# QUANTUM MECHANICAL CALCULATION OF OPTICAL PROPERTIES OF SILICON- OXIDE NANOPARTICLES USING BORN-MAYER POTENTIAL

## Abstract

In this work, the basic postulates of quantum mechanics are applied to calculate the optical properties of silicon oxide nanoparticle using Born-Mayer potential. The Born-Mayer potential was inputted into the time independent Schrödinger equation and the radial wave function was obtained by solving the equation using Runge-Kutta fourth and fifth order method.. The radial wave functions were then used to evaluate the expectation values of the energy of the system. The energies were used in the calculation of the optical properties such as, absorption coefficient extinction coefficient, real and imaginary dielectric constant and optical conductivity of the

system. The accuracy of the theoretical model used for calculating the optical properties of silicon oxide nanoparticles was validated by comparing the results with experimental values using the Tauc plot which was used to determine the energy band gap (???????? ) of silicon oxide nanoparticles and its optical properties empirically. The results obtained shows an oscillating radial wave function which decays

exponentially in amplitude along the direction of propagation due to the core-shell nature of silicon-oxide nanoparticles The curve of the absorption coefficient shows a band gap value of 9.0 eV which is the theoretical band gap value of silicon oxide nanoparticle. The calculated and experimental curves of the absorption, extinction coefficients and the optical conductivity almost ride on each other and also showed good agreement between the theoretical calculation and experimental data.

**Views | Downloads:**32 / 16

## Downloads

## References

Griffiths, D.J. & Schroeter, F.D. (2018). Introduction to quantum mechanics, 3 rd edition. https://doi.org/10.1080/00107514.2020.1736178

Harrison, P.: Quantum Wells, Wires and Dots, 2nd edn. Willey, New York (2005)

Prasad, P.N.: Nanophotonics, 1st edn. Wiley, Hoboken (2004)

Sadeghi, J., & Pourhassan, B.,(2008) .The Exact Solution of the Non – Central Modified Kratzer Potential Plus a Ring – Shaped like potential by the factorization method, Electronic Journal of Theoretical Physics,17,(4), 193 – 202.

Nayyar, J.S., (2010). Theoretical estimation of optical absorption and photoluminescence in nanostructured silicon; an approach to improve efficiency in nanostructured solar cells. Michigan Tech. http://digitalcommons.mtu.edu/etds/50.

Anande,K.,Durdu, G.,&Ankit V.,(2013).Optical Absorption in NanoStructures: Classical and Quantum Models, . Journal of Quantum Computing,43,(38),1-3

Néstor, D.E., Francisco, J. F., José, A.L., Carlos, R.G., Alfredo, M.S., José, L.S., David, H.L., & Francisco, M.M., (2013).Ab Initiomolecular orbital calculation for optical and electronic properties evaluation of small and medium size silicon nano-clusters found in silicon rich oxide films.Journal of Modern Physics, 4,1-26. http://dx.doi.org/10.4236/jmp.2013.411A2001.

Ikot, 1.A.N., Akpabio, L.E., &Umoren, E.B., (2011).Exact solution of Schrödinger equation with inverted Woods- Saxon and manning-rosen potential. Journal of Scientific Research, 1, 25-33.www.banglajol.info/index.php/JSR.

Okon, I.B., Ituen, E.E., Popoola, O., & Antia, A.D. (2013). Analytical solution of Schrödinger equation with Mie–Type potential using factorisation method. International Journal of Recent advances in Physics (IJRAP), 2(2), 7.

Hasan, H.E.,(2013). A Simple General Solution of the radial Schrodinger equation for spherically symmetric potential, International Journal of Recent advanced in physics,57,(27),12-23.

Iorngbough, Echi, Onoja and Tikyaa J. of NAMP Muzzammi, A.B., Gautam, J.,& Pandy, N.k., (2010). Soliton solution in nonlinear lattice with nearest neighbor Born-Mayer interaction. Journal of Taibah University for Science, 6,(15), 25-56.

Koushki1, A., Koushki2, E. & Gholizadeh, A. (2016). Solution of Schrodinger equation and optical susceptibility for core–shell nanoparticles using Runge–Kutta method. Springer Science+Business Media New York,48(53). DOI 10.1007/s11082-015-0264

Binesh, A., Mowlavi, A.A., & Arabshahi, H.,(2010). Suggestion of proper boundary condition to solving Schrodinger equation for different potential by Runge-Kutta method. Research Journal of Applied Science, 6, 383- 387.

En Naciri, A., Miska, P., Keita, A.S., Battie, Y., Rinnert, H., & Vergnat, M. (2013). Optical properties of uniformly sized silicon nanocrystals within a single silicon oxide layer. Springer Science+Business Media Dordrecht. DOI 10.1007/s11051-013-1538-0.

Kulkarni, A., Guney, G., & Vora, A., (2013). Optical Absorption in Nanostructures: classical and quantum models. Hindawi International Scholarly Research Journal. http://dx.doi.org/10.1155/2013/504341.

Meier, C.,Gondorf, A., Stephan Lüttjohann, S., Lorke, A., & Wigger, H.(2007). Silicon nanoparticles: absorption, emission, and the nature of the electronic bandgap. Journal of Applied Physics,ID 101,302112. http://dx.doi.org/10.1063/1.2720095

Vergen, M. (2013). Optical Properties of uniformly sized silicon nanocrystals within a single silicon oxide layer. Springer Science+Business Media NewYork, 36,(17), 21-45. 1

Abdulla, J.S. (2017). Numerical solution of the Schrödinger Equation for a short-Range singular potential with l????angular momentum. Journal of Applied Physics, 53 (37), 47..

Matsumoto, N.,&Kumabe ,K., (2002). Effect of hydrogen incorporation during deposition by sputtering for amorphous Gallium Phosphide films, Japanese Journal of Applied Physics, 18,(5), 1011

Tshipa, M., (2014). Oscillator strength transitions in a cylindrical quantum wire with an inverse parabolic confining electric potential, Indian Journal of Physics,88,(8),849-853

Nestor, D. (2016). Theoretical and experimental characterization of Silicon Nanocluster embedded in silicon rich oxide, Journal of Modern Physics. 34, (46).37-78.

Nestor,D.,(2016).Theoretical and experimental-characterization-of-silicon-nanocluster-embedded-in-silicon-rich oxide, Journal of Modern Physics, 13,(4),1-28.

Mandal, A., Sarkar,S., Ghosh, P., & Ghosh,M.,(201 [24] Wang,M.J.,Yue,F.Y., & Guo,F.M.,(2015). Photoelectric characteristics of double barrier quantum dots-quantum well photodetector, Advanced in Condensed Matter Physics,ID 920805.

Yilmaz, S.,& Sahin,M.,(2010).Third-order absorption spectra of an impurity in a spherical quantum dot with different confining potential, Physica Status Solidi(b)-Basic Solid State Physics,247,(2),371-374.

Akinlami, J.O. & Olateju, I.O. (2015). Investigation of complex index of refraction of gallium nitride GaN.Journal of Natural Science, Engineering and Technology, 2,(45) ,29-39.

Akinlami, J.O. & Olatunji, A.O. (2014).Optical properties of Gallium Phosphide Gap. Journal of Natural Science, Engineering and Technology,13,18-27.

Ita, 1.B.I, Louis, 1.H., Akakuru,1.O.U., Magu,1.T.O., Joseph, .I., Tchoua, A. P.I., Effiong, I., & Nzeata,1.N.A. (2018). Bound state solutions of the Schrodinger equation for the more general exponential screened coulomb potential plus Yukawa (mgescy) potential using nikiforov-uvarov method. Journal of Quantum Information Science,8, 24-25. http://www.scirp.org/journal/jqis.

Abbas, K., Ehgan, K, & Abdollah, G. (2015). Schrödinger Equation and optical susceptibility for core-shell nanoparticles using Runge-Kutta. Springer Science+ Business media New York, 56,(46),43-57.

Khordad, R. & Mirhossein, B. (2015).Application of Tietz potential to study optical properties of spherical quantum dots. Pramana Journal of Physics,85(4),723-737.

Wang, H., Liu, X., & Zhang, M.Z, (2013). Absorption coefficients of crystalline silicon at wavelengths from 500 nm to 1000 nm. Springer Science+Business Media NewYork, 34, 213-225. DOI 10.1007/s10765-013-1414-2.

Kumari,P., Sinha,S.,& Misha,K.,(2017). A theoretical evalution of changes of refractive index as a function of photon energy for different incident optical intensities and fixed length of quantum wire, Journal of Pure and Applied and Industrial Physics.7,(6),264-274.

Belouifa, A., Bensaad, Z., Soudini, B.,Sekkal, N., Bensaad, A.,& Abid, H., (2009)..First principles calculations of the structural and electronic properties of AlN, GaN,InN, AlGaN and InGaN, Int. J. Nanoelectronicsand Materials 2,(1), 11.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2024 The Journals of the Nigerian Association of Mathematical Physics

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

## How to Cite

*The Journals of the Nigerian Association of Mathematical Physics*,

*63*, 55 –66. https://nampjournals.org.ng/index.php/home/article/view/110