Determination of thickness and refractive index of SiO2 thin films using the cross-entropy global optimization method




Global optimization; Optical characterization; Thin film; Cauchy model; SiO2; Cross-entropy; Bootstrapping.


Silicon dioxide (SiO2) is a material that is abundant in nature and has wide application in semiconductor and insulating devices. In this work, a set of six SiO2 samples were grown on a Sigma-Aldrich Silicon substrate, varying the growth time and temperature. This set of samples were grown using times of 10 and 12h and temperatures of 800, 900, and 1000 ºC, under ambient atmosphere. After film growth, reflectance measurements were performed on the films and the substrate, using the Stellarnet UV-VIS-NIR spectrophotometer between 194 and 1081.5 nm. These measurements were modeled using a global optimization method, called Cross-entropy, together with the Bootstrapping resampling technique, seeking to robustly and statistically determine the thin film refractive index as a function of the wavelength and its thickness. To estimate the refractive index of the SiO2 thin film, the Cauchy model was used. For the substrate, reflectance measurements were used. The method proved to be efficient, presenting thickness values that were validated according to growth parameters and literature data. This method proved to be an important and low-cost tool, compared to traditional methods, to help in the steps of building thin films for semiconductor and insulating devices, thus improving their physical properties and enabling the development of new devices.


Batsanov, S. S., Ruchkin, E. D., & Poroshina, I. A. (2016). Refractive Indices of Solids. In Springer Briefs in Applied Sciences and Technology. Springer Singapore.

Black, R. D., Arthur, S. D., Gilmore, R. S., Lewis, N., Hall, E. L., & Lillquist, R. D. (1988). Silicon and silicon dioxide thermal bonding for silicon‐on‐insulator applications. Journal of Applied Physics, 63(8), 2773–2777.

Boos, D. D. (2003). Introduction to the Bootstrap World. Statistical Science, 18(2).

Chakravarty, S., Teng, M., Safian, R., & Zhuang, L. (2021). Hybrid material integration in silicon photonic integrated circuits. Journal of Semiconductors, 42(4), 041303.

Curran, A., Gocalinska, A., Pescaglini, A., Secco, E., Mura, E., Thomas, K., Nagle, R. E., Sheehan, B., Povey, I. M., Pelucchi, E., O’Dwyer, C., Hurley, P. K., & Gity, F. (2021). Structural and Electronic Properties of Polycrystalline InAs Thin Films Deposited on Silicon Dioxide and Glass at Temperatures below 500 °C. Crystals, 11(2), 160.

Eckertová, L. (2012). Physics of Thin Films. Estados Unidos: Springer US.

El-Bindary, A., Anwar, Z., & El-Shafaie, T. (2021). Effect of silicon dioxide nanoparticles on the assessment of quercetin flavonoid using Rhodamine B Isothiocyanate dye. Journal of Molecular Liquids, 323, 114607.

Gao, L., Lemarchand, F., & Lequime, M. (2013). Refractive index determination of SiO2 layer in the UV/Vis/NIR range: spectrophotometric reverse engineering on single and bi-layer designs. Journal Of The European Optical Society - Rapid Publications, 8. doi:10.2971/jeos.2013.13010

Garcia-Caurel, E., De Martino, A., Gaston, J.-P., & Yan, L. (2013). Application of Spectroscopic Ellipsometry and Mueller Ellipsometry to Optical Characterization. Applied Spectroscopy, 67(1), 1–21.

Heavens, O. S. (1991). Optical Properties of Thin Solid Films, Dover Books on Physics Series

Huanca, D. R., & Salcedo, W. J. (2015). Optical characterization of one-dimensional porous silicon photonic crystals with effective refractive index gradient in depth. Physica Status Solidi (a), 212(9), 1975–1983.

Jain, A. K., Dubes, R. C., & Chen, C.-C. (1987). Bootstrap Techniques for Error Estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-9(5), 628–633.

Jesus, J. J. de, Oliveira, A. F. , & Silva, A. P. da. (2021). Espectrômetro digital. Uma proposta de construção de um experimento de Física Moderna para o ensino remoto. Research, Society and Development, 10(8), e51410817786.

Li, J., & Wu, S.-T. (2004). Extended Cauchy equations for the refractive indices of liquid crystals. Journal of Applied Physics, 95(3), 896–901.

Liu, S., Deng, Z., Li, J., Wang, J., & Huang, N. (2019). Measurement of the refractive index of whole blood and its components for a continuous spectral region. Journal of Biomedical Optics, 24(03), 1.

Losurdo, M., Bergmair, M., Bruno, G., Cattelan, D., Cobet, C., de Martino, A., Fleischer, K., Dohcevic-Mitrovic, Z., Esser, N., Galliet, M., Gajic, R., Hemzal, D., Hingerl, K., Humlicek, J., Ossikovski, R., Popovic, Z. V., & Saxl, O. (2009). Spectroscopic ellipsometry and polarimetry for materials and systems analysis at the nanometer scale: state-of-the-art, potential, and perspectives. Journal of Nanoparticle Research, 11(7), 1521–1554.

Oliveira, A. F., Rubinger, R. M., Monteiro, H., Rubinger, C. P. L., Ribeiro, G. M., & de Oliveira, A. G. (2015). Main scattering mechanisms in InAs/GaAs multi-quantum-well: a new approach by the global optimization method. Journal of Materials Science, 51(3), 1333–1343.

Pereira, A.S., Shitsuka, D. M., Parreira, F. J., Shitsuka, R. (2018) Metodologia de pesquisa científica, UFSM

Ribeiro, L. H., Ider, J., Oliveira, A. F., Rubinger, R. M., Rubinger, C. P. L., & de Oliveira, A. G. (2021). Investigation of electronic transport in InAs/GaAs samples. A study using the metaheuristic self-adaptive differential evolution method. Physica B: Condensed Matter, 413293.

Rubinger, R. M., da Silva, E. R., Pinto, D. Z., Rubinger, C. P. L., Oliveira, A. F., & da Costa Bortoni, E. (2015). Comparative and quantitative analysis of white light-emitting diodes and other lamps used for home illumination. Optical Engineering, 54(1), 014104.

Rubinstein, R. Y. (1997). Optimization of computer simulation models with rare events. European Journal of Operational Research, 99(1), 89–112.

Rubinstein, R. Y. & Kroese, D. P. (2004) The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning (Information Science and Statistics), Springer.

Vidakis, N., Petousis, M., Velidakis, E., Tzounis, L., Mountakis, N., Korlos, A., Fischer-Griffiths, P. E., & Grammatikos, S. (2021). On the Mechanical Response of Silicon Dioxide Nanofiller Concentration on Fused Filament Fabrication 3D Printed Isotactic Polypropylene Nanocomposites. Polymers, 13(12), 2029.

Zhu, W. (1997). Making Bootstrap Statistical Inferences: A Tutorial. Research Quarterly for Exercise and Sport, 68(1), 44–55.

Zou, X., Ji, L., Ge, J., Sadoway, D. R., Yu, E. T., & Bard, A. J. (2019). Electrodeposition of crystalline silicon films from silicon dioxide for low-cost photovoltaic applications. Nature Communications, 10(1).




How to Cite

ZACCARO, S. J. V. .; OLIVEIRA, A. F.; RUBINGER, R. M.; SIQUEIRA, C. C. de .; COSTA JUNIOR, R. A. da . Determination of thickness and refractive index of SiO2 thin films using the cross-entropy global optimization method. Research, Society and Development, [S. l.], v. 10, n. 10, p. e326101019028, 2021. DOI: 10.33448/rsd-v10i10.19028. Disponível em: Acesso em: 25 oct. 2021.



Exact and Earth Sciences