A study on the refractive index of sol-gel Ba_1-xGd_xTiO₃ thin films using spectroscopic ellipsometry

ABSTRACT

Ba_1-xGd_xTiO₃ thin films have been fabricated at different Gd³⁺ ionic concentrations, film thicknesses, and annealing temperatures using the sol-gel method. The refractive index of the Ba_1-xGd_xTiO₃ films on a silicon substrate is characterized using Spectroscopic Ellipsometry (SE), where the ellipsometry angles Ψ and Δ are fitted very well with the Cauchy dispersion model. The results show that the refractive index at 632.8 nm decreases from 2.18 to 1.892 with the increase of the Gd³⁺ ratio, while it increases with film thickness and annealing temperature. This trend for refractive index variation is explained based on interatomic spacing and density densification of the films. Using Wemple–Di Domenico (WDD) model shows that the dispersion energy increases with film thickness and annealing temperature and decreases with Gd³⁺ doping. The relatively high refractive index of the samples supports the possibility of using Ba_1-xGd_xTiO₃ thin films as AR coating for solar cells.

Keywords
Ba_1-xGd_xTiO₃; Thin films; Refractive index; Ellipsometry

1. INTRODUCTION

Antireflection (AR) coatings are considered one of the most effective strategies to minimize optical loss for solar cells as illuminated by solar radiation [¹, ²]. Perovskites are promising materials to be used as the active layer in solar cells for those that have a reasonable value of energy bandgap, while using the wide bandgap perovskites type as a top antireflection layer to reduce light reflection and enhance high energy photon absorption [³, ⁴]. Among perovskites, BaTiO₃ has a wide bandgap, high ultraviolet and visible spectrum transparency, and a relatively high refractive index [⁵]. These characteristics are similar to many antireflection coating oxide materials such as ZnO, ZrO₂, and TiO₂ [³]. This means that BaTiO₃ would be an attractive option as an antireflection coating layer for solar cells, enhancing radiation absorption by lowering the incoming light reflection, boosting short circuit current (J_sc), and thereby improving the performance of solar cells [⁶]. The benefit of employing BaTiO₃ as an antireflection coating material is because of the wide range of preparation methods, which are generally inexpensive. However, much focus has lately been placed on the advancement of sol-gel technology for producing and characterizing BaTiO₃ films over the other techniques. At the same time, the sol-gel approach enables to fabrication of homogeneous large-area films and the easiness of doping [⁷].

Due to the desirable characteristics of Gd₂O₃, it has been used for various optoelectronic devices [⁸]. Therefore, it is suggested as a dopant for BaTiO₃ in this work. In Gd³⁺ doped BaTiO₃, substituting Gd³⁺ ions on the Ba²⁺ or Ti³⁺ site in the crystalline structure is proportional to the doping proportion, thus influencing the material’s electrical and optical features [⁹]. A previous study by our team found that Ba_0.9Gd_0.1TiO₃ has a significant transmittance within the visible spectrum and a relatively high energy band gap [¹⁰]. Moreover, it is reported by FASASI et al. [¹¹] that Gd³⁺ doped BaTiO₃ thin films show a high index of refractive. These properties allow using this composition for antireflection coating for different types of solar cells.

Considering the limited works in the literature concerned with studying the optical properties of Gd³⁺ doped BaTiO₃. This work attempts to investigate the impact of Gd³⁺ doping, film thickness, and annealing temperature on the refractive index of Ba_1-xGd_xTiO₃. In which films of the Ba_1-xGd_xTiO₃ formula at different Gd³⁺ doping concentrations, in addition to films prepared at different thicknesses and annealing temperatures via a sol-gel approach and discuss the refractive index of these films using Spectroscopic Ellipsometry.

2. MATERIALS AND METHODS

Ba_1-xGd_xTiO₃ (BGT) thin films with different Gd³⁺ ionic concentrations, thicknesses, and annealing temperatures on Si substrate using the same technique as performed in our previous work [¹⁰]. In order to investigate the refractive index of the films, the angle of Psi (Ψ) and Delta (Δ) are recorded using Spectroscopic Ellipsometry (Alpha-SE, J. A. Woollam Co.) within the wavelength range from 381 nm to 893 nm at room temperature. For higher accuracy of the layer modeling, the Ψ and Δ spectra were measured at three different incident angles of 65°, 70°, and 75°. A compatible fitting model built utilizing the BGT/Si sample structure is used to analyze the obtained spectra. A three-layer model structure was proposed to extract the refractive index to fit the measured spectroscopic ellipsometry data [¹²–¹⁴]. Figure 1 illustrates the three-layer model structure used for Ba_1-xGd_xTiO₃ films.

Figure 1:
Ellipsometry model for Ba_1-xGd_xTiO₃ samples characterization.

3. RESULTS AND DISCUSSIONS

The refractive index of the material beholds a mechanism for trapping the light for more absorption to occur. Theoretically, in the light-to-electricity conversion of photovoltaic applications, a higher refractive index layer adds an advantage to trapping more photons within the material for better absorption. In order to study the refractive index of the Ba_1-xGd_xTiO₃ films, the reflectance properties of the films, deposited on a silicon substrate, are studied using Spectroscopic Ellipsometry (SE). In which the polarisation state change of the sample-reflected beam is recorded. The ellipsometric Psi (Ψ) and Delta (Δ) parameters are frequently used to describe the shift in the polarization state; they are usually defined as [¹⁵]

$$\rho =\tan \left(\psi \right).e^{i\Delta }=\frac{r_p}{r_s}$$(1)

where Ψ and Δ are ellipsometry angles, and rp and rs are the complex coefficients of reflection of light that are polarized parallel and normal to the plane of impinging, respectively. Tan Ψ is the ratio of the absolute value of the p-polarized and s-polarized field components, while Δ denotes the phase difference between the reflected components.

The optical model and the best-fit parameter values are calculated using CompleteEASE software provided by J.A. Woollam Co., Inc. The goodness of fitting, denoted as Mean Squared Error (MSE), quantifies the quality of the fitting, and it is defined by [¹⁶]

$$MSE=\sqrt{\frac{1}{3n-m}{\displaystyle {\sum }_{i=1}^n\left[{\left(N_{Ei}-N_{Gi}\right)}^2+{\left(C_{Ei}-C_{Gi}\right)}^2+{\left(S_{Ei}-S_{Gi}\right)}^2\right]}}\times 1000$$(2)

where n is the number of wavelengths, m is the number of fitting values, and N = Cos(2Ψ ), C = Sin(2Ψ )Cos(Δ), and S = Sin(2Ψ)Sin(Δ). The data is evaluated using a linear regression technique, and optical parameters and film structures are identified by lowering fitting errors [¹⁷]. According to J.A. Woollam Co., the acceptable MSE value should be lower than 20.

The optical constant of Ba_1-xGd_xTiO₃ films is derived using a three-layer model (SRL/Ba_1-xGd_xTiO₃/Substrate). Since Ba_1-xGd_xTiO₃ films exhibit partial transparency over this wavelength range [¹⁰], the measured Ψ and Δ are fitted using the Cauchy dispersion model given by [¹³]

$$n\left(\lambda \right)=A_n+\frac{B_n}{{\lambda }^2}+\frac{C_n}{{\lambda }^2}$$(3)

Here, n (λ) represents the refractive index at a specific wavelength λ, whereas An, Bn, and Cn are constants. The well-fitted Cauchy dispersion model of the measured Ψ and Δ for Ba_1-xGd_xTiO₃ films with various Ba:Gd ratios, film thicknesses, and annealing temperatures are illustrated in Figures 2, 3, and 4, respectively. The figures show the good matching of the fitting model and the measured data with a Mean Squared Error (MSE) value lower than 20. The refractive index dispersion of BGT films is determined through the variable fitted parameters of An, Bn, and Cn. The values of the fitted parameters for Ba_1-xGd_xTiO₃ films at various Gd³⁺ content, film thicknesses and annealing temperature are summarized in Tables 1, 2, and 3, respectively.

Table 1:
The Cauchy-fitted parameters values for Ba_1-xGd_xTiO₃ films at different doping ratios.

Ba:Gd	MSE	A	B (nm2)	C (nm4)	n at 632.8 nm
BaTiO3	2.412	2.103 ± 0.253	0.031 ± 0.007	0.0002 ± 0.0001	2.180
0.95:0.05	1.98	2.048 ± 0.232	0.025 ± 0.005	0.0007 ± 0.0002	2.130
0.90:0.10	2.15	2.024 ± 0.221	0.020 ± 0.004	0.0002 ± 0.0001	2.076
0.85:0.15	6.99	1.998 ± 0.889	0.020 ± 0.017	–0.00002 ± 0.0001	2.048
0.80:0.20	1.85	1.926 ± 0.304	0.024 ± 0.008	0.0002 ± 0.0001	1.988
0.75:0.25	2.16	1.887 ± 0.429	0.024 ± 0.011	0.00008 ± 0.0001	1.948
0.70:0.30	13.2	1.847 ± 1.148	0.020 ± 0.026	–0.0008 ± 0.001	1.892

Table 2:
The Cauchy-fitted parameters values for Ba_0.95Gd_0.05TiO₃ films at different thicknesses.

THICKNESS (nm)	MSE	A	B (nm2)	C (nm4)	n at 632.8 nm
269.57	6.347	1.953 ± 0.528	0.019 ± 0.01	0.00147 ± 0.001	2.008
332.96	8.986	1.971 ± 0.004	0.041 ± 0.002	–0.00046 ± 0.0002	2.071
416.89	10.221	2.204 ± 0.018	0.032 ± 0.01	0.000422 ± 0.001	2.142
502.15	5.487	2.263 ± 0.045	0.016 ± 0.001	0.00447 ± 0.0001	2.332

Table 3:
The Cauchy model derived fitted parameters values for Ba_0.95Gd_0.05TiO₃ films at various temperatures.

ANNEALING TEMPERATURE	MSE	A	B (nm2)	C (nm4)	n at 632.8 nm
600 °C	7.11	1.718 ± 0.566	0.014 ± 0.011	0.0005 ± 0.0003	1.756
700 °C	3.05	1.874 ± 0.251	0.011 ± 0.003	0.0023 ± 0.0006	1.914
800 oC	2.42	2.068 ± 0.156	0.017 ± 0.002	0.0010 ± 0.0001	2.116
900 °C	5.75	1.864 ± 1.188	0.005 ± 0.007	0.0003 ± 0.0005	1.879

Figure 2:
Measured Psi and Delta spectra with the fitted Cauchy model for Ba_1-xGd_xTiO₃ films for the ratios (a) 0.95:0.05 and (b) 0.70:0.30.

Figure 3:
Measured Psi and Delta spectra with the fitted Cauchy model for Ba_0.95Gd_0.05TiO₃ films at various thicknesses (a) 325.56 nm, (b) 424.19 nm, (c) 519.24 nm, and (d) 591.54 nm.

Figure 4:
Measured Psi and Delta spectra with the fitted Cauchy model for Ba_0.95Gd_0.05TiO₃ films annealed at (a) 600 °C, (b) 700 °C, (c) 800 °C, and (d) 900 °C.

The refractive index dispersion for Ba_1-xGd_xTiO₃ films is illustrated in Figure 5. A normal dispersion behavior is observed as the refractive index is reduced with the increase of the wavelength. From Figure 5(a), it is observed that the refractive index at 632.8 nm wavelength reduces from 2.13 to 1.89 with the increase of Gd³⁺ content into the Ba_1-xGd_xTiO₃ lattice. The refractive index of polycrystalline films is often controlled by film crystallinity, electronic band structure, lattice point defect, and film density. According to SENGODAN et al. [¹⁸], the refractive index of the thin film is proportional to electronic polarization, which is also inversely proportional to the interatomic spacing. Interatomic spacing refers to the distance between atoms in the material. When the interatomic spacing reduces and forms packed crystals, the film density increases and thus increases the refractive index as well. Similarly, WANG et al. [¹⁹] reported that the increase in refractive index is correlated to the expansion of the lattice. This elucidates that the decrease in refractive index is due to the lattice contraction resulting from ionic size reduction. The lattice constants and volume for Ba_1-xGd_xTiO₃ are deeply investigated and reported in previous work [²⁰].

Figure 5:
The refractive index dispersion of Ba_1-xGd_xTiO₃ thin films at different (a) Ba:Gd ratios, (b) film thicknesses for Ba_0.95Gd_0.05TiO₃, and (c) annealing temperatures.

The refractive index at different film thicknesses is presented in Figure 5(b). Where the refractive index at 632.8 nm wavelength increases from 2.01 to 2.33 with the increase of film thickness. Generally, the increase in film thickness improves film densification with the addition of layer deposition [¹³, ²¹]. Thus, it can be concluded that the variation of the refractive index with the variation of film thickness is attributed to the increase in the film density for thicker films. Under the effect of annealing temperature, the refractive index is increased from 1.76 to 2.12 as the annealing temperature rises from 600 °C to 800 °C. A sudden drop in refractive index to 1.88 is noticed at 900 °C. The increase in refractive index with the annealing temperature is attributed to the improved densification and crystallinity of the films; however, the drop in the refractive index at 900 °C could result from the second phase formed at high annealing temperature [²⁰]. The values of the refractive index for Ba_1-xGd_xTiO₃ films show comparable values to the reported refractive index of sol-gel derived BST film [²²], BCZT film [¹⁹] and BaTiO₃ film [¹²]. However, the values are lower than the reported value for Gd-BaTiO₃ film fabricated via laser ablation [¹¹], which could be due to the fabrication method.

In order to have insight into the dispersion of the refractive index of the Ba_1-xGd_xTiO₃ films and correlate its behavior with the films’ structural order, the refractive index as a function of wavelength is further analyzed by fitting into the Wemple–Di Domenico (WDD) model. The dispersion energy parameters are extracted through the fitting of the WWD model using the following relation [²³]

$$n^2-1=\frac{E_0{E}_d}{E_0^2-E^2}$$(4)

where n is the refractive index, E0 denotes the single-oscillator energy, and Ed denotes the dispersion energy. The dispersion energy describes the average intensity of interband optical transitions and relates to variations in the structural ordering of the material. Figure 6 demonstrates the 1/(n²–1) vs E² plot using the least squares fitting of the WWD model. The excellent linear fitting of the straight line in the plot over the wavelength range of 381–893 nm confirms the validity of the WWD model for the Ba_1-xGd_xTiO₃ thin films. E0 and Ed can be evaluated from the slope and the intercept of the plot at E = 0. The estimated dispersion energy parameters for Ba_1-xGd_xTiO₃ thin films at various Gd³⁺ ratios, film thickness and annealing temperature are listed in Figure 7.

Figure 6:
The plot of (n²-1)^–1 against E² with fitted WWD model for Ba_1-xGd_xTiO₃ thin films at different (a) Ba:Gd ratios, (b) thicknesses, and (c) annealing temperatures.

Figure 7:
The dispersion energy of Ba_1-xGd_xTiO₃ films at various (a) Gd³⁺ doping ratios, (b) film thicknesses, and (c) annealing temperatures.

Figure 7(a) illustrates the dispersion energy E_d of Ba_1-xGd_xTiO₃ at different Gd³⁺ ratios. It can be noticed that the dispersion energy reduced from 24.99 to 15.13 eV as the Gd³⁺ ratio increased. According to KERMADI et al. [²⁴], dispersion energy is related to the structural order of a material. The higher order of the material leads to larger dispersion energy. Accordingly, the decrease in E_d value indicates that the order of the atomic structure in the Ba_xGd_1-xTiO₃ films reduces as more Gd³ ions are substituted into the Ba-site. To confirm this finding and give detailed information on the level of structural disorder of current films, XRD results from previous work have been employed [²⁰]. In which the estimated lattice constants and volume show decrement variation with Gd³⁺, indicating that the lattice order is degraded; this is in line with E_d values reduction. A similar observation for the dependence of the dispersion energy on the lattice size for barium strontium titanate films is reported by SINGH et al. [²⁵].

An increment in the dispersion energy parameters is observed with the film thickness, as shown in Figure 7(b). The Ed increases from 18.93 to 26.90 eV with the increase of the film thickness, indicating that the microstructure ordering is enhanced. Such increment in the structural order is correlated to the increases in grain size with thickness [²⁰].

Figure 7(c) shows that the strength of dispersion energy E_d increases from 14.77 eV to 29.15 eV with the increase of annealing temperature. When the annealing temperature increased, the order of the atomic structure of the film increased and resulted in high dispersion energy. This is in line with the XRD results reported in a previous work [¹⁰], in which it is found that the lattice volume increases with annealing temperature due to lattice growth at higher temperatures, which indicates the crystallinity improvement. Similar behaviour of dispersion energy under the effect of annealing temperature is also observed by Kuo and Tseng in barium strontium titanate thin films [²⁶]. They explained that the increment of the dispersion energy is associated with the increase in film packing density.

4. CONCLUSIONS

The refractive index of Ba_1-xGd_xTiO₃ thin films at different Gd³⁺ contents, film thicknesses, and annealing temperatures using Spectroscopic Ellipsometry. The measured Ψ and Δ for all tested samples fitted very well with the Cauchy dispersion model, which is used to evaluate the refractive index of the films. The results show that the dispersion of refractive index with wavelength shows that n at 632.8 nm decreases from 2.18 to 1.892 with the increase of Gd³⁺ ratio, while it increases from 2.01 to 2.33 with film thickness and increases from 1.76 at 600 ^°C to 2.12 at 800 ^°C then drops to 1.88 at 900 ^°C. This trend for refractive index variation is explained based on interatomic spacing and lattice volume. Besides, the films under investigation show very good fitting to the Wemple–Di Domenico (WDD) model; accordingly, the dispersion energy shows an increment with film thickness and annealing temperature, indicating that the microstructure of the films is getting more order. On the other hand, an opposite attitude with Gd³⁺ doping is obtained. Out of the results, it can be concluded that a low Gd³⁺ doping and high film thickness and annealing temperature of 800 °C are recommended to obtain a high value of the refractive index for Ba_1-xGd_xTiO₃ thin films for AR coating solar cells application.

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  » https://doi.org/10.1016/S0254-0584(99)00157-1

Publication Dates

• Publication in this collection
  04 Sept 2023
• Date of issue
  2023

History

• Received
  12 June 2023
• Accepted
  31 July 2023