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Abstract
Spectral fitting method (SFM) was proposed to obtain the refractive index (RI) and thickness of chalcogenide films based on transmission spectra. It extended the Swanepoel method to the films on the order of hundreds of nanometers in thickness. The RI and thickness of the films can be obtained quickly and accurately by using the SFM based on the transmission spectrum with only one peak and valley. The method’s reliability theoretically was evaluated by simulation analysis. The results showed that the accuracy of the RI and thickness was better than 0.2% by using the SFM regardless of thin or thick film. Finally, the RI and thickness of the new ultralow loss reversible phase-change material Sb2Se3 films were obtained experimentally by the SFM. This work should provide a useful guideline for obtaining the RI and thickness of the transparent optical films.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Chalcogenide films are an important carrier for miniaturization and integration of infrared optical devices [1]. Due to its excellent infrared transparency [2,3], large third-order optical nonlinearity [4] and good photosensitivity [5,6], it has been widely used in photonic and phase change memory devices, etc. The RI and thickness of the films are the key parameter which significantly affect the device performance [7]. Therefore, the accurate determination of the RI and thickness of the films is of critical importance for manufacturing high-quality photonic devices based on chalcogenide films [8–10].
Many methods have been presented to obtain the RI and thickness of the films, including prism coupling method [11,12], spectroscopic ellipsometry [13,14], transmission or reflection spectrum method [15–17], etc. The most common method was Swanepoel method. The RI and thickness of the films is calculated only from their transmission spectra in Swanepoel method. And the method is fast, simple and accurate, so it has been known and used widely [18–20]. However, the calculation accuracy of the method will decrease as the film becomes thinner [8], the thickness of films is usually required to be more than 0.6 μm for obtaining accurate results [21]. Therefore, a new method for getting the RI and thickness of the films on the order of hundreds of nanometers in thickness which are often used in photonic and phase change memory devices is highly desired.
In this paper, the SFM was proposed to obtain the RI and thickness of chalcogenide films quickly and accurately based on transmission spectra. Firstly, the principle of the method was described. Then, the accuracy of the method was analyzed by simulation. Thirdly, the RI and thickness of a new ultralow loss reversible phase-change material Sb2Se3 films were obtained by the SFM [22]. Finally, the accuracy and reliability of the method were analyzed experimentally.
2. Principle
The SFM is an extension of Swanepoel method for obtaining the RI and thickness of chalcogenide films. The RI and thickness of the films can be accurately obtained by optimizing the root mean square error (RMSE) between the theoretically calculated and the experimentally measured transmission spectrum. The schematic diagram of this method is shown in Fig. 1.
The detailed steps are as follows:
Step 1. Measurement of the transmittance T0, the wavelength range is 500 ∼ 2500 nm.
Step 2. Calculate the interference order m at the extrema and the preliminary thickness d. The dispersion of dielectric or semiconductor is usually small in its region of transparency [23,24]. Therefore, for two consecutive extrema in this region there is the following relationship:
From Eq. (1), the following relations are derived:
The last two consecutive extrema in the long wave band are used to calculate m which is integer at peak wavelength and half-integer at valley wavelength, which ensures the accurate acquisition of the interference order. The interference orders of all extrema are obtained based on the interference order difference of consecutive extrema is 0.5. In the transparent region, the upper envelope TM = Ts, the RI at the wavelength of interference order m2 is calculated by Eq. (3) [17].
Step 3. Obtain preliminarily the RI curve n(λ). The theoretical transmittance of a single layer optical film deposited on transparent substrate is as follows:
where A=16n2s, B=(n+1)3(n + s2), C=2(n2-1)(n2-s2), D=(n-1)3(n-s2), φ=4πnd/λ, x = exp(-αd), λ is light wavelength, φ is phase. n is the RI of the film and α is the absorption coefficient of the film. Equation (4) is written as:For Eq. (6) the RI n is a function of the interference order m if the wavelength λ is a certain value and the thickness d is known [25]. Because of the absorption coefficient α = 0 in the transparent area of the film, substituting Eq. (6) and x = 1 into Eq. (5), the relationship between the theoretical transmittance T and the interference order m of the film in the transparent region is obtained.
In this region, the interference order of the adjacent extremum is m1 and m2 respectively, and the wavelength λ is between the two extremum points, then the range of the interference order m at wavelength λ is (m2, m1). The interference order m is obtained when the theoretical transmittance T calculated by Eq. (7) is equal to the experimental transmittance T0 by changing m. The above method is used to obtain the RI at more than 6 different wavelengths. The RI is fitted using the appropriate dispersion formula. Since the chalcogenide film is weakly absorbing in the measurement band, the Cauchy dispersion formula n = a + b/λ2 is selected.
Step 4. Obtain the RMSE between theoretical and experimental transmission spectrum in transparent regions. The RI curve n(λ) and preliminary thickness d, x = 1 are substituted to Eq. (5) to obtain the theoretical transmission spectrum of the transparent region. Because of the error of preliminary thickness d, then the RI curve n(λ) is not accurate, resulting in errors between theoretical and experimental transmission spectrum. The error is evaluated using Eq. (8), where N is the number of wavelengths in the region.
Step 5. Determination of the accurate thickness d. The RMSE of different thickness is obtained by changing the thickness d and repeating steps 3-4 until the RMSE converges. As the RMSE decreases, the thickness approaches the true value. The film thickness is determined until the RMSE is minimum.
Step 6. Obtain the accurate RI of the films by repeating step 3.
3. Simulation and analysis
The calculation accuracy and reliability of the SFM for obtaining RI and thickness were analyzed by using two films with known parameters, these two films were marked as Film 1 and Film 2 respectively. The properties of the two films were as follows: s=1.51, ${n_{\textrm{tr}}} = \frac{{3 \times {{10}^5}}}{{{\lambda ^2}}} + 2.6$, $\lg \alpha = \frac{{1.5 \times {{10}^6}}}{{{\lambda ^2}}} - 8$, d1=128.46 nm, d2=1000.46 nm. The simulation transmittance curve T of the film was shown as the black full curve in Fig. 2(a) by use of the Eq. (5). The curve contained 0.1% noise which was the measurement error from the spectrophotometer.
Fig. 2. The simulation results (a) Transmission spectrum; (b) RMSE at different thicknesses; (c) The RI dispersion curve
The simulation results showed that the interference order of the two films at the position of Tm were 0.5 and 2.5 respectively, and the preliminary thickness of two films were 121.07 nm and 998.87 nm. The curves of the RMSE with thickness were obtained. The scanning step length of the film thickness was 1 nm. As shown in Fig. 2(b), the RMSE converged near the thickness d were 128.00 nm and 999 nm respectively. The illustrations in Fig. 2(b) were obtained by reducing the thickness range and improving scanning accuracy. The RMSE was the smallest when the two films thickness were 128.30 nm and 999.40 nm respectively, and the calculation error was less than 0.13% compared with the true thickness. The interference order m at multiple wavelengths in the transparent region were obtained by using Eq. (7). Then the RI n of the corresponding wavelength were calculated according to Eq. (6). The results were listed in Table 1 and Table 2.
Table 1. The RI of Film 1 at multiple wavelengths.
Table 2. The RI of Film 2 at multiple wavelengths.
The discrete RI n were fitted by Cauchy dispersion formula as shown in Fig. 2(c). The Cauchy fitting equation were as follows:
The results in Table 1 and Table 2 showed that the RI calculated by the SFM were in good agreement with the real value. As shown in Fig. 3(a) and (b), the absolute error of the $\triangle$nCauchy between the fitting value and the real value was only about 0.0025, and the error of RI were less than 0.1%. Although there was weak absorption in the calculated band, the simulation results show that it will not affect the results.
Fig. 3. The RI error of two films by the SFM. (a) The RI error of Film 1; (b) The RI error of Film 2
The thickness error of the films had a greater impact on the calculation accuracy of the RI in the SFM. The influence of thickness error on RI accuracy was analyzed by simulating different thickness errors. The results were listed in Table 3.
Table 3. The influence of thickness error on the accuracy of RI
In Table 3, der was the absolute error of thickness, $\bar{\sigma }$ was the average value of RI error between calculated discrete value and real value, and $\overline {{\sigma _{Cauchy}}} $ was the average value of RI error between Cauchy fitted value and real value. The data in Table 3 were fitted linearly, as shown in Fig. 4. The results showed that the error of RI increased with the increase of der, thus RMSE eventually converged when the thickness d approached the true value. The accuracy of RI was better than 0.2% when the absolute error of thickness der was less than 1 nm. Furthermore, the results showed that the Cauchy fitting of experimental values reduced errors, as shown in Fig. 3 and Fig. 4(a).
Fig. 4. The calculation accuracy analysis. (a) The influence of thickness error der on refractive index accuracy; (b) The influence of thickness non-uniformity $\triangle$d on the accuracy
In addition, the thickness non-uniformity of the film affected the transmission spectrum, which affected the accuracy of the SFM. Therefore, the influence of thickness non-uniformity $\triangle$d on the accuracy of the SFM was analyzed through simulation, and the result was shown in Fig. 4(b). As the non-uniformity of the thickness increased, the accuracy of the SFM decreased. When the non-uniformity$\triangle$d is higher than 6 nm, the error increases significantly. Therefore, the thickness non-uniformity threshold in this method is 6 nm which is an inflection point in the error curve.
The simulation results showed that the overall accuracy of RI by using the SFM was better than 0.2% regardless of thin or thick film and the RI and thickness of the films could be obtained accurately by using the SFM based on the transmission spectrum with at least one peak and valley in the transparent region.
4. Experiment and analysis
4.1 Preparation and characterization of thin films
Sb2Se3 films were deposited by radio frequency (RF) magnetron sputtering the targets prepared by using conventional melt-quenching vacuum method onto clean glass slide substrate. The purity of the target was 99.99%, and the size was Φ76.2mm×3 mm. The principle of magnetron sputtering is the exchange of atomic momentum between the bombarding particles and the target surface. Because of the large kinetic energy of the particle (1-10 eV), the adhesion of the film layer and the substrate is large. During the coating process there is no melting-gaseous state transition. The difference in composition between the film and target was relatively small and the purity was high enough. The chamber pressure was evacuated to lower than 5 × 10−5 Pa, and as Ar gas was introduced and then the sputtering pressure was set to 0.5 Pa. The sputtering was maintained at 40 W RF power. The substrate upon which was deposited was rotated at a speed of 20 r/min and maintained at 25℃ during the deposition. The sputtering time of the two films were 15 and 75 minutes respectively. These two films were marked as Film 1 and Film 2 respectively. When the two films were prepared, the transmission spectrum of the films was obtained by spectrophotometer (PerkinElmer Lambda 950) immediately. The thickness of the two films was measured by profilometer (Veeco Dektak 150).
4.2 Calculation and analysis of film thickness and RI
The measured transmission spectrum was affected by high-frequency noise. The Savitzky-Golay [26–27] filter was used to filter out the noises. The smooth transmission curve was shown in Fig. 5(a). In order to improve the accuracy of calculation the cubic spline interpolation was used for processing the discrete experimental data [28]. The transparent region was determined according to the maximum transmittance of the films equal to that of the substrate. The preliminary thickness of two films calculated by SFM were 174.06 nm and 909.85 nm respectively. The relationship between the thickness and the RMSE was obtained which was shown in Fig. 5(b).
Fig. 5. The RI and thickness of Sb2Se3 films obtained by SFM. (a) Transmission spectrum (b) RMSE at different thicknesses; (c) The RI dispersion curve
The results showed that the thickness d of the two films were 176.80 nm and 911.00 nm respectively. The interference order m and the RI n at multiple wavelengths in the transparent region of the two films were calculated by Eq. (7) and (6) respectively. The results were listed in Table 4 and Table 5.
Table 4. The RI of Film 1 was obtained by SFM
Table 5. The RI of Film 2 was obtained by SFM
The relationship between n and λ of the films was obtained by fitting the RI n in Table 4 and Table 5 with Cauchy dispersion formula.
The nCauchy of the film at multiple wavelengths in Table 4 and Table 5 was calculated by Eq. (10). The result showed that the RI of Film 2 was slightly larger than that of Film 1. In order to explain this phenomenon, the components of the two films were measured. The results showed that the Sb content of Film 2 was slightly higher than that of Film 1. The more Sb content made the RI of the Film 2 larger since the polarizability of Sb was higher than that of Se. In addition, the transmission spectrum of the Film 2 has a red shift, which was consistent with the results of the Ref. [29,30].
Furthermore, the accuracy of the experimental results was verified by comparing the measured and calculated results. Two films thickness measured by profilometer were 177.15 nm and 909.78 nm respectively, and the roughness of the two films were 0.85 nm and 4.46 nm respectively, as shown in Fig. 6. The thickness deviation between measured value and calculated value was less than 0.2%.
Fig. 6. The thickness of the two films was measured by profilometer. (a) The measured thickness of Film 1; (b) The measured thickness of Film 2
The measured value and calculated value of transmission spectrum in transparent region were compared. As shown in Fig. 7, $\triangle$T was the deviation between the calculated and the measured value, the calculated values of the transmittance of the two films were consistent with the measured values, and the overall deviation of the transmittance in the transparent region was less than 2%, the average absolute deviation of the transmittance of the two films was only 0.1% and 0.6%, respectively. The calculation accuracy of the RI was better than 0.2%. The RI of the films calculated was consistent with the value in Ref. [22]. In addition, the results calculated by Levenberg–Marquardt (LM) [31] algorithm and SFM were compared, the results were consistent. Compared to the LM algorithm, the SFM does not require the initial values of each parameter, and accurate results are obtained through step-by-step calculations. In addition, the complicated dispersion models will not increase the computational difficulty of the SFM and affect its accuracy.
Fig. 7. Comparison of transmission spectra calculated and measured. (a1) the transmittance of Film 1; (b1) the transmittance of Film 2; (a2) the deviation between the calculated and measured transmittance of Film 1; (b2) the deviation between the calculated and measured transmittance of Film 2
5. Conclusion
The SFM was presented to obtain the RI and thickness of chalcogenide films accurately based on transmission spectra. It enables the Swanepoel method to be extended to the films on the order of hundreds of nanometers in thickness. The RI and thickness of the films could be accurately obtained by optimizing the RMSE between the theoretical calculation spectrum and experimental transmission spectrum in the SFM regardless of thin or thick film, which was extremely meaningful for the preparation of high-quality micro-nano optical devices. The simulation results showed that the calculated values were in good agreement with the real values. Furthermore, the influence of thickness error on the calculation accuracy was evaluated. The results showed that the calculation error of RI increased with the increase of thickness error and the accuracy of RI was better than 0.2% when the thickness error was less than 1 nm and the thickness non-uniformity of the film was less than 10 nm. Finally, the RI and thickness of the new ultralow loss reversible phase-change material Sb2Se3 films were obtained by the SFM. The results showed that the measured thickness was consistent with the calculated value and the deviation was about 1 nm. Therefore, according to the effect of thickness error on the accuracy of the RI, the measurement accuracy of the RI was about 0.2%. This work should help facilitate further research and application into accurate acquisition of the RI and thickness of the different kinds of transparent optical films.
Funding
K. C. Wong Magna Fund in Ningbo University; Natural Science Foundation of Zhejiang Province (LY18F050003, LY19F050003); National Natural Science Foundation of China (61935006, 62075107, 62075109).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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