Abstract

A half-wave method of measurement of wafer birefringence that is based on interference fringes recorded from a uniaxial wafer by use of a standard phase-modulated spectroscopic ellipsometer is investigated. The birefringence of uniaxial wafers is calculated from the extremal points in the recorded oscillating intensities. A formalism is developed to incorporate the change in birefringence with wavelength as a correction factor. The correction explains the overestimation of the birefringence from previous similar research on thick uniaxial sapphire substrates. The enhanced derivative of the birefringence that is due to polarization-dependent intraconduction band transitions is detected. Furthermore, for well-characterized wafers it is shown that this method can be used in wafer-thickness mapping of 4H-SiC and similar uniaxial high-bandgap semiconductors.

© 2000 Optical Society of America

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References

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  1. S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
    [CrossRef]
  2. R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
    [CrossRef]
  3. I. J. Hodgkinson, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), Chaps. 7 and 8.
  4. J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).
  5. N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
    [CrossRef]
  6. A. Zuber, H. Jancher, N. Kaiser, “Perpendicular-incidence photometric ellipsometry of biaxial anisotropic thin films,” Appl. Opt. 35, 5553–5556 (1996).
    [CrossRef] [PubMed]
  7. M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).
  8. G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. F. Durrant, V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65–78 (1998).
    [CrossRef]
  9. See, for example, W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull. 22, 25–29 (1997).
  10. B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).
  11. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands, 1977), pp. 358, 491.
  12. J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
    [CrossRef]
  13. M. Kildemo, O. Hunderi, “Spectroscopic Fourier methods for thickness measurements of thick uniaxial wafers with dispersive birefringence using polarimetric techniques,” J. Opt. A: Pure Appl. Opt. (to be published).
  14. E. Biedermann, “The optical absorption bands and their anisotropy in the various modifications of SiC,” Solid State Commun. 3, 343–346 (1965).
    [CrossRef]
  15. S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
    [CrossRef]
  16. S. Logothedis, J. Petalas, “Dielectric function and reflectivity of 2C-silicon carbide and the component perpendicular to the c axis of 6H-silicon carbide in the energy region 1.5–9.5 eV,” J. Appl. Phys. 80, 1768–1772 (1996).
    [CrossRef]
  17. W. J. Choyke, E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
    [CrossRef]
  18. M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using an incoherent reflection model and spectroscopic UV–visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).
  19. S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
    [CrossRef]

2000 (2)

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[CrossRef]

1999 (2)

S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
[CrossRef]

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

1998 (2)

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).

G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. F. Durrant, V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65–78 (1998).
[CrossRef]

1997 (3)

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using an incoherent reflection model and spectroscopic UV–visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

See, for example, W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull. 22, 25–29 (1997).

1996 (2)

S. Logothedis, J. Petalas, “Dielectric function and reflectivity of 2C-silicon carbide and the component perpendicular to the c axis of 6H-silicon carbide in the energy region 1.5–9.5 eV,” J. Appl. Phys. 80, 1768–1772 (1996).
[CrossRef]

A. Zuber, H. Jancher, N. Kaiser, “Perpendicular-incidence photometric ellipsometry of biaxial anisotropic thin films,” Appl. Opt. 35, 5553–5556 (1996).
[CrossRef] [PubMed]

1994 (1)

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
[CrossRef]

1993 (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).

1965 (1)

E. Biedermann, “The optical absorption bands and their anisotropy in the various modifications of SiC,” Solid State Commun. 3, 343–346 (1965).
[CrossRef]

Adolph, N.

N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands, 1977), pp. 358, 491.

Baranauskas, V.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands, 1977), pp. 358, 491.

Bechstedt, F.

N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Biedermann, E.

E. Biedermann, “The optical absorption bands and their anisotropy in the various modifications of SiC,” Solid State Commun. 3, 343–346 (1965).
[CrossRef]

Chen, J. G.

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Choyke, W. J.

See, for example, W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull. 22, 25–29 (1997).

W. J. Choyke, E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

Crean, G.

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

Drévillon, B.

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).

Duda, E.

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Durrant, S. F.

Elman, J. F.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).

Gavrilenko, V. I.

N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Ghiner, A. V.

Greener, J.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).

Herzinger, C. M.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).

Hilfiker, J. N.

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Hodgkinson, I. J.

I. J. Hodgkinson, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), Chaps. 7 and 8.

Hunderi, O.

M. Kildemo, O. Hunderi, “Spectroscopic Fourier methods for thickness measurements of thick uniaxial wafers with dispersive birefringence using polarimetric techniques,” J. Opt. A: Pure Appl. Opt. (to be published).

Jancher, H.

Johs, B.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).

Kaiser, N.

Kelly, P.

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

Kildemo, M.

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using an incoherent reflection model and spectroscopic UV–visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

M. Kildemo, O. Hunderi, “Spectroscopic Fourier methods for thickness measurements of thick uniaxial wafers with dispersive birefringence using polarimetric techniques,” J. Opt. A: Pure Appl. Opt. (to be published).

Lambrecht, W. R. L.

S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
[CrossRef]

Lekner, J.

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
[CrossRef]

Limpijumnong, S.

S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
[CrossRef]

Logothedis, S.

S. Logothedis, J. Petalas, “Dielectric function and reflectivity of 2C-silicon carbide and the component perpendicular to the c axis of 6H-silicon carbide in the energy region 1.5–9.5 eV,” J. Appl. Phys. 80, 1768–1772 (1996).
[CrossRef]

Mooney, M.

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[CrossRef]

Nee, S.-M. F.

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

Ossikovski, R.

R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using an incoherent reflection model and spectroscopic UV–visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

Palik, E. D.

W. J. Choyke, E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

Pensl, G.

See, for example, W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull. 22, 25–29 (1997).

Petalas, J.

S. Logothedis, J. Petalas, “Dielectric function and reflectivity of 2C-silicon carbide and the component perpendicular to the c axis of 6H-silicon carbide in the energy region 1.5–9.5 eV,” J. Appl. Phys. 80, 1768–1772 (1996).
[CrossRef]

Rashkeev, S. N.

S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
[CrossRef]

Segall, B.

S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
[CrossRef]

Stchakovsky, M.

R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using an incoherent reflection model and spectroscopic UV–visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

Sudre, C.

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

Surdutovich, G. I.

Tenelsen, K.

N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

Vitlina, R. Z.

Wetteroth, T.

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Wilson, S. R.

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Zollner, S.

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Zuber, A.

Appl. Opt. (3)

J. Appl. Phys. (2)

S. Logothedis, J. Petalas, “Dielectric function and reflectivity of 2C-silicon carbide and the component perpendicular to the c axis of 6H-silicon carbide in the energy region 1.5–9.5 eV,” J. Appl. Phys. 80, 1768–1772 (1996).
[CrossRef]

S. Zollner, J. G. Chen, E. Duda, T. Wetteroth, S. R. Wilson, J. N. Hilfiker, “Dielectric function of bulk 4H and 6H SiC and spectroscopic ellipsometry studies of thin SiC films on Si,” J. Appl. Phys. 83, 8353–8361 (1999).
[CrossRef]

Mater. Sci. Forum (1)

M. Kildemo, M. Mooney, C. Sudre, P. Kelly, G. Crean, “Anisotropic dielectric function of semi-insulating 4H-SiC using spectroscopic ellipsometry and an incoherent reflection model of a thick transparent anisotropic substrate,” Mater. Sci. Forum 338–342, 571–574 (2000).

MRS Bull. (1)

See, for example, W. J. Choyke, G. Pensl, “Physical properties of SiC,” MRS Bull. 22, 25–29 (1997).

Phys. Rev. B (2)

N. Adolph, K. Tenelsen, V. I. Gavrilenko, F. Bechstedt, “Optical and loss spectra of SiC polytypes from ab initio calculations,” Phys. Rev. B 55, 1422–1429 (1997).
[CrossRef]

S. Limpijumnong, W. R. L. Lambrecht, S. N. Rashkeev, B. Segall, “Optical absorption bands in the 1–3-eV range in n-type SiC polytypes,” Phys. Rev. B 59, 12890–12899 (1999).
[CrossRef]

Prog. Cryst. Growth Charact. (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).

Pure Appl. Opt. (1)

J. Lekner, “Optical properties of a uniaxial layer,” Pure Appl. Opt. 3, 821–837 (1994).
[CrossRef]

Solid State Commun. (1)

E. Biedermann, “The optical absorption bands and their anisotropy in the various modifications of SiC,” Solid State Commun. 3, 343–346 (1965).
[CrossRef]

Thin Solid Films (2)

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using an incoherent reflection model and spectroscopic UV–visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1997).

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially stretched plastic films by generalized ellipsometry,” Thin Solid Films 313–314, 814–818 (1998).

Other (5)

I. J. Hodgkinson, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), Chaps. 7 and 8.

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands, 1977), pp. 358, 491.

W. J. Choyke, E. D. Palik, “Silicon carbide (SiC),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 587.
[CrossRef]

M. Kildemo, O. Hunderi, “Spectroscopic Fourier methods for thickness measurements of thick uniaxial wafers with dispersive birefringence using polarimetric techniques,” J. Opt. A: Pure Appl. Opt. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Recorded PMSE intensities I s and I c (solid curves) including the half-wave points (open triangles and squares) of double-side-polished 4H-SiC wafers: (a) I c and (c) I s of the semi-insulating higher-quality polished wafer. (b) I c and (d) I s of the n-type doped wafer with a lower-quality surface polish.

Fig. 2
Fig. 2

Recorded spectroscopic ellipsometric intensities I s , I c , and I c3 of a double-side-polished 4H-SiC wafer at a 65° angle of incidence. The dotted and the dashed curves represent the values of I s and I c , respectively, that result from the spectroscopic fitting. The uppermost curve (diamonds) represents the degree of polarization.

Fig. 3
Fig. 3

Simulated values of the difference in phase Δη at 1.5 eV plotted as a function of the tilt angle θ between the surface normal and the c axis. In the simulation the c axis is assumed to be in the incidence plane, as shown in the schematic inset.

Fig. 4
Fig. 4

Birefringence calculated by use of the half-wave method and Eq. (9) with no correction (filled squares), Eq. (10) with an analytical approximated correction (open circles), and Eq. (8) with an accurate numerical correction (open triangles). The birefringence of 4H-SiC (solid curve) and 6H-SiC (dotted curve) that results from the Sellmeier parameters are also shown.

Fig. 5
Fig. 5

(a) Approximate simulation of the polarization-dependent intraconduction band transitions represented by simple Lorentzian line shapes. (b) Birefringence calculated from (a), which is the left-hand axis, whereas the right-hand axis [in (b)] shows the correction factor obtained by the introduction of both the first- and the second-order derivatives of the birefringence that corresponds to (a).

Fig. 6
Fig. 6

Uncorrected (filled squares) and corrected (open triangles) values of the birefringence Δn of the n-type doped 4H-SiC wafer, corresponding to the measurements in Fig. 1(c). The birefringence of 6H-SiC is shown for comparison. The arrows indicate the positions of the intraconduction band transitions P 1 and P 4.

Fig. 7
Fig. 7

Thickness measurements that correspond to the intensities shown in Figs. 1(a) and 2. The filled squares represent uncorrected measurements; the open triangles represent dispersion-corrected measurements; the solid curve represents the real thickness.

Tables (3)

Tables Icon

Table 1 Parameters of the Sellmeier Dispersion Models [Eq. (12)] Used in this Studya

Tables Icon

Table 2 Birefringence of 4H- and 6H-SiC Calculated from Table 1

Tables Icon

Table 3 Comparison between Mechanically and Optically Measured Wafer Thicknesses

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Is=2Imrprs*rsrs*+rprp*,  Ic=2Rerprs*rsrs*+rprp*,  Ic3=rsrs*-rprp*rsrs*+rprp*,
rxry*=r01xr01y*+t01xt01y*t10xt10y*r12xr12y*exp2iβx-βy*1-r10xr10y*r12xr12y*exp2iβx-βy*,
βs=2πDσηo=2π Dλno2-n2 sin2 ϕo1/2,
βp=2πDσηe=2π Dλnonene2-n2 sin2 ϕo1/2,
βsσ2-βpσ2-βsσ1-βpσ1=π.
ηoσ1+dηoσ1dσ yσ2-ηeσ1+dηeσ1dσ yσ2-ηoσ1+ηeσ1=12D
ηoσ1-ηeσ1-dηoσ1dσ-dηeσ1dσσ2=12Dy.
ηoσ1-ηeσ1=Δησ1=12Dy1+σ2Δησ1dηoσ1dσ-dηeσ1dσ.
Δn0n0n02-n2 sin2 ϕo1/22Dn2 sin2 ϕo1σ2-σ1,
Δn1Δn01+σ2noΔn0dΔndσΔn01-σ2noΔn0dΔndσ.
D=12Δησ1y1+σ2Δησ1dΔησ1dσ.
ΔDD=0.02 eV2 eV=1%,
qE=nq2E=+fqEq2Eq2-E2,
q=ordinary, direction perpendicular to the c axisextraordinary, direction parallel to the c axis.
Δn0Δn¯01±ξ2n¯02-n2 sin2 ϕon¯on¯02-n2 sin2 ϕo.
ηe=none0+Δ cos2 θ0+Δ cos2 θ-n2 sin2 ϕo1/2,

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