Composite medium with silver nanoparticles as an anti-reflection optical coating

Appl Phys A (2011) 103: 619–622 DOI 10.1007/s00339-010-6193-z Composite medium with silver nanoparticles as an anti-reflection optical coating Sergey G. Moiseev Received: 19 January 2010 / Accepted: 3 December 2010 / Published online: 29 December 2010 © Springer-Verlag 2010 Abstract It is theoretically shown that a thin composite slab comprised of a transparent host and uniformly oriented disclike silver nanoparticles deposited onto a glass surface can act as an anti-reflection coating. The applicability of the effective-medium model for describing the optical properties of thin heterogeneous layers with moderate volume fraction of inclusions is verified using full-wave finite-element simulations. 1 Introduction The homogeneous-layer antireflection coating was the first antireflection coating and perhaps is still the one most widely used. Theoretically, it should be possible to obtain a zero reflectance at one wavelength with single dense film of refractive index n= √ na ns , (1) where na and ns are refractive indices of the air (na ≈ 1) and substrate, respectively. However, because of a lack of suitable low-index coating materials, this cannot be realized in practice for substrates with refractive indices less than about S.G. Moiseev () V.A. Kotelnikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Ulyanovsk Branch, Russia e-mail: serg-moiseev@yandex.ru S.G. Moiseev Ulyanovsk State Technical University, Ulyanovsk, Russia S.G. Moiseev Ulyanovsk State University, Ulyanovsk, Russia 1.9 [1]. For example, silica glass (SiO2 ) has a refractive index of 1.46. √ Dense materials with a suitable refractive index of n = 1.46 ≈ 1.2 do not exist. Recent developments in the understanding of materials on the nanometer scale have allowed engineers to create new composites with optical properties vastly different from that usually found in natural materials. In this connection metaldielectric nanocomposites attract considerable attention as promising materials for the development of new element bases for optoelectronics, bio-sensing, optical data storage and solar energy conversion [2, 3]. Such artificial media exhibit abnormal dispersion characteristics at optical scales and can behave as a medium with unique (negative, ultrahigh or low) effective refractive index [4–8]. Unfortunately, their application may practically be limited by absorption of incident electromagnetic radiation due to the presence of metal components. Optical properties of metal–dielectric composites are fundamentally based on surface plasmon resonance modes of metallic nanoparticles and can be efficiently tailored by nanoparticle materials, sizes, shapes, and concentration [6– 11]. All the degrees of freedom could be used to decrease the absorption of composite medium at the spectral region over which the effective refractive index is generally low. This work proposes a design of the composite material with low effective refractive index which absorption is small enough to use thin layer of this material as antireflection interference coatings with high transmittance. 2 Low refractive index of composite medium with ellipsoidal nanoparticles Let us consider the effective optical characteristics of composite medium with uniformly oriented silver spheroids. For 620 S.G. Moiseev modeling the optical properties of such material, sophisticated numerical methods are not necessary. The moderate concentration of nanoparticles permits one to proceed with a rather simple analytical analysis. Such an analysis is based on a well-known analytical Maxwell–Garnett formula for the dielectric function ε of mixture [12] εp − εh ε − εh =η , L(ε − εh ) + εh L(εp − εh ) + εh (2) where εp and εh are dielectric permittivities of nanoparticles and matrix, respectively, η is the volume concentration of nanoparticles (filling factor), L is the factor of depolarization that takes the value    1 arcsin 1 − ξ 2 L = 1−ξ  (3) 1 − ξ2 1 − ξ2 for the field directed along the axis of revolution of spheroid, L⊥ = (1 − L )/2 Fig. 1 Spectral dependences of the effective refractive index n (red line) and the extinction coefficient k (blue line) of composite medium calculated on the basis of effective-medium theory. The external field is polarized along the equatorial semi-axis of the silver discs. The computational parameters are a = 5 nm, b = 50 nm, εh = 2.25. Dot-dashed line shows the refractive index of the matrix and (4) for the field directed perpendicular to the axis of revolution. Here, ξ = a/b is the ratio of the length of polar semi-axis a and equatorial semi-axis b of spheroid; ξ < 1 corresponds to disc-like nanoparticles, ξ > 1—to needle-like nanoparticles, ξ = 1—to spherical nanoparticles. The spatial positions of the nanoparticles do not have to be periodic and can be random. As is well known, the frequency of the plasmon resonance depends on the form of the nanoparticles and their axes orientation relative to the electric field vector [9]. Hence, the plasmon resonance frequency can be changed by using nanoparticles with different geometric factors ξ . Using relations (2)–(4) one can show that for the glass matrix (εh = 2.25), the frequency of plasmon resonance is happened to be in the visible range for the field directed along polar semi-axis of needle-like spheroids or along equatorial semi-axis of disc-like spheroids. Taking into consideration the rotation symmetry of ellipsoidal nanoparticles, it is easy to conclude that the composite medium with proper optical anisotropy is realized by using the matrix with uniformly oriented disc-like silver spheroids which axes of revolution are all parallel to the propagation direction of light wave. In this case the transmitted wave corresponds to the ‘ordinary’ ray in a uniaxial crystal. So, to realize an anti-reflection coating at normal incidence, the axes of revolution of disc-like nanoparticles have to be directed perpendicularly to the plane interface between media. The effective parameters of  (Re(ε))2 + (Im(ε))2 + Re(ε) n= 2 k=  (Re(ε))2 + (Im(ε))2 − Re(ε) 2 from this design are shown in Fig. 1. Here, the effective refractive index n and the effective extinction coefficient k as functions of the frequency of the incident field are calculated using the Maxwell–Garnett formula (2). Size-dependence of the dielectric function εp of small silver nanoparticles is taken into account in terms of the model of limitation of the electron mean free path [13]. One can see from Fig. 1 that due to plasmon resonance of silver nanoparticles the effective refractive index of composite medium differs considerably from the one of matrix. For the parameters used in this work, the plasmon resonance occurs in long-wavelength range, but its left wing reduces effective refractive index n over the whole visible range. The effective refractive index possesses values from 1 to 1.4 in the spectral range 400–550 nm. In this spectral range the effective extinction coefficient of composite is relatively small, and the penetration depth of light exceeds hundreds of nanometers. It makes it possible to use such composite layer as an optical interference coating. 3 Reducing reflection with a thin heterogeneous film According to (1), the suitable refractive index n of a planeparallel plate placed between vacuum and semi-infinite substrate with refractive index of ns = 1.5 should be approximately equal to 1.225. As follows from the results presented in Sect. 2 (see Fig. 1), composite film with silver nanoparticles fits enough to be quarter-wave anti-reflection coating adjusted to λ ≈ 480–500 nm. Assuming that the nanoparticles are sufficiently small, we can, in first approximation, consider the composite Composite medium with silver nanoparticles as an anti-reflection optical coating 621 Fig. 2 Reflection and transmittance spectra of quarter-wave composite layer on glass substrate. Solid lines show the results of simulations according to finite elements method, and dashed lines show the results of calculation according to electrostatic theory. Dot-dashed lines correspond to the case of the uncoated surface. The thickness of layer is equal to 93 nm, εs = 2.25, the angle of incidence is normal to the layer. Other parameters are the same as in Fig. 1 medium optically homogeneous (at least for ‘ordinary’ ray). Therefore, the intensity reflection coefficient R and the intensity transmission coefficient T of plane-parallel plate made of a composite medium can be determined from the well-known Airy equation [14]. This approach allows us to easily control the parameters of a system and predictably change its optical properties, expressing the necessary conditions in an analytical form. However, the results of calculation may be inaccurate because of the effective optical parameters of thin composite layer differ generally from those in the bulk of composite medium away from the interface. This conclusion can be drawn easily by analogy with the propagation of an electromagnetic wave through an ultrathin layer of conventional material taking the discrete atomic structure into account [15]. In present work, we use three-dimensional fullwave simulations with the commercial finite-element solver COMSOL MULTIPHYSICS [16] to verify the applicability of the Maxwell–Garnett relation (2) for describing the optical properties of thin composite layer. Spectral characteristics of thin composite layer on transparent substrate obtained from the analytical expressions and on the basis of full-wave finite-element numerical analysis are shown in Fig. 2. One can see that the distinction between calculation results is mainly quantitative. So, the effective-medium approach provides a satisfactory qualitative description of the reflection and transmission spectra in thin composite layers and confirms their anti-reflective behavior. It follows from Fig. 2 that the reflectance in a broad spectral region (∼ 150 nm) never rising above that of the uncoated glass surface. The lowest reflectance calculated is about 0.2%. Unfortunately, there is no evident improvement of the transmittance because of the absorption of electromagnetic radiation and the transition of the energy of electromagnetic radiation into the thermal energy of the nanoparticles. Nevertheless, no less than 96% of the light actually enters the glass substrate in a relatively broad spectral region (∼80 nm) around the wavelength of the lowest reflectance. It should be emphasised that by selecting proper shape and distribution of the nanoparticles it is possible to achieve better characteristics of anti-reflection coating. This will be presented in detail in a forthcoming paper. 4 Conclusions The obtained results show that metal-dielectric composite media have good perspectives of application as an antireflection optical coating with high transmittance. Indeed, by selecting the proper shape of silver nanoparticles embedded in glass it is possible to achieve a very useful broadband reduction in reflection from air side to glass substrate. Meanwhile, no less than 96% of the light actually enters the glass substrate in a broad spectral region around the wavelength of the lowest reflectance. This work explores the utility of effective-medium representations to simplify the electromagnetic analysis of composite system, and demonstrates the use of this simplification in solving of the boundary problem under consideration. This approach allows us to easily control the parameters of a system and predictably change its optical properties, expressing the necessary conditions in an analytical form. 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