A general and usual approach to fit the dielectric function is to use of Lorentz oscillators, this model assume that the dielectric permittivity can be described by a sum of multiple of resonance Lorentzian functions. This method is often used to have a smooth analytical representation of the dielectric function and is given by,

where E denote the photon energy, and, A_i, Γ_i are the central photon energy, the amplitude, and the broadening of the ith oscillator, respectively. In Equation (1), the dielectric function is described as the sum of different oscillators [14] .

The Sellmeier model originated with the Lorentz model, expressed in terms of wavelengths, and replacing the plasma and resonant frequencies by empirical values. The Sellmeier model corresponds to a region where ε₂ ~ 0 in the Lorentz model and this model can be derived by assuming Γ tends to zero at, where ω₀ is resonant frequency of the electron. In this condition, if we transform dielectric constant using ω/c = 2π/λ, we obtain

where m_e and e are the mass and charge of the electron, N_e is number of electrons per unit volume, c is speed of light in free space, ε₀ is free space permittivity, ω is angular frequency and λ is wavelength of light.

On the other hand, Cauchy formula is a simplified version of the Sellmeier one, applicable to transparent material or spectral regions far from absorption lines; the Cauchy model is given by

The above equation can be obtained from the series expansion of Equation (2). Cauchy model is an equation relative to the refractive index n, an approximate function of the Sellmeier model [14] .

The Tauc-Lorentz oscillator is generally used for amorphous materials, for example for a-Si layers. The absorption band in the imaginary part of the dielectric function is defined as:

where A_TL, E_{n_TL} and B_TL, denotes the amplitude, position, broadening of the TL oscillator, respectively, and E_g is the band gap. The real part of the dielectric function is calculated using the Kramers-Kronig relation [15] .

Figure 2 shows spectral transmittance of SiO_x thin films fabricated a different evaporation rate, from 300 nm to 1100 nm range. Films obtained at low vacuum pressure have similar spectrums with a slightly decrease in transmission for higher evaporation rates, see Figure 2(a). On the other hand, spectral transmittance shown in Figure 2(b), obtained at high vacuum pressure, have the typical interference oscillations related to thin film thickness; from which more oscillations appears when increase film thickness [16] . Spectral transmission curves of SiO_x films, presented in Figure 2, exhibit a transmission shift towards longer wavelengths; this shift can be associated to the amount of silicon (Si) in the film [8] . The spectral transmission of SiO_x films, that had values of x close to 2, approaches to SiO₂ spectrum, as expected. Otherwise when stoichiometry value of SiO_x films is x close to 1, spectrum approaches to the SiO. Clearly, we can observe that when evaporation rate increase, transmittance curve have a shift towards higher wavelengths as consequence of more Si presence in SiO_x films.

Here is considered that SiO_x stoichiometry (1 < x < 2) can be tuned by means of evaporation rate of the film during the deposition process. The ellipsometry provides information of the refractive index dispersion curves and the physical thickness by means of polarization state changes described as the ratio (Ψ) between the amplitude reflection coefficients r_p and r_s and the difference between their phases (Δ). There are several models that use dispersion relations to obtain the thickness and the refractive index (real part n and imaginary part k). Normally, experimental data of Ψ and ∆ in spectral range are compared and adjusted with values of a particular model. Model parameters are adjusted to fit closely experimental and theoretical data. The fitting process includes a choice of initial values for the unknown parameters and the minimization of the function MSE (MSE mean squared error) by subsequent iterations [17] .

MSE function, which is essentially the sum of the squares of the differences between the measured and calculated data for each (∆, Ψ) pair, is given by

where N is the number of measured Y and D pairs, M is the total number of variable parameters and σ is the standard deviations. The superscript Theo means theoretical model and the Exp means the experimental data. Nevertheless, obtaining a low MSE does not guarantee accurate thicknesses and optical constants.

Films that have a refractive index that varies with depth can be approximated as a set of layers, each one with an individual refraction index value. Furthermore, surface roughness can be taken account using the Bruggeman effective medium approximation (EMA) [18] . Physical interpretation of EMA theory involves small particles of one material suspended within a host material. Under this approximation, the optical constants can be mixed to satisfy

electromagnetic equations: the Lorentz Lorenz equation for a mixture of material whose complex dielectric constants and volume fractions are e_i and f_i, and the Bruggeman formula for computing e in terms of e_i and f_i is,

In this model, a single, planar layer, with thickness varying to provide the best approximation of the surface properties, can represent roughness.

The ellipsometry software Complete EASE (J. A. Woollam Co., Inc.) was used to fit experimental and theoretical spectroscopy ellipsometry curves. With this software it is possible to select a physical model and fit with data acquired by means spectroscopic ellipsometry. A diffuse tape was used on substrate backside to eliminate or reduce secondary reflection, which can be a noise signal during measurements. Figure 3 shows dispersion curves obtained for substrate from 250 to 1000 nm with mean square error MSE of 1.88 and surface roughness of 2.7 nm.

During the fitting process, first SiO_x samples were characterized by B-spline or Cauchy model, and with these initial values a second fitness process is realized using other models that have physical significance. In our work, Cauchy, Lorentz, and Tauc-Lorentz models were used due that are more appropriate to amorphous materials. Here, to obtain a better fit, a graded layer, with ten slices, was considered. Typical spectroscopic ellipsometry data for a SiO_x film are shown in Figure 4. In this case, SiO_x film was deposited at 0.35 nm/s and Figure 4(a) shows a fit with Cauchy model with values of MSE = 22.78, thickness 456.20 ± 0.48 nm and rugosity of 2.65 ± 0.13 nm and Figure 4(b) includes a gradient structure of the film with values MSE = 19.60, thickness 453.19 ± 0.43 nm and rugosity of 1.96 ± 0.12 nm. Figure 4(c) shows a fit for a SiO_x film deposited at 0.35 nm/s using Lorentz model with MSE = 9.41, thickness 461 ± 0.23 nm and rugosity of 1.49 ± 0.06 nm and Figure 4(d) include a gradient structure of the film with MSE = 7.44, thickness 450.19 ± 0.54 nm and rugosity of 0.43 ± 0.07 nm.

Table 2 show results of calculated refractive indices for SiO_x films using different models. Cauchy and Lorentz models were used to fit SiO_x films obtained with low vacuum evaporation process, and Lorentz and Tauc- Lorentz models were used to fit SiO_x films obtained with high vacuum evaporation process.

Figure 5 shows dispersion curves for SiO_x films evaporated with different deposition conditions, presented in Table 1. From the last results, it is possible to note the influence of evaporation parameters on dispersion curves. In particular, SiO_x films obtained at low vacuum condition have refractive index in the range of n ~ 1.4 - 1.53, and films obtained at high vacuum condition have higher values of refractive index i.e. n ~ 2.0. The evaporation rate has a notable influence on dispersion curves, and their contribution is described below. In general, we can observe little discrepancies on dispersion curves obtained from the different used models. B-spline model do not have a physical meaning and Cauchy Model is used on transparent films modeling. In order to obtain a best description of dispersion curves, a Lorentz model (oscillators) and Tauc-Lorentz (considering oscillators and density of states) models were used, including a refraction index gradient supposition.

Note^*: Lv is Low vacuum, Hv is high vacuum, C is Cauchy model, L is Lorentz model, and T-L is Tauc-Lorentz model.

From dispersion curves shown in Figure 5 for SiO_x films, it is possible to see a relationship between refractive index of the films and their evaporation rates; refraction index increase as evaporation rate increase. Previous works on the synthesis of SiO_x films, have shown a direct relation between the stoichiometry x and the refractive index of the films [7] - [10] .

Figure 6 shows the refraction index of SiO_x films at 633 nm as a function of evaporation rate, when deposited with high vacuum process of about ~10⁻⁶ Torr. Refractive index values of SiO_x films have values in a range from 1.6 - 1.95; higher n values were obtained at higher evaporation rate, as is expected. At higher evaporation rates, the oxidation in the films is avoided and therefore approach to the stoichiometry and refractive index of

SiO films with x = 1. However, refractive index of SiO_x films evaporated with low vacuum process of about ~10⁻⁴ Torr has lower values of n ~ 1.45 which is close to a SiO₂ stoichiometry with x = 2. When the vacuum in the chamber is low, the film tends to oxide even for high evaporation rates. Furthermore, it can be appreciated that for higher evaporation rates (i.e. 1.5 nm/s) a higher refractive index is obtained, even at this vacuum. From the results presented it is possible to foresee the potential of modulation of refractive index of SiO_x films as function of evaporation rate and vacuum pressure for application in integrated optical devices.