Microstructured anti-reflection surface design for the omni-directional solar cells

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1 Microstructured anti-reflection surface design for the omni-directional solar cells Li Chen, Hongjun Yang, Men Tao and Weidong Zhou Department of Electrical Engineering, NanoFAB Center, University of Texas at Arlington, TX 7619, USA ABSTRACT In this paper, a new process for the formation of hemispherical structures as an omni-directional anti-reflection (omni-ar) coating in solar cell is reported. We also demonstrated the simulation results of the angular and spectral dependences of the total reflectivity on various micro-structured surfaces. Close to zero reflection can be achieved in some micro-structured surfaces over an extended spectral region for large ranges of incident light angles. Daily generated current in such hemispherical solar cells hence enhanced to 1.5 times of bulk silicon solar cells. The impact of feature size, density, shape and refractive index has all been investigated. The experimental results agree reasonably well with the theoretical work. Such an omni-ar structure offers an attractive solution to current bulk crystalline silicon solar cells, as well as thin film, organic and future quantum based solar cells. Keywords: coating, anti-reflection, surface texturing, solar cells. 1. INTRODUCTION For bulk silicon solar cells, optical loss is mainly due to the front surface reflection. A flexible optical design for light collection is vital in achieving high performance solar cells.[1] An ideal anti-reflection (AR) structure should lead to zero reflection loss on solar cell surfaces over an extended solar spectral range for all angles of incidence. Such a coating can eliminate the need for a mechanical tracking device for proper optical alignment of the solar cell with respect to incident sunlight.[1] The perfect omni-directional AR (omni-ar) structure has been a subject of intensive research in thin film optics and most importantly, for solar cell applications. A series of papers were published by Dobrowolski and Poitras et al. in search for the perfect AR structure.[2-4] In theory, there are three major types of non-absorbing omni-ar structures. The first group is the single or multilayer quarter-wavelength film stacks.[5, 6] Close to zero reflection over a certain spectral range for certain ranges of incident angles can be achieved by controlling the refractive index and thickness of each individual layer in the multilayer structure. The major challenges for this approach are the availability of materials with the right indexes and good optical transparency, as well as the precise control of film thickness. In solar cell applications, a single layer thin film AR coating, e.g. silicon nitride (SiNx) thin film for silicon [6] solar cells, is often used as a cost effective approach. Such a single layer AR coating reduces reflectivity only in a limited spectral range under surface normal incident conditions. The second group is the graded index (GRIN) AR structures,[7] where the refractive index of the structure changes gradually from the top to the bottom.[8, 9] The index profile can follow different mathematical functions, such as linear, cubic or quintic. With a GRIN AR structure, extremely low reflection can be achieved over a broad spectral range for a wide range of incident angles, especially with the quintic index profile.[7] The difficulty associated with the choice of materials for practical control of index profiles has prevented this structure from practical applications. Sol-gel based approaches with either porosity control and/or high index dopant introduction were reported.[1-12] Recently a SiO2/TiO2 GRIN AR coating was reported by Xi et al.,[13] by oblique-angle physical vapor deposition, where the refractive index can vary from 1. to 2.. The third group is the textured surface AR structures. The most successful example is the anisotropic etching of single crystal Si(1) surface in a solution containing potassium hydroxide (KOH).[9, 14-17] It has led to record solar cell efficiencies in the lab and has been the standard AR structure on all commercial single crystal Si solar cells.[9, 14- Optical Modeling and Measurements for Solar Energy Systems II, edited by Benjamin K. Tsai, Proc. of SPIE Vol. 746, 7468, (28) X/8/$18 doi: / Proc. of SPIE Vol

2 17] Close to zero reflection over a wide spectral range can be achieved when an additional layer of low index film (e.g. SiNx) is coated on these microscale pyramidal structures on Si surface. However, the microscale surface texture involves anisotropic etching of the Si substrate, which does not apply to poly-crystal Si and non-si solar cells. In this case, anisotropic etching becomes too unreliable and photolithography-defined surface textures too expensive. In this paper, we will introduce a new process for microscale surface texturing for omni-ar coatings. The process is solution based, which is compatible with large scale manufacturing and flexible for integration on various solar cell substrates. It is inherently low-cost and energy efficient, without complicated large vacuum systems. A detailed theoretical investigation has been carried out to understand the optical performance of the microscale surface textures for AR applications. We compared hemispherical structures to pyramidal and conical structures. The impact of refractive index, feature size and density is also reported. Finally experimental results from omni-ar coatings on quartz substrates are compared with simulation results. 2. OMNI-AR COATINGS FROM SPHERICAL PARTICLES The basic structure of the omni-ar coating is shown in Figure 1. The coating comprises an array of, in an ideal example, hemispherical and microscale dielectric particles. The hemispherical particles are formed by partially immersing microscale spherical particles into a dielectric film of the same refractive index. Both the particle array and the dielectric film can be prepared from solutions containing microscale particles and precursors for the dielectric film. The performance of the coating can be controlled by tuning the size and packing density of the dielectric particles and the thickness and refractive index of the dielectric film, as discussed in this paper. Shown in Figure 1 are scanning electron micrographs of processed omni-ar structures on quartz substrates. A closely packed monolayer of 2 µm silica spheres were first deposited on the substrate, followed by coating of a spin-on-glass (SOG) film with a desired thickness, with Figure 1 sowing a.2 µm SOG film. Finally an omni-ar structure is formed after baking to cure the SOG film. Detailed experimental conditions can be found in Ref [18]. (i) Monolayer self assembly of spherical particles Substrate (ii) Spin coating of sol-gel or SOG glass based thin films Substrate (iii) Formation of omni-directional anti-reflection coating Omni-AR Coating Substrate Figure 1 omni-directional anti-reflection (omni-ar) coating based on monolayer of spherical particles and spin-on-glass (SOG) film; Scanning electron micrographs of fabricated omni-ar coatings: (i) top view; (ii) cross-sectional view before SOG film and (iii) cross-sectional view after SOG film. 3.1 Simulation Approach 3. DESIGN AND SIMULATION The geometrical structure under consideration is depicted in Figure 2. A square lattice of hemispheres is constructed on an optional dielectric film, which sits on a substrate. The hemispherical omni-ar structure, optional dielectric film and substrate may have the same or different refractive indices n 1, n 2 and n s, respectively. The structural parameters, as Proc. of SPIE Vol

3 shown in the top view and cross-sectional view of the omni-ar structure in Figure 2, are variables in our simulations to investigate their impact on reflectivity based on rigorous coupled-wave analysis (RCWA).[19] The substrate material is Si in most cases, though the structure proposed here can be applied to other substrate materials with different optical properties. In some cases quartz substrates are used in order to be able to measure the total transmissivity over the visible spectral region. Shown in Figure 3 is the dispersion curves for the refractive index and extinction coefficient for Si used in our simulations.[2] The complex index is expressed as n = n + ik, where n and k are the real and imaginary part of the refractive index, respectively. Note the refractive index of Si changes with the wavelength over the spectral region of interest. For practical designs, the impact of absorption from the Si substrate should be considered, especially when the k value is comparable to the index n value, e.g., in the UV region (for photon energy greater than 3eV or wavelength below 4 nm, as shown in Figure 3). We did not consider absorption in our simulation, as it will not alter significantly the reflectivity results in our simulation, for the wavelength region from 4 nm to 16 nm, where the extinction coefficient for Si substrate is relatively small (k <.4), as compared to the real part of the index (n > 3.5). The primary focus of our work is on light propagation through an omni-ar structure on a Si substrate, where the omni-ar structure has no or little absorption. The refractive index profiles for Si and silicon dioxide (SiO 2 ) used in our simulation in the wavelength region of interest is shown in Figure 3. 2 :b) <i F! OnI-i-\R cos n! 2!L2IFle (CI Figure 2 A hemispherical grating as the basic anti-reflection structure for simulation: Three-dimensional view of the square lattice; top view and (c) cross-sectional view of the basic structure WaYelenqth ( nm S 2 U (-) C (-) C x LU fl(sic) - / 1.46 'fl(8i) c3 S I 44 C r ('3 Energt (ev) Energt (ev) Figure 3 Crystal silicon complex index dispersion curve n = n + ik from ultraviolet to infrared and refractive index dispersion curve of silicon and silicon dioxide used in our simulation. Proc. of SPIE Vol

4 3.2 Multi-Layer Thin Film Structures For comparison, we first considered quarter wavelength multi-layer thin film AR structures. Assuming ideal indices for all the layers and with optical thicknesses targeting 6 nm wavelength, we simulated different AR structures of single and triple layers, as shown in Figure 4. The ideal indices are chosen to achieve close to zero reflection at the target wavelength (e.g. 6 nm in this case). For single layer AR coating (AR), the ideal index for the coating layer n 1 follows the equation of n 12 = n n s, where n and n s are the indices of top (air) and bottom (Si) layers, respectively (inset of Figure 4 ). Similarly, for multi-layer ARC, the idea indices for the double layer and triple layer ARC are (n 2 /n 1 ) 2 = n s /n and (n 1 n 3 / n 2 ) 2 = n n s, respectively (inset of Figure 4 ). As expected, for the single layer AR structure, close to zero reflectivity can be achieved only at the target wavelength with normal incident direction. By increasing the number of layers, it is possible to achieve relatively wider spectral coverage with close to zero reflection in a reasonably wide range of incident angles (up to 6º). Air n=l n1=l.1; d1=ls6 nm n1=l.5; d1=l nm n3=2.72; d1 =55 nm R Silicon TripIe layer e = 75?oo Wavelength nm) Wavelen gth (nm) Figure 4 Simulated quarter wavelength multi-layer thin film AR structures with ideal indices and thicknesses for single and triple layers. 3.3 Hemispherical Surface and omni-ar Structures We then considered hemispherical structures with different structural parameters. In what follows, we assume the substrate material is Si, and the coating material is SiO 2, with a refractive index dispersion curve shown in Figure 3 (n SiO2 ~ 1.45). Starting with a simple case in Figure 5, where a hemispherical structure is coated on a Si substrate, with the hemispherical radius R = 1µm and lattice spacing a = 2.5µm. The reflectivity is reduced by ~5% for incident angles up to 6º over the entire spectral region of simulation, as compared to the reflectivity for a bare Si substrate at surface normal incident conditions. The reflectivity for incident angle of 75º is also reduced in the shorter wavelength regime. The result is encouraging, as it proves the incident angle independence, thus omni-directional, of the hemispherical structure we investigated here. It is worth pointing out here that the hemispherical structure we considered here is not ideally closely-packed, which could lead to slightly under-estimated results as compared to ideally close-packed structure (with a = 2R), especially at surface-normal or small incident angle directions. This will be discussed later on the packing density impact section (Figure 7 and Figure 9). The parameters chosen here seem to have little impact on the reflectivity as compared to the ideal close packed case. The parameters chosen in our simulation are better representation of the actual experimental conditions, where spherical particles may not be closely packed together to form a mono-layer in a large area domain. The low index film between substrate and hemispherical structure, as shown in Figure 5, is inherently present in the structure we proposed in Section II. We investigated the impact of such thin film (h 1 ) on reflectivity, as shown in Figure 5 for different incident angles. Compared to Figure 5, we see insignificant changes in reflectivity. The oscillations in Figure 5 are most likely due to interferences at different interface, which is to be investigated. Proc. of SPIE Vol

5 5 4 3 > U o 2 ' (-ic\ f (\ n3 substrate. "-... 6E 4 >3 > C) Wavelength ( nm) Figure 5 Simulation results for hemispherical structures with index n 1 = 1.5 on silicon substrate: Square lattice hemispherical structure with radius R = 1 µm and lattice constant a = 2.5 µm and omni-ar structure with hemispherical structure on top of a thin film with the same index n1 and thickness h1 = R = 1 µm. eo 2 - e3o n1 Om ni-ar. e6o n2 coating n substrate WavelengTh nm Si absorption window SiNon Si 2Omni-AR > V o 2 Si absorption window e = 75. e=6o SiNon Si /._.f N = 6 %f 8 12 (C) OmnLAR Figure 6 Simulation results for omni-ar structure on Si substrate with a SiNx thin film in between (n 2 = 2. and h 2 = 75 nm); Other simulation parameters are the same as those in Figure 5. The reflectivity for small incident angles (θ = and 3 ) and large incident angles (θ = 6 and 75 ), for two cases: SiNx coated Si and omni-ar on SiNx coated Si. We further considered another scenario where the omni-ar structure is an add-on to current polycrystal Si solar cells, where a single layer of SiNx is coated on top of the solar cells. The results are shown in Figure 6. Significantly reduced reflectivity can be achieved over a wide spectral range for incident angles from º to 6º. A comparison is given in Figure 6 and (c) for SiNx coated Si substrates with and without omni-ar coatings. We see the improvement in reflectivity is relatively small for small incident angles of ºand 3º (Figure 6). However, significant improvement in reflectivity can be achieved for large incident angles of 6º and 75º (Figure 6(c)). This is the most attractive feature of the omni-ar structure, which provides angle independent reflectivity over a wide spectral range. The impact of packing density and particle size was investigated as well. Shown in Figure 7 is the simulated reflectivity with different packing densities. With fixed radius R = 1 µm, the packing density can be varied by changing the lattice constant a. Shown in Figure 7 and are two examples with a = 2 µm, and a = 5 µm,, respectively. The surface normal reflectivity for different packing densities is shown in Figure 7(c). It is worth noting that the simulated reflectivity does not change significantly over a relatively large range of packing densities. The only notable feature is the increased oscillation for smaller packing densities. This could mainly be due to the dominance of the interferences in the layer under the particle (h 1 layer as shown in the inset of Figure 6), as the exposed flat surface area increases with the reduction of packing density. 16 Proc. of SPIE Vol

6 e 3 ii a 2 ) e7s 2i 3 > V aj _ eoo P = WavelengTh nm o 5um ---.-e 6 3 e_ Figure 7 Packing density impact for the proposed hemispherical omni-ar structure similar to the one in with R = 1 µm, n 1 = 1.5 and lattice constant a = 2 µm and a = 5 µm. (c) Surface normal reflectivity for different packing densities. 5 flflfl eo \2 \1 e3o 2um s.3 e7& 2\ (C) 4 > V WavelengTh nm 5, WavelengTh nm e = 7 RMO.4 P O5tm 5km (C) Figure 8 Spherical radius impact for the proposed hemispherical omni-ar structure similar to the one in Fig. 6 with R/a =.4, n 1 = 1.5 and spherical radius R =.5 µm and R = 2 µm. (c) Surface normal reflectivity and (d) large incident angle reflectivity for different spherical radius. Figure 8 is the simulated reflectivity for different particle sizes. Here the packing fraction is fixed, i.e., the ratio between radius R and lattice constant a is fixed at.4. Shown in Figure 8 and are two examples with R =.5 µm, and R = 2 µm, respectively. The surface normal reflectivity and large incident angle reflectivity (θ = 75º) for different particle size is shown in Figure 8(c) and (d), respectively. Again, we do not see significant differences in the simulated reflectivity, especially at small incident angles up to 6. These results suggest large process windows in manufacturing the proposed structure for solar cell applications. However, for larger incident angle (θ = 75º), we see slightly reduced reflectivity with the increase of the particle size, which is due to the thicker transition layer at large incident angles. In all simulations shown in Figure 8, the thickness of h1 layer (1 µm) as shown in the inset of Figure 6, and the packing (d) Proc. of SPIE Vol

7 fraction (proportional to the ratio of R/a =.4), are kept the same. The fixed packing fraction for non-close-packed structure can lead to the actual open flat areas between the hemispheres to be increased with the increase of the particle sizes, which may be the cause of the more profound interference oscillation features observed for larger particle sizes. 3.4 Pyramidal, Conical vs. Hemispherical Structures Surface structures of different shapes were also investigated. Shown in Figure 9 are the structures and simulation results for three different shapes: pyramids, cones and (c) hemispheres. All the structures under simulation have similar structural parameters. The structure material is SiO2 and the substrate is Si. The lattice constant a = 2.5 µm and the film thickness h1 = 1 µm. The pyramid base and height are µm. The cone base radius is R = 1 µm and the height is also 1 µm. The hemisphere radius is R = 1 µm. We also simulated these structures with ideal close-packed case where the base size equals to the lattice constant a (2 µm), and the feature sizes remain the same. Based on the simulated results, the hemispherical structure has slightly better performance overall, especially at large incident angles, while the pyramid structure has better performance at small incident angles and at shorter wavelength regime. This could be due to the enhanced second strike effect for pyramid structure at small incident angles,[16] as compared to the case for hemispherical structures. Pyramid-Shaped Cone-Shaped (c) Hemisphere-Shaped n 1 Omni-AR n s substrate n 1 Omni-AR n s substrate R n 1 Omni-AR h 1 n s substrate a=2.5µm, d=2 µm a=2.5µm, R=1µm a=2.5µm, R=1µm Reflectivity (%) θ = θ=6 θ = θ=6 θ = θ=6 4 θ =3 θ =75 4 θ =3 θ =75 θ =3 θ =75 Reflectivity (%) Reflectivity (%) Wavelength ( nm ) Wavelength ( nm ) Wavelength ( nm ) Figure 9 The structures and simulation results for three differently shaped structures: pyramids, cones and (c) hemispheres. 4. SOME SOLAR CELL PERFORMANCE The solar cell performance was evaluated based on the simulated optical characteristics. From the reflection results, the total absorpted photon numbers and the corresponding solar cell short current can be derived according to the experimental internal quantum efficiency of Si. The total generated daily electricity can thus be determined based on the solar spectral intensity information available from NREL. We used the solar spectral intensity data for Dallas, Texas with AM1.5 normalized conditions. An outline of the simulation procedure is shown in Figure 1, with more detailed description to be presented. The simulated results for solar electricity generation are shown in Figure 11. Compared to the bulk Si solar cells, the total daily electricity generated from omni-ar based Silicon solar cells are 1.5 times higher. Proc. of SPIE Vol

8 Solar Photon Flux:, ) SPECTRAL 2 Data: solar spectra For various times through the day Reflectivity Calculations: ) Transfer matrix method (TMM) and Coupled wave diffraction method Internal Quantum Efficiency Solar Cells Measured: 1QW) L Incident Photon Flux Calculations ) =, ) [1 R(X, )] L Solar iveighted Reflqction SWR JIqE(2);(2)2)dzfiIqE(2);(2)dz J Generation Rate Calculations] G(A) IQE(A)F(A)[l - I Short Cqrrent Density Calculations] J. qig(2)uqfiqe(2)p(2)[l R2)]U Solar Cell Performance: Power Generation over the Day Total Power Generation Over a Day Figure 1 Simulation procedure for solar cell performance evaluation based on the optical characteristics of solar cell. # of absorbed photons per day [/m 2 ] 4 x Case # Total current per day [A/m 2 ] 5.5 x Case # Norm.Absorpted Photons Per Day Case # (c) Norm.Total Current Per Day Case # (d) Figure 11 Simulated solar cell performance in terms of total daily photon absorption and electricity generation for three cases discussed here: (1) Bulk si; (2) Si under omni-ar ; and (c) Si under omni-ar with SiNx. 5. EXPERIMENTAL RESULTS Since the purpose of an anti-reflection coating is to maximize light transmission, total transmissivity of quartz wafers with various coatings has been measured using a JASCO V-57 spectrophotometer. The coating was done on one side of the quartz wafer based on the spin coating processes.19 An integrating sphere was used in the measurement, which collects transmitted light through a sample from all directions. Total transmissivity measurements at different incident angles have also been performed using a home-built monochromator-based spectroscopic setup with an integrating sphere. The white light from a 1-W Oriel quartz tungsten halogen lamp was focused on the sample at different incident angles through a flexible liquid light guide. Care was taken to ensure the focused light can be coupled into the integrating sphere. Due to limitations in the setup, the maximum incident angle is limited to ~3º. Figure 12 shows a normal-incidence total transmissivity measurement of a quartz with an omni-ar coating, which comprises a monolayer of 2 µm, spherical silica particles immersed in a SOG film of.2 µm, thick. For comparison, the total transmissivity of a quartz wafer without any coating, a quartz wafer coated with.2 µm SOG only and a quartz wafer coated with a monolayer of 2 µm spherical silica particles only were also measured. The omni-ar coating improves the transmissivity from ~88% to ~92% at 4 nm and from ~9% to ~92% at 1,1 nm, demonstrating its broad-spectrum effect. Since the bare quartz wafer already has a high transmissivity above 88%, the improvement by the Proc. of SPIE Vol

9 omni-ar coating is tainted by the high background transmissivity. A.2 µm SOG film alone slightly improves the transmissivity at short wavelengths by ~1%, possibly due to its smaller refractive index (~1.39), than that of quartz (~1.55). The reduced surface roughness could be another contributing factor here. A monolayer of 2 µm silica spherical particles alone decreases the transmissivity by ~1% in the entire spectral range of interest. This is likely due to reflection from the multiple surfaces in the monolayer particle coating: the top surface of the particles, the bottom surface of the particles and the surface of the quartz wafer. Light was scattered away on these particles surfaces, which could lead to reduced transmittance. Therefore, the SOG film not only serves as a glue to hold together the coating, but also reduces the number of internal surfaces which cause reflection. Measured Transmittance (%) 96 Ground quartz coated with: 94 Omni-AR 92 9 SOG only Quartz 88 Particles only Wavelength (nm) Figure 12 Total transmissivity under normal incidence of quartz wafers with various surface coatings. 96 Omni-AR MgI d p nd nttotl TrnimIIhnci Cu re me ft S etu p Power detector Quart only 86 4U Wavelength (nm) Figure 13 Angle-dependent transmissivity of a quartz wafer before and after an omni-ar coating; Schematic of experimental setup for angle dependent total transmittance setup. An incident-angle dependent transmissivity measurement for the omni-ar coating on quartz wafer, along with the schematic of experimental setup, is shown in Figure 13. Within the limited range of incident angle (~3 ), the omni-ar coating improves the total transmissivity from ~87% to ~92% at 5 nm and from ~89% to ~92% at 1, nm for an incident angle of 3º. Note under surface-normal incident light condition, the measured transmittances show similar spectral dependences for quartz and quartz with omni-ar coating. However, the absolute transmittance values are different. It is mostly due to the different experimental setup we used and variations in the calibration processes. Figure 14 is simulated transmissivity of quartz wafers with and without an omni-ar coating at different incident angles. To match the experimental conditions, the coating in the simulation consists of a.2 µm, SOG film with 2 µm, hemispherical particles on top (particle radius R= 1 µm). The wavelength-dependent refractive index is assumed to be that of silica, as shown in Figure 3. However, the simulation did not take into account the reflection from the back surface of the quartz wafer, which could leads to slight reduce in the actual transmitted power. The transmissivity increases from ~95% to ~98% with the coating at small incident angles of º and 3º. The results agree reasonably well with experiments in Figure 12 and Figure 13. The simulated results for large incident angle are also shown for Proc. of SPIE Vol

10 completeness. With a large incident angle of 6º, the transmissivity increases from ~9% to ~96%. Work is undertaken to have an experimental setup capable of measuring large incident angle transmissions. 1 With Omni-AR Quartz 98 3 Simulated Transmittance (%) Quartz only 6 Quartz only Wavelength (nm) Figure 14 Simulated transmissivity at different incident angles for quartz wafers with and without an omni-ar coating. 6. CONCLUSIONS A detailed analysis was presented to understand the reflectivity of micro-structured surfaces for solar cell AR coatings. It was found that omni-directionality (incident angle independent) anti-reflection can be achieved in various microstructured surfaces. Coupled with index matching to the substrate, close to zero reflectivity can be achieved on Si substrates with silica omni-ar coatings. Therefore the daily generated current in omni-ar based solar cell can be raised up to 1.5 time of bulk silicon solar cell. Experimental results agree reasonably well with the theory. The results suggest that the proposed omni-ar structure is a promising and cost effective solution for current and future generation solar cells. ACKNOWLEDGEMENTS The authors would like to thank Mr. K. Han, G. Song and Z. Qiang for their help with transmissivity measurements. The work was supported by Air Force Office of Scientific Research and National Science Foundation. REFERENCE [1] Luque, A., and Hegedus, S., [Handbook of Photovoltaic Science and Engineering], Wiley New York, (23). [2] Dobrowolski, J. A., Poitras, D., Ma, P., Vakil, H., and Acree, M., "Toward Perfect Antireflection Coatings: Numerical Investigation," Applied Optics 41, 375 (22) [3] Poitras, D., and Dobrowolski, J. A., "Toward Perfect Antireflection Coatings. 2. Theory," Applied Optics 43, 1286 (24) [4] Dobrowolski, A., Guo, Y., Tiwald, T., Ma, P., and Poitras, D., "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Applied Optics. 45, 1555 (26) [5] Yariv, A., and Yeh, P., [Optical Waves in Crystals: Propagation and Control of Laser Radiation], Wiley-Interscience, (22). [6] Bouhafs, D., Moussi, A., Chikouche, A., and Ruiz, J. M., " Design and simulation of antireflection coating systems for optoelectronic devices: Application to silicon solar cells - Appl. Phys.," Solar Energy Materials and Solar Cells 52, 79 (1998) [7] Southwell, W. H., "Gradient-index antireflection coatings," Optics Letters 8, 584 (1983) [8] Yablonovitch, E., "Statistical ray optics," Journal of the Optical Society of America 72, 899 (1982) [9] Campbell, P., and Green, M. A., "Light trapping properties of pyramidally textured surfaces," J. Appl. Phys. 62, 243 (1987) [1] Bautista, M. C., and Morales, A., "Silica antireflective films on glass produced by the sol-gel method," Solar Energy Materials and Solar Cells 8, 217 (23) [11] Lee, D., Rubner, M. F., and Cohen, R. E., "All-Nanoparticle Thin-Film Coatings," Nano Letter 6, 235 (26) Proc. of SPIE Vol

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