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A Sellmeier extended empirical model for the spectral dependence of the refractive index applied to the case of thin-film silicon and some of its more common alloys

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Abstract

Through modeling thin-film silicon’s refractive index’s spectral dependence, a hybridized empirical approach has been developed. Noting that the empirical polynomial fit paradigm of Moghaddam and O’Leary (J Mater Sci: Mater Electron 30:1637–1646, 2019) has a spectral range of validity that is limited to the exact spectral range over which the primary experimental data set is available, we suggest a means whereby this spectral range may be extended. In particular, by splining a first-order Sellmeier equation onto the long-wavelength side of such a polynomial fit, we have been able to increase the spectral range over which our empirical approach to modeling the refractive index’s spectral dependence is valid. The Sellmeier coefficients corresponding to this first-order Sellmeier ‘extension’ are selected so as to ensure continuity in the refractive index’s spectral dependence, and in that of its derivative, across the boundary between the two regimes, i.e., the polynomial fit regime and the Sellmeier ‘extension’ regime. We then apply this approach to a number of thin-film silicon cases, as well as to two of its more common alloys with other materials, and have found that it adequately captures the experimentally observed spectral dependencies. The coefficients associated with these polynomial fits, as well as that associated with the corresponding first-order Sellmeier equations, are presented in tabular form. We believe that the simplicity of this model offers the thin-film silicon community with a distinct advantage when compared with other approaches to modeling thin-film silicon’s refractive index’s spectral dependence.

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Notes

  1. The long-wave-length limit, i.e., \({\text {E}}_{\text{ph}}\rightarrow \) 0 eV, represents a theoretical ideal that aims to capture the limiting value of the refractive index at long-wave-lengths. As literally written, it is expressed drawing upon the precepts and traditions of calculus. In practice, as may be seen from Fig. 2a, b, the refractive index’s long-wave-length limit is typically achieved with photon energies, \({\text {E}}_{\text{ph}}\), less than 1 eV.

  2. The fact that the optical resonances typically occur in the spectral range over which the primary experimental data is available, or at photon energies above it, precludes the application of this first-order approach to photon energies above the lowest photon energy over which the primary experimental data set is available. Accurate spectral dependence modeling of \(n \left( {\text {E}}_{\text{ph}}\right) \) at such photon energies may require the use of additional terms in the Sellmeier expression, i.e., additional resonant frequencies, and possibly consideration of the corresponding damping coefficients. Given the complexities involved, and given the paucity of high photon energy experimental data, we did not pursue this line of analysis within the framework of the current article.

  3. Our recognition of the limitations to the polynomial fit approach of Moghaddam and O’Leary [56] arose when we tried to apply this approach to a stacked structure. This recognition is what motivated us to seek an extension to our polynomial fit approach in the first place. This recognition occurred several months after the publication of the polynomial fit approach itself, i.e., Moghaddam and O’Leary [56].

  4. There are alternate forms for the Sellmeier’s equation, where a fitting coefficient is introduced in place of the unity immediately following the equality sign depicted in Eq. (2). For the case of this analysis, however, Eq. (2) is employed, no additional element of physicality being added through the introduction of this additional coefficient.

  5. We do not want to consider the use of the Sellmeier expression at shorter wave-lengths, i.e., ones in which a multitude of optical resonances are at play, as the complexities involved in such a spectral fit requires the use of the primary experimental data set corresponding to the particular sample under investigation. Instead, we cover this spectral range with the polynomial fit itself, only considering the Sellmeier expression for an extension into the longer wave-length range, where only one resonance is dominant. Essentially, in our approach, the polynomial fit approach forms the backbone of this analysis, the first-order Sellmeier expression providing for the required spectral extension. We believe that this greatly simplifies the required analysis.

  6. Exact continuity, in \(n({\text {E}}_{\text{ph}})\) and its derivative, can not be achieved on a microscopic scale owing to practical constraints. Nevertheless, our tabulated and plotted results are sufficiently continuous that they may be used, the discontinuities present being relatively minor in nature.

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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 Advanced Micro Biosciences, Inc., are also acknowledged. We would also like to thank the anonymous referees, whose constructive criticisms helped shape the final form of this manuscript.

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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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  1. Saeed Moghaddam
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Correspondence to Stephen K. O’Leary.

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Moghaddam, S., O’Leary, S.K. A Sellmeier extended empirical model for the spectral dependence of the refractive index applied to the case of thin-film silicon and some of its more common alloys. J Mater Sci: Mater Electron 31, 212–225 (2020). https://doi.org/10.1007/s10854-019-02457-9

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