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Reflective properties of nanoparticle-arrayed surfaces
Thin Solid Films 518 (2010) 6015–6021
Contents lists available at ScienceDirect
Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Reflective properties of nanoparticle-arrayed surfaces Bo-Tau Liu ⁎, Wei-De Yeh Department of Chemical and Materials Engineering, National Yunlin University of Science and Technology, Yunlin, Taiwan 64002
a r t i c l e
i n f o
Article history: Received 12 October 2009 Received in revised form 9 June 2010 Accepted 14 June 2010 Available online 21 June 2010 Keywords: Reflectance Rigorous coupled-wave analysis Effective medium theory Effective single layer
a b s t r a c t The reflective properties of nanostructured surfaces, fabricated by laying nanoparticles in a square arrangement on the substrate, were investigated theoretically. Maxwell's equations relating to the subwavelength structure were solved by a rigorous coupled-wave analysis (RCWA), and the results were compared with those of two approximation methods, i.e., the effective medium theory (EMT) and the effective single-layer model (ESL), which have been used extensively in the prior literature. The study presents an analysis of the reflectance characteristics subject to the influences of the size of the nanoparticles, their refractive index, and the surface nanoparticle density. The calculated results revealed that, disregarding variations in the refractive index of the nanoparticles and the surface nanoparticle density, a nanostructured surface with particles of ∼ 120 nm in diameter yields the optimal performance for antireflection with respect to the visible-light region. It was found that, for large particle sizes and refractive indices of the nanoparticles, the results calculated by EMT and ESL may result in considerable deviations relative to those calculated by RCWA and that regarding the nanoparticle-arrayed structure as a gradient refractive-index layer or as a homogeneous layer is inappropriate. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Surface reflection is a significant problem in applications such as liquid crystal displays, optical components, and solar cells [1] because it reduces the contrast and clarity of the image and the transmittance of the incident light. The antireflection technologies presented in the literature to date are found in two categories: the multilayer type, with varying refractive indices [2–4], and the nanostructured surface [5–11]. The former requires both low-refractive-index materials and high-refractive-index materials to produce interference in the light reflected from the interfaces between the differently refracting layers. In the case of a single-layer coating composed of a low-refractiveindex material, the reflectance is usually larger than 1.5% and limited to a narrow wavelength region. To produce broad-band antireflection with less than 1.0% reflectance, the substrate must be coated with several layers having different refractive indices. The antireflective performance usually rises with an increased number of layers, leading to time-consuming, laborious, and expensive fabrication. The antireflection of nanostructured surfaces has been found in the cornea of night-flying moths, and so is called the “moth-eye” effect [12]. This characteristic is also found in other insects, such as the compound eye of a fly [9]. Such surfaces can be made by phaseseparation [5,6], reaction-patterning [7,8], and lithography techniques [11]. The phase-separation and reaction-pattern methods require the
⁎ Corresponding author. Tel.: +886 5 534 2601; fax: +886 5 531 2071. E-mail address: [email protected] (B.-T. Liu). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.06.044
Fig. 1. Schematic representation of the identical spherical particles with diameter D arrayed regularly on the substrate for the problem under consideration. (a) Top view, (b) cross-sectional view, and (c) effective layer stack for EMT.
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B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
use of solvents to remove the soluble resins or unreactive monomers, resulting in a limited range of applicable materials. The pattern quality, reproducibility, and pollution may be significant issues in the washing process. The lithography method is suitable only for small substrates and is expensive. Recently, nanostructured surfaces have been fabricated from nanoparticles through the electrostatic attraction between charged colloidal particles and charged polyelectrolyte multilayers [13–18]; this method is applicable for large and/or curved substrates and can be produced by an inexpensive roll-to-roll method. The optical analysis of multilayer structures with varying refractive-index materials has been well developed by the calculation of the recursive Fresnel formula or the admittance method [19]. Compared with relevant analyses on multilayer structures, those on nanostructured surfaces are very limited, especially for nanoparticle-arrayed surfaces. In the literature to date, although the antireflection of arraystructured surfaces has been analyzed by grating theory [20–23], that of nanoparticle surfaces has not yet been analyzed theoretically, and the nanoparticle layer is usually regarded as a homogenous layer [24]. Considering the need for gathering basic information on antireflection in nanoparticle-arrayed surfaces for practical applications,
performing a systematic analysis is highly desirable. In this study, the effects of the size of the nanoparticles, the surface nanoparticle density, and the refractive index of the nanoparticles on the antireflection of the nanoparticle-arrayed surface were analyzed. An attempt was made to find the optimal conditions with relation to antireflection performance. The validities of the assumptions of a homogenous layer and a gradient-index layer were also evaluated. 2. Analysis To simplify the present analysis, we considered identical spherical particles with diameter D arrayed on the substrate as illustrated in Fig. 1. Here, a is the period for the square arrangement, and n0, ns, and np are the refractive indices of the free space, the substrate and the nanoparticles, respectively, and Λ denotes the ratio of the particle diameter to the period, where Λ = 1 indicates that the nanoparticles are touching each other. The imaginary parts of the refractive indices, i.e., the extinction coefficients, were neglected in the simulation, as, for most optical materials, they are relatively small in the wavelength range of visible light (400–800 nm). The refractive indices were assumed to be constants in our simulation because the refractive
Fig. 2. Variation of the reflectance versus the incident wavelength for various combinations of Λ and D at np = 1.41. (a) D = 80 nm; (b) D = 100 nm; (c) D = 120 nm; (d) D = 140 nm.
B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
indices for the optical materials in common use are nearly constant over the wavelength range of interest in this study, 400–800 nm. For instance, the refractive index of silicone dioxide is ∼1.46–1.48 over the visible-light region [25]. The incident angle was set at zero for the present study, leading to the same results for both transverse-electric and transverse-magnetic polarizations. 2.1. Rigorous coupled-wave analysis The present simulation is based on the rigorous coupled-wave analysis (RCWA), which analyzes the diffraction of planar waves on a periodic grating structure [26]. The surface-relief region is characterized by a relative permittivity, expanded in a sum of spatial harmonics, i.e., a Fourier expansion. A set of coupled ordinary differential equations can be derived from Maxwell's equations. The number of coupled equations is dependent on the spatial harmonics; as the number of spatial harmonics is increased, the diffraction efficiencies through RCWA will converge to the exact solution. These results can approach an arbitrary level of accuracy as no extra approximations are made to Maxwell's equations. The number of
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spatial harmonics and the refractive-index resolution used in this study were sufficient to control the simulation to within an error of 5%. In this study, all of order diffracted fields were included in the RCWA calculation. In other words, the light scattering effect was also considered in the calculation. If only the zero-order diffracted field is to propagate and the higher-order diffracted fields are forced to be evanescent, the limitation on the grating period for operation at all angles are expressed by Λ 1 b : λ maxðns ; ni Þ + ni
ð1Þ
When D and/or Λ is increased, the effect of the higher-order diffracted fields becomes important; the more spatial harmonics and computing time are needed. For most cases in this study, the higher-order effect is not significant due to the prevention of the light scattering in the practical application. However, the RCWA model is not limited to only the zero-order diffracted field, i.e., no light scattering.
Fig. 3. Variation of the reflectance versus the incident wavelength for various combinations of Λ and D, for the case of Fig. 2 except that here np = 1.46.
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2.2. Effective medium theory
In this expression, the filling factor f is given by
Although RCWA is valid for the periodic surface-relief structure, it requires considerable computer time to determine reflected and transmitted power. If a surface-relief structure does not lose energy to diffracted orders, the interaction of light with the subwavelength structures can be described by a non-structured layer with a gradientindex profile, in which the refractive index varies with profile depth, in a model known as effective medium theory (EMT). For 1-D grating surfaces with subwavelength surface periodicity, a closed-form integral solution of the field reflection coefficients can be found by the application of EMT. However, a closed-form solution is not found for a 2-D subwavelength grating structure. If the ratio of period to incident wavelength is small, EMT can be formulated by making a zero-order or a static approximation solution that assumes that electromagnetic fields do not vary spatially within a given material. The effective index is given by the expression [27]
f = w= a
2
2
2
ne = fnp + ð1−f Þn0 :
ð2Þ
2
ð3Þ
where w is the cross-sectional area of a particle at a height z above the substrate. The reflection coefficient for the effective inhomogeneous layer can be calculated by subdividing the layer into N plane-parallel homogeneous sub-layers, as shown in Fig. 1(c). These sub-layers are perpendicular to the z-axis and are numbered from the ambient to the substrate. Each sub-layer has the same thickness, d, and the refractive index of the sub-layer i is denoted by ni, where i = 1, 2, …, N. The reflectance of the sub-layer stack can be calculated exactly by making use of the recursive formula [19] ρi =
ri + ρi + 1 expðj2βi + 1 Þ 1 + ri ρi + 1 expð j2βi + 1 Þ
ð4Þ
where ρi denotes the total reflection coefficient of the ith sub-layer and all the sub-layers beneath it, ri is the Fresnel refraction coefficient
Fig. 4. Variation of the reflectance versus the incident wavelength for various combinations of Λ and D, for the case of Fig. 2 except that here np = 1.59.
B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
for the interface between the ith and i + 1th sub-layers, j = the propagation function through the ith layer is βi =
2πni d cos θi : λ
pffiffiffiffiffiffiffiffi −1, and
ð5Þ
In this expression, λ is the incident wavelength and θi is the incident angle at the ith sub-layer, determined by Snell's law. The reflective properties based on EMT can be obtained by the solution of Eq. (4) subject to Eq. (2). 2.3. Effective single layer In the literature, a nanoparticle coating usually is regarded as a porous homogeneous layer [24], in which the nanostructure is not taken into consideration. The refractive index of the effective single layer (ESL) can be expressed as 2
2
2
nes = Fnp + ð1−FÞn0 ;
ð6Þ
where F is the volume fraction of the nanoparticles in the layer with thickness d.
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decrease of λm. This indicates that, as seen in Figs. 2–4, the wavelength where the minimal Rm occurs is independent of the refractive index of the nanoparticles and dependent only on the size of nanoparticles. For the particle diameter of 100 nm, for example, the minimal Rm occurs at about the incident wavelength of 450–500 nm for np = 1.41, 1.46, and 1.59. Therefore, the responding incident wavelength to the minimal Rm mainly depends on the size of the nanoparticles. We can observe in Figs. 2–4 that for nanoparticle-arrayed surface the particle size for the optimum antireflection in the visible-light range is ∼120 nm, resulting in the minimal reflectance approaching zero at the incident wavelength of ∼ 550–600 nm. The same result is also observed in the disorder single-layer nanoparticle coating [29]. The deviation of the reflectance calculated from EMT or ESL for different Λ, D, and np was evaluated according to the following equation: Error = Rapp −Racc
ð7Þ
where Rapp is the approximate reflectance calculated from EMT or ESL, and Racc denotes the accurate reflectance simulated from RCWA. Figs. 5 and 6 show the errors of the reflectance with respect to the
3. Results and discussion For illustration, we assume that ns = 1.5 and N = 100 in the subsequent discussion. The 100 sub-layers for EMT are sufficient to assure convergence to the desired accuracy. Figs. 2–4 show the simulated variation of the reflectance, R, of the nanoparticle-arrayed surface calculated by RCWA as a function of the incident wavelength, λ. These figures reveal that for fixed values of Λ , the wavelength where the minimum reflectance occurs, λm, moves from a short wavelength to a longer wavelength with an increase of the particle size, D, and/or the refractive index of the particle, np. This result is similar to that of a homogeneous single-layer coating, in which the value of λm increases with an increase of the thickness and/ or of the refractive index of the coating film. There are many kinds of materials that have been used for nanoparticles applied to an antireflection structure, such as porous materials, silicon dioxide, polystyrene, etc. To cover the variety of nanoparticles in our simulation, the choices of np = 1.41, 1.46, and 1.59, respectively, represent low, medium, and high refractive indices of the nanoparticles simulated in Figs. 2–4. Referring to these figures, λm, for all three nanoparticle refractive indices, shifted from a long wavelength to a short wavelength with the increase of Λ. For np = 1.41, as shown in Fig. 2, a smaller Λ yields a lower minimum reflectance, Rm, that is, the minimum reflectance can be reduced by raising the surface nanoparticle density. However, the situation was quite different for np = 1.46 and 1.59, shown in Figs. 3 and 4. These figures reveal that the minimum reflectance decreased with an increase in Λ , reached a minimum, and then increased with a further increase in Λ . This may explain why the reflectance of an amorphous monolayer (loose packing), formed from silica particles, is lower than that of a crystalline monolayer (close packing), as found in the work of Dimitrov et al. [28], and why the transmittance rose when reducing the poly(methyl methacrylate) (PMMA) particle density on the surface of the substrate, approached a maximum, and decreased with a further reduction of the PMMA particle density, as revealed in the results of Jiang et al. [16]. Similar results are also found in our recent experiments for the disorder single-layer nanoparticle coating [29]. Hence, the optimal density of nanoparticles on the surface of the substrate corresponding to a minimal reflectance, as related to Λ , depends on the refractive index of the nanoparticles. Although λm increased with the rise in np, referring to the aforementioned discussion, the value of Λ for the minimal Rm also increased with the rise of np. The increase of Λ also results in the
Fig. 5. Variation of the deviation of the reflectance calculated by EMT and ESL, Error, as a function of the incident wavelength for various combinations of Λ and D at np = 1.41. ——: EMT; —: ESL. (a) D = 80 nm; (b) D = 140 nm.
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B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021 Table 1 The minimum reflectance and the corresponding wavelength calculated from RCWA, EMT, and ESL. np
1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59
D, nm
80 80 80 80 80 140 140 140 140 140 80 80 80 80 80 140 140 140 140 140
Λ
1.0 1.2 1.4 1.6 2.0 1.0 1.2 1.4 1.6 2.0 1.0 1.2 1.4 1.6 2.0 1.0 1.2 1.4 1.6 2.0
ne
1.23 1.14 1.09 1.06 1.03 1.23 1.14 1.09 1.06 1.03 1.34 1.21 1.14 1.09 1.05 1.34 1.21 1.14 1.09 1.05
Rm (%)@λm (μm) RCWA
EMT
ESL
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
antireflective tapered two-dimensional subwavelength structures [23] analyzed by EMT. 4. Summary
Fig. 6. Variation of the deviation of the reflectance calculated by EMT and ESL, Error, as a function of the incident wavelength for various combinations of Λ and D, for the case of Fig. 5 except that here np = 1.59.
surface nanoparticle density over the incident wavelength range from 400 to 800 nm for the cases where the refractive index and the size of the nanoparticles were 1.41 and 80 nm (141/80), 1.41 and 140 nm (141/140), 1.59 and 80 nm (159/80), and 1.59 and 140 nm (159/140), respectively. The results indicate that, with decreasing surface nanoparticle density, roughly speaking, the deviation of the reflectance as calculated by EMT or ESL trends from a positive value to a negative one. Referring to Figs. 5 and 6 for the cases of 141/80, 141/140, 159/80, and 159/140, the maximum values of the deviations in the reflectance calculated by EMT are around 0.004, 0.006, 0.01, and 0.015, respectively, and by ESL are 0.0025, 0.01, 0.005, and 0.03, respectively. In comparing the approximate methods to each other, the results calculated by ESL are more accurate, basically, than those calculated by EMT. The results are also observed in the minimum reflectance shown in Table 1. Notice that the deviation of reflection calculated by EMT or ESL increases both with a higher refractive index and a larger size of the nanoparticles. In other words, EMT and ESL are more appropriate for the cases of lower nanoparticle refractive indices and smaller sizes, which is similar to the results for the antireflective twodimensional subwavelength binary grating structures [22] and the
In this study, we examined the reflective properties of nanoparticle-arrayed surfaces using RCWA. The effects of the particle size, the refractive index of the nanoparticles, and the surface nanoparticle density on the reflectance were investigated. We show that, disregarding variations in the refractive index of the nanoparticles and surface nanoparticle density, the optimal performance for antireflection over the visible-light region occurs for a nanostructured surface with nanoparticles of about 120 nm in diameter. In comparison with the approximation methods, EMT and ESL, the results calculated by ESL are more accurate than those calculated by EMT. The calculated results indicate that, with decreasing surface nanoparticle density, the deviation of the reflectance as calculated by EMT or ESL trends from a positive value to a negative one. Acknowledgments This work was partially financially supported by the National Science Council of the Republic of China (NSC 97-2221-E-224-022) and the Ministry of Economic Affairs of Taiwan (97-EC-17-A-08-S1015). References [1] Y.J. Lee, D.S. Ruby, D.W. Peters, B.B. McKenzie, J.W.P. Hsu, Nano Lett. 8 (2008) 1501. [2] K. Takematsu, H. Katagiri, U.S. Patent No. 6207263, 27 Mar. 2001. [3] T. Taruishi, U.S. Patent No. 6649271, 26 Jul. 2003. [4] S. Shoshi, Y. Onozawa, S. Maruoka, Y. Takesako, U.S. 6841272, 1 Nov. 2005. [5] S. Walheim, E. Schaffer, J. Mlynek, U. Steiner, Science 283 (1999) 520. [6] U. Steiner, S. Walheim, E. Schaffer, S. Eggert, J. Mlynek, US 6605229, 8 Dec. 2003. [7] M. Ibn-Elhaj, M. Schadt, Nature 410 (2001) 796. [8] H. Seiberle, M. Schadt, M. Ibn-Elhaj, C. Benecke, K. Schmitt, W.O. Patent No. 01/29148, 26 Apr. 2001. [9] J. Huang, X. Wang, Z.L. Wang, Nanotechnology 19 (2008) 025602. [10] W. Zhou, M. Tao, L. Chen, H. Yang, J. Appl. Phys. 102 (2007) 103105. [11] T. Glaser, A. Ihring, W. Morgenroth, N. Seifert, S. Schroter, V. Baier, Microsyst. Technol. 11 (2005) 86. [12] C.G. Bernhard, Endeavour 26 (1967) 79. [13] H. Hattori, Adv. Mater. 13 (2001) 51. [14] G. Decher, Science 277 (1997) 29. [15] X. Liu, H. He, J. Colloid Interface Sci. 314 (2007) 341.
B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021 [16] H. Jiang, K. Yu, Y. Wang, Opt. Lett. 32 (2007) 575. [17] J.S. Ahn, P.T. Hammond, M.F. Rubner, I. Lee, Colloids Surf. A 259 (2005) 45. [18] X.T. Zhang, O. Sato, M. Taguchi, Y. Einaga, T. Murakami, A. Fujishima, Chem. Mater. 17 (2005) 696. [19] H.A. Macleod, Thin Film Optical Filters, Vol. 3, Institute of Physics Pub., Bristol and Philadelphia, 2001. [20] D.H. Raguin, M. Morris, Appl. Opt. 32 (1993) 1154. [21] D.H. Raguin, M. Morris, Appl. Opt. 32 (1993) 2582.
[22] [23] [24] [25] [26] [27] [28] [29]
E.B. Grann, M.G. Moharam, D.A. Poment, J. Opt. Soc. Am. A 11 (1994) 2695. E.B. Grann, M.G. Moharam, D.A. Poment, J. Opt. Soc. Am. A 12 (1995) 333. Y. Zhao, G. Mao, J. Wang, Opt. Lett. 30 (2005) 1885. M.J. Weber, Handbook of Optical Materials, CRC Press, Cleveland, 2003. T.K. Gaylord, M.G. Moharam, Proc. IEEE 73 (1985) 894. W.H. Southwell, J. Opt. Soc. Am. A 8 (1991) 549. A.S. Dimitrov, T. Miwa, K. Nagayama, Langmuir 15 (1999) 5257. B.T. Liu, W.D. Yeh, Colloids Surf. A 356 (2010) 145.
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Contents lists available at ScienceDirect
Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Reflective properties of nanoparticle-arrayed surfaces Bo-Tau Liu ⁎, Wei-De Yeh Department of Chemical and Materials Engineering, National Yunlin University of Science and Technology, Yunlin, Taiwan 64002
a r t i c l e
i n f o
Article history: Received 12 October 2009 Received in revised form 9 June 2010 Accepted 14 June 2010 Available online 21 June 2010 Keywords: Reflectance Rigorous coupled-wave analysis Effective medium theory Effective single layer
a b s t r a c t The reflective properties of nanostructured surfaces, fabricated by laying nanoparticles in a square arrangement on the substrate, were investigated theoretically. Maxwell's equations relating to the subwavelength structure were solved by a rigorous coupled-wave analysis (RCWA), and the results were compared with those of two approximation methods, i.e., the effective medium theory (EMT) and the effective single-layer model (ESL), which have been used extensively in the prior literature. The study presents an analysis of the reflectance characteristics subject to the influences of the size of the nanoparticles, their refractive index, and the surface nanoparticle density. The calculated results revealed that, disregarding variations in the refractive index of the nanoparticles and the surface nanoparticle density, a nanostructured surface with particles of ∼ 120 nm in diameter yields the optimal performance for antireflection with respect to the visible-light region. It was found that, for large particle sizes and refractive indices of the nanoparticles, the results calculated by EMT and ESL may result in considerable deviations relative to those calculated by RCWA and that regarding the nanoparticle-arrayed structure as a gradient refractive-index layer or as a homogeneous layer is inappropriate. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Surface reflection is a significant problem in applications such as liquid crystal displays, optical components, and solar cells [1] because it reduces the contrast and clarity of the image and the transmittance of the incident light. The antireflection technologies presented in the literature to date are found in two categories: the multilayer type, with varying refractive indices [2–4], and the nanostructured surface [5–11]. The former requires both low-refractive-index materials and high-refractive-index materials to produce interference in the light reflected from the interfaces between the differently refracting layers. In the case of a single-layer coating composed of a low-refractiveindex material, the reflectance is usually larger than 1.5% and limited to a narrow wavelength region. To produce broad-band antireflection with less than 1.0% reflectance, the substrate must be coated with several layers having different refractive indices. The antireflective performance usually rises with an increased number of layers, leading to time-consuming, laborious, and expensive fabrication. The antireflection of nanostructured surfaces has been found in the cornea of night-flying moths, and so is called the “moth-eye” effect [12]. This characteristic is also found in other insects, such as the compound eye of a fly [9]. Such surfaces can be made by phaseseparation [5,6], reaction-patterning [7,8], and lithography techniques [11]. The phase-separation and reaction-pattern methods require the
⁎ Corresponding author. Tel.: +886 5 534 2601; fax: +886 5 531 2071. E-mail address: [email protected] (B.-T. Liu). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.06.044
Fig. 1. Schematic representation of the identical spherical particles with diameter D arrayed regularly on the substrate for the problem under consideration. (a) Top view, (b) cross-sectional view, and (c) effective layer stack for EMT.
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B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
use of solvents to remove the soluble resins or unreactive monomers, resulting in a limited range of applicable materials. The pattern quality, reproducibility, and pollution may be significant issues in the washing process. The lithography method is suitable only for small substrates and is expensive. Recently, nanostructured surfaces have been fabricated from nanoparticles through the electrostatic attraction between charged colloidal particles and charged polyelectrolyte multilayers [13–18]; this method is applicable for large and/or curved substrates and can be produced by an inexpensive roll-to-roll method. The optical analysis of multilayer structures with varying refractive-index materials has been well developed by the calculation of the recursive Fresnel formula or the admittance method [19]. Compared with relevant analyses on multilayer structures, those on nanostructured surfaces are very limited, especially for nanoparticle-arrayed surfaces. In the literature to date, although the antireflection of arraystructured surfaces has been analyzed by grating theory [20–23], that of nanoparticle surfaces has not yet been analyzed theoretically, and the nanoparticle layer is usually regarded as a homogenous layer [24]. Considering the need for gathering basic information on antireflection in nanoparticle-arrayed surfaces for practical applications,
performing a systematic analysis is highly desirable. In this study, the effects of the size of the nanoparticles, the surface nanoparticle density, and the refractive index of the nanoparticles on the antireflection of the nanoparticle-arrayed surface were analyzed. An attempt was made to find the optimal conditions with relation to antireflection performance. The validities of the assumptions of a homogenous layer and a gradient-index layer were also evaluated. 2. Analysis To simplify the present analysis, we considered identical spherical particles with diameter D arrayed on the substrate as illustrated in Fig. 1. Here, a is the period for the square arrangement, and n0, ns, and np are the refractive indices of the free space, the substrate and the nanoparticles, respectively, and Λ denotes the ratio of the particle diameter to the period, where Λ = 1 indicates that the nanoparticles are touching each other. The imaginary parts of the refractive indices, i.e., the extinction coefficients, were neglected in the simulation, as, for most optical materials, they are relatively small in the wavelength range of visible light (400–800 nm). The refractive indices were assumed to be constants in our simulation because the refractive
Fig. 2. Variation of the reflectance versus the incident wavelength for various combinations of Λ and D at np = 1.41. (a) D = 80 nm; (b) D = 100 nm; (c) D = 120 nm; (d) D = 140 nm.
B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
indices for the optical materials in common use are nearly constant over the wavelength range of interest in this study, 400–800 nm. For instance, the refractive index of silicone dioxide is ∼1.46–1.48 over the visible-light region [25]. The incident angle was set at zero for the present study, leading to the same results for both transverse-electric and transverse-magnetic polarizations. 2.1. Rigorous coupled-wave analysis The present simulation is based on the rigorous coupled-wave analysis (RCWA), which analyzes the diffraction of planar waves on a periodic grating structure [26]. The surface-relief region is characterized by a relative permittivity, expanded in a sum of spatial harmonics, i.e., a Fourier expansion. A set of coupled ordinary differential equations can be derived from Maxwell's equations. The number of coupled equations is dependent on the spatial harmonics; as the number of spatial harmonics is increased, the diffraction efficiencies through RCWA will converge to the exact solution. These results can approach an arbitrary level of accuracy as no extra approximations are made to Maxwell's equations. The number of
6017
spatial harmonics and the refractive-index resolution used in this study were sufficient to control the simulation to within an error of 5%. In this study, all of order diffracted fields were included in the RCWA calculation. In other words, the light scattering effect was also considered in the calculation. If only the zero-order diffracted field is to propagate and the higher-order diffracted fields are forced to be evanescent, the limitation on the grating period for operation at all angles are expressed by Λ 1 b : λ maxðns ; ni Þ + ni
ð1Þ
When D and/or Λ is increased, the effect of the higher-order diffracted fields becomes important; the more spatial harmonics and computing time are needed. For most cases in this study, the higher-order effect is not significant due to the prevention of the light scattering in the practical application. However, the RCWA model is not limited to only the zero-order diffracted field, i.e., no light scattering.
Fig. 3. Variation of the reflectance versus the incident wavelength for various combinations of Λ and D, for the case of Fig. 2 except that here np = 1.46.
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B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
2.2. Effective medium theory
In this expression, the filling factor f is given by
Although RCWA is valid for the periodic surface-relief structure, it requires considerable computer time to determine reflected and transmitted power. If a surface-relief structure does not lose energy to diffracted orders, the interaction of light with the subwavelength structures can be described by a non-structured layer with a gradientindex profile, in which the refractive index varies with profile depth, in a model known as effective medium theory (EMT). For 1-D grating surfaces with subwavelength surface periodicity, a closed-form integral solution of the field reflection coefficients can be found by the application of EMT. However, a closed-form solution is not found for a 2-D subwavelength grating structure. If the ratio of period to incident wavelength is small, EMT can be formulated by making a zero-order or a static approximation solution that assumes that electromagnetic fields do not vary spatially within a given material. The effective index is given by the expression [27]
f = w= a
2
2
2
ne = fnp + ð1−f Þn0 :
ð2Þ
2
ð3Þ
where w is the cross-sectional area of a particle at a height z above the substrate. The reflection coefficient for the effective inhomogeneous layer can be calculated by subdividing the layer into N plane-parallel homogeneous sub-layers, as shown in Fig. 1(c). These sub-layers are perpendicular to the z-axis and are numbered from the ambient to the substrate. Each sub-layer has the same thickness, d, and the refractive index of the sub-layer i is denoted by ni, where i = 1, 2, …, N. The reflectance of the sub-layer stack can be calculated exactly by making use of the recursive formula [19] ρi =
ri + ρi + 1 expðj2βi + 1 Þ 1 + ri ρi + 1 expð j2βi + 1 Þ
ð4Þ
where ρi denotes the total reflection coefficient of the ith sub-layer and all the sub-layers beneath it, ri is the Fresnel refraction coefficient
Fig. 4. Variation of the reflectance versus the incident wavelength for various combinations of Λ and D, for the case of Fig. 2 except that here np = 1.59.
B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021
for the interface between the ith and i + 1th sub-layers, j = the propagation function through the ith layer is βi =
2πni d cos θi : λ
pffiffiffiffiffiffiffiffi −1, and
ð5Þ
In this expression, λ is the incident wavelength and θi is the incident angle at the ith sub-layer, determined by Snell's law. The reflective properties based on EMT can be obtained by the solution of Eq. (4) subject to Eq. (2). 2.3. Effective single layer In the literature, a nanoparticle coating usually is regarded as a porous homogeneous layer [24], in which the nanostructure is not taken into consideration. The refractive index of the effective single layer (ESL) can be expressed as 2
2
2
nes = Fnp + ð1−FÞn0 ;
ð6Þ
where F is the volume fraction of the nanoparticles in the layer with thickness d.
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decrease of λm. This indicates that, as seen in Figs. 2–4, the wavelength where the minimal Rm occurs is independent of the refractive index of the nanoparticles and dependent only on the size of nanoparticles. For the particle diameter of 100 nm, for example, the minimal Rm occurs at about the incident wavelength of 450–500 nm for np = 1.41, 1.46, and 1.59. Therefore, the responding incident wavelength to the minimal Rm mainly depends on the size of the nanoparticles. We can observe in Figs. 2–4 that for nanoparticle-arrayed surface the particle size for the optimum antireflection in the visible-light range is ∼120 nm, resulting in the minimal reflectance approaching zero at the incident wavelength of ∼ 550–600 nm. The same result is also observed in the disorder single-layer nanoparticle coating [29]. The deviation of the reflectance calculated from EMT or ESL for different Λ, D, and np was evaluated according to the following equation: Error = Rapp −Racc
ð7Þ
where Rapp is the approximate reflectance calculated from EMT or ESL, and Racc denotes the accurate reflectance simulated from RCWA. Figs. 5 and 6 show the errors of the reflectance with respect to the
3. Results and discussion For illustration, we assume that ns = 1.5 and N = 100 in the subsequent discussion. The 100 sub-layers for EMT are sufficient to assure convergence to the desired accuracy. Figs. 2–4 show the simulated variation of the reflectance, R, of the nanoparticle-arrayed surface calculated by RCWA as a function of the incident wavelength, λ. These figures reveal that for fixed values of Λ , the wavelength where the minimum reflectance occurs, λm, moves from a short wavelength to a longer wavelength with an increase of the particle size, D, and/or the refractive index of the particle, np. This result is similar to that of a homogeneous single-layer coating, in which the value of λm increases with an increase of the thickness and/ or of the refractive index of the coating film. There are many kinds of materials that have been used for nanoparticles applied to an antireflection structure, such as porous materials, silicon dioxide, polystyrene, etc. To cover the variety of nanoparticles in our simulation, the choices of np = 1.41, 1.46, and 1.59, respectively, represent low, medium, and high refractive indices of the nanoparticles simulated in Figs. 2–4. Referring to these figures, λm, for all three nanoparticle refractive indices, shifted from a long wavelength to a short wavelength with the increase of Λ. For np = 1.41, as shown in Fig. 2, a smaller Λ yields a lower minimum reflectance, Rm, that is, the minimum reflectance can be reduced by raising the surface nanoparticle density. However, the situation was quite different for np = 1.46 and 1.59, shown in Figs. 3 and 4. These figures reveal that the minimum reflectance decreased with an increase in Λ , reached a minimum, and then increased with a further increase in Λ . This may explain why the reflectance of an amorphous monolayer (loose packing), formed from silica particles, is lower than that of a crystalline monolayer (close packing), as found in the work of Dimitrov et al. [28], and why the transmittance rose when reducing the poly(methyl methacrylate) (PMMA) particle density on the surface of the substrate, approached a maximum, and decreased with a further reduction of the PMMA particle density, as revealed in the results of Jiang et al. [16]. Similar results are also found in our recent experiments for the disorder single-layer nanoparticle coating [29]. Hence, the optimal density of nanoparticles on the surface of the substrate corresponding to a minimal reflectance, as related to Λ , depends on the refractive index of the nanoparticles. Although λm increased with the rise in np, referring to the aforementioned discussion, the value of Λ for the minimal Rm also increased with the rise of np. The increase of Λ also results in the
Fig. 5. Variation of the deviation of the reflectance calculated by EMT and ESL, Error, as a function of the incident wavelength for various combinations of Λ and D at np = 1.41. ——: EMT; —: ESL. (a) D = 80 nm; (b) D = 140 nm.
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B.-T. Liu, W.-D. Yeh / Thin Solid Films 518 (2010) 6015–6021 Table 1 The minimum reflectance and the corresponding wavelength calculated from RCWA, EMT, and ESL. np
1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59
D, nm
80 80 80 80 80 140 140 140 140 140 80 80 80 80 80 140 140 140 140 140
Λ
1.0 1.2 1.4 1.6 2.0 1.0 1.2 1.4 1.6 2.0 1.0 1.2 1.4 1.6 2.0 1.0 1.2 1.4 1.6 2.0
ne
1.23 1.14 1.09 1.06 1.03 1.23 1.14 1.09 1.06 1.03 1.34 1.21 1.14 1.09 1.05 1.34 1.21 1.14 1.09 1.05
Rm (%)@λm (μm) RCWA
EMT
ESL
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
antireflective tapered two-dimensional subwavelength structures [23] analyzed by EMT. 4. Summary
Fig. 6. Variation of the deviation of the reflectance calculated by EMT and ESL, Error, as a function of the incident wavelength for various combinations of Λ and D, for the case of Fig. 5 except that here np = 1.59.
surface nanoparticle density over the incident wavelength range from 400 to 800 nm for the cases where the refractive index and the size of the nanoparticles were 1.41 and 80 nm (141/80), 1.41 and 140 nm (141/140), 1.59 and 80 nm (159/80), and 1.59 and 140 nm (159/140), respectively. The results indicate that, with decreasing surface nanoparticle density, roughly speaking, the deviation of the reflectance as calculated by EMT or ESL trends from a positive value to a negative one. Referring to Figs. 5 and 6 for the cases of 141/80, 141/140, 159/80, and 159/140, the maximum values of the deviations in the reflectance calculated by EMT are around 0.004, 0.006, 0.01, and 0.015, respectively, and by ESL are 0.0025, 0.01, 0.005, and 0.03, respectively. In comparing the approximate methods to each other, the results calculated by ESL are more accurate, basically, than those calculated by EMT. The results are also observed in the minimum reflectance shown in Table 1. Notice that the deviation of reflection calculated by EMT or ESL increases both with a higher refractive index and a larger size of the nanoparticles. In other words, EMT and ESL are more appropriate for the cases of lower nanoparticle refractive indices and smaller sizes, which is similar to the results for the antireflective twodimensional subwavelength binary grating structures [22] and the
In this study, we examined the reflective properties of nanoparticle-arrayed surfaces using RCWA. The effects of the particle size, the refractive index of the nanoparticles, and the surface nanoparticle density on the reflectance were investigated. We show that, disregarding variations in the refractive index of the nanoparticles and surface nanoparticle density, the optimal performance for antireflection over the visible-light region occurs for a nanostructured surface with nanoparticles of about 120 nm in diameter. In comparison with the approximation methods, EMT and ESL, the results calculated by ESL are more accurate than those calculated by EMT. The calculated results indicate that, with decreasing surface nanoparticle density, the deviation of the reflectance as calculated by EMT or ESL trends from a positive value to a negative one. Acknowledgments This work was partially financially supported by the National Science Council of the Republic of China (NSC 97-2221-E-224-022) and the Ministry of Economic Affairs of Taiwan (97-EC-17-A-08-S1015). References [1] Y.J. Lee, D.S. Ruby, D.W. Peters, B.B. McKenzie, J.W.P. Hsu, Nano Lett. 8 (2008) 1501. [2] K. Takematsu, H. Katagiri, U.S. Patent No. 6207263, 27 Mar. 2001. [3] T. Taruishi, U.S. Patent No. 6649271, 26 Jul. 2003. [4] S. Shoshi, Y. Onozawa, S. Maruoka, Y. Takesako, U.S. 6841272, 1 Nov. 2005. [5] S. Walheim, E. Schaffer, J. Mlynek, U. Steiner, Science 283 (1999) 520. [6] U. Steiner, S. Walheim, E. Schaffer, S. Eggert, J. Mlynek, US 6605229, 8 Dec. 2003. [7] M. Ibn-Elhaj, M. Schadt, Nature 410 (2001) 796. [8] H. Seiberle, M. Schadt, M. Ibn-Elhaj, C. Benecke, K. Schmitt, W.O. Patent No. 01/29148, 26 Apr. 2001. [9] J. Huang, X. Wang, Z.L. Wang, Nanotechnology 19 (2008) 025602. [10] W. Zhou, M. Tao, L. Chen, H. Yang, J. Appl. Phys. 102 (2007) 103105. [11] T. Glaser, A. Ihring, W. Morgenroth, N. Seifert, S. Schroter, V. Baier, Microsyst. Technol. 11 (2005) 86. [12] C.G. Bernhard, Endeavour 26 (1967) 79. [13] H. Hattori, Adv. Mater. 13 (2001) 51. [14] G. Decher, Science 277 (1997) 29. [15] X. Liu, H. He, J. Colloid Interface Sci. 314 (2007) 341.
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