PRISA: a user-friendly software for determining refractive index, extinction co-efficient, dispersion energy, band gap, and thickness of semiconductor and dielectric thin films

Figure 1(a) shows the thin film/substrate structure assigned with corresponding optical parameters and thickness. The substrate is transparent and the film can be semitransparent or transparent. The n, k, and d are refractive index, extinction co-efficient and thickness of the thin film, while n_s , k_s = 0 and d_s are that of the substrate. The refractive index of air is assumed to be n₀ = 1. If the film thickness is uniform, then the interference effects give rise to a transmission spectrum as shown in figure 1(b). These interference fringes can be used to determine the optical constants and thickness of thin films using Envelope method, which has been developed and refined by several workers after the work of Manifacier et al [36]. This method uses the interference maxima and minima to derive n, k and d of the film.

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The extremes in a transmission spectrum occur according to the interference condition

$$\begin{eqnarray}&&2nd\,=\,m\lambda \end{eqnarray} \tag{ 1 }$$

where m is an integer for maxima and half integer for minima, and λ is the corresponding wavelength. The curve T_M and T_m define the envelopes passing tangentially through the maxima and minima in the spectrum as shown in figure 1(b). Generally, transmission spectrum has two regions as per absorption in the film: region of weak and medium absorption, and strong absorption region. The first approximation of refractive index of thin film at wavelength of different extreme points is given by

$$\begin{eqnarray}&&n\,=\,\sqrt{N\,+\,\sqrt{{N}^{2}-{n}_{s}^{2}}}\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&N\,=\,2{n}_{s}\displaystyle \frac{{T}_{M}-{T}_{m}}{{T}_{M}{T}_{m}}+\displaystyle \frac{{n}_{s}^{2}\,+1}{2}\end{eqnarray} \tag{ 3 }$$

If n₁ and n₂ are the refractive indices at two adjacent maxima (or minima) at wavelengths λ₁ and λ₂ with λ₁ > λ₂, then the film thickness using equation (1) can be estimated by the relation:

$$\begin{eqnarray}&&d\,=\,\displaystyle \frac{{\lambda }_{1}{\lambda }_{2}}{2\left[{\lambda }_{1}n\left({\lambda }_{1}\right)-{\lambda }_{2}n\left({\lambda }_{2}\right)\right]}\end{eqnarray} \tag{ 4 }$$

Exact smooth envelopes of T_M and T_m are always difficult to generate and may have little errors. As a result, the initial estimated values of refractive index (n_in) and thickness (d_in) of the films derived from equations (2) and (4) could be inaccurate. In order to get accurate values of refractive index and thickness of the thin film, following optimized algorithm should be executed. At first, use the average value of d_in obtained from each two adjacent extremes in equation (1) to calculate order number (m_in) for each extreme and round off the obtained m_in to the nearby integer or half integer. These rounded values are accepted as the exact order number m_fn corresponding to each maxima or minima. Then, the values of m_fn and n_in are used in equation (1) to estimate the accurate thickness d_fn for each extreme excluding first maximum and minimum generally present in the strong absorption region. The average value of d_fn is finally accepted as true thickness of the film. Finally, the true thickness and the exact m values are used in equation (1) to derive the accurate refractive index n_fn for each extreme. Once n(λ) and d of the film are known, then the value of k as a function of wavelength λ can be calculated as follows:

$$\begin{eqnarray}&&k\,=\,-\displaystyle \frac{\lambda }{4\pi d}\,\mathrm{ln}\left(x\right)\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}&&x\,=\,\displaystyle \frac{F-\sqrt{{F}^{2}-\left[{\left({n}^{2}-1\right)}^{3}\left({n}^{2}-{n}_{s}^{4}\right)\right]}}{{\left(n-1\right)}^{3}\left(n-{n}_{s}^{2}\right)}\end{eqnarray} \tag{ 5a }$$

and

$$\begin{eqnarray}&&F\,=\,4{n}^{2}{n}_{s}\displaystyle \frac{{T}_{M}\,+\,{T}_{m}}{{T}_{M}{T}_{m}}\end{eqnarray} \tag{ 5b }$$

The value of n(λ) and k(λ) can be extrapolated by fitting the estimated data using suitable dispersion models such as Cauchy or Sellimier dispersion equations [37]. It is noteworthy to mention that Envelope method works fine and gives very accurate results for relatively thick films as they exhibit large number of interference fringes in their transmission spectrum, while it does not work for metallic or highly absorbing thin films due to absence of interference fringes. It is also not effective for thin films having local absorption bands in between interference fringes. The accuracy of the method becomes poor for lower thickness films. Because the number of fringe extremes gets lower and the spacing between the fringe extremes gets wider with decreasing thickness, which makes interpolation between extremes more difficult [38]. Instead of these shortcoming, Envelope method still remains popular as a routine analysis technique for estimation of optical constants of transparent or semiconducting thin/thick films.

The transmission (T) of a single layer homogeneous thin film deposited on a thick transparent substrate, whose schematic is shown in figure 1(a), is given by [39]:

$$\begin{eqnarray}&&T\,=\,\displaystyle \frac{{n}_{s}{\left|{E}_{t}\right|}^{2}\left(1-{r}_{3}^{2}\right)}{{n}_{0}\left(1-{\left|{r}_{2}{r}_{3}\right|}^{2}\right)}\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&{E}_{t}\,=\,\displaystyle \frac{{t}_{1}{t}_{2}{e}^{-i\beta }}{1\,+\,{r}_{1}r{}_{2}e^{-2i\beta }}\end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray}\begin{array}{lll}{r}_{1}\,=\,\displaystyle \frac{{n}_{0}-\tilde{n}}{{n}_{0}+\tilde{n}} & {r}_{2}\,=\,\displaystyle \frac{\tilde{n}-{n}_{s}}{\tilde{n}+{n}_{s}} & {r}_{3}\,=\,\displaystyle \frac{{n}_{s}-{n}_{0}}{{n}_{s}+{n}_{0}}\end{array}\end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray}\begin{array}{l}\begin{array}{l}\begin{array}{lll}{t}_{1}\,=\,\displaystyle \frac{2{n}_{0}}{{n}_{0}\,+\,\tilde{n}} & {t}_{2}\,=\,\displaystyle \frac{2\tilde{n}}{\tilde{n}\,+\,{n}_{s}} & {t}_{3}\,=\,\displaystyle \frac{2{n}_{s}}{{n}_{s}\,+\,{n}_{0}}\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 6c }$$

$$\begin{eqnarray}&&\beta \,=\,\displaystyle \frac{2\pi \tilde{n}d}{\lambda }\end{eqnarray} \tag{ 6d }$$

Here $\tilde{n}\,=\,n-ik$ is the complex refractive index of thin film. (r₁, t₁), (r₂, t₂₎, and (r₃, t₃) are the amplitudes of (reflection, transmission) co-efficients of air-film, film-substrate, and substrate-air interfaces, respectively. Equation (6) can be used to check the reliability of the n, k, and d values obtained using Envelope/Swanepoel method. These optical parameters and thickness values will be used to simulate or retrieve the transmission spectrum. If the theoretically generated/retrieved spectrum overlaps the original measured spectrum, then it confirms the reliability of the obtained n, k, and d. If it does not then either the thin film is inhomogeneous for which the adopted Swanepoel method does not work or the given refractive index for the substrate is wrong or it could be both.

Wemple and DiDomenico have shown that single oscillator model can successfully describe the dielectric response function of various materials. Using time-dependent perturbation theory and considering only a single group of valence and conduction bands, the frequency dependent real part of the dielectric permittivity of a material can be written as [21]

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(\omega \right)\,=\,1\,+\,\displaystyle \frac{4\pi {e}^{2}}{m{\rm{\Omega }}}\displaystyle \displaystyle \sum _{c,v}\displaystyle \frac{{f}_{cv}^{\alpha }\left(\mathop{k}\limits^{\longrightarrow}\right)}{{\omega }_{cv}^{2}\left(\mathop{k}\limits^{\longrightarrow}\right)-{\omega }^{2}}\end{eqnarray} \tag{ 7 }$$

where, e and m are charge and mass of electron, respectively. Ω is the volume of the crystal ${f}_{cv}^{\alpha }\left(\mathop{k}\limits^{\longrightarrow}\right)$ is the interband oscillator strength, and the sum extends over only valence and conduction bands. If we consider the dispersive dielectric permittivity is due to interband transitions that are described by individual oscillators, and each valence electron contributes one such oscillator, then equation (7) can be approximately given by

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(\omega \right)\,=\,1\,+\,{\omega }_{p}^{2}\displaystyle \displaystyle \sum _{s}\displaystyle \frac{{f}_{s}}{{\omega }_{s}^{2}-{\omega }^{2}}\end{eqnarray} \tag{ 8 }$$

Here, ω_p is the plasma frequency, f_s is the electric-dipole oscillator strength associated with transitions at frequency ω_s . For ω < ω_s , the above equation can be written as

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(\omega \right)\,=\,1\,+\,{\omega }_{p}^{2}\left[\displaystyle \frac{{f}_{1}}{{\omega }_{1}^{2}-{\omega }^{2}}+\displaystyle \displaystyle \sum _{s\ne 1}\displaystyle \frac{{f}_{s}}{{\omega }_{s}^{2}}\left(1+\displaystyle \frac{{\omega }^{2}}{{\omega }_{s}^{2}}\right)\right]\end{eqnarray} \tag{ 9 }$$

Equation (9) can be approximated to a single oscillator by adding all the higher order contributions with the 1st resonant oscillator (s = 1) and retaining terms to order ω², which yields the following equation

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(\omega \right)\approx 1\,+\,\displaystyle \frac{F}{{E}_{0}^{2}-{\left(\hslash \omega \right)}^{2}}\end{eqnarray} \tag{ 10 }$$

where the parameters E₀, and F are directly related to all the f_s and ω_s in equation (9). Wemple and DiDomenico have experimentally shown that the above two parameters are related to each other through a simple equation F = E₀ E_d, where E_d is the so-called dispersion energy, and E₀ is the single oscillator energy. Finally, equation (10) can be expressed as

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(E\right)={n}^{2}\left(E\right)=1+\displaystyle \frac{{E}_{d}{E}_{0}}{{E}_{0}^{2}-{E}^{2}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{n}^{2}\left(E\right)-1}=\left(\displaystyle \frac{-1}{{E}_{d}{E}_{0}}\right){E}^{2}+\left(\displaystyle \frac{{E}_{0}}{{E}_{d}}\right)\end{eqnarray} \tag{ 12 }$$

where E = ћω (eV) is the energy of the light. By plotting (n²−1)⁻¹ versus E² and fitting it with a straight line, the value of the two meaningful parameters E_d and E₀ can be obtained from the slope -1/E_d E₀ and the intercept E₀/E_d, whose physical significances are discussed as follows.

The dispersion energy E_d is a measure of the strength of interband optical transitions, and is related to the charge distribution within unit cell and hence with the chemical bonding. It is directly related to physical parameters of the material through the empirical relation E_d = βN_c Z_a N_e, where N_c is the effective co-ordination number of the cation nearest neighbor to the anion, Z_a is the formal chemical valence of the anion, N_e is the effective number of valence electrons per anion (cores excluded), and β is an energy constant expressing the covalent or ionic character of the solid through the values 0.37 ± 0.04 and 0.26 ± 0.03 eV, respectively [40]. The empirical relation indicates that the dispersion energy parameter E_d is strongly related to the atomic/electronic structure environment in the thin film. Therefore, any variation in structure (chemical bonding, lattice structure, etc.) of thin films can be reflected in the value of E_d. The single oscillator energy E₀ is generally considered as an 'average' energy-gap parameter equivalent to the energy difference between the 'centres of gravity' of the valence and conduction bands. This is like the energy parameter ('Penn gap'), used by Penn' for the determination of the static refractive index (n₀) in semiconductors [4]. The parameter E₀ is empirically related to the lowest direct band gap (E_g ) of the material through the equation E₀ ≈ 1.5E_g [41]. The static dielectric constant of any material is defined as ${\varepsilon }_{0}=\mathop{\mathrm{lim}}\limits_{{\rm{E}}\to {\rm{0}}}{n}^{2}(E)={n}_{0}^{2}.$ The static refractive index (n₀) can be expressed in terms of dispersion parameters using equation (11) as $\varepsilon {}_{0}={n}_{0}^{2}=1+({E}_{d}/{E}_{0}).$ So, the knowledge of dispersion parameters allows us to determine the static dielectric constant of materials.

Absorption co-efficient (α) can be calculated from the interference-free transmission spectrum T_α using the well-known method suggested by Connel and Lewis [42]. The simplified equations to determine α given in the reference [43] and appendix of the reference [16] are as follows:

$$\begin{eqnarray}&&\alpha \,=\,-\displaystyle \frac{1}{d}\,\mathrm{ln}\left(\displaystyle \frac{P+\sqrt{{P}^{2}\,+\,2Q{T}_{\alpha }\left(1-{R}_{2}{R}_{3}\right)}}{Q}\right)\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}&&P\,=\,\left({R}_{1}-1\right)\left({R}_{2}-1\right)\left({R}_{3}-1\right)\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}&&Q\,=\,2{T}_{\alpha }\left({R}_{1}{R}_{2}\,+\,{R}_{3}{R}_{1}-2{R}_{1}{R}_{2}{R}_{3}\right)\end{eqnarray} \tag{ 13b }$$

$$\begin{eqnarray}\begin{array}{lll}{R}_{1}\,=\,{r}_{1}^{2} & {R}_{2}\,=\,{r}_{2}^{2} & {R}_{3}\,=\,{r}_{3}^{2}\end{array}\end{eqnarray} \tag{ 13c }$$

Where r₁, r₂, and r₃, are the Fresnel reflection co-efficients of the air-film, film-substrate, substrate-air interfaces, respectively as illustrated in equation (6b) earlier. Since, the value of imaginary part of the complex refractive index i.e. k is much less than its real part (n) in the region for which the value of α ≤ 10⁵cm⁻¹, therefore the value of k is neglected in the expression of reflectivity at all interfaces. The geometric mean of transmission spectrum having interference fringes in the transparent region is T_α = (T_M T_m )^1/2. But in the strong absorption region, the fringes vanish and all the three curves T_α , T_M , and T_m , converge to a single curve.

The optical band gap (E_g ) can be derived from the absorption co-efficient spectrum (α) in the high absorption region (α ≥ 10⁻⁴ cm⁻¹) using the methods proposed by Tauc et al [44], which was further developed by Mott and Davis [45]. They reported that the optical absorption strength is proportional to the difference between photon energy and band gap as follows:

$$\begin{eqnarray}&&{\left(\alpha E\right)}^{1/M}\,=\,A\left(E-{E}_{g}\right)\end{eqnarray} \tag{ 14 }$$

where E = hν is the photon's energy (h is Planck's constant, and ν is the photon's frequency), E_g is the band gap, and A is a proportionality constant. The value of the exponent denotes the nature of the electronic transition. The value of M = 1/2 and 2 represents direct and indirect allowed transitions, respectively while M = 3/2 and 3 represents direct and indirect forbidden transitions, respectively [46]. Usually, the allowed transitions dictate the fundamental absorption processes, giving either M = 1/2 or M = 2, for direct and indirect transitions, respectively. Thus, the basic procedure for a Tauc analysis is to acquire optical absorbance data for the sample in question that spans a range of energies from below the band-gap transition to above it. Plotting the (αE)^1/2 versus E and (αE)² versus E is a matter of testing M = 1/2 or M = 2 to compare which provides the better linear fit and thus identifies the correct transition type. Generally, absorption becomes saturate at higher energy and the curve deviates from linear in case of both direct and indirect bandgap thin films. The linear region for extrapolation should be wisely selected considering the reasons for the deviations in the lower and higher energy regions. Defect states near the band edge are responsible for the deviation in the low energy end, while saturation of transition states is responsible in higher energy end. The tail or edge associated with the defect states are popularly known as 'Urbach tail' [47] and its distribution of density of sates is represented by an exponential function which can be evident from absorption curve of numerous thin films.

The transmission spectrum of a-Si:H thin film is numerically generated using the optical properties as mentioned in the references [16, 49]. The n, and k of the film are expressed by the following equations:

$$\begin{eqnarray}&&{\rm{Refractive}}\,{\rm{index}}:n=\displaystyle \frac{3\,\times \,{{\rm{10}}}^{5}}{{\lambda }^{2}}\,+\,2.6\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\rm{Absorption}}\,\mathrm{co} \mbox{-} \mathrm{efficient}:{\mathrm{log}}_{10}\alpha \,=\,\displaystyle \frac{1.5\,\times \,{10}^{6}}{{\lambda }^{2}}-8\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\rm{Extinction}}\,\mathrm{co} \mbox{-} \mathrm{efficient}:k\,=\,\displaystyle \frac{\alpha \lambda }{4\pi }\end{eqnarray} \tag{ 17 }$$

The above expression for n and k matches to the experimentally measured values for a-Si:H [43]. Equation (6) is used to generate numerical transmission spectra of a-Si:H thin films of three different thicknesses d = 1000 nm, 700 nm, and 300 nm as shown in figure 3(a). Now, these transmission spectra are analyzed using PRISA. The refractive index spectra obtained from the analysis are plotted in figure 3(b) along with the n(λ) data of equation (15), and that obtained using PARAV software. This figure clearly shows that PRISA gives much accurate results as compared to PARAV, and almost overlaps with the original n(λ) data. The n and k values obtained using Envelope method are extrapolated using the dispersion relations mentioned earlier in section 3. These extrapolated n & k, and d are used in in equation (6) to retrieve the transmission spectrum in the medium-weak absorption and transparent region. The retrieved spectra for a-Si:H thin films for different thicknesses exactly overlap the original spectra as shown in figure 3(a), which confirms the reliability of the obtained value of n, k, and d using PRISA. Accuracy of Envelope method primarily depends on the number of fringes in the transmission spectrum. More the number of fringes, better is the accuracy. Therefore, the software is tested for a-Si:H thin films from lower to higher number of interference fringes. Higher the film thickness, larger is the number of fringes in the spectrum. PRISA is implemented to analyses numerically generated transmission spectra of a-Si:H thin films with film thicknesses varying from 300–1500 nm. The obtained n and k values are plotted in figure (4) along with their numerical data obtained from equations (15) and (17), respectively. The figure shows that PRISA performs well for smaller to larger thickness films. The extinction co-efficient k(λ) spectra does not match to the numerical data in the transparent region, which shows that the reliability of k value becomes poor above ∼10⁻⁴, which is one of the limitations of Envelope method. Now, the dispersive refractive index spectra for different thicknesses of a-Si:H thin films are analyzed using equation (12) to obtain their corresponding E_d , E₀, and n₀. The derived dispersion energy parameters with varying thicknesses are plotted in figure 4(c). It can be clearly seen that the values obtained for higher thickness films are closer to the actual values of the dispersion parameters, while the values are slightly deviated for the lower thickness values. This is primarily due to lower number of data points in the 1/n²−1 versus E plot for lower thickness films, consequently the linear fitting to that data with the equation (13) gets poor. As a result, the parameters E_d , and E₀ derived from the fitting parameters of slope and intercept becomes deviated from the actual value for lower thickness films.

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The thickness of a-Si:H thin films obtained from PRISA and PARAV softwares are listed in table 1 along with the thickness data used to generate the numerical spectrum. It is clearly evident that PRISA gives more accurate thickness result as compared to that of PARAV for a given thin film sample. The thickness value derived using PRISA closely matches to the true thickness value within error bar, while PARAV gives thickness values with large deviation from the original value. Since the refractive index value obtained from PARAV largely deviates in the lower wavelength region of the spectrum as shown in figure 3(b), which subsequently affects the thickness values as per equation (1). The algorithm in PRISA avoids 1st maximum and 1st minimum in the strong absorption region to get accurate thickness value of the film. As per equation (4), at least two peak maxima are required for thickness estimation. Therefore, minimum three numbers of peaks in transmission spectrum are necessary for the optical analysis using PRISA. For a-Si:H thin films thickness below 200 nm, the number of peaks in the transmission spectrum goes below three, therefore the software fails to estimate the optical constants of the films. Hence, the analysis is done up to a-Si:H thin films of thickness 300 nm.

A set of HfO₂ thin films were prepared using reactive electron beam evaporation at different O₂ pressure, and their optical transmission spectra were measured using UV–vis-NIR spectrophotometer, whose details can be found in the reference [3]. A representative transmission spectrum of HfO₂ thin films (sample H-3) along with that of fused silica substrate is plotted in figure 5(a). Transmission spectra of all the films have been analyzed using PRISA (see supplementary material for complete analysis of the representative sample H-3, figure S1 available online at stacks.iop.org/NANOX/2/010008/mmedia). Thickness of the HfO₂ thin films obtained using PRISA, and PARAV are listed in table 2 along with the actual values which is estimated using inverse synthesis of transmission spectrum as mentioned in the reference [3]. The table shows that PRISA gives thickness values close to the actual values within error bar as compared to PARAV. The refractive index spectra of the sample H-3 are plotted in figure 5(b), which shows that PRISA gives more accurate refractive index spectrum as compared to PARAV. In the lower wavelength region, PARAV fails to match the true refractive index value. The n(λ) curve obtained using PRISA shows slight deviation from the true values, which is primarily due to assumed refractive index value of the fused silica substrate. The HfO₂ thin film is deposited on fused silica substrate. Since the fused silica has refractive index variation of 1.54 to 1.46 in the wavelength region of 200–1200 nm, therefore, the assumption of refractive index value of 1.47 for the substrate gives slightly deviated values of refractive index values and thickness. Envelope method provides accurate results for substrate having less dispersive refractive index which means transmission of the substrate is almost parallel to the wavelength axis.

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Subsequently, dispersion energy parameters of the films are derived from the refractive index spectra using PRISA are listed in table 2. The value of E_d decreases from 22.68 eV to 20.12 eV, the values E₀ increases from 8.961 eV to 9.481 eV, and the value of n₀ decreases with increasing O₂ pressure. The variation of n₀ with O₂ pressure indicates that porosity in HfO₂ thin film increases with increasing O₂ pressure during deposition, as refractive index is proportional to the film density [50]. Algorithm in PRISA for estimating absorption co-efficient (α) of thin films at the interference-free absorption region uses equation (13). Subsequently, direct and indirect band gap of the films have been determined from the values of α ≥ 10⁴ cm⁻¹ by fitting the Tauc plots using equation (14) for M = 1/2 and 2, respectively. The data of α versus E, (αE)^1/2 versus E, and (αE)² versus E, for the representative sample H-3 are plotted in figure 5(c), are obtained using PRISA. The derived value of E_g is 5.62 eV for indirect and 6.19 eV for direct band gap of the HfO₂ thin film (sample H-3). The bandgap E_g of all the HfO₂ thin films are listed in table 2. The value of E_g increases with increasing O₂ pressure, and has the same trend like E₀, which validates the proportional correlation between them. The variation of n₀ and E_g with O₂ pressure can be correlated via Moss rule: n₀ ⁴ E_g = constant [51].

PRISA gives accurate value of n, k, d, E_d , E₀, n₀, and E_g , for any kind of homogeneous thin films deposited on weakly dispersive transparent substrate and transmission spectrum of the film must have at least three interference maxima. Absorption in the substrate affects the transmission of the thin film/substrate system. If the absorption is significant, then PRISA gives erroneous results (see supplementary material figures S2 and S3 available online at stacks.iop.org/NANOX/2/010008/mmedia) for such spectrum. However, the accuracy of the parameters can be significantly improved by analysing only the interference fringes data in the spectrum where the substrate seems to be transparent (see supplementary material figure S4 available online at stacks.iop.org/NANOX/2/010008/mmedia). Refractive index inhomogeneity across the layer thickness influences the overall transmission spectrum of thin films (see supplementary material figure S5 available online at stacks.iop.org/NANOX/2/010008/mmedia). It is found that PRISA provides reliable results of n, k, and d for thin films having very small inhomogeneity (see supplementary material figure S6), while it fails for significantly inhomogeneous thin films (see supplementary material figure S7 available online at stacks.iop.org/NANOX/2/010008/mmedia). Other than metallic thin films and very low thickness fringe free thin films, substrate absorption and film inhomogeneity are the other two factors that limit the use of PRISA for optical analysis of thin films. Though, there exist an error or uncertainty in the derived values using different methods in the software, however they are within the acceptable range. The error in the parameters like d n, and E_g are very crucial, as they significantly affect the optical properties of a thin layer. The error in the d value arises primarily due to error or uncertainty in the transmission measurement, error in numerically determining peak value and peak position using algorithm, error in estimating fringe order number, and error in defining sharp edge of strong absorption region. In case of a-Si:H thin films, the results show that the error in d lies in the range of 0.2%–0.4% for the thickness range of 300–1500 nm. The error is very small and is acceptable. Almost the same factors are responsible for error in n value. It is found that a relative error of 0.5% in transmission (T) gives a relative error of about 0.2% in n. The error in n will increase to 0.3% if the relative error in T is 1% [18]. The maximum uncertainty in the bandgap E_g values is 0.01 eV, which is rational for Tauc plot fitting procedure to estimate E_g [46].