M. M. Ibrahim and N. M. Bashara, "Parameter-Correlation and Computational Considerations in Multiple-Angle Ellipsometry*," J. Opt. Soc. Am. 61, 1622-1629 (1971)

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.

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First derivatives of Δ and ψ with respect to N_{F} (film refractive index) and D (film thickness) for a thin oxide film on silicon at an angle of incidence of 70°.^{a}

Wavelength

D

∂Δ/∂N_{F}

∂ψ/∂N_{F}

∂Δ/∂D

∂ψ/∂D

5461 Å

20 Å

−5.96

0.078

−0.298

0.005

4358 a

20 Å

−6.47

0.113

−0.316

0.008

100 Å

−29.08

1.91

−0.272

0.026

2967 Å

20 Å

−7.22

0.325

−0.323

0.026

The substrate and film optical constants are the same as those used in Table VI.

Table II

The variation with angle of incidence of the normalized derivatives of Δ and ψ for Si–SiO_{2} at a 5461-Å wavelength and 20-Å film thickness.

ϕ

∂Δ/∂K_{S} (×34)

∂Δ/∂N_{F} (×6)

∂Δ/∂D (×0.3)

∂ψ/∂N_{S} (×6.8)

∂ψ/∂K_{F}(×1.2)

∂Δ/∂N_{S} (×3.3)

∂Δ/∂K_{F} (×0.25)

55°

0.280

0.281

0.282

0.586

0.587

0.066

−0.15

60

0.386

0.386

0.387

0.720

0.721

0.124

−0.13

65

0.568

0.569

0.569

0.864

0.860

0.285

0.0

70

1.000

1.000

1.000

1.000

1.000

1.000

1.000

75

3.760

3.767

3.689

0.945

0.959

22.35

41.22

80

−1.186

−1.169

−1.176

−1.015

0.991

2.56

6.20

Table III

The variation with angle of incidence of the derivatives of ψ and the relative derivatives of Δ for Si–SiO_{2} and 20-Å film thickness.

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.

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

Specimen S92

Specimen S107

Wavelength

N_{S}

K_{S}

N_{S}

K_{S}

4358 Å

4.85

0.136

4.85

0.139

4047 Å

5.47

0.30

5.46

0.26

3650 Å

6.67

2.75

6.69

2.70

3131 Å

4.97

3.54

4.94

3.52

3023 Å

4.92

3.99

4.92

3.97

2967 Å

4.78

4.49

4.80

4.46

N_{S} and K_{S} are substrate refractive index and extinction coefficient, respectively, MAI angles are 55°, 60°, 65°, 70°, and 80°. The initial estimates used in the least-squares analyses are discussed in the text.

Table VII

DEHM values for Si–SiO_{2} at a wavelength of 4358 Å.^{a}

G_{0} is the initial sum of squares residual, G_{F} is the residual when iteration was terminated.
The minus signs indicate that an incorrect value was calculated.

Tables (8)

Table I

First derivatives of Δ and ψ with respect to N_{F} (film refractive index) and D (film thickness) for a thin oxide film on silicon at an angle of incidence of 70°.^{a}

Wavelength

D

∂Δ/∂N_{F}

∂ψ/∂N_{F}

∂Δ/∂D

∂ψ/∂D

5461 Å

20 Å

−5.96

0.078

−0.298

0.005

4358 a

20 Å

−6.47

0.113

−0.316

0.008

100 Å

−29.08

1.91

−0.272

0.026

2967 Å

20 Å

−7.22

0.325

−0.323

0.026

The substrate and film optical constants are the same as those used in Table VI.

Table II

The variation with angle of incidence of the normalized derivatives of Δ and ψ for Si–SiO_{2} at a 5461-Å wavelength and 20-Å film thickness.

ϕ

∂Δ/∂K_{S} (×34)

∂Δ/∂N_{F} (×6)

∂Δ/∂D (×0.3)

∂ψ/∂N_{S} (×6.8)

∂ψ/∂K_{F}(×1.2)

∂Δ/∂N_{S} (×3.3)

∂Δ/∂K_{F} (×0.25)

55°

0.280

0.281

0.282

0.586

0.587

0.066

−0.15

60

0.386

0.386

0.387

0.720

0.721

0.124

−0.13

65

0.568

0.569

0.569

0.864

0.860

0.285

0.0

70

1.000

1.000

1.000

1.000

1.000

1.000

1.000

75

3.760

3.767

3.689

0.945

0.959

22.35

41.22

80

−1.186

−1.169

−1.176

−1.015

0.991

2.56

6.20

Table III

The variation with angle of incidence of the derivatives of ψ and the relative derivatives of Δ for Si–SiO_{2} and 20-Å film thickness.

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.

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

Specimen S92

Specimen S107

Wavelength

N_{S}

K_{S}

N_{S}

K_{S}

4358 Å

4.85

0.136

4.85

0.139

4047 Å

5.47

0.30

5.46

0.26

3650 Å

6.67

2.75

6.69

2.70

3131 Å

4.97

3.54

4.94

3.52

3023 Å

4.92

3.99

4.92

3.97

2967 Å

4.78

4.49

4.80

4.46

N_{S} and K_{S} are substrate refractive index and extinction coefficient, respectively, MAI angles are 55°, 60°, 65°, 70°, and 80°. The initial estimates used in the least-squares analyses are discussed in the text.

Table VII

DEHM values for Si–SiO_{2} at a wavelength of 4358 Å.^{a}

G_{0} is the initial sum of squares residual, G_{F} is the residual when iteration was terminated.
The minus signs indicate that an incorrect value was calculated.