Abstract

An absence of correlation between parameters, indicated by invariance of the normalized ratio of the first derivatives of Δ, makes it possible to make optimal use of the overdetermined set of equations, which are available from multiple-angle measurements. Accurate estimates of the parameters are not needed for the correlation test so that experimental conditions can be chosen to minimize correlation. Also, the second derivatives of the least-squares residuals are useful in deciding on the best method of searching for a solution, in error analysis and in illustrating the critical importance of initial estimates of the unknown parameters in obtaining accurate least-squares solutions.

© 1971 Optical Society of America

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References

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  1. F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurements of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natl. Bur. Std. (U.S.) Misc. Publ. 256 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.
  2. D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964).
    [Crossref]
  3. This is not to be construed to mean that MAI cannot be used to characterize film and/or substrate properties.
  4. D. G. Schueler, Surface Sci. 16, 104 (1969); also in Proceedings on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, 1969).
    [Crossref]
  5. W. G. Oldham, Surface Sci. 16, 97 (1969).
    [Crossref]
  6. J. A. Johnson and N. M. Bashara, J. Opt. Soc. Am. 61, 457 (1971).
    [Crossref]
  7. John R. Rice, in Numerical Solutions of Nonlinear Problems (Computer Sci. Center, Univ. of Maryland, College Park, 1970), p. 80.
  8. J. Kowalk and M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).
  9. H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
    [Crossref]
  10. R. J. Archer, Ellipsometry (Gaertner Scientific Corporation, Chicago, 1968).
  11. L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961); J. Phys. Chem. 66, 2719 (1962).
    [Crossref]
  12. D. L. Marquardt, J. Soc. Indus. Appl. Math. 2, 431 (1963).

1971 (1)

1969 (2)

D. G. Schueler, Surface Sci. 16, 104 (1969); also in Proceedings on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, 1969).
[Crossref]

W. G. Oldham, Surface Sci. 16, 97 (1969).
[Crossref]

1964 (1)

1963 (1)

D. L. Marquardt, J. Soc. Indus. Appl. Math. 2, 431 (1963).

1961 (1)

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961); J. Phys. Chem. 66, 2719 (1962).
[Crossref]

1960 (1)

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

Archer, R. J.

R. J. Archer, Ellipsometry (Gaertner Scientific Corporation, Chicago, 1968).

Bartell, L. S.

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961); J. Phys. Chem. 66, 2719 (1962).
[Crossref]

Bashara, N. M.

Bennett, H. E.

Burge, D. K.

Churchill, D.

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961); J. Phys. Chem. 66, 2719 (1962).
[Crossref]

Colson, J. P.

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurements of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natl. Bur. Std. (U.S.) Misc. Publ. 256 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.

Johnson, J. A.

Kowalk, J.

J. Kowalk and M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

Marquardt, D. L.

D. L. Marquardt, J. Soc. Indus. Appl. Math. 2, 431 (1963).

McCrackin, F. L.

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurements of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natl. Bur. Std. (U.S.) Misc. Publ. 256 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.

Oldham, W. G.

W. G. Oldham, Surface Sci. 16, 97 (1969).
[Crossref]

Osborne, M. R.

J. Kowalk and M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

Philipp, H. R.

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

Rice, John R.

John R. Rice, in Numerical Solutions of Nonlinear Problems (Computer Sci. Center, Univ. of Maryland, College Park, 1970), p. 80.

Schueler, D. G.

D. G. Schueler, Surface Sci. 16, 104 (1969); also in Proceedings on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, 1969).
[Crossref]

Taft, E. A.

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. Chem. (1)

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961); J. Phys. Chem. 66, 2719 (1962).
[Crossref]

J. Soc. Indus. Appl. Math. (1)

D. L. Marquardt, J. Soc. Indus. Appl. Math. 2, 431 (1963).

Phys. Rev. (1)

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

Surface Sci. (2)

D. G. Schueler, Surface Sci. 16, 104 (1969); also in Proceedings on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, 1969).
[Crossref]

W. G. Oldham, Surface Sci. 16, 97 (1969).
[Crossref]

Other (5)

This is not to be construed to mean that MAI cannot be used to characterize film and/or substrate properties.

R. J. Archer, Ellipsometry (Gaertner Scientific Corporation, Chicago, 1968).

John R. Rice, in Numerical Solutions of Nonlinear Problems (Computer Sci. Center, Univ. of Maryland, College Park, 1970), p. 80.

J. Kowalk and M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurements of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natl. Bur. Std. (U.S.) Misc. Publ. 256 (U. S. Govt. Printing Office, Washington, D. C., 1964), p. 61.

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Figures (3)

Fig. 1
Fig. 1

Illustrates the errors of estimation of the refractive index and thickness of a film due to a small error of ψ when measurements are made at one angle of incidence. Note the difference of scale for ψ and Δ.

Fig. 2
Fig. 2

Cross sections of the error function G from Eq. (18). Curve 1 is for a thick film and 2 for a thin film. The dashed curve |F| is discussed in the text. If higher-order terms are needed in Eq. (18), bending of the error function at values of NF far from the minimum would be observed.

Fig. 3
Fig. 3

The relative error R = δNS/NF of estimation of the substrate refractive index compared to the film refractive index as a function of film thickness. The solid line is for Si–SiO2 and the broken line for an absorbing film on a chromium substrate, both at 4358-Å wavelength. The optical constants of the film on chromium were those used in Ref. 1, 2.5–i0.5.

Tables (8)

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Table I First derivatives of Δ and ψ with respect to NF (film refractive index) and D (film thickness) for a thin oxide film on silicon at an angle of incidence of 70°.a

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Table II The variation with angle of incidence of the normalized derivatives of Δ and ψ for Si–SiO2 at a 5461-Å wavelength and 20-Å film thickness.

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Table III The variation with angle of incidence of the derivatives of ψ and the relative derivatives of Δ for Si–SiO2 and 20-Å film thickness.

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Table IV Correlation between parameters for SiO2–Si.

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Table V The variations with angle of incidence of the derivative of ψ and the relative derivatives of Δ for a contamination film on silver for 37-Å film thickness.

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Table VI MAI determination of the optical constants of boron-doped silicon (ρ ≈ 0.1 Ω cm) covered with a natural oxide.a

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Table VII DEHM values for Si–SiO2 at a wavelength of 4358 Å.a

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Table VIII Effect of initial guesses and experimental errors on the final solutions for the Si–SiO2 system for 4358 Å.

Equations (35)

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δ Δ = Δ D δ D + Δ N F δ N F
δ ψ = ψ D δ D + ψ N F δ N F .
δ D = ( ψ N F δ Δ - Δ N F δ ψ ) / S
δ N F = ( Δ D δ ψ - ψ D δ Δ ) / S ,
S = ψ N F Δ D - ψ D Δ N F .
ψ i = ψ ( N S , K S , N F , K F , D , ϕ i )
Δ i = Δ ( N S , K S , N F , K F , D , ϕ i ) ,
G ( B ) = i = 1 m { [ Δ i - Δ ( B , ϕ i ) ] 2 + [ ψ i - ψ ( B , ϕ i ) ] 2 } .
i = 1 m [ ψ i - ψ ( B , ϕ i ) ] 2
i = 1 m [ Δ i - Δ ( B , ϕ i ) ] 2 .
ψ ( B , ϕ i ) = ψ ( B 0 , ϕ i ) + j = 1 k ψ i b j δ b j ,
Δ ( B , ϕ i ) = Δ ( B 0 , ϕ i ) + j = 1 k Δ i b j δ b j .
j = 1 k ψ i b j δ b j = 0 ,
j = 1 k Δ i b j δ b j = 0 ,
ψ i N F δ N F + ψ i K F δ K F + ψ i D δ D + ψ i N S δ N S + ψ i K S δ K S = 0 ,
Δ i N F δ N F + Δ i K F δ K F + Δ i D δ D + Δ i N S δ N S + Δ i K S δ K S = 0 , i = 1 , 2 , m .
δ N F + C δ D = 0 ,
C = ( Δ i / D ) ( Δ i / N F )
( Δ i / D ) ( Δ j / D ) = ( Δ i / N F ) ( Δ j / N F ) ,
Δ / b 1 Δ / b 2
H ( B ) ( 2 G ( B ) b i b j ) ,             i , j = 1 , k
G ( B ) - G ( B 0 ) = 1 2 ( B - B 0 ) T H ( B 0 ) ( B - B 0 ) + = 1 2 i = 1 k j = 1 k 2 G b i b j Δ b i Δ b j + .
G ( b i ) - G ( B 0 ) = 1 2 ( b i - b i 0 ) 2 [ 2 G ( B ) / b i 2 ] + ,
δ b j = ( 2 δ G 2 G / b j 2 ) 1 2 .
δ G = i = 1 { ( δ Δ i ) 2 + ( δ ψ i ) 2 } .
B ( n + 1 ) = B ( n ) - α g ( B ( n ) ) ,
α 2 / H - ,
F = tan ψ i e j Δ i - ρ ( B , ϕ i ) ,
δ N F + 0.53 δ D = 0 , N F = 1.47 - 0.053 ( D - 20 ) .
δ K S + 0.009 δ D = 0 , K S = 0.03 - 0.009 ( D - 20 ) .
Δ / N S Δ / N F             and             Δ / K F Δ / N F
δ b 1 = δ b 2 ( Δ / b 2 ) ( Δ / b 1 ) .
δ b k = [ i = 1 l - 1 δ b i ( Δ b i ) ] / ( Δ b k ) .
( Δ / D ) ( Δ / N F )
R = δ N S Δ N F = ( 2 G / N F 2 2 G / N S 2 ) 1 2 .