Abstract

A quantitative study of sensitivity of multiple-angle-of-incidence ellipsometry in determining several parameters of stratified structures is presented. The principles of determination of parameter errors caused by random and systematic errors in measured ellipsometric angles and by the use of incorrect values of fixed parameters are given. An appropriate choice of angles of incidence is discussed. The sensitivity for dielectric films on low-loss substrates is studied in detail, and large variations with the parameters, especially with film thickness-to-wavelength ratio, are observed.

© 1985 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. M. M. Ibrahim, N. M. Bashara, “Parameter-correlation and computational considerations in multiple-angle ellipsometry,”J. Opt. Soc. Am. 61, 1622 (1971).
    [CrossRef]
  3. G. H. Bu-Abbud, N. M. Bashara, “Parameter correlation and precision in multiple-angle ellipsometry,” Appl. Opt. 20, 3020 (1981).
    [CrossRef] [PubMed]
  4. W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).
  5. Y. Gaillyová, E. Schmidt, J. Humlíček, “Multiple-angle ellipsometry of Si–SiO2polycrystalline Si system,” J. Opt. Soc. Am. A 2, 723–726 (1985).
    [CrossRef]
  6. J. Humlíček, “Evaluation of derivatives of reflectance and transmittance by stratified structures and solution of the reverse problem of ellipsometry,” Opt. Acta 30, 97–105 (1983).
    [CrossRef]
  7. J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation, Linear Algebra (Springer-Verlag, Heildelberg, 1972).

1985 (1)

1983 (1)

J. Humlíček, “Evaluation of derivatives of reflectance and transmittance by stratified structures and solution of the reverse problem of ellipsometry,” Opt. Acta 30, 97–105 (1983).
[CrossRef]

1981 (1)

1971 (1)

Azzam, R. M. A.

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

Bashara, N. M.

Bu-Abbud, G. H.

Dryard, D.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Eadie, W. T.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Gaillyová, Y.

Humlícek, J.

Y. Gaillyová, E. Schmidt, J. Humlíček, “Multiple-angle ellipsometry of Si–SiO2polycrystalline Si system,” J. Opt. Soc. Am. A 2, 723–726 (1985).
[CrossRef]

J. Humlíček, “Evaluation of derivatives of reflectance and transmittance by stratified structures and solution of the reverse problem of ellipsometry,” Opt. Acta 30, 97–105 (1983).
[CrossRef]

Ibrahim, M. M.

James, F. E.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Reinsch, C.

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation, Linear Algebra (Springer-Verlag, Heildelberg, 1972).

Roos, M.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Sadoulet, B.

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

Schmidt, E.

Wilkinson, J. H.

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation, Linear Algebra (Springer-Verlag, Heildelberg, 1972).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

J. Humlíček, “Evaluation of derivatives of reflectance and transmittance by stratified structures and solution of the reverse problem of ellipsometry,” Opt. Acta 30, 97–105 (1983).
[CrossRef]

Other (3)

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation, Linear Algebra (Springer-Verlag, Heildelberg, 1972).

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

W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971).

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Figures (7)

Fig. 1
Fig. 1

Solid lines show the dependence of the ellipsometric angle ψ on the film thickness d1 to wavelength λ ratio for the substrate (N0 = n0 = 4)–film (N1 = n1 = 1.5)–ambient (na = 1) system. The dotted line adjacent to the ψ axis represents the Gaussian probability density of ψ with the mean of 72.03° and the standard deviation of 0.07°. Dotted lines adjacent to the vertical axis are the resulting densities of the probability of d ^ 1/λ [Eq. (2)]. Dashed lines indicate the solution of ellipsometric equation for the mean value of ψ. Note the difference in the d1/λ scales.

Fig. 2
Fig. 2

Standard deviations [Eqs. (11) and (12)] and the factor R [Eq. (13)] of the refractive index n1 in the system: substrate (N0 = 4)–film (d1, N1 = 1.5)–ambient (na = 1). Angles of incidence are φ1 = 50°, φ2 = 55°, …, φ7 = 80°; standard deviations of ψ and Δ: δψ1 = … = δψ7 = 0.1°, δΔ1 = … = δΔ7 = 0.2°. a, δ n ^ 1 for n0, k0, d1, n1 estimated (—); k0, d1, n1 estimated, n0 fixed (– –); d1, n1 estimated, n0, k0 fixed (- · -). b, δ0 n ^ 1, i.e., the standard deviation of n ^ 1 with all remaining parameters fixed. c, R = δ n ^ 1 / δ 0 n ^ 1, where δ n ^ 1, is the standard deviation for n0, k0, d1, n1 estimated; the product 0 yields the solid line in a.

Fig. 3
Fig. 3

Upper bounds (solid lines) and actual magnitudes (crosses and circles) of systematic errors of a, n1; b, d1/λ; c, k0; d, n0 for the system from Fig. 2. The same set of N = 7 angles of incidence was used. The actual systematic errors were computed for δψk = δψk and δ′Δk = δΔk (circles) or δ′Δk = −δΔk (crosses), with δψk = 0.0270, δΔk = 0.0540, k = 1, …, 7.

Fig. 4
Fig. 4

Reduction of standard deviations of parameter estimates owing to the addition of one measurement at the angle of incidence φ to the set of seven angles 50°, 55°, …, 80°, with δΔk = 2δψk. The parameters n0 (—), k0 (— — —), d1 (— · — — ·), and n1 (— – — –) estimated for the system from Fig. 2. Results for the three d1/λ ratios are shown: a, 0.43; b, 0.3; c, 0.22.

Fig. 5
Fig. 5

As for Fig. 4, but for the following system: substrate (N0 = 3.88 + i0.02)–first film (d1, N1 = 1.46)–second film (d2/λ = 0.19, N2 = 3.9 + i0.04)–ambient (na = 1). The four parameters d1 (—), d2 (— — —), n2 (— · — · —), k2 (— – — –) estimated for the d1/λ values: a, 0.5; b, 0.44; c, 0.35.

Fig. 6
Fig. 6

Standard deviations of a, n1; b, d1/λ; c, k0; d, n0 for the substrate–dielectric-film–ambient system. The refractive index of the substrate is n0 = 3 (—), 4 (— — —), 5 (— · — · —), 6 (— – — –), the absorption coefficient k0 ≪ 1 (k0 = 0 used in the computation), na = 1. The refractive index of the film is A, n1 = 1.2; B, 1.5; C, 1.8; D, 2. Angles of incidence and standard deviations of ψ and Δ are the same as in Fig. 2.

Fig. 7
Fig. 7

As for Fig. 6, but for n0 =4, n1 = 1.5, and k0 = 0(—), 0.1 (— — —), 0.2 (— · — · —), 0.3 (— – — –).

Equations (36)

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ψ k = ψ ( λ , n a , φ k ; N 0 , d 1 , N 1 , , d m , N m ) , Δ k = Δ ( λ , n a , φ k ; N 0 , d 1 , N 1 , , d m , N m ) ,
g ( d ^ 1 / λ ) = f ( ψ e ) d ψ - 1 ( ψ e ) / d ψ e = f [ ψ ( d ^ 1 / λ ) ] d ψ ( d ^ 1 / λ ) d ( d ^ 1 / λ ) .
f ( x ) = 1 2 ( π ) δ ( x ) exp [ - ( x - x 0 ) 2 2 δ 2 ( x ) ] ,
δ ( d ^ 1 / λ ) = δ ( ψ e ) | d ψ ( d ^ 1 / λ ) d ( d ^ 1 / λ ) | - 1 .
S ( q 1 , , q L ) = k = 1 N { [ ψ e k - ψ k ( q 1 , , q L ) δ ψ k ] 2 + [ Δ e k - Δ k ( q 1 , , q L ) δ Δ k ] 2 } .
ψ k ( q 1 , , q L ) ψ 0 k + i = 1 L ψ k q i ( q i - q 0 i ) , Δ k ( q 1 , , q L ) Δ 0 k + i = 1 L Δ k q i ( q i - q 0 i ) ,
i = 1 L H j i ( q i - q 0 i ) = g j ,
H j i = k = 1 N [ 1 ( δ ψ k ) 2 ψ k q i ψ k q j + 1 ( δ Δ k ) 2 Δ k q i Δ k q j ] ,
g j = k = 1 N [ ψ e k - ψ 0 k ( δ ψ k ) 2 ψ k q j + Δ e k - Δ 0 k ( δ Δ k ) 2 Δ k q j ] ,             i , j = 1 , , L .
H i j = ½ 2 S q i q j ,             i , j = 1 , , L ,
q ^ i = q 0 i + j = 1 L D i j g j ;
m = 1 L D i m H m j = δ i j ,
D ( ψ e k - ψ 0 k δ ψ k , ψ e l - ψ 0 l δ ψ l ) = D ( Δ e k - Δ 0 k δ Δ k , Δ e l - Δ 0 l δ Δ l ) = δ k l , D ( ψ e k - ψ 0 k δ ψ k , Δ e l - Δ 0 l δ Δ l ) = 0 ,             k , l = 1 , , N .
D ( g i , g j ) = k = 1 N l = 1 N ( 1 δ ψ k ψ k q i 1 δ ψ l ψ l q j + 1 δ Δ k Δ k q i 1 δ Δ l Δ l q j ) δ k l = H i j .
D ( q ^ i , q ^ j ) = m = 1 L n = 1 L D i m D j n D ( g m , g n ) = D i j .
δ q ^ i = ( D i i ) 1 / 2 ,             i = 1 , , L .
δ 0 q ^ i = 1 ( H i i ) 1 / 2 = { k = 1 N [ ( 1 δ ψ k ψ k q i ) 2 + ( 1 δ Δ k Δ k q i ) 2 ] } - 1 / 2 .
δ q ^ i = ( D i i H i i ) 1 / 2 δ 0 q ^ i = 1 ( 1 - ρ i 2 ) 1 / 2 δ 0 q ^ i = R i δ 0 q ^ i ,
R i = 1 ( 1 - ρ i 2 ) 1 / 2 ,             ρ i = [ 1 - ( D i i H i i ) - 1 ] 1 / 2 0 , 1 .
δ q ^ i q j fixed = ( 1 - ρ i j 2 ) 1 / 2 δ q ^ i ,
ρ i j = D i j ( D i i D j j ) 1 / 2 - 1 , 1
σ ^ = [ S 0 / ( 2 N - L ) ] 1 / 2 ,
δ ψ k α k δ ψ k ,             δ Δ k β k δ Δ k ,             k = 1 , , N ,
S ( q 1 , , q L ) = S ( q 1 , , q L ) + ½ i = 1 L j = 1 L 2 S q i q j ( q i - q i ) ( q j - q j ) = S ( q 1 , , q L ) + i = 1 L j = 1 L H i j ( q i - q i ) ( q j - q j ) .
S ( q 01 , , q 0 L ) = k = 1 N [ ( δ ψ k δ ψ k ) 2 + ( δ Δ k δ Δ k ) 2 ] k = 1 N ( α k 2 + β k 2 ) .
i = 1 L j = 1 L H i j ( q 0 i - q i ) ( q 0 j - q j ) k = 1 N ( α k 2 + β k 2 ) ,
F ( δ q 1 , , δ q L ) = i = 1 L j = 1 L H i j δ q i δ q j = f ,
F / ( δ q i ) = 0 ,             i = 1 , , m - 1 , m + 1 , , L ,
δ q i = const . D i m ,             i = 1 , , L ,
δ q m [ D m m k = 1 N ( α k 2 + β k 2 ) ] 1 / 2 ,             m = 1 , , L .
δ q m ( 2 N ) 1 / 2 K ( D m m ) 1 / 2 = ( 2 N ) 1 / 2 K δ q m ,
j = 1 L - 1 H i j ( q 0 j - q 0 j ) = - H i L ( q 0 L - q 0 L ) ,             i = 1 , , L - 1.
j = 1 L H i j ( q 0 j - q 0 j ) = { 0 for i = 1 , , L - 1 q 0 L - q 0 L D L L for i = L .
q 0 j - q 0 j = D j L q 0 L - q 0 L D L L = ρ i L D j j q 0 L - q 0 L D L L ,             j = 1 , , L - 1.
S δ = w 1 ( δ q ^ 1 ) 2 + + w L ( δ q ^ L ) 2 ,
δ ψ k = [ ( δ ψ 1 k ) 2 + ( d ψ d φ | φ = φ k δ φ k ) 2 ] 1 / 2 , δ Δ k = [ ( δ Δ 1 k ) 2 + ( d Δ d φ | φ = φ k δ φ k ) 2 ] 1 / 2 ,

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