## Abstract

This paper aims at modeling bi-periodic micro-structured silicon surfaces exhibiting broadband antireflection properties in the infrared range using Rigorous Coupled-Wave Analysis (RCWA). These structures of pyramidal shape, which typical dimensions are smaller than the wavelength, are not in the Effective Medium Theory (EMT) validity domain. The influence of various opto-geometrical parameters such as period, depth, shape of the pattern is examined. The antireflective properties of such biperiodic patterned surfaces are then discussed using the photonic crystal theory and photonic band diagrams description. Correlations between the density of Bloch modes, their localizations with respect to the incident medium light line and the surface reflectance are presented.

©2008 Optical Society of America

## 1. Introduction

Broadband antireflection properties of material surfaces are of primary interest for a wide variety of applications: to enhance the efficiency of photovoltaic cells, to increase the sensitivity of photodetectors, to improve the performance of light emitting diodes, etc. Conventional antireflection coatings consist of a single layer, a quarter-wavelength thick, which has a refractive index between that of the adjacent materials. But, it is well known that such single layer antireflection coatings do not operate well over a broad range of wavelengths. This deficiency can be overcome by fabricating antireflection coatings in which the refractive index is graduated from the low refractive index material on one side to the high refractive-index material on the other [1]. Recently very low refractive index materials in thin film form have been fabricated [2] and can be used to improve the efficiency of broadband antireflection multilayer coatings. In this paper, we will focus our attention on microstructured surfaces which constitute other type of antireflective surfaces and which are competitors of thin film antireflective coatings.

The history begins in 1973 when P.H. Clapham et al. [3] showed that some night insects have micrometric conical structures on the surface of their eyes. These structures create an antireflective effect and thus provide them enhanced night vision capabilities. Very recently, arrays of tapered pillars which can greatly minimize the reflectivity have also been found on cicada wings [4]. Some have explained this phenomenon by continuous index fitting from external to internal medium. This approach is based on the “effective medium theory” (EMT) [5,6] and is valid when the structuration period P is much lower than the wavelength (P<λ/10). In the 90’s, thanks to improved computing capabilities, rigorous resolution of the Maxwell equations has been possible, and has allowed a better understanding of physics phenomena. Then, theoretical optimizations and experimental realizations have been performed [7-10].

The purpose of our work is to study antireflecting bi-periodic micro-structured surfaces working in the infrared domain with typical dimensions smaller than the wavelength, but not in the EMT validity domain. In a previous paper [11], our preliminary results showed that the optimization of the optical properties of such media needs a complete description of light propagation phenomena by using rigorous computation methods. In this paper, we define microstructure parameters allowing efficient antireflective effect to be obtained, according to the characteristics of the incident wave (spectral range, incidence angle, polarization). Applications include enhancement of the sensitivity of infrared detectors [12]. Furthermore, by simply adjusting the sizes of the surface structuration pattern with respect to the incident wavelength, the principle of such antireflective microstructured surface may also be useful for the improvement of light coupling in photovoltaic cells [13,14].

We will first present the general shape of the studied structure, the computation tools and then the methodology applied. Finally, we will describe the results and propose an original approach using photonic band diagrams to further explain the origins of the antireflective properties of such microstructured surfaces.

## 2. Structure of interest

The structure of interest exhibits a bi-periodic arrangement of pyramids as shown on the Fig. 1(a). This elementary pattern can be truncated at its top (Fig. 1(b)). Variable parameters of the pyramids are adjustable when optimizing the surface to reach the most efficient antireflective effect: “P” the period of the bi-dimensional (2D) grating, “T” the height of the pyramid, “M” the width of the flat part at the top.

The main goals of the work presented in this paper are:

∗ To define the optimal shape of the structure which allows the lowest reflectance to be obtained for wavelengths ranging between 3 and 5µm (infrared domain band II).

∗ To obtain the most efficient antireflective effect in this wavelength range for any polarizations and for incident angles θ_{inc} ranging between −45° and +45°.

∗ To show and explain the influence of the flat part at the top of pyramids on the antireflective effect efficiency.

∗ To deduce from the previous optimization and the study of the pyramidal shape the guidelines allowing microstructured antireflective surfaces to be designed in a general case.

We will restrict our study to pyramidal shaped bi-periodic structures exhibiting only a transmitted and reflected specular beam when illuminated in the 3-5 µm wavelength range. More precisely, no order will be diffracted either in the incident medium (the air in our study) or in the silicon substrate (refractive index n_{sub}=3.42 (+/-0.02)). This restriction is necessary when considering that, due to the surface structuration, light has to be coupled in the silicon substrate as efficiently as possible, that is to say light must not be diffracted in the air (n_{inc}). Furthermore, such surface microstructuration will be used in pixelized imaging sensors and thus no diffracted order is allowed in the substrate to avoid cross talk from one pixel to another. By simply using the grating Eq. (1):

We obtain the maximal value of the period of the bi-periodic pyramidal shaped grating at the substrate surface: P_{max}=1.2µm (table 1). Moreover, by taking into account technological considerations, we deduce the range of parameters to be explored (table 1).

Indeed, the grating period and the size of the flat part at the top of pyramids are not too small to allow an easy fabrication process using standard photolithography. The grating depth is limited also to 1.5 µm with the aim of using standard wet TMAH (TetraMethylAmmonium Hydroxide) chemical etching to perform the surface structuration. Due to the range of parameters shown in the table 1 and the wavelength range of interest (3-5 µm), the studied microstructured surface is clearly outside the validity domain of the EMT. Thus, the computation of the optical properties (electromagnetic fields, reflectance and transmittance) of such microstructured surfaces will be performed by using rigorous method: the RCWA [15, 16] (“Rigorous Coupled Wave Analysis”) and the FDTD [17] (“Finite-Difference Time Domain”) method. Comparable results given by RCWA and FDTD are obtained when using: 6 modes or more for the RCWA and a spatial step of P/150 (P: Period of the 2D grating of pyramids) and a temporal step of 2.10^{-16} s for the FDTD method.

## 3. Simulation results

It has to be underlined that a bare plane interface between air and silicon shows a reflectance of around 30% in normal incidence and in the infrared domain (λ between 3 and 5 µm). Our aim is to decrease this reflectance as far as possible in the whole infrared domain. When considering the curves (solid lines) of Fig. 2 in the case of pyramids without top-flat (M=0µm), the higher is the pyramid, the better is the antireflective effect, until an optimal value of T=1µm. Beyond T=1µm, the reflectance remains at the same level, that is between 5% and 10%. According to this result, it seems that the best antireflective effect is obtained using the slowest varying refractive index profile. But, when considering a discontinuous refractive index profile, such as that obtained using a flat part on top of the pyramids, we obtain a more efficient and broader antireflective effect. Indeed, if we consider the dotted curves of Fig. 2, corresponding to the computed reflectances of the microstructured surfaces with several pyramid heights (ranging between 0.5 µm and 1 µm) and exhibiting a top-flat of size M=0.25 µm, we clearly demonstrate that for a given pyramid height the adjunction of the flat parts at the top of the pyramids strongly decreases the reflectance. Besides, we must notice that, opposite to the EMT predictions, in our case, the fill factor is not the main parameter describing the behavior of such subwavelength gratings, but the shape of the pattern plays an important role too. Finally, Fig. 2 shows an optimized silicon pyramid shape (P=1µm, T=1µm, M=0.25µm) exhibiting a reflectance lower than 2.5%, in normal incidence, without diffracted orders, and in the 3 to 5µm wavelength range. By a simple homothetic down- or up-scaling of the pyramid parameters (P, T, M), one can easily deduce a new set of parameters allowing an efficient antireflective effect to be obtained in another spectral range, for example in the visible range for photovoltaic cell applications.

The reflectance in normal incidence of the optimized pyramidal pattern (P=1µm, T=1µm, M=0.25µm) is shown in Fig. 3(a) and can be easily compared to the reflectance obtained in the case of a pyramidal pattern without top-flat part (M=0µm). The presence of the top-flat allows the reflectance in normal incidence to be decreased by a factor 2 to 4 for wavelengths higher than 3.5µm. Moreover, when considering antireflection coatings, crucial parameters are the incidence angle and the polarization states of light. As shown in Figs. 3(b) and 3(c), respectively at 30° and 45° of incidence and in s and p polarizations, top-flat pyramids with a size M=0.25µm allow reflectance values to be decreased on a broad wavelength range. The optimized pattern (P=1µm, T=1µm, M=0.25µm) enables the reflectance to be kept below 10% at 45° of incidence, in the 3 to 5µm wavelength range and for the two polarization states. In the same illumination conditions, the reflectance of a bare flat silicon surface is 42% and 18% respectively in s and p polarizations.

(a). Normal incidence.

(b). 30° of incidence (s and p polarizations).

(c). 45° of incidence (s and p polarizations).

It is of high interest to understand why the shape of the pattern has such an influence on the reflectance of a subwavelength structured surface. As it was already noticed, the height of the pattern and the graded refractive index profile are not sufficient to explain the results. Then, better antireflective effects due to the flat parts need to be further studied and explained.

## 4. Discussion

Photonic band diagrams are a powerful tool to investigate the optical properties of periodically structured surfaces. In the following, we will compare bands diagrams of different patterns and their corresponding spectral reflectance curves. Band diagrams are computed using the BandSOLVE® software. In our study, all band diagrams are computed using a plane waves theory. Figure 4(a) depicts the band diagram of a structured silicon surface such as that of Fig. 1(a), with pyramid parameters P=1µm, T=1µm and without flat top (M=0µm). In this diagram, for all frequencies higher than 0.292µm^{-1} (which correspond to wavelengths λ < 3.42µm), the structure exhibits diffracted orders which correspond to a continuum of Bloch modes. This part of the diagram is not in the scope of our study as we are mainly concerned with antireflective effects on pixelized photodetectors. The interesting part (represented by the rectangle in the Fig. 4(a)) is the spectral range between 3.42µm and 6µm (that is 0.167µm^{-1}< frequency < 0.292µm^{-1}). Thus, in the following, we will only present a small part of the band diagrams corresponding to this frequency range (as shown in Fig. 4(b), 4(c), and 4(d)). In the band diagrams of Figs. 4(b), 4(c) and 4(d), computed with P=1µm, T=1µm and M respectively equals 0µm, 0.25µm and 0.5µm, only the Bloch modes localized above the light line of the incident medium (bold line) will be coupled out of the structure and will contribute to increase the reflectance. So, to reach a minimal reflectance, one has to set the pyramid parameters (P, T, M) with the aim of minimizing the number of modes in the area localized above the light line. Especially, the Γ direction, corresponding to the direction perpendicular to the surface, is always above the light line whatever the frequency. Thus, a minimal number of Bloch modes is required in the Γ direction to obtain the best antireflective effect. For example, in Fig. 4(d), when Bloch modes appear near the Γ direction at 1/λ=0,28µm^{-1} (see the dotted circle), the reflectance increases to 5.5 %. On the contrary, if we compare Fig. 4(c) and Fig. 4(b) around 1/λ=0.24µm^{-1}, when some Bloch modes are under the light line in Fig. 4(c) and not in Fig. 4(b) (see the dotted circles), the reflectance at this frequency is much lower in Fig. 4(c) (close to 0) than in Fig. 4(b) (around 4%). We have here a clear demonstration of the positive influence on the reflectance decrease of the flat top of the pyramids (M=0.25µm in the case of Fig. 4(c) and M=0µm in Fig. 4(b)). The flat top allows a shift of some Bloch modes below the light line preventing them to interact with the external medium (the air) and thus decreasing the reflectance of the surface.

## 5. Conclusion

To conclude, this paper shows that periodic structures showing pronounced antireflective effects can be described, studied and optimized using their photonic band diagrams. We have demonstrated that a strong reduction of the reflectance is possible by the presence of a flat part at the top of the periodic pyramid pattern. So, the pattern geometry of the grating plays an important role in the antireflective effect. If, by adjusting the pattern parameters, one is able to remove all Bloch modes localized above the light line in the wavelength range of interest, a very efficient antireflective effect is obtained. Our study paves the way to further explorations of the relations between the positions of Bloch modes respective to the light line and the geometry of the structure replicated at the substrate surface.

## Acknowledgment

This work was supported by the French DGA (Délégation Générale pour l’Armement) under Grant number 05.34.014.

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