Abstract

The pseudodielectric function is often used to represent ellipsometric data and corresponds to the actual dielectric functions of materials when there is no surface overlayer and the material is isotropic. If a uniaxial material is oriented such that the optic axis is in the plane of incidence or is perpendicular to the plane of incidence, then the cross-polarization terms are zero and appropriate pseudodielectric functions can be determined from the ellipsometry data. We calculate the pseudodielectric functions for uniaxial crystals in three primary symmetry directions: (1) the optic axis is perpendicular to the plane of incidence, (2) the optic axis is in the plane of the sample surface and parallel to the plane of incidence, and (3) the optic axis is in the plane of the sample surface and perpendicular to the plane of incidence. These results are expanded in terms of the difference in the ordinary and extraordinary dielectric functions and compared with the approximation of Aspnes [J. Opt. Soc. Am. 70, 1275 (1980)] . Comparisons are made with experimental results on oriented crystals of rutile (TiO2), and a simple procedure is presented to determine the complex dielectric function from standard ellipsometry techniques.

© 2006 Optical Society of America

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References

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  1. D. E. Aspnes, "Approximate solution of ellipsometric equations for optically biaxial crystals," J. Opt. Soc. Am. 70, 1275-1277 (1980).
    [CrossRef] [PubMed]
  2. V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).
  3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  4. J. Lekner, "Reflection ellipsometry of uniaxial crystals," J. Opt. Soc. Am. A 14, 1359-1362 (1997).
    [CrossRef]
  5. J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys.: Condens. Matter 3, 6121-6133 (1991).
    [CrossRef]
  6. G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett. 22, 1808-1810 (1997).
    [CrossRef]
  7. G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
    [CrossRef]
  8. M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, "Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2," J. Opt. Soc. Am. A 13, 875-883 (1996).
    [CrossRef]
  9. T. E. Tiwald and M. Schubert, "Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 µm using generalized ellipsometry," in Proc. SPIE 4103, 19-29 (2000).
    [CrossRef]
  10. G. E. Jellison, Jr., and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
    [CrossRef]
  11. G. E. Jellison, Jr., and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
    [CrossRef]
  12. G. E. Jellison, Jr., "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
    [CrossRef]
  13. G. E. Jellison, Jr., "Data analysis for spectroscopic ellipsometry," Thin Solid Films 234, 416-422 (1993).
    [CrossRef]
  14. D. A. G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen," Ann. Phys. (Leipzig) 24, 636-679 (1935).

2003 (1)

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

2000 (1)

T. E. Tiwald and M. Schubert, "Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 µm using generalized ellipsometry," in Proc. SPIE 4103, 19-29 (2000).
[CrossRef]

1998 (1)

G. E. Jellison, Jr., "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

1997 (4)

1996 (1)

1994 (1)

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

1993 (1)

G. E. Jellison, Jr., "Data analysis for spectroscopic ellipsometry," Thin Solid Films 234, 416-422 (1993).
[CrossRef]

1991 (1)

J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys.: Condens. Matter 3, 6121-6133 (1991).
[CrossRef]

1980 (1)

1935 (1)

D. A. G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen," Ann. Phys. (Leipzig) 24, 636-679 (1935).

Aspnes, D. E.

Azzam, R. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Boatner, L. A.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett. 22, 1808-1810 (1997).
[CrossRef]

Bruggeman, D. A.

D. A. G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen," Ann. Phys. (Leipzig) 24, 636-679 (1935).

Budai, J. D.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Filippov, V. V.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

Herzinger, C. M.

Jellison, G. E.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

G. E. Jellison, Jr., "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

G. E. Jellison, Jr., and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
[CrossRef]

G. E. Jellison, Jr., and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurement of the optical functions of uniaxial materials by two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett. 22, 1808-1810 (1997).
[CrossRef]

G. E. Jellison, Jr., "Data analysis for spectroscopic ellipsometry," Thin Solid Films 234, 416-422 (1993).
[CrossRef]

Jeong, B. S.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Johs, B.

Konstantinova, A. F.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

Lekner, J.

J. Lekner, "Reflection ellipsometry of uniaxial crystals," J. Opt. Soc. Am. A 14, 1359-1362 (1997).
[CrossRef]

J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys.: Condens. Matter 3, 6121-6133 (1991).
[CrossRef]

Modine, F. A.

Norton, D. P.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Rheinlander, B.

Schubert, M.

Tiwald, T. E.

T. E. Tiwald and M. Schubert, "Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 µm using generalized ellipsometry," in Proc. SPIE 4103, 19-29 (2000).
[CrossRef]

Tronin, A. Yu.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

Woollam, J. A.

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen," Ann. Phys. (Leipzig) 24, 636-679 (1935).

Appl. Opt. (2)

Crystallogr. Rep. (1)

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

J. Appl. Phys. (1)

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B. S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys.: Condens. Matter (1)

J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys.: Condens. Matter 3, 6121-6133 (1991).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

T. E. Tiwald and M. Schubert, "Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 µm using generalized ellipsometry," in Proc. SPIE 4103, 19-29 (2000).
[CrossRef]

Thin Solid Films (2)

G. E. Jellison, Jr., "Spectroscopic ellipsometry data analysis: measured versus calculated quantities," Thin Solid Films 313-314, 33-39 (1998).
[CrossRef]

G. E. Jellison, Jr., "Data analysis for spectroscopic ellipsometry," Thin Solid Films 234, 416-422 (1993).
[CrossRef]

Other (1)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

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Figures (8)

Fig. 1
Fig. 1

Raw generalized ellipsometry data, expressed in the ρ representation, taken on rutile with the optic axis in the plane of the sample surface, oriented at + 45 ° with respect to the plane of incidence.

Fig. 2
Fig. 2

Optical functions of rutile determined from generalized ellipsometry measurements with the optic axis of the crystal oriented in the plane of the sample surface at + 45 ° with respect to the plane of incidence. The data have been corrected for a surface overlayer. The difference between ε e and ε o is also plotted.

Fig. 3
Fig. 3

Raw ellipsometry data, expressed in terms of real and imaginary parts of the complex reflection ratio ρ from a rutile crystal with the optic axis perpendicular to the surface plane, taken at several different angles of incidence.

Fig. 4
Fig. 4

Several plots of the complex dielectric function of a rutile crystal with the optic axis perpendicular to the sample surface ( OA surf ) . Dashed curves show the raw ellipsometric data, expressed in terms of the PDF, taken at ϕ = 64.92 ° . Narrow solid curves show the raw data, corrected as described in Subsection 3A. This to be compared with the ordinary dielectric function (heavy solid curves), also shown in Fig. 2. The calculated PDF [using Eq. (5)] is shown by the dotted curves.

Fig. 5
Fig. 5

PDF raw data (uncorrected for surface overlayer) for the optic axis in the plane of the sample, parallel and perpendicular to the plane of incidence ( OA surf , POI ) , taken for different angles of incidence. The data are shown for two different angles of incidence.

Fig. 6
Fig. 6

Several plots of the complex dielectric function of a rutile crystal with the optic axis parallel to the sample surface and parallel to the plane of incidence. The PDF data, corrected for the overlayer and angle of incidence as described in Subsection 3A, are shown by the dotted curves. The PDF calculated using Eq. (11) is shown by the thin solid curves. This is compared with the actual extraordinary dielectric function in the thick solid curves.

Fig. 7
Fig. 7

Several plots of the complex dielectric function of a rutile crystal with the optic axis parallel to the sample surface and perpendicular to the plane of incidence ( OA surf , POI ) . The PDF data, corrected for the overlayer and angle of incidence as described in the text, are shown by the dotted curves. The PDF calculated using Eq. (15) is shown by the thin solid curves. This is compared with the actual ordinary dielectric function in the thick solid curves.

Fig. 8
Fig. 8

Complex dielectric functions of rutile determined using two different techniques. The dark solid curves represent the data from Fig. 2 and are determined by utilizing the generalized ellipsometry data for the optic axis in the plane of the sample, oriented at γ = ± 45 ° with respect to the plane of incidence. The dotted curves are the complex dielectric functions determined using the normal ellipsometry data for the optic axis in the plane of the sample, oriented parallel and perpendicular to the plane of incidence, utilizing the procedure outlined in Subsection 3F.

Equations (28)

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ξ = sin ( ϕ ) ,
η = cos ( ϕ ) .
η o = ( ε ξ 2 ) 1 2 ,
Δ ε = ε e ε o .
ε = ξ 2 [ 1 + ξ 2 η 2 ( 1 ρ ) 2 ( 1 + ρ ) 2 ] .
η e = [ ε o ( ε e ξ 2 ) ε e ] 1 2 .
r p = ε e η η e ε o η + η e ,
r s = η η o η + η o ,
ε = ξ 2 [ 1 + ξ 2 ( ε o η o η e ε o η 2 η o η e ) 2 ] .
ε ε o Δ ε ε o 1 + Δ ε 2 4 ε o ( ε o 1 ) 2 4 ε o 2 ε o 3 ε o ξ 2 + ξ 2 ε o ξ 2 ,
ε ε o + Γ 2 ξ 2 ( 1 + Γ ) 2 Γ 2 ξ 2 ,
Γ = Δ ε 2 ε e ( ε o 1 ) .
η e = [ ε e ( ε o ξ 2 ) ε o ] 1 2 = ε ave η o ε o ,
r p = ε ave η η o ε ave η + η o ,
r s = η η o η + η o .
ε = ξ 2 [ 1 + ξ 2 η o 2 ( ε ave 1 ε ave η 2 η o 2 ) 2 ] .
ε ε o + Δ ε ( ε o ξ 2 ) ξ 2 ( ε o 1 ) + Δ ε 2 ( 3 ε o η 2 + ξ 2 ) ( ε o ξ 2 ) 4 ε o ξ 4 ( ε o 1 ) 2 ;
ε ε o + ( ε e ε o ) ξ 2 = ( ε e ε o η 2 ) ξ 2 .
η e = [ ε e ξ 2 ] 1 2 ,
r p = ε o η η o ε o η + η o ,
r s = η η e η + η e .
ε = ξ 2 [ 1 + ξ 2 ( ε o η e η o ε o η 2 η e η o ) 2 ] .
ε ε o Δ ε ( ε o ξ 2 ε o ξ 2 ) ξ 2 ( ε o 1 ) + Δ ε 2 ( 3 ε o 4 ξ 2 ) ( ε o ξ 2 ε o ξ 2 ) 4 ξ 4 ( ε o 1 ) 2 ( ε o ξ 2 ) .
ε ε o + Δ ε 1 ξ 2 ( ε o 1 ) .
ρ = r p p r s s = tan ( ψ ) e i Δ = C + i S 1 + N ,
ρ p s = r p s r s s = tan ( ψ p s ) e i Δ p s = C p s + i S p s 1 + N ,
ρ s p = r s p r s s = tan ( ψ s p ) e i Δ s p = C s p + i S s p 1 + N ,
FOM = 1 N ( PDF PDF approx ) ( PDF PDF approx ) * ,

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