Absolute and random error analysis of the dynamic imaging microellipsometry technique

Ralph F. Cohn; James W. Wagner

doi:10.1364/AO.28.003187

Applied Optics
Vol. 28,
Issue 15,
pp. 3187-3198
(1989)
•https://doi.org/10.1364/AO.28.003187

Absolute and random error analysis of the dynamic imaging microellipsometry technique

Ralph F. Cohn and James W. Wagner

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Ralph F. Cohn and James W. Wagner, "Absolute and random error analysis of the dynamic imaging microellipsometry technique," Appl. Opt. 28, 3187-3198 (1989)

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Abstract

A summary of the rapid high resolution full field ellipsometric imaging technique termed dynamic imaging microellipsometry is presented. The theoretical basis of a linear sensitivity coefficient approach to the coupling of individual error sources to ellipsometric absolute error is derived. Numerically computed sensitivity coefficient maps are presented with a discussion of their application to ellipsometric data calibration. An analysis of random error is presented and verified using comparisons between the theoretical estimations and system measurements.

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Error source	Estimated error	Case 1 A = 29° C = 116°				Case 2 A = 45° C = 45°
		Δ = 91°	ψ = 25°	Δ = 80	ψ = 47.5°	Δ = 91°	ψ = 25°	Δ = 80°	ψ = 47.5°

		Δ_coef	ψ _coef	Δ_coef	ψ _coef	Δ_coef	ψ _coef	Δ_coef	ψ _coef
ρ _H	1.2°	−0.65	−0.18	−0.37	−0.53	−0.71	−0.09	−0.61	−0.27
ρ _C	1.2°	0.01	−0.05	0.01	−0.07	0.0	0.0	0.0	0.0
α _P	3.2 × 10⁻³	5.7	−5.0	−97.0	−23.1	96.1	1.0	−9.9	−9.9
α _A	1 × 10⁻⁵	21.2	−43.0	58.1	−55.9	−2.0	0.0	20.2	0.0
P	0°	0.1	−0.08	−1.7	−0.40	1.68	0.02	−0.17	−0.17
C	0.13°	2.73	−0.88	2.73	−1.15	−2.0	−0.77	−2.0	−1.0
A	0.13°	−2.5	−0.08	−2.5	−0.1	2.0	0.0	2.0	0.0
H	0.05°	−0.2	0.18	3.4	0.81	−3.4	−0.03	0.34	0.35
AOI	0.7°	−4.7	−0.05	−4.0	−0.24	−4.7	−0.05	−4.0	−0.24
Worst case error		4.8°	0.46°	4.4°	1.2°	5.1°	0.25°	4.1°	0.67°
Root sum of squares		3.4°	0.25°	2.8°	0.72°	3.4°	0.13°	2.9°	0.39°
Precision alignment root sum of squares		0.80°	0.19°	0.55°	0.53°	0.85°	0.089°	0.73°	0.27°

Run	Oxide (nm)	C (deg)	A (deg)	Quad	$\bar{Δ}$	$\bar{ψ}$	$σ_{Δ}^{-}$	$σ_{ψ}^{-}$	Spatial		Temporal
Run	Oxide (nm)	C (deg)	A (deg)	Quad	$\bar{Δ}$	$\bar{ψ}$	$σ_{Δ}^{-}$	$σ_{ψ}^{-}$	σ _Δ	σ _ψ	σ _Δ	σ _ψ
1.0	1.5	87	7	1	169.81	10.40	0.476	0.052	0.491	0.048	0.150	0.020
2.0	1.5	87	7	4	168.41	10.38	0.478	0.046	0.478	0.034	0.210	0.021
3.0	1.5	88	8	1	176.93	11.35	0.112	0.090	0.135	0.023	0.097	0.018
4.0	1.5	88	8	4	176.11	11.31	0.179	0.081	0.151	0.026	0.100	0.018
5.0	55		45	1	91.06	24.99
5.1	110		45	1	80.56	50.51
6.0	55		45	4	91.08	24.51
6.1	110		45	4	79.91	50.52
7.0	110		45	1	82.63	49.04	0.166	0.258	0.125	0.088	0.108	0.054
8.0	55		45	1	91.27	24.76	0.142	0.122	0.162	0.134	0.156	0.085
9.0	55		45	4	92.98	23.86	0.085	0.115	0.165	0.117	0.160	0.087
10.0	55		45	1	85.85	26.32
10.1	110		45	1	77.91	51.94
11.0	55	88	45	1		25.36				0.141
12.0	110	88	45	1		50.63				0.191
13.0	110	141	57	1	80.53	50.18	0.104	0.054	0.170	0.059	0.142	0.055
14.0	55	141	57	1	91.20	24.00	0.477	0.207	0.249	0.189	0.223	0.128
15.0	55	65	157	1	75.11	24.83	0.359	0.209	0.096	0.034	0.086	0.032
16.0	110	65	157	1	85.62	51.81	0.099	0.371	0.222	0.241	0.208	0.178
17.0	110	116	29	1	80.06	52.54	0.323	0.311	0.387	0.233	0.250	0.165
18.0	55	116	29	1	90.00	25.91	0.232	0.128	0.105	0.038	0.094	0.036
19.0	55	130	45	1	92.84	23.42	0.616	0.277	0.192	0.141	0.174	0.086
20.0	110	130	45	1	84.48	49.11	0.162	0.130	0.162	0.085	0.143	0.073

Run	Δ Random error			ψ Random error
Run	Theory	Spatial	Temporal	Theory	Spatial	Temporal
7	0.087	0.126	0.108	0.043	0.088	0.054
8	0.143	0.162	0.156	0.083	0.134	0.085
11	13.00			0.097	0.141
12	1.070			0.066	0.191
13	0.109	0.170	0.142	0.047	0.059	0.055
14	0.188	0.249	0.223	0.122	0.189	0.128
15	0.082	0.096	0.086	0.033	0.034	0.032
16	0.169	0.222	0.208	0.147	0.241	0.178
17	0.189	0.387	0.250	0.134	0.233	0.165
18	0.081	0.105	0.094	0.033	0.038	0.036
19	0.125	0.192	0.174	0.062	0.141	0.086
20	0.110	0.162	0.143	0.053	0.085	0.073

Error source	Estimated error	Case 1 A = 29° C = 116°				Case 2 A = 45° C = 45°
		Δ = 91°	ψ = 25°	Δ = 80	ψ = 47.5°	Δ = 91°	ψ = 25°	Δ = 80°	ψ = 47.5°

		Δ_coef	ψ _coef	Δ_coef	ψ _coef	Δ_coef	ψ _coef	Δ_coef	ψ _coef
ρ _H	1.2°	−0.65	−0.18	−0.37	−0.53	−0.71	−0.09	−0.61	−0.27
ρ _C	1.2°	0.01	−0.05	0.01	−0.07	0.0	0.0	0.0	0.0
α _P	3.2 × 10⁻³	5.7	−5.0	−97.0	−23.1	96.1	1.0	−9.9	−9.9
α _A	1 × 10⁻⁵	21.2	−43.0	58.1	−55.9	−2.0	0.0	20.2	0.0
P	0°	0.1	−0.08	−1.7	−0.40	1.68	0.02	−0.17	−0.17
C	0.13°	2.73	−0.88	2.73	−1.15	−2.0	−0.77	−2.0	−1.0
A	0.13°	−2.5	−0.08	−2.5	−0.1	2.0	0.0	2.0	0.0
H	0.05°	−0.2	0.18	3.4	0.81	−3.4	−0.03	0.34	0.35
AOI	0.7°	−4.7	−0.05	−4.0	−0.24	−4.7	−0.05	−4.0	−0.24
Worst case error		4.8°	0.46°	4.4°	1.2°	5.1°	0.25°	4.1°	0.67°
Root sum of squares		3.4°	0.25°	2.8°	0.72°	3.4°	0.13°	2.9°	0.39°
Precision alignment root sum of squares		0.80°	0.19°	0.55°	0.53°	0.85°	0.089°	0.73°	0.27°

Run	Oxide (nm)	C (deg)	A (deg)	Quad	$\bar{Δ}$	$\bar{ψ}$	$σ_{Δ}^{-}$	$σ_{ψ}^{-}$	Spatial		Temporal
Run	Oxide (nm)	C (deg)	A (deg)	Quad	$\bar{Δ}$	$\bar{ψ}$	$σ_{Δ}^{-}$	$σ_{ψ}^{-}$	σ _Δ	σ _ψ	σ _Δ	σ _ψ
1.0	1.5	87	7	1	169.81	10.40	0.476	0.052	0.491	0.048	0.150	0.020
2.0	1.5	87	7	4	168.41	10.38	0.478	0.046	0.478	0.034	0.210	0.021
3.0	1.5	88	8	1	176.93	11.35	0.112	0.090	0.135	0.023	0.097	0.018
4.0	1.5	88	8	4	176.11	11.31	0.179	0.081	0.151	0.026	0.100	0.018
5.0	55		45	1	91.06	24.99
5.1	110		45	1	80.56	50.51
6.0	55		45	4	91.08	24.51
6.1	110		45	4	79.91	50.52
7.0	110		45	1	82.63	49.04	0.166	0.258	0.125	0.088	0.108	0.054
8.0	55		45	1	91.27	24.76	0.142	0.122	0.162	0.134	0.156	0.085
9.0	55		45	4	92.98	23.86	0.085	0.115	0.165	0.117	0.160	0.087
10.0	55		45	1	85.85	26.32
10.1	110		45	1	77.91	51.94
11.0	55	88	45	1		25.36				0.141
12.0	110	88	45	1		50.63				0.191
13.0	110	141	57	1	80.53	50.18	0.104	0.054	0.170	0.059	0.142	0.055
14.0	55	141	57	1	91.20	24.00	0.477	0.207	0.249	0.189	0.223	0.128
15.0	55	65	157	1	75.11	24.83	0.359	0.209	0.096	0.034	0.086	0.032
16.0	110	65	157	1	85.62	51.81	0.099	0.371	0.222	0.241	0.208	0.178
17.0	110	116	29	1	80.06	52.54	0.323	0.311	0.387	0.233	0.250	0.165
18.0	55	116	29	1	90.00	25.91	0.232	0.128	0.105	0.038	0.094	0.036
19.0	55	130	45	1	92.84	23.42	0.616	0.277	0.192	0.141	0.174	0.086
20.0	110	130	45	1	84.48	49.11	0.162	0.130	0.162	0.085	0.143	0.073

Abstract

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Tables (3)

Equations (38)

Applied Optics