\widehat{r}}_q^{RRT}.$$

(20)

The symbol ${\widehat{r}}_q^0$ represents the reflection coefficients corresponding to the polymer-like thin films with transition layers having smooth boundaries, and $\Delta {\widehat{r}}_q^{RRT}$ denotes the corrections of the second order, expressed in the following form:

$$\Delta {\widehat{r}}_q^{RRT}={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\sum _{i,j=1}^3{W}_{ij}(K_x,K_y){\widehat{f}}_{ij,q}(K_x,K_y)\mathrm{d}{K}_x\mathrm{d}{K}_y,$$

(21)

where $i,j=1,2,3$ label the individual boundaries (1, 2 and 3 correspond to the upper boundary, the boundary between the polymer-like film and transition layer and the lower boundary between the transition layer and substrate, respectively). The quantity ${\widehat{f}}_{ij,q}(K_x,K_y)$ is the complicated function of $K_x$, $K_y$, $\lambda$, the optical constants, the mean thicknesses of the layers and the angle of incidence ${\theta }_0$ (these functions are not dependent on the parameters of roughness). The formulae for functions ${\widehat{f}}_{ij,q}(K_x,K_y)$ are presented in [69]. The symbols $K_x$ and $K_y$ represent the spatial frequencies. The symbol $W_{ij}(K_x,K_y)$ corresponds to the power spectral density function (PSDF). It is necessary to point out that Formula (21) includes the effects corresponding to the correlation of the roughness among the boundaries within the RRT.

It is assumed that the roughness with moderate and high spatial frequencies is described by the PSDFs modeled by the Gaussian function for all boundaries. It is also assumed that the autocorrelation length ${\tau }_{RRT}$ is identical for all PSDFs. Moreover, although the heights of the roughness can be different, it is assumed that the roughness of different boundaries is fully correlated (all cross-correlation coefficients are equal to one). These assumptions imply the PSDFs in the following form:

$$W_{ij}(K_x,K_y)={\sigma }_i{\sigma }_j\pi {\tau }_{RRT}^{2}exp\left\{-{\displaystyle \frac{1}{4}}{\tau }_{RRT}^2\left(K_x^2+K_y^2\right)\right\},$$

(22)

where ${\sigma }_1={\sigma }_{RRT}^U$, ${\sigma }_2={\sigma }_3={\sigma }_{RRT}^L$ and ${\tau }_{RRT}$ is the autocorrelation length.

The foregoing assumption concerning one value of the autocorrelation length for all of the power spectral density functions can be explained in the following way.

The topography of the rough boundaries of the polymer-like thin films is close to each other (see Figure 3) and so the values of the auto- and cross-correlation lengths are mutually close. Moreover, it is known that, within the RRT, the sensitivity of the optical quantities of the randomly rough thin films to these lengths is considerably smaller compared to the sensitivity to the rms values of the heights. It is thus reasonable to set the values of all of the correlation lengths to the same value, i.e., ${\tau }_{ij}={\tau }_{RRT}$ for all values of indices i and j (see Equation (22)).

Note that Formula (20) for the reflection coefficients ${\widehat{r}}_q$ corresponds to coherently (specularly) reflected light from films with randomly rough boundaries.

4.3.4. Combination of Scalar Diffraction Theory and Rayleigh–Rice Theory

If the random roughness of the boundaries of the polymer-like thin films exhibits low, moderate and high spatial frequencies, then it is necessary to combine the SDT and RRT when calculating the optical quantities of these thin films (see, e.g., [33,70]). This combination consists of the expression of the local reflection coefficients occurring in the integrand of the HK integral by means of the reflection coefficients calculated using the RRT. As mentioned in Section 4.1, it is assumed that the roughness with low spatial frequencies, which corresponds to the part described by the SDT, is identical for all boundaries. After calculating this HK integral, the following formula for the reflection coefficients ${\widehat{r}}_q$ of the rough polymer-like films with the transition layers corresponding to coherently reflected light is obtained:

$${\widehat{r}}_q=\left({\widehat{r}}_q^0+\Delta {\widehat{r}}_q^{RRT}\right)exp\left(-{\displaystyle \frac{8{\pi }^2{n}_0^2{\sigma }^2{cos}^2{\theta }_0}{{\lambda }^2}}\right).$$

(23)

The detailed description of the derivation of Formula (23) is presented in [33,70]. The correction $\Delta {\widehat{r}}_q^{RRT}$ is given by (21) and (22). It is necessary to emphasize that Formula (23) is only valid when the slopes of the rough boundaries belonging to the low frequencies are negligible. Then, the random roughness with low spatial frequencies does not influence the ellipsometric parameters measured in coherent light because these ellipsometric parameters are ratio quantities. This reality is evident from Formula (23) and the formulae expressing the ellipsometric parameters in reflected light:

$$\begin{array}{cccccc}\hfill I_{\mathrm{s}}& =\mathrm{i}{\displaystyle \frac{{\widehat{r}}_p{\widehat{r}}_s^{\ast }-{\widehat{r}}_p^{\ast }{\widehat{r}}_s}{|{\widehat{r}}_s{|}^2+{\left|{\widehat{r}}_p\right|}^2}},\hfill & \hfill \hspace{1em}\hspace{1em}{I}_{\mathrm{c}}& =-{\displaystyle \frac{{\widehat{r}}_p{\widehat{r}}_s^{\ast }+{\widehat{r}}_p^{\ast }{\widehat{r}}_s}{|{\widehat{r}}_s{|}^2+{\left|{\widehat{r}}_p\right|}^2}},\hfill & \hfill \hspace{1em}\hspace{1em}{I}_{\mathrm{n}}& ={\displaystyle \frac{|{\widehat{r}}_s{|}^2-{\left|{\widehat{r}}_p\right|}^2}{|{\widehat{r}}_s{|}^2+{\left|{\widehat{r}}_p\right|}^2}}.\hfill \end{array}$$

(24)

This means that the ellipsometric parameters corresponding to coherent light are only affected by moderate and high spatial frequencies, which implies that the ellipsometric parameters of the rough polymer-like thin films must be calculated using the RRT.

The coherent reflectance is affected by all of the spatial frequencies, including the low frequencies, even if the roughness with low frequencies exhibits negligible slopes. It is calculated using the following formula:

$$R_{\mathrm{c}}={\displaystyle \frac{|{\widehat{r}}_p{|}^2+{\left|{\widehat{r}}_s\right|}^2}{2}}.$$

(25)

4.3.5. Coherent and Incoherent Reflectance for Shorter Wavelengths

It was found that light scattering was registered by the spectrophotometer from a certain wavelength in the ultraviolet range. It was also found that, for the wavelengths corresponding to light scattering, the values of the total reflectance have to be processed using the SDT. This is due to the fact that the wavelengths are so short that the roughness of the film boundaries fulfils the geometrical conditions corresponding to the SDT (see below). Since the roughness of the upper and lower boundaries is mutually very close, some simplifications can be performed in the formulae for both the coherent and incoherent reflectance presented above. The closeness of the rough boundaries enables us to set $C_{UL}=1$ and ${\tau }_U={\tau }_L={\tau }_{UL}={\tau }_{SDT}$. The symbol ${\tau }_{SDT}$ denotes the common value of the autocorrelation lengths and cross-correlation lengths.

From the foregoing, it is evident that the following form of the characteristic function must be used in the expression of the coherent reflectance $R_{\mathrm{c}}$ and incoherent reflectance $R_{\mathrm{i}}$, i.e.,

$$\overline{\chi }({\widehat{B}}_m,{\widehat{M}}_m)=exp\left\{-{\displaystyle \frac{1}{2}}{\left({\widehat{B}}_m{\sigma }_U+{\widehat{M}}_m{\sigma }_L\right)}^2\right\}.$$

(26)

Formula (18) for the incoherent reflectance $R_{\mathrm{i}}$ is then rewritten in the following way:

$$\begin{array}{c}{R}_{\mathrm{i}}=\sum _{m=0}^{\infty }\sum _{l=0}^{\infty }{\widehat{c}}_m{\widehat{c}}_l^{\ast }\left[\overline{\chi }({\widehat{B}}_m,{\widehat{M}}_m)\right]{\left[\overline{\chi }({\widehat{B}}_l,{\widehat{M}}_l)\right]}^{\ast }\times \hfill \\ \hspace{1em}\hspace{1em} \times \sum _{n=1}^{\infty }{\displaystyle \frac{{\left[{\sigma }_U^2{\widehat{B}}_m{\widehat{B}}_l^{\ast }+{\sigma }_L^2{\widehat{M}}_m{\widehat{M}}_l^{\ast }+{\sigma }_U{\sigma }_L\left({\widehat{B}}_m{\widehat{M}}_l^{\ast }+{\widehat{B}}_l^{\ast }{\widehat{M}}_m\right)\right]}^n}{n!}}\left[1-exp\left(-{\displaystyle \frac{{\pi }^2{n}_0^2{\tau }_{SDT}^2{\alpha }_0^2}{n{\lambda }^2}}\right)\right].\hfill \end{array}$$

(27)

The combination RRT+SDT described in Section 4.3.4 treated the roughness with high and moderate spatial frequencies separately from the roughness with low spatial frequencies. The heights of the roughness ${\sigma }_{RRT}^U$ and ${\sigma }_{RRT}^L$ were used within the RRT and the heights $\sigma$ were used within the SDT. However, when all of the roughness is included using the SDT, the parameters ${\sigma }_U$ and ${\sigma }_L$ correspond to the total heights of the roughness, which can be related to the parameters used in the combination RRT+SDT as

$$\begin{array}{cccc}\hfill {\sigma }_U& =\sqrt{{\left({\sigma }_{RRT}^U\right)}^2+{\sigma }^2},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill & \hfill {\sigma }_L& =\sqrt{{\left({\sigma }_{RRT}^L\right)}^2+{\sigma }^2}.\hfill \end{array}$$

(28)

5. Data Processing

The ellipsometric and reflectometric data of the selected sample were processed simultaneously by means of the least-squares method in the entire spectral range of interest. The spectral dependencies of the optical constants of the silicon single-crystal substrate were fixed to the values published in [71]. Two values for the thickness of the rough polymer-like thin films were determined. One of the thickness values was obtained by processing the ellipsometric data and the latter value corresponded to the reflectance data. These thickness values should be distinguished because of the different systematic errors occurring in the ellipsometric and reflectometric measurements. Moreover, the ellipsometric and reflectometric data are not measured at identical light spots on the sample. In principle, the film characterized can exhibit a certain degree of thickness non-uniformity [46,51]. This can also cause a certain small difference in the thickness values of the films obtained using ellipsometry and reflectometry.

The ellipsometric data satisfied the RRT within the entire spectral range. It was not necessary to include the influence of scattered light on the ellipsometry. This was confirmed by the values of the polarization degree $P={(I_{\mathrm{s}}^2+I_{\mathrm{c}}^2+I_{\mathrm{n}}^2)}^{1/2}$ (see [72]) being equal to unity within the experimental accuracy in the entire spectral range.

The reflectometric data had to be processed with the combination of the SDT and RRT for photon energies under 3.5 eV (see Equation (23)). For photon energies above 3.5 eV, the reflectance data were compatible with the SDT only. Thus, these data were processed using the sum $R_{\mathrm{T}}=R_{\mathrm{c}}+R_{\mathrm{i}}$ within the spectral range above 3.5 eV.

The optical constants and thicknesses of the polymer-like film and transition layer, as well as the parameters ${\sigma }_{RRT}^U$, ${\sigma }_{RRT}^L$, ${\tau }_{RRT}$, $\sigma$, ${\alpha }_0{\tau }_{SDT}$ characterizing the roughness of the boundaries, were sought in the processing of the experimental data. An overview of the used theoretical approaches is provided in

Table 1. Overview of used theoretical models of roughness.

	Ellipsometry	Reflectance IR+VIS	Reflectance UV
model	RRT (non-identical roughness of boundaries)	combination of RRT (non-identical roughness of boundaries) and SDT (identical roughness of boundaries)	SDT (non-identical roughness of boundaries) including contribution of incoherent reflectance
modeled quantities	ellipsometric quantities $I_{\mathrm{s}}$, $I_{\mathrm{c}}$, $I_{\mathrm{n}}$	reflectance $R_{\mathrm{c}}$	reflectance $R_{\mathrm{T}}=R_{\mathrm{c}}+R_{\mathrm{i}}$
relevant formulae	(20)–(22), (24)	(20)–(23), (25)	(4), (5), (13), (16), (26)–(28)
roughness parameters	${\sigma }_{RRT}^U$, ${\sigma }_{RRT}^L$, ${\tau }_{RRT}$	${\sigma }_{RRT}^U$, ${\sigma }_{RRT}^L$, ${\tau }_{RRT}$, $\sigma$	${\sigma }_U={[{\left({\sigma }_{RRT}^U\right)}^2+{\sigma }^2]}^{1/2}$, ${\sigma }_L={[{\left({\sigma }_{RRT}^L\right)}^2+{\sigma }^2]}^{1/2}$, ${\alpha }_0{\tau }_{SDT}$

.

6. Results and Discussion

The values of the structural parameters of the selected polymer-like thin film determined by the optical method presented in

Table 2. The values of the structural parameters corresponding to the polymer-like thin film and transition layer.

Quantity		Value
mean thickness of the film (ellipsometry)	${\overline{d}}_{1\mathrm{e}}$ [nm]	$441.3\pm 0.3$
mean thickness of the film (reflectance)	${\overline{d}}_{1r}$ [nm]	$438.9\pm 0.2$
mean thickness of the transition layer	${\overline{d}}_T$ [nm]	$17.6\pm 0.2$
rms value of the heights at the upper boundary (RRT)	${\sigma }_{RRT}^U$ [nm]	$10.7\pm 0.2$
rms value of the heights at the lower boundary (RRT)	${\sigma }_{RRT}^L$ [nm]	$13.7\pm 0.2$
autocorrelation length (RRT)	${\tau }_{RRT}$ [nm]	$302\pm 5$
rms value of the heights (SDT)	$\sigma$ [nm]	$12.4\pm 0.2$
total rms value of the heights at the upper boundary	${\sigma }_U$ [nm]	$16.4\pm 0.2$
total rms value of the heights at the lower boundary	${\sigma }_L$ [nm]	$18.5\pm 0.2$
product of acceptance half-angle and autocorr. length (SDT)	${\alpha }_0{\tau }_{SDT}$$[\mathrm{rad}·\mathrm{nm}]$	$26\pm 1$

are divided into three groups. The first group consists of the mean thicknesses of the rough polymer-like thin films corresponding to ellipsometric data ${\overline{d}}_{1\mathrm{e}}$ and reflectometric data ${\overline{d}}_{1r}$ and the mean thickness of the transition layer ${\overline{d}}_T$.

The second group consists of the roughness parameters determined using the combination of the RRT and SDT applied to the reflectance below 3.5 eV and using the RRT applied to the ellipsometric parameters in the entire spectral range. These parameters are the rms values of the heights of the upper boundary ${\sigma }_{RRT}^U$ and lower boundary ${\sigma }_{RRT}^L$ corresponding to moderate and high spatial frequencies, the autocorrelation lengths of both boundaries ${\tau }_{RRT}$ belonging to the moderate and high spatial frequencies and the rms value of the heights corresponding to the low spatial frequencies $\sigma$, which is identical for both boundaries.

The third group consists of the roughness parameters determined using the SDT applied to the reflectance above 3.5 eV. These parameters are the total rms values of the heights for the upper boundary ${\sigma }_U$ and lower boundary ${\sigma }_L$ and the product of the acceptance angle ${\alpha }_0$ and autocorrelation length ${\tau }_{SDT}$ at low spatial frequencies. Note that the total rms values of the heights ${\sigma }_U$ and ${\sigma }_L$ are not independent parameters, but they are calculated from $\sigma$, ${\sigma }_{RRT}^U$ and ${\sigma }_{RRT}^L$, as described in Equation (28). Apart from these two quantities, the other parameters listed in Table 2 represent structural parameters sought within the data processing.

From Table 2, it is seen that the total rms value of the heights of the upper boundary ${\sigma }_U$ is smaller than the rms value ${\sigma }_L$ of the lower boundary, which is in agreement with the AFM results. This supports the reality that, during film growth, the smoothing of the upper boundary occurs. It is evident that there is excellent agreement concerning the corresponding rms values of the heights determined by AFM and the optical method for both boundaries. This supports the correctness in determining the rms values ${\sigma }_U$ and ${\sigma }_L$ by the optical method.

Note that the value of the autocorrelation length ${\tau }_{RRT}$ is surprisingly large. The values of this parameter for rough surfaces and rough boundaries with high and moderate spatial frequencies are typically smaller (see, e.g., [33]). The values of this parameter are usually in the order of tens of nanometers for the type of roughness studied here. This large value of ${\tau }_{RRT}$ may be caused by the neglect of a certain statistical correlation between the low spatial frequencies on one side and high and moderate spatial frequencies on the other side in the theoretical approach.

The acceptance angle ${\alpha }_0$ of the detector of the used spectrophotometer was estimated as a value of ${\alpha }_0=0.056\mathrm{rad}$. This value enabled us to estimate the value of the autocorrelation length ${\tau }_{SDT}=450\mathrm{nm}$. The rms values of the slopes can be calculated using the formula $tan{\beta }_0^v=\sqrt{2}{\sigma }_v/{\tau }_{SDT}$, where $v=U,L$. One obtains the following values: $tan{\beta }_0^U=0.0511$ and $tan{\beta }_0^L=0.0576$. These small values justify the assumption that the influence of the slopes can be neglected within the SDT.

The determined values of the dispersion parameters of the polymer-like thin film and transition layer are introduced in

Table 3. The determined values of the dispersion parameters. Arrows indicate that parameters were linked together during fitting.

		Polymer-like	Transition Layer
Quantity		Film	1st Term	2nd Term	3rd Term
Transition strength	$N_{\mathrm{vc}}$$\left[{\mathrm{eV}}^2\right]$	$328\pm 9$	$29\pm 4$	$602\pm 39$	$81\pm 11$
Band gap	$E_{\mathrm{g}}$ [eV]	$12\pm 3$	→	$2.11\pm 0.02$	←
Peak position	$E_{\mathrm{c}}$ [eV]	$13\pm 1$	$3.52\pm 0.01$	$6.7\pm 0.1$	$4.31\pm 0.01$
Peak width	B [eV]	1 (fixed)	$0.61\pm 0.03$	$9.2\pm 0.8$	$1.09\pm 0.04$

. The band gap $E_g$ is searched for all three terms of the dispersion model of the transition layer at an identical value. The polymer-like film was found to be non-absorbing within the entire spectral range. This means that the absorption structures of electron excitations lie above the studied spectral range; therefore, only their influence on the refractive index through the Kramers–Kronig relation can be observed. For this reason, the correct shape of these absorption structures cannot be determined. The value of B could not be determined and therefore its value was fixed at a value of 1 eV.

The spectral dependency of the refractive index of the polymer-like thin film with the randomly rough boundaries calculated using the determined values of the dispersion parameters is presented in Figure 4. The spectral dependency of the refractive index of the polymer-like thin film with the smooth boundaries, prepared under the same technological conditions, is also plotted for comparison. It is seen that both spectral dependencies are very close to each other. This supports the correctness of the determined spectral dependency of the refractive index of the film with the rough boundaries. In the right panel of Figure 4, the spectral dependencies of the optical constants of the transition layer are depicted. In this panel, the spectral dependencies of the optical constants of crystalline silicon are introduced for comparison. From the similarity of the optical constants of the transition layer and substrate, it can be deduced that the transition layer corresponds to the damaged layer on the surface of the substrate created by the pre-treatment in Ar discharge. A relatively complex model with three Campi–Coriasso terms was used to describe the optical constants of the transition layer. It may seem excessive to use such a model for this thin layer because it could be expected that its optical constants and thickness and the parameters describing the roughness of its boundaries are correlated. However, it is evident that the structures corresponding to critical points in the silicon band structure were found correctly. It would not be possible to correctly describe these structures if a simpler model was utilized.

In Figure 5, the fits of the reflectance and ellipsometric parameters corresponding to the selected rough polymer-like thin film containing the transition layer are plotted. One can see that these fits coincide with the experimental data very well. This indicates that both models, i.e., the dispersion and structural models, are constructed in a satisfactory way.

7. Conclusions

A method based on processing the experimental data obtained using variable-angle spectroscopic ellipsometry and spectroscopic reflectometry is described for the optical characterization of homogeneous isotropic non-absorbing thin films exhibiting correlated but differently rough boundaries with wide intervals within the spatial frequencies. This method is illustrated by means of a sample of a polymer-like thin film prepared by PECVD onto a randomly rough silicon single-crystal substrate. It is shown that a combination of the SDT and RRT must be used to process the reflectance data for photon energies under 3.5 eV, and the SDT must be employed for photon energies above 3.5 eV. Moreover, it is found that the ellipsometric data can be processed by the RRT alone within the entire spectral range of interest. The transition layer can be identified as the region of the substrate with structural damage. The determined spectral dependency of the refractive index of the polymer-like film with rough boundaries is in good agreement with that of the polymer-like film having smooth boundaries, prepared using the same technological conditions as the rough film. This supports the correctness of the determined values of the refractive index for the rough polymer-like film. The values of the mean thickness of the polymer-like film and transition layer are determined with high accuracy. The rms values of the heights of the upper and lower boundaries determined by the optical method are in excellent agreement with the corresponding rms values determined by AFM. The optical method and AFM found that the rms value of the height of the upper boundary was evidently smaller than that of the lower boundary. This can be explained by the smoothing of the roughness of the upper boundary with the film’s growth. The very good agreement between the experimental and theoretical data also supports the correctness of the achieved results. The optical method based on spectroscopic ellipsometry and reflectometry described here gives the possibility to perform the optical characterization of thin films fulfilling the presented structural model. This method will be useful particularly if the differences between the values of the roughness parameters of the upper and lower boundaries are relatively small. Thus, the described optical method and its application to a concrete sample represent the novelty of this work. Another novelty lies in the derivation of the new formula for the incoherent reflectance of a film with differently rough boundaries.