Determination of properties of wedged, nonuniformly thick, and absorbing thin films by using a new numerical method

Jonghoon Baek; Desiderio Kovar; John W. Keto; Michael F. Becker

doi:10.1364/AO.45.001627

Applied Optics
Vol. 45,
Issue 7,
pp. 1627-1639
(2006)
•https://doi.org/10.1364/AO.45.001627

Determination of properties of wedged, nonuniformly thick, and absorbing thin films by using a new numerical method

Jonghoon Baek, Desiderio Kovar, John W. Keto, and Michael F. Becker

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Jonghoon Baek, Desiderio Kovar, John W. Keto, and Michael F. Becker, "Determination of properties of wedged, nonuniformly thick, and absorbing thin films by using a new numerical method," Appl. Opt. 45, 1627-1639 (2006)

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Abstract

Nonuniformity in the thickness of thin films can severely distort their transmission spectra as compared with those of flat, smooth films. Methods that extract properties such as refractive index, thickness, and extinction coefficient of such films can suffer inaccuracies when they are applied to wedged or nonuniformly thick films. To accurately extract optical properties of nonuniform films, we have developed a novel numerical method and efficient constitutive relations that can determine film properties from just the transmission spectrum for films that are locally smooth with negligible scattering loss. This optimum parameter extraction (OPE) method can accommodate films with two-dimensional thickness variation that would result in significant errors in the values of refractive index and film thickness if not considered. We show that for carefully chosen test cases and for actual pulsed-laser-deposition AlN thin films, properties such as refractive index, extinction coefficient, and film thickness were very accurately determined by using our OPE method. These results are compared with previous techniques to determine the properties of thin films, and the accuracy of and applicable conditions for all these methods are discussed.

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	d (nm)
Optical Parameters	1000			2000
k	10⁻⁴	5 × 10⁻⁴	10 × 10⁻⁴	0	1 × 10⁻⁴	5 × 10⁻⁴	10 × 10⁻⁴
χ at 633 nm	0.998	0.990	0.980	1.000	0.996	0.980	0.961
Error (%) of n	1.5	2.9	4.4	0.0	1.1	5.0	11.5
Error (%) of d	−1.5	−2.8	−4.2	−0.1	−1.1	−4.8	−17.4
Error (%) of Δd	5.5	8.1	12.7	1.1	5.8	18.9	31.8

	Nonuniformity
Value	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 10 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 5 Fitted Parameter	Δd_y = 10 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 20 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 10 Fitted Parameter	Δd_y = 20 Δr_y = 0 Error (%)
p	0.999	−0.1%	0.999	−0.1%	0.998	−0.2%	0.997	−0.3%
k (×10⁻⁴)	10.30	3.0%	10.03	0.3%	10.11	1.1%	10.14	1.4%
d (nm)	2000.6	0.03%	2000.7	0.03%	2002.2	0.11%	2002.7	0.14%
Δd (nm)	40.8	2.0%	41.1	2.6%	43.3	8.2%	44.0	10.1%
	Nonuniformity
	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 30 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 20 Fitted Parameter	Δd_y = 30 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 40 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 10 Fitted Parameter	Δd_y = 40 Δr_y = 0 Error (%)
p	0.996	−0.4%	0.994	−0.6%	0.993	−0.7%	0.993	−0.7%
k (×10⁻⁴)	10.22	2.2%	10.30	3.0%	10.35	3.5%	10.36	3.6%
d (nm)	2004.4	0.22%	2006.2	0.31%	2006.9	0.34%	2007.2	0.36%
Δd (nm)	47.3	18.1%	50.0	25.0%	52.8	31.9%	53.4	33.4%

Film Sample	T (°C)	n at 633 nm	k at 633 nm (×10⁻⁴)	d (nm)	Δd (nm)	RMS meas. (nm)	E_g (ev)
High fluence (3–12 J∕cm²)	100	2.06	6.38	2264	73	1.6	4.40
	300	2.08	4.55	1881	68	2.3	5.05
	500	2.09	4.81	1499	45	1.5	5.21
Wedge-shaped method, χ = 0.986	500	2.15	−0.9	1456	57	—	—
Conventional method	500	1.91	10.8	1974	NA	—	—
	700	2.07	8.20	1849	46	2.6	5.23
Low fluence (0.7–3.0 J∕cm²)	300	2.22	9.9	491	13	0.5	5.62
Wedge-shaped method, χ = 0.990	300	2.28	−3.9	477	28	—	—
Conventional method	300	2.17	13.9	502	NA	—	—
	500	2.15	15.0	575	20	2.5	5.81
	600	2.15	17.9	392	8	1.0	5.88
	800	2.18	22.6	553	15	2.4	5.82

	d (nm)
Optical Parameters	1000			2000
k	10⁻⁴	5 × 10⁻⁴	10 × 10⁻⁴	0	1 × 10⁻⁴	5 × 10⁻⁴	10 × 10⁻⁴
χ at 633 nm	0.998	0.990	0.980	1.000	0.996	0.980	0.961
Error (%) of n	1.5	2.9	4.4	0.0	1.1	5.0	11.5
Error (%) of d	−1.5	−2.8	−4.2	−0.1	−1.1	−4.8	−17.4
Error (%) of Δd	5.5	8.1	12.7	1.1	5.8	18.9	31.8

	Nonuniformity
Value	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 10 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 5 Fitted Parameter	Δd_y = 10 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 20 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 10 Fitted Parameter	Δd_y = 20 Δr_y = 0 Error (%)
p	0.999	−0.1%	0.999	−0.1%	0.998	−0.2%	0.997	−0.3%
k (×10⁻⁴)	10.30	3.0%	10.03	0.3%	10.11	1.1%	10.14	1.4%
d (nm)	2000.6	0.03%	2000.7	0.03%	2002.2	0.11%	2002.7	0.14%
Δd (nm)	40.8	2.0%	41.1	2.6%	43.3	8.2%	44.0	10.1%
	Nonuniformity
	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 30 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 20 Fitted Parameter	Δd_y = 30 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 0 Fitted Parameter	Δd_y = 40 Δr_y = 0 Error (%)	Δd_x = 40 Δr_x = 10 Fitted Parameter	Δd_y = 40 Δr_y = 0 Error (%)
p	0.996	−0.4%	0.994	−0.6%	0.993	−0.7%	0.993	−0.7%
k (×10⁻⁴)	10.22	2.2%	10.30	3.0%	10.35	3.5%	10.36	3.6%
d (nm)	2004.4	0.22%	2006.2	0.31%	2006.9	0.34%	2007.2	0.36%
Δd (nm)	47.3	18.1%	50.0	25.0%	52.8	31.9%	53.4	33.4%

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Applied Optics