A Whale Optimization Algorithm-Based Data Fitting Method to Determine the Parameters of Films Measured by Spectroscopic Ellipsometry

Abstract

A data-fitting method based on the whale optimization algorithm (WOA) is proposed to determine the thickness and refractive index of films measured by spectroscopic ellipsometry (SE). To demonstrate this method, tin oxide (SnO₂) films with transparent wavelength coverage (400–800 nm) are evaluated. The comparative analysis indicates that Psi and Delta parameter curves based on WOA fitting align more closely with those obtained through experiments. Furthermore, the thickness and refractive index of films obtained by WOA are in nearly agreement with the results from the well-known Levenberg–Marquardt (LM) algorithm. This validation confirms that it has great potential in the determination of film parameters in ellipsometry data fitting.

1. Introduction

Spectroscopic ellipsometry (SE), as a non-contact, non-destructive, and pollution-free optical measurement technique, is commonly employed to measure substrates and films. The measurement principle of SE obtains the change in polarization state after the interaction between the light and the sample. Based on that, multiple measurement parameters, e.g., optical constant (refractive index $n$ and extinction coefficient $k$), thickness, band gap, and surface roughness, can be obtained in one measurement [1,2,3]. SE possesses high precision and accuracy in characterizing the properties of materials. Hence, SE measurement technology has become a powerful tool to obtain optical properties of perovskite solar cells [4,5].

SE measurement technology, as an indirect measurement technology, has some limitations. The optical and geometrical properties of the substrate or the film cannot be directly obtained from the measured spectrum. Instead, it requires a complex process of ellipsometry data analysis, involving a parametric optimization procedure. One of the most commonly used algorithms is the linear regression method based on the gradient-based Levenberg–Marquardt (LM) algorithm. However, the LM algorithm is strongly dependent on good initial values of parameters and easily trapped in local optimality. To address the problem, metaheuristic optimization algorithms have been taken into account. Among them, the whale optimization algorithm (WOA) is not only random but also a complex learning process; hence, it does not dependent on good initial values of parameters, compared to the LM algorithm [6,7]. When the WOA is applied to ellipsometry data fitting, it can realize global searching and does not depend on the initial parameter values. Therefore, the WOA is applied to determine the thickness and refractive index of SnO₂ films in 400–800 nm (transparent band).

Tin oxide (SnO₂), as a kind of n-type semiconductor, has excellent physics and chemistry properties, such as high direct band gap in the range of (3.6–4.31 eV), high electron mobility (250 cm²V⁻¹s⁻¹), low resistivity (2.8 × 10⁻³ Ω cm), natural abundance, non-toxicity, and stable chemical properties [8,9]. The extinction coefficient is close to 0 in the visible and infrared regions due to the high transparency (80–90%) of SnO₂ films, and it has absorption in the ultraviolet region [10]. Due to significant potential application value, SnO₂ material, as the electron transfer layer, has been widely applied in perovskite solar cells [11], and perovskite materials also have wide spectral absorption (about 300–800 nm).

In this paper, WOA is applied to align generated Psi and Delta data by the established model with experimental data to obtain the thickness and refractive index of SnO₂ films, consistently achieving satisfactory results through multiple iterations. The thickness values can be determined in the transparent band of the materials; the data of 400–800 nm is used in the simulation. Simultaneously, the refractive index of the films will be obtained. Based on that, the complexity of fitting can be reduced and the fitting speed sped up. In addition, the refractive index and thickness of SnO₂ films on two types of substrates can be analyzed by the LM with a single Tauc–Lorentz oscillator model.

The structure of this paper is as follows. Section 2 introduces the measurement and fitting principle of spectroscopic ellipsometry. In Section 3, a data-fitting method based on the WOA is proposed to determine the thickness and refractive index of films. In Section 4, the main results are derived from the proposed fitting method. Conclusions and future directions are presented in Section 5.

2. Spectroscopic Ellipsometry Measurement Technology

SE requires a process of measurement, modelling, and fitting to determine the properties of samples. As shown in Figure 1, based on measuring the polarization state of light, SE is used to measure the polarization change information when polarized light interacts with the sample. Based on that, the ellipsometry parameters (i.e., amplitude ratio Ψ/phase difference Δ) can be obtained. Then, the measured data can be fitted by establishing appropriate structural models and optical models of films. Subsequently, optical properties of samples, such as the thickness, refractive indices, or dielectric constants, will be obtained with high precision and accuracy. Usually, it is essential to understand the properties of film material. The fundamental ellipsometry equation is as follows [12]:

$${\displaystyle \frac{R_p}{R_s}}=\tan \mathsf{\Psi }\cdot e^{i\mathsf{\Delta }}$$

(1)

where $R_p$ and $R_s$ are Fresnel reflection coefficients of p-polarized and s-polarized light, respectively.

For an SnO₂ film sample, it consists of three parts, namely the air/SnO₂ film/glass substrate model, as shown in Figure 2. There are two interfaces for the optical model. One is the air/film interface and the other is the film/substrate interface. Hence, the Fresnel reflection coefficient cab be described as follows:

$$R_p={\displaystyle \frac{r_{p,01}+r_{p,12}{e}^{-i2\beta }}{1+r_{p,01}{r}_{p,12}{e}^{-i2\beta }}}$$

(2)

$$R_s={\displaystyle \frac{r_{s,01}+r_{s,12}{e}^{-i2\beta }}{1+r_{s,01}{r}_{s,12}{e}^{-i2\beta }}}$$

(3)

where $r_{p,01}={\displaystyle \frac{N_1\cos {\theta }_0-N_0\cos {\theta }_1}{N_1\cos {\theta }_0+N_0\cos {\theta }_1}}$, $r_{s,01}={\displaystyle \frac{N_0\cos {\theta }_0-N_1\cos {\theta }_1}{N_0\cos {\theta }_0+N_1\cos {\theta }_1}}$. $\beta =2\pi \left({\displaystyle \frac{d}{\lambda }}\right)N_1\cos {\theta }_1$. Similarly, $r_{p,12}$ and $r_{s,12}$ can be obtained. $r_p$ and $r_s$ are reflection coefficients. $d$ is the thickness of the film. The complex refractive index can be described as $N=n+ik$, where $n$ is the refractive index and $k$ is the extinction coefficient.

The process of ellipsometry data fitting involves a parametric optimization procedure. It is necessary to use the fitting algorithm to match the data measured (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$) by SE and the data generated (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) by the model in the entire spectral range. The iterative fitting process is usually completed by the nonlinear LM algorithm. However, the LM algorithm is strongly dependent on good initial values of parameters and is easily trapped in local optimality. Herein, the WOA is utilized to fit measured data and model data. It has the ability to search globally and does not depend on the selection of the initial point. The mean square error parameter (MSE) is usually used to evaluate the fitting effect. Additionally, MSE is regarded as the evaluation function in the WOA. The MSE is defined as follows [13]:

$$MSE=\sqrt{{\displaystyle \frac{1}{2a-b}}{\displaystyle \sum _{i=1}^a\left[{\left({\psi }_{\exp }-{\psi }_{\mathrm{mod}}\right)}^2+{\left({\mathsf{\Delta }}_{\exp }-{\mathsf{\Delta }}_{\mathrm{mod}}\right)}^2\right]}}\times 1000$$

(4)

where $a$ is the number of test points and $b$ is the number of all parameters involved in the fitting process. ${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$ represent measured data, and ${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$ are theoretically modelled data.

Note that some dispersion models are used to describe the optical properties of the materials during the data fitting process. In this paper, the fused quartz (FQ) substrate and soda-lime (SL) substrate employ a Cauchy model [14] and Cauchy–Urbach model [15], respectively. The Cauchy model is obtained from the following equation:

$$n(\lambda )=A+{\displaystyle \frac{B}{{\lambda }^2}}+{\displaystyle \frac{C}{{\lambda }^4}}$$

(5)

where $n$ is refractive index, $\lambda$ is wavelength, and $A$, $B$, and $C$ are the parameters known as the Cauchy coefficients.

The Cauchy–Urbach model adds a second function to describe the absorption tail, with the extinction coefficient as a function of wavelength given by

$$k=\alpha \exp \beta \left(12400\left({\displaystyle \frac{1}{\lambda }}-{\displaystyle \frac{1}{\gamma }}\right)\right)$$

(6)

To obtain the thickness of films, the SnO₂ film layer is modelled with the Tauc–Lorentz model. The fitting procedure of films can be described in three steps. In the first step, the Cauchy model is used to describe the transparent region of films, and can obtain the initial values of thickness. In the second step, the B-spline function is used to fit the whole spectrum curve of films; simultaneously, it satisfies Kramers–Kronig. In the last step, the fitting curve is parameterized by a Tauc–Lorentz model. Based on the above three steps, a better fitting effect and minimization of the MSE can be obtained. The Tauc–Lorentz model has five fitting parameters, namely ${\epsilon }_1(\infty )$, $A$, $C$, $E_g$, and $E_{n0}$, and it can be described as follows [16]:

$${\epsilon }_1={\epsilon }_{\infty }+{\displaystyle \frac{2}{\pi }}P{\displaystyle {\int }_{E_{\mathrm{g}}}^{\infty }\frac{\xi {\epsilon }_2\left(\xi \right)}{{\xi }^2-E^2}}d\xi$$

(7)

$${\epsilon }_2=\{\begin{array}{ll}\frac{A{E}_{n0}C{(E_n-E_g)}^2}{{(E_n^2-E_{n0}^2)}^2}·\frac{1}{E_n}& E_n>E_g\\ 0& E_n\le E_g\end{array}$$

(8)

where $A$ represents amplitude parameter, $C$ is a broadening parameter, $E_g$ represents band gap energy, $E_{n0}$ is peak transition energy, and ${\epsilon }_1(\infty )$ is a parameter associated with high frequency.

3. A Data-Fitting Method Based on the WOA

The WOA is a swarm intelligence optimization search method proposed by Mirjalili and Lewis in 2016, and it is inspired by the hunting of humpback whales [17]. The WOA can realize the purpose of optimal search through the process of whale group search, encircling, hunting, and attacking prey. The WOA is based on meta-heuristics, and the advantages satisfy [18] the following: (1) the searching process is random and (2) the local optimal solutions can be prevented. Details about the optimization process of the WOA can be found in some studies [19,20].

In ellipsometry fitting, the properties of films are unknown and there are two parameters (thickness and refractive index) that need to be determined. In this paper, the WOA is demonstrated for the determination of the thickness and refractive index of SnO₂ film on soda-lime (SnO₂-SL) and SnO₂ film on fused quartz (SnO₂-FQ) in the 400–800 nm (transparent band) range. According to Formula (5), the refractive index of films is described using the Cauchy model, where two parameters ($A$ and $B$) need to be determined, with $C$ set to 0. The WOA will explore a three-dimensional space. Firstly, the WOA will initialize the random solution and constantly update the location. Then, it can find the optimal solution by learning, which is the desired parameter for films. The performance of WOA fitting will be determined based on the fitness function, as defined in Formula (4). The fitness function represents the error between the data measured (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$) by SE and the data calculated (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) by the model. A smaller MSE value between the measured and model data indicates a better fitting result. The (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) can be obtained, and from Formulas (1)–(3) and (5) the values of $A$ and $B$ can be determined.

The framework of the data-fitting method is based on the WOA method, as shown in Figure 3, comprising four distinct parts. Based on that, the desired parameters of films can be obtained. It is worth noting that some parameters of the proposed method must be set. Herein, the maximum number of iterations is set to 60, and the number of individuals is set to 30.

Four steps of the WOA for ellipsometry data fitting are given as follows:

Step 1: The films are measured by SE to obtain ellipsometry parameter curve data (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$).

Step 2: The models of films are built to obtain parameter curve data (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) of samples.

Step 3: The WOA is used to fit the parameter curve data measured (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$) and the model’s parameter curve data built (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$).

Step 4: The thickness and refractive index of the films can be obtained when the value of MSE is relatively small.

4. Experimental and Results

In this section, experiments are performed based on SnO₂ films. To verify the effectiveness of the data-fitting method based on the WOA for ellipsometry data, the thickness and refractive index of SnO₂ films are determined with the transparent wavelength coverage (400–800 nm). During data fitting, the optical model of the samples is considered to be an air/SnO₂ film/glass substrate model.

(1)

Fabrication of films

Two kinds of SL and FQ glasses are selected as the substrates, with a size of 20 × 0 mm. The thickness of the SL and FQ substrates is about 0.7 mm and 1 mm, respectively. Also, the two substrates are transparent in the visible spectrum, which is conducive to the analysis of SE. One side of the substrates is polished, and the other side is roughened to avoid backside reflections in ellipsometry measurement. It is well known that the cleanliness of the substrate can affect the performance of the film. Firstly, the substrates are washed ultrasonically for 15 min. After that, the substrates are washed with detergent, deionized water, acetone, ethyl alcohol, and isopropanol for 30 min, and then blow-dried. Finally, the substrates are cleaned with ultraviolet ozone (UVO) for 30 min. SnO₂ colloidal dispersion (15 wt% in H₂O) was purchased from Alfa Aesar. SnO₂ colloidal solution is diluted by deionized water to SnO₂ nanocrystal solution. The SnO₂ nanocrystal solution is obtained by diluting the SnO₂ colloidal solution with deionized water (v/v = 1:2). The SnO₂ nanocrystal solution is spin-coated on SL and FQ substrates with 4000 rpm for 30 s, and then the films are annealed on a hotplate at 150 °C for 15 min.

X-ray photoelectron spectroscopy (XPS) is used to analyze the elemental composition of the films. Figure 4a,b indicate the XPS survey spectra of both FQ-SnO₂ and SL-SnO₂ films, which can clearly show the peaks of Sn element, O element, and C element in all SnO₂ films. Figure 4c,d show the detailed XPS spectra of Sn3d and O1s. The detailed XPS spectra of Sn3d include Sn3d_5/2 and Sn3d_3/2, and they are found at around 486.5 eV and 494.9 eV, which is very close to the binding energy in the literature [21]. The detailed XPS spectra of O1s are found at around 530.4 eV and 532.0 eV, and are consistent with the article [22]. Based on that, it is indicated that SnO₂ films have been successfully prepared.

When the SnO₂ films have been prepared, the surface morphology of the films is studied using the 3D profilometer (Zygo). Figure 5a,b show the Zygo results of two samples. The surface roughness Sa for nine areas of the films is measured using Zygo. The average values of surface roughness Sa of the SL-SnO₂ and the FQ-SnO₂ are 2.06 nm and 1.03 nm, respectively, which indicates that films have an extremely smooth surface. Based on that, the roughness of the SnO₂ film may be neglected in ellipsometry fitting processing.

Due to the fact that the roughness of SnO₂ films measured by Zygo is relatively small, it may be ignored in the fitting process of ellipsometry. So, to further explore the effect of surface roughness on the optical properties of SnO₂ films, the relationship between the thickness of the rough layer and the ellipsometry curves (Psi and Delta) is simulated. Assuming that the optical constants of two glass substrates are unchanged, the surface roughness of the films is set as 0 nm, 1 nm, 2 nm, and 3 nm, respectively. As shown in Figure 6a,b, it can be clearly observed that the Psi curves and Delta curves hardly change when the values of the rough layer change. In addition, the Delta curve has a slight change compared to the Psi curve. Usually, Psi is related to the optical constant of the film, and Delta is usually related to the thickness of the film. Therefore, the result shows that the film’s optical constant cannot change. In the follow-up ellipsometry fitting, a surface roughness layer of SnO₂ is not taken into consideration for the optical model. On the other hand, it can also avoid introducing too many parameters into the fitting process, which could lead to parameter coupling and negatively affect the SE analysis results.

The thicknesses of SnO₂-FQ and SnO₂-SL films are determined using the LM, and the measurements are performed at light incidence angles of 55°, 60°, and 65°. The SnO₂ films are modelled using the Tauc–Lorentz model. As shown in Figure 7a–d, the measured ellipsometry spectra curves (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) and model curves (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) agree well over the entire spectral range with smaller values of MSE. Subsequently, the thickness of the films is obtained. The thickness and MSE of SnO₂-SL are found to be 36.90 nm and 2.015, respectively, while the thickness and MSE of SnO₂-FQ are 33.94 nm and 1.275, respectively.

(2)

Validation of the WOA for ellipsometry data fitting

The Cauchy model and the Cauchy–Urbach model are utilized to characterize the optical properties of the FQ substrate and SL substrate, respectively. The optical constants of the two substrates are determined, and their parameters remain fixed during the fitting process. To verify the suitability of the WOA, the same SnO₂ films are subjected to ellipsometry measurements and LM fitting. The obtained values of thickness and refractive index of prepared SnO₂ films are compared with the corresponding results obtained using LM fitting on the same SnO₂ films.

Firstly, the WOA is applied to determine the thickness and refractive index of SnO₂-FQ. Figure 8a,b illustrate the experimentally measured ellipsometry parameters (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$) using SE and their corresponding fitted ellipsometry parameters (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$) derived from the model. Here, the symbols of $\psi$ and $\mathsf{\Delta }$ represent Psi curves and Delta curves, respectively. The model data (blue dotted line) are generated using a single Cauchy model applied to the film, while the experimentally measured data (red line) are obtained through SE. The comparison between the two curves indicates a significantly better fitting result. The experimental curve exactly overrides the fitted curve. Based on this, parameters $A$ and $B$ of the Cauchy model are determined, with $C$ set to 0. Further, the refractive index and thickness of the films can be determined. As illustrated in Figure 8c, the refractive index curve (green dotted line) of the SnO₂ film is obtained using the WOA. For the same sample, the corresponding refractive index curve (red dotted line) derived from LM is also shown in Figure 8c. It is clear from the figure that both curves demonstrate a consistent overall trend. These results prove the potentiality of the present approach. Figure 8d presents the convergence curve based on the WOA, showing that the best particle approaches a smaller fitness value after 12 iterations. The thickness of the SnO₂-FQ obtained by the WOA is found to be 34.52 nm.

Similarly, the WOA is applied to SnO₂-SL to determine the thickness and refractive index of SnO₂ films. As shown in Figure 9a,b, it is clearly shown that the measurement ellipsometry curves’ data (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$) fit very well with the corresponding curves generated by the model (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$). The experimental parameter curve exactly overrides the fitted curve. Based on that, to verify the suitability of the WOA, the same SnO₂ film is subjected to LM fitting. In Figure 9c, the refractive index curve (green dotted line) of SnO₂-SL is obtained using the WOA. The corresponding refractive index curve (red dotted line) derived from LM is also presented, demonstrating a consistent overall trend across the entire wavelength range. Hence, the WOA can successfully obtain the refractive index of the SnO₂ film. Figure 9d is the convergence curve of the fitness function based on the WOA, and the best particle is near the smaller fitness value after nine iterations. The thickness of SnO₂-SL obtained by the WOA is found to be 36.66 nm.

The thickness values of two samples obtained by the WOA and LM algorithm are compared, as listed in

Table 1. The thickness of films from LM and the WOA.

Fitting Algorithm	Thickness of SnO2-FQ (nm)	Thickness of SnO2-SL (nm)
LM	33.94	36.90
WOA	34.52	36.66

. The thickness of films obtained by the WOA and LM remains consistent. Hence, as a search algorithm, the WOA can effectively obtain the thickness and refractive index of SnO₂ (transparent band). These results prove the potentiality of the present approach, and they are still very much within the acceptable range for most optical applications. On the other hand, the thickness and refractive index of samples from the WOA can serve as initial values for the LM algorithm, potentially reducing the number of iterations and enhancing the speed of ellipsometry analysis. The algorithm can be successfully applied to obtain a variety of results for the thickness and refractive index of transparent films.

5. Conclusions

In this paper, a data-fitting method based on the WOA is proposed to determine the refractive index and thickness of films in ellipsometry data fitting. The WOA does not depend on the initial parameter values, in comparison with the most commonly used ellipsometry fitting algorithms. The results reveal that the WOA can effectively align experimental data (${\psi }_{\exp }$ and ${\mathsf{\Delta }}_{\exp }$) and model data (${\psi }_{\mathrm{mod}}$ and ${\mathsf{\Delta }}_{\mathrm{mod}}$). The obtained thickness values and refractive index of films are similar under the different methods, confirming the utility of the proposed algorithm in ellipsometry data fitting. The algorithm effectively determines the required parameters of films. As a result, it can be applied to obtain a variety of the thickness and refractive index of transparent films. The WOA allows for the automatization of the ellipsometry data analysis process, which will help in its application by non-experienced users.