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 NF (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/∂NF
∂ψ/∂NF
∂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–SiO2 at a 5461-Å wavelength and 20-Å film thickness.
ϕ
∂Δ/∂KS (×34)
∂Δ/∂NF (×6)
∂Δ/∂D (×0.3)
∂ψ/∂NS (×6.8)
∂ψ/∂KF(×1.2)
∂Δ/∂NS (×3.3)
∂Δ/∂KF (×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–SiO2 and 20-Å film thickness.
Wavelength
ϕ
5461 Å
60°
−0.04 × 0.08
0.16 × 0.005
0.0526
0.0092
−5.65
0.016
70°
1.0 × 0.08
1.0 × 0.005
0.0525
0.0091
−1.81
−0.049
80
3.8 × 0.08
2.6 × 0.005
0.0528
0.0090
0.82
0.26
2967 Å
60
−0.09 × 0.35
0.20 × 0.025
0.0423
−0.75
−1.16
−0.021
70
1.0 × 0.35
1.0 × 0.025
0.0420
−0.26
−1.17
0.10
80
6.9 × 0.35
4.6 × 0.025
0.0403
−0.064
−1.22
0.64
Table IV
Correlation between parameters for SiO2–Si.
Wavelength
D
δNF/δD
δKS/δD
δNS/δNF
δKF/δNF
5461 Å
20 Å
0.053
0.009
Uncorrelated
Uncorrelated
4358 Å
20 Å
0.049
0.017
Uncorrelated
Uncorrelated
100 Å
0.01
0.017
Uncorrelated
Uncorrelated
4047 Å
20 Å
0.047
0.024
Uncorrelated
Uncorrelated
2967 Å
20 Å
0.042
Uncorrelated
−1.16
Uncorrelated
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.
Wavelength
ϕ
5461 Å
55°
0.08
−0.006
0.0378
0.052
65
0.123
−0.008
0.038
0.063
75
0.170
−0.007
0.038
0.076
2967 Å
55
−1.58
0.01
0.020
−0.46
65
0.35
0.046
0.014
+0.107
75
1.56
0.051
0.009
0.62
Table VI
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
NS
KS
NS
KS
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
NS and KS 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–SiO2 at a wavelength of 4358 Å.a
G0 is the initial sum of squares residual, GF 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 NF (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/∂NF
∂ψ/∂NF
∂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–SiO2 at a 5461-Å wavelength and 20-Å film thickness.
ϕ
∂Δ/∂KS (×34)
∂Δ/∂NF (×6)
∂Δ/∂D (×0.3)
∂ψ/∂NS (×6.8)
∂ψ/∂KF(×1.2)
∂Δ/∂NS (×3.3)
∂Δ/∂KF (×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–SiO2 and 20-Å film thickness.
Wavelength
ϕ
5461 Å
60°
−0.04 × 0.08
0.16 × 0.005
0.0526
0.0092
−5.65
0.016
70°
1.0 × 0.08
1.0 × 0.005
0.0525
0.0091
−1.81
−0.049
80
3.8 × 0.08
2.6 × 0.005
0.0528
0.0090
0.82
0.26
2967 Å
60
−0.09 × 0.35
0.20 × 0.025
0.0423
−0.75
−1.16
−0.021
70
1.0 × 0.35
1.0 × 0.025
0.0420
−0.26
−1.17
0.10
80
6.9 × 0.35
4.6 × 0.025
0.0403
−0.064
−1.22
0.64
Table IV
Correlation between parameters for SiO2–Si.
Wavelength
D
δNF/δD
δKS/δD
δNS/δNF
δKF/δNF
5461 Å
20 Å
0.053
0.009
Uncorrelated
Uncorrelated
4358 Å
20 Å
0.049
0.017
Uncorrelated
Uncorrelated
100 Å
0.01
0.017
Uncorrelated
Uncorrelated
4047 Å
20 Å
0.047
0.024
Uncorrelated
Uncorrelated
2967 Å
20 Å
0.042
Uncorrelated
−1.16
Uncorrelated
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.
Wavelength
ϕ
5461 Å
55°
0.08
−0.006
0.0378
0.052
65
0.123
−0.008
0.038
0.063
75
0.170
−0.007
0.038
0.076
2967 Å
55
−1.58
0.01
0.020
−0.46
65
0.35
0.046
0.014
+0.107
75
1.56
0.051
0.009
0.62
Table VI
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
NS
KS
NS
KS
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
NS and KS 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–SiO2 at a wavelength of 4358 Å.a
G0 is the initial sum of squares residual, GF is the residual when iteration was terminated.
The minus signs indicate that an incorrect value was calculated.
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