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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

  • Saeed Moghaddam
  • Stephen K. O’LearyEmail author
Article
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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

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.

References

  1. 1.
    G. D. Cody, in Hydrogenated Amorphous Silicon, Semiconductors and Semimetals Vol. 21, edited by J. I. Pankove (Academic Press, New York, 1984), Pt. B, p. 11-82Google Scholar
  2. 2.
    M. Stuckelberger, R. Biron, N. Wyrsch, F.-J. Haug, C. Ballif, Renew. Sust. Energ. Rev. 76, 1497–1523 (2017)CrossRefGoogle Scholar
  3. 3.
    S. Wagner, MRS Bull. 43, 617–622 (2018)CrossRefGoogle Scholar
  4. 4.
    D.T. Pierce, W.E. Spicer, Phys. Rev. B 5, 3017–3029 (1972)CrossRefGoogle Scholar
  5. 5.
    W.B. Jackson, S.M. Kelso, C.C. Tsai, J.W. Allen, S.-J. Oh, Phys. Rev. B 31, 5187–5198 (1985)CrossRefGoogle Scholar
  6. 6.
    F. Stern, Phys. Rev. B 3, 2636–2645 (1971)CrossRefGoogle Scholar
  7. 7.
    G.D. Cody, T. Tiedje, B. Abeles, B. Brooks, Y. Goldstein, Phys. Rev. Lett. 47, 1480–1483 (1981)CrossRefGoogle Scholar
  8. 8.
    Z. Shi, M.A. Green, Prog. Photovolt. Res. Appl. 6, 247–257 (1998)CrossRefGoogle Scholar
  9. 9.
    C. Beneking, B. Rech, S. Wieder, O. Kluth, H. Wagner, W. Frammelsberger, R. Geyer, P. Lechner, H. Rübel, H. Schade, Thin Solid Films 351, 241–246 (1999)CrossRefGoogle Scholar
  10. 10.
    A. Terakawa, Sol. Energ. Mater. Sol. Cells 119, 204–208 (2013)CrossRefGoogle Scholar
  11. 11.
    T. Tsukada, J. Non-Cryst. Solids 164–166, 721–726 (1993)CrossRefGoogle Scholar
  12. 12.
    Y. Yamamoto, Jpn. J. Appl. Phys. 51, 060001-1-12 (2012)CrossRefGoogle Scholar
  13. 13.
    N. Wyrsch, C. Ballif, Semicond. Sci. Technol. 31, 103005-1-15 (2016)CrossRefGoogle Scholar
  14. 14.
    M.H. Brodsky, R.S. Title, K. Weiser, G.D. Pettit, Phys. Rev. B 1, 2632–2641 (1970)CrossRefGoogle Scholar
  15. 15.
    E.C. Freeman, W. Paul, Phys. Rev. B 20, 716–728 (1979)CrossRefGoogle Scholar
  16. 16.
    R.E. Viturro, K. Weiser, Philos. Mag. B 53, 93–103 (1986)CrossRefGoogle Scholar
  17. 17.
    R.V. Kruzelecky, C. Ukah, D. Racansky, S. Zukotynski, J.M. Perz, J. Non-Cryst. Solids 103, 234–249 (1988)CrossRefGoogle Scholar
  18. 18.
    H.V. Nguyen, Y. Lu, S. Kim, M. Wakagi, R.W. Collins, Phys. Rev. Lett. 74, 3880–3883 (1995)CrossRefGoogle Scholar
  19. 19.
    L. Jiao, I. Chen, R.W. Collins, C.R. Wronski, N. Hata, Appl. Phys. Lett. 72, 1057–1059 (1998)CrossRefGoogle Scholar
  20. 20.
    A. Kondilis, Phys. Rev. B 62, 10526–10534 (2000)CrossRefGoogle Scholar
  21. 21.
    B.J. Fogal, S.K. O’Leary, D.J. Lockwood, J.-M. Baribeau, M. Noël, J.C. Zwinkels, Solid State Commun. 120, 429–434 (2001)CrossRefGoogle Scholar
  22. 22.
    S.K. O’Leary, B.J. Fogal, D.J. Lockwood, J.-M. Baribeau, M. Noël, J.C. Zwinkels, J. Non-Cryst. Solids 290, 57–63 (2001)CrossRefGoogle Scholar
  23. 23.
    D.J. Lockwood, J.-M. Baribeau, M. Noël, J.C. Zwinkels, B.J. Fogal, S.K. O’Leary, Solid State Commun. 122, 271–275 (2002)CrossRefGoogle Scholar
  24. 24.
    L.-L. Tay, D.J. Lockwood, J.-M. Baribeau, M. Noël, J.C. Zwinkels, F. Orapunt, S.K. O’Leary, Appl. Phys. Lett. 88, 121920-1-3 (2006)Google Scholar
  25. 25.
    F. Orapunt, S.K. O’Leary, J. Mater. Sci.: Mater. Electron. 20, 1033–1038 (2009)Google Scholar
  26. 26.
    E. Malainho, M.I. Vasilevskiy, P. Alpuim, S.A. Filonovich, J. Appl. Phys. 106, 073110-1-9 (2009)CrossRefGoogle Scholar
  27. 27.
    J. Müllerová, L. Prus̆áková, M. Netrvalová, V. Vavrun̆ková, P. Š̆utta, Appl. Surf. Sci. 256, 5667–5671 (2010)CrossRefGoogle Scholar
  28. 28.
    F. Yuan, Z. Li, T. Zhang, W. Miao, Z. Zhang, Nanoscale Res. Lett. 9, 173-1-5 (2014)Google Scholar
  29. 29.
    F. Orapunt, L.-L. Tay, D.J. Lockwood, J.-M. Baribeau, M. Noël, J.C. Zwinkels, S.K. O’Leary, J. Appl. Phys. 119, 065702-1-12 (2016)CrossRefGoogle Scholar
  30. 30.
    J. Tauc, R. Grigorovici, A. Vancu, Phys. Status Solidi 15, 627–637 (1966)CrossRefGoogle Scholar
  31. 31.
    W.-C. Chen, B.J. Feldman, J. Bajaj, F.-M. Tong, G.K. Wong, Solid State Commun. 38, 357–363 (1981)CrossRefGoogle Scholar
  32. 32.
    S. John, C. Soukoulis, M.H. Cohen, E.N. Economou, Phys. Rev. Lett. 57, 1777–1780 (1986)CrossRefGoogle Scholar
  33. 33.
    S.K. O’Leary, S. Zukotynski, J.M. Perz, Phys. Rev. B 51, 4143–4149 (1995)CrossRefGoogle Scholar
  34. 34.
    S.K. O’Leary, S. Zukotynski, J.M. Perz, Phys. Rev. B 52, 7795–7797 (1995)CrossRefGoogle Scholar
  35. 35.
    S.K. O’Leary, S.R. Johnson, P.K. Lim, J. Appl. Phys. 82, 3334–3340 (1997)CrossRefGoogle Scholar
  36. 36.
    S.K. O’Leary, P.K. Lim, J. Appl. Phys. 82, 3624–3626 (1997)CrossRefGoogle Scholar
  37. 37.
    S.M. Malik, S.K. O’Leary, Appl. Phys. Lett. 80, 790–792 (2002)CrossRefGoogle Scholar
  38. 38.
    S.K. O’Leary, S.M. Malik, J. Appl. Phys. 92, 4276–4282 (2002)CrossRefGoogle Scholar
  39. 39.
    S.K. O’Leary, J. Appl. Phys. 96, 3680–3686 (2004)CrossRefGoogle Scholar
  40. 40.
    F. Orapunt, S.K. O’Leary, J. Appl. Phys. 104, 073513-1-14 (2008)CrossRefGoogle Scholar
  41. 41.
    J.J. Thevaril, S.K. O’Leary, J. Appl. Phys. 107, 083105-1-6 (2010)CrossRefGoogle Scholar
  42. 42.
    J.J. Thevaril, S.K. O’Leary, Solid State Commun. 151, 730–733 (2011)CrossRefGoogle Scholar
  43. 43.
    J.J. Thevaril, S.K. O’Leary, J. Appl. Phys. 120, 135706-1-15 (2016)CrossRefGoogle Scholar
  44. 44.
    S.K. O’Leary, Appl. Phys. Lett. 82, 2784–2786 (2003)CrossRefGoogle Scholar
  45. 45.
    S.B. Minar, S. Moghaddam, S.K. O’Leary, J. Mater. Sci.: Mater. Electron. 30, 9964–9972 (2019)Google Scholar
  46. 46.
    A.R. Forouhi, I. Bloomer, Phys. Rev. B 34, 7018–7026 (1986)CrossRefGoogle Scholar
  47. 47.
    A.R. Forouhi, I. Bloomer, Phys. Rev. B 38, 1865–1874 (1988)CrossRefGoogle Scholar
  48. 48.
    S. Adachi, Phys. Rev. B 43, 12316–12321 (1991)CrossRefGoogle Scholar
  49. 49.
    S. Adachi, J. Appl. Phys. 70, 2304–2308 (1991)CrossRefGoogle Scholar
  50. 50.
    W.A. McGahan, T. Makovicka, J. Hale, J.A. Woollam, Thin Solid Films 253, 57–61 (1994)CrossRefGoogle Scholar
  51. 51.
    G.E. Jellison Jr., F.A. Modine, Appl. Phys. Lett. 69, 371–373 (1996)CrossRefGoogle Scholar
  52. 52.
    A.S. Ferlauto, G.M. Ferreira, J.M. Pearce, C.R. Wronski, R.W. Collins, X. Deng, G. Ganguly, J. Appl. Phys. 92, 2424–2436 (2002)CrossRefGoogle Scholar
  53. 53.
    G.K. Hubler, C.N. Waddell, W.G. Spitzer, J.E. Fredrickson, S. Prussin, R.G. Wilson, J. Appl. Phys. 50, 3294–3303 (1979)CrossRefGoogle Scholar
  54. 54.
    J.P. Lehan, P.C. Yu, D.L. Backfisch, J.P. Chambers, J. Appl. Phys. 92, 3608–3614 (2002)CrossRefGoogle Scholar
  55. 55.
    S.K. O’Leary, J. Mater. Sci.: Mater. Electron. 15, 401–410 (2004)Google Scholar
  56. 56.
    S. Moghaddam, S.K. O’Leary, J. Mater. Sci.: Mater. Electron. 30, 1637–1646 (2019)Google Scholar
  57. 57.
    R.H. Klazes, M.H.L.M. van den Broek, J. Bezemer, S. Radelaar, Philos. Mag. B 45, 377–383 (1982)CrossRefGoogle Scholar
  58. 58.
    D.R. McKenzie, N. Savvides, R.C. McPhedran, L.C. Botten, R.P. Netterfield, J. Phys. C: Solid State Phys. 16, 4933–4944 (1983)CrossRefGoogle Scholar
  59. 59.
    A.S. Ferlauto, Ph.D. Thesis, The Pennsylvania State University, 2001Google Scholar
  60. 60.
    H.R. Philipp, J. Phys. Chem. Solids 32, 1935–1945 (1971)CrossRefGoogle Scholar
  61. 61.
    T.M. Mok, S.K. O’Leary, J. Appl. Phys. 102, 113525-1-9 (2007)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of EngineeringThe University of British ColumbiaKelownaCanada

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