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Effect of surface roughness on the optical constants of bulk polycrystalline gold samples
Optik 125 (2014) 1085–1087
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Effect of surface roughness on the optical constants of bulk polycrystalline gold samples N.N. Nagib a,∗ , N.A. Mahmoud a , L.Z. Ismail b , M.A. Amer a , Kh. Abd-El-Sabour b a b
National Institute for Standards, Giza, Egypt Faculty of Science, Cairo University, Giza, Egypt
a r t i c l e
i n f o
Article history: Received 21 March 2013 Accepted 28 July 2013
Keywords: Bulk polycrystalline gold Surface roughness Optical constants
a b s t r a c t Four bulk polycrystalline samples of gold were subjected to different polishing treatments using diamond pastes of grain size 10, 6, 3 and 1 m. The effect of surface roughness on the optical constants n and k is studied by 45◦ angle-of-incidence ellipsometry at 632.8 nm. Results for n and k are extrapolated to the case of an ideal surface which we believe to be highly representative of gold. Comparison with published results for the optical constants of gold thin films is presented. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction The effect of surface roughness on the optical constants of metals was investigated by several authors (see for example [1–4]). Almost in all studies, measurements were performed on thin metal films. Several factors are then expected to affect the optical properties including surface roughness, method and conditions of deposition, film thickness, trapped contaminating molecules and thermal treatment after deposition. This explains the widely distinct published data for the optical constants of metals. More over, deposited films show evidence of aggregation and porosity which make their optical properties different from those of bulk materials [5]. Only a pure bulk sample of perfect smooth and clean surface is expected to show the intrinsic optical characteristics of the material which cannot be realized in practice. In this work, four samples of pure bulk polycrystalline gold subjected to different polishing treatments are studied by ellipsometry to show the effect of surface roughness on the optical constants n and k (refractive index and extinction coefficient). 2. Samples The investigated samples (S1, S2, S3 and S4) are in the form of gold sheets of purity 99.999%, thickness 0.05 mm and surface dimensions ∼30 mm × 30 mm. The samples were subjected to different polishing treatments using diamond pastes of different grain
∗ Corresponding author. E-mail address: nabil [email protected] (N.N. Nagib). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.07.127
sizes. The final polishing step for S1, . . ., S4 was with pastes of grain size 10, 6, 3 and 1 m respectively. The samples of different polishing treatments were then cleaned with a stream of distilled water and dried. 3. Experimental The fundamental equation of ellipsometry for a metal of refractive index n and extinction coefficient k is expressed as [6] = tan
ei = f (n, k)
(1)
where is the complex reflectance ratio, = |rp |/|rs | and = (ıp − ıs ) express the differential changes in amplitude and phase experienced upon reflection by the p and s components. The ellipsometric system used in the work is shown in Fig. 1. A laser beam S ( = 632.8 nm) falls successively on the polarizer P, a quartz quarterwave phase plate C oriented with its fast axis at 45◦ and the sample M. The angle of incidence ϕ is 45◦ . The reflected beam passes through the analyzer A and the detecting system D (photomultiplier, power supply and micro-voltmeter). By simultaneous adjustments of P and A, the reflected beam could be extinguished. Two extinction, pairs (p1 , a1 ) and (p2 , a2 ) could be recorded such that p2 = p1 ± 90◦ ,
(2a)
a2 = a1 ± 90◦ ,
(2b)
where p, a are the readings of P and A and angles are measured in a CCW sense from the positive X-axis by an observer receiving the radiation with respect to the transmission axes of the polarizing
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N.N. Nagib et al. / Optik 125 (2014) 1085–1087
Fig. 1. The ellipsometric system. S – laser source (632.8 nm), P and A – polarizing prisms, C – quarterwave phase plate and D – detector. The angle of incidence ϕ is 45◦ .
prisms and the fast axis of the phase plate. At extinction, the relations between ( , ) and (p, a) are = 2p1 − 90◦ = 2p2 − 270◦ , = a1 = −a2 .
Fig. 2. Variations of
and with the grain size of the polishing paste.
(3a) (3b)
The parameters and are related to the optical constants by the relations [7]
2
n =
k=
A ± (A2 + B2 ) 2
1/2
,
B 2
[A ± (A + B2 )
1/2
.
(4a)
(4b)
]
where, for the complex refractive index n, we used the formula n = n(1 − ik)
(5)
and A=
CE + DF , E2 + F 2
(6a)
B=
CF − DE , E2 + F 2
(6b)
C = 1 + tan2 cos2, 2
D = 2tan
sincos,
(6c) (6d)
E = 1 + 2tan cos + tan2 cos2,
(6e)
F = 2tan sin(1 + tan cos).
(6f)
We calculated the ellipsometric parameters and for each of our four samples and the optical constants n and k were then calculated. 4. Results and discussion Fig. 2 shows the variations of and for the samples of different surface treatments. Results indicate that as the sample surface becomes more smooth, both of and increase. This behaviour is the same as that observed in [4] for thin films of gold samples of different surface RMS values. However, variations of and with roughness are more appreciable in thin films. Results for n and k are presented in Fig. 3 and Table 1 where n decreased from 0.3047 to 0.2737 for samples polished with diamond pastes of reduced grain sizes from 10 to 1 m. The corresponding values for k increased from 7.8375 to 8.9160. Note that in some works, the expressions for the complex refractive index are considered as n = n − ik
or n = n + ik
(7)
Fig. 3. Variations of n and k with the grain size of the polishing paste.
Table 1 Optical constants of bulk polycrystalline gold samples subjected to different polishing treatments. Polish grain size, m
n
k
1 3 6 10
0.2737 0.2802 0.2907 0.3047
8.916 8.6281 8.2762 7.8375
Results in other works using these expressions are changed to agree with our formula when mentioned in the text. Extrapolating our results to the case of a perfect surface assuming that such surface corresponds to the intersections of n and k curves with the vertical axis, we found for the optical constants of gold at 632.8 nm the following results n = 0.2703,
(8a)
k = 9.0559.
(8b)
To compare our results with previous works on thin films at the same wavelength value, we list in Table 2 some of these results.
N.N. Nagib et al. / Optik 125 (2014) 1085–1087 Table 2 Values of n and k in this work and from other sources for gold at 632.8 nm. Sample type
n
k
Ref.
Bulk polycrystalline Thin filma Thin filma Thin film Thin filma Thin filmb
0.2703 0.20–0.55 0.20–0.26 0.1829 0.21–0.50 0.18
9.0559 4.5–17.5 12–16 18.5265 6.200–16.429 19.44
This work [4] [8] [9] [10] [11]
a Range of values for n and k corresponding to different thin film samples in the same work. b Values of n and k at 632.8 nm obtained by interpolation.
5. Conclusion We have studied the effect of surface roughness on the optical constants of four bulk polycrystalline gold samples subjected to different polishing treatments. Our results show that the effect of roughness on the ellipsometric parameters and and on the optical constants n and k is not strong as for thin films. The presented values for n and k (Eq. (8)) are believed to be highly representative for gold.
1087
References [1] D.E. Aspnes, E. Kinsborn, D.D. Bacon, Optical properties of Au: sample effects, Phys. Rev. 21 (8) (1980) 3290–3299. [2] C.A. Fenstermaker, F.L. McCrackin, Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface, Surf. Sci. 16 (1969) 85–96. [3] S.A. Kovalenko, R.D. Fedorovych, Optical properties of thin gold films, Semicond. Phys. Quantum Electron. Optoelectron. 3 (2000) 388. [4] V. Sirtori, L. Magagnin, E. Saglia, P.L. Cavallotti, Calculation model of rough gold optical constants, Surf. Sci. 554 (2004) 119–124. [5] G.D. Scott, Optical constants of thin-film materials, J. Opt. Soc. Am. 45 (3) (1955) 176–179. [6] R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam/New York/Oxford, 1977. [7] D. Goldstein, Polarized Light, Marcel Dekker, Inc., New York, NY, 2003. [8] T.A. McMath, Transmission measurement of the optical constants of thin solid Films, Simon Fraser Univ, Canada, 1978 (Ph.D. Thesis). [9] Y. Cui, R.M.A. Azzam, Applications of the normal-incidence rotating-sample ellipsometer to high-and-low-spatial-frequency gratings, Appl. Opt. 35 (13) (1996) 2235–2238. [10] I. Ohlidal, F. Lukes, Optical analysis of thin gold films by combined reflection and transmission ellipsometry, Thin Solid Films 85 (1981) 181–190. [11] P.B. Johnson, R.W. Christy, Optical constants of the noble metals, Phys. Rev. B 6 (12) (1972) 4370–4379.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Effect of surface roughness on the optical constants of bulk polycrystalline gold samples N.N. Nagib a,∗ , N.A. Mahmoud a , L.Z. Ismail b , M.A. Amer a , Kh. Abd-El-Sabour b a b
National Institute for Standards, Giza, Egypt Faculty of Science, Cairo University, Giza, Egypt
a r t i c l e
i n f o
Article history: Received 21 March 2013 Accepted 28 July 2013
Keywords: Bulk polycrystalline gold Surface roughness Optical constants
a b s t r a c t Four bulk polycrystalline samples of gold were subjected to different polishing treatments using diamond pastes of grain size 10, 6, 3 and 1 m. The effect of surface roughness on the optical constants n and k is studied by 45◦ angle-of-incidence ellipsometry at 632.8 nm. Results for n and k are extrapolated to the case of an ideal surface which we believe to be highly representative of gold. Comparison with published results for the optical constants of gold thin films is presented. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction The effect of surface roughness on the optical constants of metals was investigated by several authors (see for example [1–4]). Almost in all studies, measurements were performed on thin metal films. Several factors are then expected to affect the optical properties including surface roughness, method and conditions of deposition, film thickness, trapped contaminating molecules and thermal treatment after deposition. This explains the widely distinct published data for the optical constants of metals. More over, deposited films show evidence of aggregation and porosity which make their optical properties different from those of bulk materials [5]. Only a pure bulk sample of perfect smooth and clean surface is expected to show the intrinsic optical characteristics of the material which cannot be realized in practice. In this work, four samples of pure bulk polycrystalline gold subjected to different polishing treatments are studied by ellipsometry to show the effect of surface roughness on the optical constants n and k (refractive index and extinction coefficient). 2. Samples The investigated samples (S1, S2, S3 and S4) are in the form of gold sheets of purity 99.999%, thickness 0.05 mm and surface dimensions ∼30 mm × 30 mm. The samples were subjected to different polishing treatments using diamond pastes of different grain
∗ Corresponding author. E-mail address: nabil [email protected] (N.N. Nagib). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.07.127
sizes. The final polishing step for S1, . . ., S4 was with pastes of grain size 10, 6, 3 and 1 m respectively. The samples of different polishing treatments were then cleaned with a stream of distilled water and dried. 3. Experimental The fundamental equation of ellipsometry for a metal of refractive index n and extinction coefficient k is expressed as [6] = tan
ei = f (n, k)
(1)
where is the complex reflectance ratio, = |rp |/|rs | and = (ıp − ıs ) express the differential changes in amplitude and phase experienced upon reflection by the p and s components. The ellipsometric system used in the work is shown in Fig. 1. A laser beam S ( = 632.8 nm) falls successively on the polarizer P, a quartz quarterwave phase plate C oriented with its fast axis at 45◦ and the sample M. The angle of incidence ϕ is 45◦ . The reflected beam passes through the analyzer A and the detecting system D (photomultiplier, power supply and micro-voltmeter). By simultaneous adjustments of P and A, the reflected beam could be extinguished. Two extinction, pairs (p1 , a1 ) and (p2 , a2 ) could be recorded such that p2 = p1 ± 90◦ ,
(2a)
a2 = a1 ± 90◦ ,
(2b)
where p, a are the readings of P and A and angles are measured in a CCW sense from the positive X-axis by an observer receiving the radiation with respect to the transmission axes of the polarizing
1086
N.N. Nagib et al. / Optik 125 (2014) 1085–1087
Fig. 1. The ellipsometric system. S – laser source (632.8 nm), P and A – polarizing prisms, C – quarterwave phase plate and D – detector. The angle of incidence ϕ is 45◦ .
prisms and the fast axis of the phase plate. At extinction, the relations between ( , ) and (p, a) are = 2p1 − 90◦ = 2p2 − 270◦ , = a1 = −a2 .
Fig. 2. Variations of
and with the grain size of the polishing paste.
(3a) (3b)
The parameters and are related to the optical constants by the relations [7]
2
n =
k=
A ± (A2 + B2 ) 2
1/2
,
B 2
[A ± (A + B2 )
1/2
.
(4a)
(4b)
]
where, for the complex refractive index n, we used the formula n = n(1 − ik)
(5)
and A=
CE + DF , E2 + F 2
(6a)
B=
CF − DE , E2 + F 2
(6b)
C = 1 + tan2 cos2, 2
D = 2tan
sincos,
(6c) (6d)
E = 1 + 2tan cos + tan2 cos2,
(6e)
F = 2tan sin(1 + tan cos).
(6f)
We calculated the ellipsometric parameters and for each of our four samples and the optical constants n and k were then calculated. 4. Results and discussion Fig. 2 shows the variations of and for the samples of different surface treatments. Results indicate that as the sample surface becomes more smooth, both of and increase. This behaviour is the same as that observed in [4] for thin films of gold samples of different surface RMS values. However, variations of and with roughness are more appreciable in thin films. Results for n and k are presented in Fig. 3 and Table 1 where n decreased from 0.3047 to 0.2737 for samples polished with diamond pastes of reduced grain sizes from 10 to 1 m. The corresponding values for k increased from 7.8375 to 8.9160. Note that in some works, the expressions for the complex refractive index are considered as n = n − ik
or n = n + ik
(7)
Fig. 3. Variations of n and k with the grain size of the polishing paste.
Table 1 Optical constants of bulk polycrystalline gold samples subjected to different polishing treatments. Polish grain size, m
n
k
1 3 6 10
0.2737 0.2802 0.2907 0.3047
8.916 8.6281 8.2762 7.8375
Results in other works using these expressions are changed to agree with our formula when mentioned in the text. Extrapolating our results to the case of a perfect surface assuming that such surface corresponds to the intersections of n and k curves with the vertical axis, we found for the optical constants of gold at 632.8 nm the following results n = 0.2703,
(8a)
k = 9.0559.
(8b)
To compare our results with previous works on thin films at the same wavelength value, we list in Table 2 some of these results.
N.N. Nagib et al. / Optik 125 (2014) 1085–1087 Table 2 Values of n and k in this work and from other sources for gold at 632.8 nm. Sample type
n
k
Ref.
Bulk polycrystalline Thin filma Thin filma Thin film Thin filma Thin filmb
0.2703 0.20–0.55 0.20–0.26 0.1829 0.21–0.50 0.18
9.0559 4.5–17.5 12–16 18.5265 6.200–16.429 19.44
This work [4] [8] [9] [10] [11]
a Range of values for n and k corresponding to different thin film samples in the same work. b Values of n and k at 632.8 nm obtained by interpolation.
5. Conclusion We have studied the effect of surface roughness on the optical constants of four bulk polycrystalline gold samples subjected to different polishing treatments. Our results show that the effect of roughness on the ellipsometric parameters and and on the optical constants n and k is not strong as for thin films. The presented values for n and k (Eq. (8)) are believed to be highly representative for gold.
1087
References [1] D.E. Aspnes, E. Kinsborn, D.D. Bacon, Optical properties of Au: sample effects, Phys. Rev. 21 (8) (1980) 3290–3299. [2] C.A. Fenstermaker, F.L. McCrackin, Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface, Surf. Sci. 16 (1969) 85–96. [3] S.A. Kovalenko, R.D. Fedorovych, Optical properties of thin gold films, Semicond. Phys. Quantum Electron. Optoelectron. 3 (2000) 388. [4] V. Sirtori, L. Magagnin, E. Saglia, P.L. Cavallotti, Calculation model of rough gold optical constants, Surf. Sci. 554 (2004) 119–124. [5] G.D. Scott, Optical constants of thin-film materials, J. Opt. Soc. Am. 45 (3) (1955) 176–179. [6] R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam/New York/Oxford, 1977. [7] D. Goldstein, Polarized Light, Marcel Dekker, Inc., New York, NY, 2003. [8] T.A. McMath, Transmission measurement of the optical constants of thin solid Films, Simon Fraser Univ, Canada, 1978 (Ph.D. Thesis). [9] Y. Cui, R.M.A. Azzam, Applications of the normal-incidence rotating-sample ellipsometer to high-and-low-spatial-frequency gratings, Appl. Opt. 35 (13) (1996) 2235–2238. [10] I. Ohlidal, F. Lukes, Optical analysis of thin gold films by combined reflection and transmission ellipsometry, Thin Solid Films 85 (1981) 181–190. [11] P.B. Johnson, R.W. Christy, Optical constants of the noble metals, Phys. Rev. B 6 (12) (1972) 4370–4379.