Abstract

Dielectric thin films often present microstructures that give rise to a variation of the refractive index with the distance from the substrate. We propose a method of analysis of ellipsometric data for homogeneous and slightly inhomogeneous films that are deposited on transparent substrates. Assuming a linear refractive-index gradient, we are able to determine not only the average index and the thickness but also the degree of inhomogeneity of the films by spectroscopic ellipsometry at variable angles of incidence. We apply this method to titanium dioxide films deposited on glass, which present different degrees of inhomogeneity depending on the preparation conditions.

© 1992 Optical Society of America

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References

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  1. J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
    [CrossRef] [PubMed]
  2. H. K. Pulker, G. Paesold, E. Ritter, “Refractive indices of TiO2 films produced by reactive evaporation of various titanium—oxygen phases,” Appl. Opt. 15, 2986–2991 (1976).
    [CrossRef] [PubMed]
  3. M. Harris, “Structure related optical properties of thin films,” J. Vac. Sci. Technol. A 14, 418–422 (1986).
  4. M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
    [CrossRef]
  5. M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
    [CrossRef]
  6. J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3373–3779 (1985).
    [CrossRef]
  7. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
    [CrossRef] [PubMed]
  8. E. Pelletier, P. Roche, et B. Vidal, “Détermination automatique des constantes optiques et de l’epaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353–362 (1976).
    [CrossRef]
  9. F. Abelès, “Quelques proprietés optiques des milieux stratifiés symétriques,” Opt. Acta 4, 42–43 (1957).
    [CrossRef]
  10. C. K. Carniglia, “Ellipsometric calculations for nonabsorbing thin films with linear refractive-index gradients,” J. Opt. Soc. Am. A 7, 848–856 (1990).
    [CrossRef]
  11. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.
  12. Spectroscopic ellipsometer Model ES3G is a product of SOPRA, 68 rue P. Joigneaux, 92270 Bois-Colombes, France.
  13. F. Abelés, “Recherche sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiès—application aux couches minces,” Thèse (Facuté des Sciences de l’Université de Paris, Paris, 1950).

1990

1989

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
[CrossRef] [PubMed]

1986

M. Harris, “Structure related optical properties of thin films,” J. Vac. Sci. Technol. A 14, 418–422 (1986).

1985

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3373–3779 (1985).
[CrossRef]

1982

1979

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

1976

H. K. Pulker, G. Paesold, E. Ritter, “Refractive indices of TiO2 films produced by reactive evaporation of various titanium—oxygen phases,” Appl. Opt. 15, 2986–2991 (1976).
[CrossRef] [PubMed]

E. Pelletier, P. Roche, et B. Vidal, “Détermination automatique des constantes optiques et de l’epaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353–362 (1976).
[CrossRef]

1957

F. Abelès, “Quelques proprietés optiques des milieux stratifiés symétriques,” Opt. Acta 4, 42–43 (1957).
[CrossRef]

Abelés, F.

F. Abelés, “Recherche sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiès—application aux couches minces,” Thèse (Facuté des Sciences de l’Université de Paris, Paris, 1950).

Abelès, F.

F. Abelès, “Quelques proprietés optiques des milieux stratifiés symétriques,” Opt. Acta 4, 42–43 (1957).
[CrossRef]

Albrand, G.

Allen, T. H.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

Bennett, J. M.

Blanco, J. R.

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3373–3779 (1985).
[CrossRef]

Borgogno, J. P.

Boulesteix, C.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Carniglia, C. K.

Flory, F.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Galindo, R.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Guenther, K. H.

Harris, M.

M. Harris, “Structure related optical properties of thin films,” J. Vac. Sci. Technol. A 14, 418–422 (1986).

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Lazarides, B.

Lottiaux, M.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Macleod, H. A.

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

McMarr, P. J.

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3373–3779 (1985).
[CrossRef]

Nihoul, G.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Ogura, S.

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Paesold, G.

Pelletier, E.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
[CrossRef] [PubMed]

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
[CrossRef] [PubMed]

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

E. Pelletier, P. Roche, et B. Vidal, “Détermination automatique des constantes optiques et de l’epaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353–362 (1976).
[CrossRef]

Pulker, H. K.

Ritter, E.

Roche, P.

E. Pelletier, P. Roche, et B. Vidal, “Détermination automatique des constantes optiques et de l’epaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353–362 (1976).
[CrossRef]

Saxer, A.

Schmell, R. A.

Tuttle-Hart, T.

Varnier, F.

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Vedam, K.

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3373–3779 (1985).
[CrossRef]

Vidal, B.

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Vidal, et B.

E. Pelletier, P. Roche, et B. Vidal, “Détermination automatique des constantes optiques et de l’epaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353–362 (1976).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. A

M. Harris, “Structure related optical properties of thin films,” J. Vac. Sci. Technol. A 14, 418–422 (1986).

Nouv. Rev. Opt.

E. Pelletier, P. Roche, et B. Vidal, “Détermination automatique des constantes optiques et de l’epaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353–362 (1976).
[CrossRef]

Opt. Acta

F. Abelès, “Quelques proprietés optiques des milieux stratifiés symétriques,” Opt. Acta 4, 42–43 (1957).
[CrossRef]

Thin Solid Films

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

M. Lottiaux, C. Boulesteix, G. Nihoul, F. Varnier, F. Flory, R. Galindo, E. Pelletier, “Morphology and structure of TiO2 thin layers: thickness and substrate temperature,” Thin Solid Films 170, 107–126 (1989).
[CrossRef]

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

Spectroscopic ellipsometer Model ES3G is a product of SOPRA, 68 rue P. Joigneaux, 92270 Bois-Colombes, France.

F. Abelés, “Recherche sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiès—application aux couches minces,” Thèse (Facuté des Sciences de l’Université de Paris, Paris, 1950).

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Figures (6)

Fig. 1
Fig. 1

Ellipsometric parameters (a) tan Ψ and (b) cos Δ versus wavelength at different angles of incidence as calculated for a transparent film (n = 2.4, d = 1500 Å) deposited upon glass (n = 1.5).

Fig. 2
Fig. 2

Variation of cos Δ as a function of the angle of incidence, calculated in the same conditions as in Fig. 1, at λo = 0.44 μm (QW fringe) and λo ± 15 Å.

Fig. 3
Fig. 3

Experimental results (a) tan Ψ and (b) cos Δ for a TiO2 homogeneous film (d = 1145 Å) deposited on glass in the neighborhood of a HW fringe at angles of incidence around the Brewster angle of the substrate (θB = 57.3°).

Fig. 4
Fig. 4

Angular variation of cos Δ measured on a TiO2 film at three wavelengths, 6360, 6365, and 6355 Å, that correspond to the position of a QW fringe with an uncertainty of 5 Å.

Fig. 5
Fig. 5

Schematic of a linear refractive-index gradient that decreases through the film from n1 at the substrate interface to n2 at the film surface.

Fig. 6
Fig. 6

Variation of (a) tan Ψ and (b) cos Δ as a function of the angle of incidence as measured on a TiO2 inhomogeneous film (d = 1090 Å) at λ = 5070 Å on a HW fringe as compared with the bare glass substrate.

Equations (17)

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ρ = r p / r s = ( r p / r s ) exp [ i ( δ p - δ s ) ] = tan Ψ exp ( i Δ ) .
M = [ cos β i N sin β i N sin β cos β ] ,
r = ( N o - N s ) cos β + i ( N o N s N - N ) sin β ( N o + N s ) cos β + i ( N o N s N + N ) sin β ,
( n o cos θ o - n s cos θ s ) 2 cos 2 β + ( n o n s cos θ n cos θ o cos θ s - n cos θ ) 2 sin 2 β = 0.
n o cos θ o - n s cos θ s = 0.
tan θ o = n s n o ,
n o n s cos θ n cos θ o cos θ s - n cos θ = 0.
n 2 = n o n s 2 cos θ o cos θ s × [ 1 + ( 1 - n o n s cos θ o cos θ s sin 2 θ o ) 1 / 2 ] .
M = [ cos ( 2 m + 1 ) β - 2 u sin β δ n n i cos θ n sin ( 2 m + 1 ) β i n cos θ sin ( 2 m + 1 ) β cos ( 2 m + 1 ) β + 2 u sin β δ n n ] ,
= n 2 - 2 n o 2 sin 2 θ o n 2 - n o 2 sin 2 θ o = 2 - 1 cos 2 θ < 1 , β = 2 π n d i λ cos θ ,             u = p = 1 m p sin 2 p β ,
n o n s cos θ n cos θ o cos θ s - n cos θ = 0.
[ ( - 1 ) k - 2 u sin β δ n n ] n o cos θ o = n s cos θ s [ ( - 1 ) k + 2 u sin β δ n n ] .
( - 1 ) k 4 u sin β = - ( 2 m + 1 ) .
n 1 - n 2 n = 2 m δ n n = ( 1 - n s cos θ o n o cos θ s ) 1 ( 2 - 1 cos 2 θ ) .
n > 2 n o n s ( 1 + n s 2 ) ,
2 - 1 cos 2 θ
( 1 - n s cos θ o n o cos θ s ) .

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