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A single-oscillator long-wave-length limit Sellmeier equation based fitting approach applied to the case of thin-film silicon and some of its more common alloys

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Abstract

A procedure for fitting a single-oscillator long-wave-length limit Sellmeier equation to refractive index experimental results is developed. This procedure involves employing a linear least-squares fit to the spectral dependence of the square of the refractive index over the spectral range for which the experimental data are both valid and sufficiently far away from the corresponding optical resonances. From the parameters acquired from these linear fits, the single-oscillator long-wave-length limit Sellmeier equation coefficients may be determined. Following its development, we then apply this approach to the case of thin-film silicon and two of its more common alloys with other materials. A contrast between the classical single-oscillator and double-oscillator Sellmeier equation based fits, as well as that offered through the use of a Cauchy equation based fit, is also provided as a coda to this analysis.

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Notes

  1. A variety of alternate Sellmeier equation forms have been presented in the literature. In this analysis, however, we will just focus on the form depicted in Eq. (1).

  2. An attempt to directly fit \({n^{2} \left( {{E}_{{\text {ph}}}}\right) }\) to the form depicted in Eq. (6), i.e., to a parabolic form, may also be attempted, but it is not as clear and unambiguous as the previously mentioned linear least-squares fit approach.

  3. As no obvious “orthogonality” in the underlying functions exists, this systematic sweep approach appears to be the only way through which optimal coefficient values may be acquired, the usual Sellmeier equation coefficient selection approaches, i.e., educated guesses of the coefficient values and evaluations of the corresponding fit error, are typically sub-optimal.

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Acknowledgements

The authors gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada and MITACS. Inspiring discussions with Mr. Calum Hughes, of Applied Corp., are also acknowledged.

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  1. School of Engineering, The University of British Columbia, 3333 University Way, Kelowna, BC, V1V 1V7, Canada

    Saeed Moghaddam & Stephen K. O’Leary

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Correspondence to Stephen K. O’Leary.

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Moghaddam, S., O’Leary, S.K. A single-oscillator long-wave-length limit Sellmeier equation based fitting approach applied to the case of thin-film silicon and some of its more common alloys. J Mater Sci: Mater Electron 32, 397–419 (2021). https://doi.org/10.1007/s10854-020-04789-3

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