Evaluation and Analysis of Optical and Electrical Constants of Thin Films as Functions of Reflectance and Transmission Data by Electronic Digital Computation*

Louis Harris; Arthur L. Loeb

doi:10.1364/JOSA.45.000179

Journal of the Optical Society of America
Vol. 45,
Issue 3,
pp. 179-188
(1955)
•https://doi.org/10.1364/JOSA.45.000179

Evaluation and Analysis of Optical and Electrical Constants of Thin Films as Functions of Reflectance and Transmission Data by Electronic Digital Computation

Louis Harris and Arthur L. Loeb

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Louis Harris and Arthur L. Loeb, "Evaluation and Analysis of Optical and Electrical Constants of Thin Films as Functions of Reflectance and Transmission Data by Electronic Digital Computation*," J. Opt. Soc. Am. 45, 179-188 (1955)

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Abstract

Whirlwind I, an electronic digital computer at M.I.T., has been used to calculate accurately the optical and electrical constants of thin metal films on nonabsorbing backings and of aluminum oxide films, and to check approximate relations which may be conveniently used in the absence of a digital computer. The conductance per square of film is found to be the principal physical property determining the behavior of conducting films toward incident infrared radiation. The variation with wavelength is related to the finite relaxation time of the free electrons and to imperfections in the films acting as condensers. In the visible wavelength region the two sets of optical constants were evaluated for different thicknesses of gold deposits and compared to optical constants previously reported.

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Sample	Thickness	Wavelength	First estimates				Discrepancies^a
Sample	Thickness	Wavelength	n	k	μcσa	∊/∊₀	ΔR	ΔT
Sb 6	170A	21.5μ	9.259	2.508	0.230	79.4	0.0000	0.0001
		32.8μ	10.523	2.881	0.197	102.4	0.0000	0.0000
		52.0μ	14.411	2.645	0.156	200.7	0.0000	0.0000
		91.0μ	17.588	3.419	0.141	297.7	0.0000	0.0000

Sample	Thickness	Wavelength	Results of approximate calculation				Results of accurate calculation
Sample	Thickness	Wavelength	n′=k′	n′k′	ΔR	ΔT	n₁	k₁	n₁k₁	n₂	k₂	n₂k₂
Al 19	40A	21.5μ	12.27	151	0.0175	0.0130	8.186	18.40	151	18.44	8.167	151
		32.8μ	13.02	170	0.0167	0.0133	7.982	21.23	170	21.27	7.967	170
		52.0μ	16.91	287	0.0152	0.0119	13.58	22.06	296	22.09	13.56	299
Al 12	40A	32.8μ	31.61	1000	0.0019	0.0007	29.87	33.45	1000	33.46	29.86	1000
		52.0μ	40.02	1600	0.0195	0.0076	33.54	47.75	1600	47.77	33.53	1600

Sample	Thickness	Wavelength	μcσa	2πa/λ	$\frac{μ c σ a}{(2 π a / λ)}$	∊/∊₀
Al 19	40A	21.5μ	0.352	0.00117	300	232
		32.8μ	0.259	0.000768	337	387
		52.0μ	0.276	0.000484	570	591
Al 12	40A	32.8μ	1.531	0.000768	2000	215
		52.0μ	1.548	0.000484	3200	1150

Sample	Thickness	Wavelength		n^a	k^a	$\frac{\partial n^{a}}{\partial R}$	$\frac{\partial n^{a}}{\partial T}$	$\frac{\partial k^{a}}{\partial R}$	$\frac{\partial k^{a}}{\partial T}$		μcσa	$\frac{∊}{∊_{0}} - 1$	ρ_dc^b
Al 19	40A	21.5μ		8.19	18.4	−117	−50.0	189	14.8		0.352	−272	5×10⁻²
		21.5μ		18.4	8.17	189	14.9	−117	−50.8		0.352	+272
		32.8μ		7.98	21.2	−135	−55.2	252	12.8		0.259	−388
		32.8μ		21.3	7.97	252	12.9	−134	−55.1		0.259	+388
		52.0μ		13.6	22.1	−475	−11.8	670	62.5		0.289	−303
		52.0μ		22.1	13.6	671	62.6	−473	−118		0.289	+303
		91.0μ		11.1	31.7	−258	−90.6	535	20.9		0.195	−883
		91.0μ		31.7	11.1	535	20.9	−257	−90.5		0.195	+883
Al 12	40A	21.5μ	Data violate inequality 11
		32.8μ		29.9	33.4	−650	−556		659	449	1.53	−227	1×10⁻⁴
		32.8μ		33.5	29.9	658	448		−648	−554	1.53	+227
		52.0μ		33.5	47.8	−256	−272		264	134	1.55	−1156
		52.0μ		47.8	33.5	264	134		−256	−272	1.55	+1156
		91.0μ		⋯	⋯	⋯	⋯		⋯	⋯	⋯	⋯
		91.0μ		63.9	38.3	248	83.9		−231	−246	1.35	+2617
Al 9	50A	21.5μ		43.9	32.7	211	297		−235	−626	4.20	+852	6×10⁻⁵
		32.8μ		54.2	40.5	261	367		−290	−773	4.20	+1301
		52.0μ		68.3	50.9	329	462		−366	−974	4.20	+2065
		91.0μ		90.3	67.4	435	612		−485	−1290	4.20	+3615
Ag 9	60A	21.5μ		4.25	24.6	−26.2	−25.4		39.7	−6.19	0.366	−589	4×10⁻¹
		21.5μ		24.7	4.24	39.7	−6.16		−26.2	−25.4	0.366	+589
		32.8μ		3.17	25.6	−30.5	−26.3		62.8	−5.40	0.186	−646
		32.8μ		25.6	3.17	62.8	−5.38		−30.4	−26.3	0.186	+646
		52.0μ		2.17	30.9	−32.9	−29.4		80.0	−7.40	0.097	−954
		52.0μ		31.0	2.17	80.0	−7.39		−32.8	−29.3	0.097	+954
		91.0μ		1.99	33.7	−50.0	−41.7		172	−6.89	0.055	−1130
		91.0μ		33.7	1.99	172	−6.88		−50.0	−41.7	0.055	+1130
		21.5μ		7.07	25.4	−29.6	−38.3		31.0	−8.36	0.943	−594	5×10⁻¹
Ag 7	90A	21.5μ		25.4	7.07	31.1	−8.28		−29.5	−38.2	0.944	+594
		32.8μ		6.66	27.2	−32.2	−34.6		40.6	−6.58	0.624	−697
		32.8μ		27.2	6.65	40.7	−6.54		−32.2	−34.5	0.624	+697
		52.0μ		6.05	31.4	−35.3	−34.3		52.0	−7.43	0.413	−951
		52.0μ		31.4	6.05	52.0	−7.41		−35.3	−34.3	0.413	+951
		91.0μ		6.30	34.2	−51.3	−41.2		97.7	−4.83	0.267	−1128
		91.0μ		34.2	6.30	97.7	−4.81		−51.2	−41.2	0.267	+1128
Ag 1	135A	21.5μ		33.8	18.3	67.3	20.6		−89.3	−324	4.88	+804	6×10⁻⁵
		32.8μ		41.7	22.6	83.1	25.2		−111	−401	4.87	+1228
		52.0μ		52.5	28.4	104	31.4		139	−505	4.87	+1948
		91.0μ		69.4	37.6	138	41.5		−185	−668	4.86	+3411
Sb 6	170A	21.5μ		2.53	9.16	−33.8	−17.9		70.9	1.68	0.230	−78.4	5.1×10⁻³
		21.5μ		9.26	2.51	70.5	1.77		−33.1	−17.7	0.230	+78.4
		32.8μ		2.91	10.4	−48.1	−22.6		107	2.71	0.197	−101
		32.8μ		10.5	2.88	106	2.80		−47.2	−22.4	0.197	+101
		52.0μ		2.66	14.3	−39.0	−23.7		99.9	−0.106	0.156	−200
		52.0μ		14.4	2.65	99.6	−0.063		−38.7	−23.6	0.156	+200
		91.0μ		3.43	17.5	−60.6	−32.9		163	1.06	0.141	−297
		91.0μ		17.6	3.42	163	1.10		−60.3	−32.8	0.141	+297
Sb 7	170A	21.5μ		7.11	7.07	−10796	−2732		10710	2683	0.499	−0.418	1.4×10⁻³
		21.5μ		7.15	7.04	9289	2325		−9153	−2318	0.499	+1.558
		32.8μ		6.17	11.5	−84.7	−38.0		119	14.5	0.461	−94.5
		32.8μ		11.5	6.14	119	14.7		−83.7	−37.8	0.461	+95.5
		52.0μ		8.70	15.1	−111	−53.5		147	22.3	0.540	−155
		52.0μ		15.2	8.67	147	22.4		−110	−53.4	0.540	+155
		91.0μ		10.9	21.5	−110	−61.2		152	19.6	0.549	−345
		91.0μ		21.5	10.9	152	19.7		−110	−61.1	0.549	+345
Sb 4	140A	21.5μ		8.31	12.3	−92.4	−57.0		106	29.0	0.836	−83.4	0.42×10⁻³
		21.5μ		12.4	8.27	107	29.2		−91.4	−56.6	0.836	+83.3
		91.0μ		21.9	31.7	−177	−166		187	81.6	1.34	−522
		91.0μ		31.7	21.9	187	81.6		−176	−166	1.34	+521
Sb 5	180 A	21.5μ		7.02	13.2	−44.3	−38.9		51.5	10.6	0.975	−126	0.4×10⁻³
		21.5μ		13.3	6.99	51.6	10.7		−43.8	−38.7	0.976	+126
		91.0μ		20.3	33.3	−113	−145		114	47.1	1.677	−699
		91.0μ		33.3	20.3	114	47.2		−113	−144	1.678	+699
Bi 2	240A	21.5μ		5.09	10.6	−34.3	−26.6		43.3	6.00	0.758	−88.2	3.0×10⁻⁴
		21.5μ		10.7	5.06	43.4	6.15		−33.8	−26.4	0.759	+88.2
		52.0μ		11.1	19.4	−65.4	−67.3		71.0	20.4	1.25	−256
		52.0μ		19.5	11.1	71.1	20.5		−65.0	−67.1	1.25	+255
		21.5μ		2.45	8.61	−10.7	−12.9		11.6	−2.42	0.863	−69.1	3.9×10⁻⁴
Bi 6	700A	21.5μ		8.73	2.44	11.8	−2.23		−10.5	−12.9	0.871	+69.3
		32.8μ		5.95	8.17	−55.6	−49.0		57.8	26.2	1.30	−32.3
		32.8μ		8.24	5.91	57.7	26.3		−53.5	−47.6	1.31	+32.1
		52.0μ		8.41	13.7	−45.3	−66.4		43.7	20.6	1.95	−118
		52.0μ		13.8	8.40	43.8	20.7		−44.3	−65.4	1.96	+118
Bi 12	1700A	21.5μ		1.66	9.70	−9.79	−20.2		5.02	−18.9	1.60	−92.3	1.9×10⁻⁴
		21.5μ		10.2	1.85	6.10	−20.0		−10.6	−23.5	1.87	+98.9
		32.8μ		3.79	9.37	−15.2	−32.3		11.7	−4.45	2.31	−74.3
		32.8μ		9.53	3.93	12.8	−3.07		−15.2	−33.6	2.44	+74.4
		52.0μ		6.31	19.3	−39.1	−175		19.5	−82.5	5.00	−332
		52.0μ		19.6	6.80	20.9	−74.7		−37.2	−164	5.49	+339
		91.0μ		⋯	⋯	⋯	⋯		⋯	⋯	⋯	⋯
		91.0μ		37.8	14.2	64.6	−571		−132	−1203	12.6	+1229

First estimates k	0.45	0.35	0.30	0.25	0.15
4.5	18 sec/10 sec	17 sec/7 sec	17 sec/9 sec	20 sec/10 sec	19 sec/7 sec
3.8	14 sec/6 sec	14 sec/9 sec	15 sec/10 sec	11 sec/8 sec	12 sec/6 sec
3.3	10 sec/9 sec	9 sec/9 sec	10 sec/6 sec	8 sec/10 sec	11 sec/6 sec
2.8	11.5 sec/6 sec	11 sec/9 sec	12 sec/9 sec	12 sec/9 sec	11 sec/6 sec
2.2	11 sec/6 sec	16 sec/—	15 sec/9 sec	16 sec/9 sec	15 sec/6 sec

Metal film thickness (A)	Krautkraemer’s results		Results of computation on Whirlwind I
Metal film thickness (A)	n	k	n₁	k₁	(μcσa)₁	∊₁/∊₀	n₂	k₂	(μcσa)₂	∊₂/∊₀
281	0.20	4.37	f 6.568	0.410	1.361	42.98	0.202	4.372	0.446	−19.08
281	0.20	4.37	b 6.550	0.427	1.412	42.73	0.207	4.369	0.458	−19.05
208	0.33	4.58	f 5.570	0.476	0.991	30.80	0.334	4.507	0.562	−20.21
208	0.33	4.58	b 5.574	0.471	0.981	30.85	0.328	4.510	0.553	−20.24
116	0.67	4.63	f 5.059	0.672	0.709	25.14	0.636	4.545	0.602	−20.26
116	0.67	4.63	b 5.066	0.666	0.703	25.22	0.630	4.552	0.597	−20.32
72	1.06	4.97	f 5.308	1.038	0.712	27.11	1.042	4.930	0.664	−23.22*
72	1.06	4.97	b 5.317	1.031	0.709	27.21	1.035	4.938	0.661	−23.32
54	2.9	5.2	f 5.254	2.923	1.489	19.07	2.975	5.022	1.448	−16.37
54	2.9	5.2	b 5.429	2.778	1.462	21.77	2.817	5.187	1.416	−18.97
43	3.3	5.0	f 5.045	3.316	1.291	14.45	3.413	4.838	1.274	−11.75
43	3.3	5.0	b 5.227	3.165	1.277	17.30	3.250	5.011	1.257	−14.55
24	4.7	3.1	f 4.431	3.279	0.626	8.883	3.444	4.208	0.624	−5.851
24	4.7	3.1	b 4.624	3.126	0.622	11.61	3.279	4.395	0.621	−8.562
20	4.6	1.9	f 4.241	2.103	0.320	13.56	2.258	3.939	0.319	−10.42
20	4.6	1.9	b 4.541	1.934	0.315	16.88	2.065	4.240	0.314	−13.72
17	4.8	1.6	f 4.454	1.747	0.237	16.79	1.875	4.140	0.236	−13.63
17	4.8	1.6	b 4.812	1.580	0.232	20.67	1.682	4.506	0.231	−17.48

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Equations (23)

Journal of the Optical Society of America