Precise Calculation of the Optical Constants of Self-standing Nanoporous Silicon Layers

The precise knowledge of the values of the optical constants (index of refraction, n, and extinction coefficient, k) for nanostructured porous silicon (nanoPS) is a necessary condition to predict the behavior of any optical and photonic devices based on this material. With this objective in mind, a simulation computational program based on the matrix method was used to determine the values of the optical constants in the visible range of self-standing nanoPS films from their experimental reflectance and transmittance spectra. Furthermore, the spectral absorption coefficient (α) was determined from the spectral k values, which motivated to the determination of the values and type of bandgap (direct or indirect) for different porosities.


Introduction
Thin film optical coatings are used to modify the response of photonic devices. The index of refraction of the individual thin films, number of layers, surface roughness and their thickness are the main parameters that affect the response of multilayer optical coatings. The optical coatings might be in bulk thin film form or nanostructured layer grown by different physical and chemical methods [1][2][3].
Within this context, the precise determination of the optical properties of thin films is essential for the subsequent development of optical devices. Three key parameters are the index of refraction (n), extinction coefficient (k), and energy bandgap (E g ). The optical constants of thin films can be determined from optical measurements, i.e., transmission and reflection [4], or by ellipsometry [5]. For the design of optical coatings with the required properties it is essential a great accuracy in the determination of the optical constants since a minimum variation of these values would lead to a deterioration of the behavior of the whole coating.
Nanostructured porous silicon (nanoPS) is one of the common materials applied in a variety of photonic devices due to the facile management of its optical response depending on its morphology [6][7][8]. NanoPS can be described as a mixture of air, amorphous silicon, silicon nanocrystals and silicon dioxide [9,10], strongly depend on the structure of the fabricated layer [11]. The structure of this material can be widely adjusted through the fabrication parameters. NanoPS layers can be fabricated by different methods. For instance, by electrochemical etching [10], electroless method [12] and by photolithography method [13]. The optical property of such material has a main function in the morphology and thickness of the nanoPS layers.
In the current work, fabrication of self-standing nanoPS layers with different porosities was carried out by electrochemical etching process followed by electropolishing. Furthermore, the values of the optical constants of nanostructured self-standing porous silicon layers of different porosities were determined with great accuracy. This would allow exploiting the unique optical properties of porous silicon for the development of such devices as light emitting diodes, filtered photodetectors, optical sensors, and photovoltaic solar cells.

Theoretical Model
A computational simulation method based on a matrix method was used to obtain a precise determination of the optical constants of self-standing nanoPS layers. The software is based on spectral reflectance and transmittance data. In this case, the materials under study is described as a plane-parallel layer of infinite extent, characterized by an index of refraction n and an extinction coefficient k [14,15]. In the general case of a multilayer coating, each interface and homogeneous layer would be described by its appropriate matrix. Therefore, the resultant matrix, M R , of a series of parallel layers would be given by M R = M 1 × M 2 × …, so the final result is expressed in terms of the product of a set of 2 × 2 matrices [16]. The accuracy of the simulated results depends on some considerations to be able to apply this formalism, which is addressed in reference [17]. Succinctly, the theoretical model used in this work overcomes the usual approximations (homogeneous layers, semi-infinite substrate, normal incidence angle for reflectance measurement, etc.) which result in the loss of solution during the numerical inversion process.

Experimental
Self-standing membranes of nanoPS were fabricated by the electrochemical etch of p-type silicon wafer of < 100 > orientation and resistivity (0.01-0.02 Ω.cm) followed by electropolishing process. The etching process was carried out at room temperature in a 1:2 mixture of HF (48%): ethanol (98%). The experimental setup of the electrochemical etching system has been described in a previous study [15]. To detach the nanoPS leyer from the underlying silicon wafer, an etching current density of 200 mA/cm 2 was applied for 10 s, a process which resulted in the generation of selfstanding nanoPS layers. Then, the samples were subjected to a thorough cleaning process in ethanol to eliminate any remaining. The thickness of the performed nanoPS layers was determined using a micrometer with an extremely accurate measurement (the possible error is around 1/200). Table 1 presents the preparation conditions used to fabricate nanoPS membranes, layer thickness, pore diameter calculation and porosity. The porosity distribution is assumed to be uniform across the thickness of the porous silicon layer.

Characterization Techniques
The optical reflectance and transmittance spectra of the selfstanding porous silicon layers were carried out in the 400 to 900 nm wavelength range using a Jasco V-560 double-beam spectrophotometer, equipped with an integrating sphere to avoid scattering losses.

Experimental Results and Discussions
For the study of the optical properties of porous silicon, self-standing layers with different porosities were studied. The optical constants (n, index of refraction and k, extinction coefficient) were determined from the experimental reflectance and transmittance spectra. Figure 1a and b show, respectively, the normalized reflectance and transmittance to the layer thickness at different porosities of porous silicon.
The increase in the porosity of porous silicon layer leads to a reduction in the optical reflectance and an increase in the transmittance as shown in Fig. 1. This behavior is attributed to the change in the morphology and the chemical composition difference between bulk silicon and porous silicon. Porous silicon can be described as an homogeneous mixture of air, amorphous silicon, silicon nanocrystallites and silicon dioxide [10]. Accordingly, increased porosity leads to increased optical transparency of the membranes as portrayed in Fig. 1b. Furthermore, the extremely low optical transmission of porous silicon in the 400 to 550 nm range is attributed to the very large absorption of porous silicon in the UV-visible region of the electromagnetic spectrum.
From the modeled R and T spectra, the optical constants n and k were calculated for self-standing porous silicon layers of different porosity using a simulation program based on the matrix method. Figure 2a and b show the spectral index of refraction and the spectral extinction coefficient as a function of wavelength of nanoPS self-standing layers with three different porosities. The reduction in the size of the Si nanocrystals which compose nanoPS with increasing etching current density leads to a reduction in the effective index of refraction and, at the same time, an increase in the extinction coefficient. This behavior is attributed to the different morphology and chemical composition of nanoPS layers grown under different fabrication parameters. This tendency is in agreement with the previously reported behavior of the optical constants of Al 2 O 3 and GaN porous layers, where a reduction in the index of refraction was observed with increasing porosity [18,19]. From the experimental results of the extinction coefficient, the absorption coefficient can be calculated with the relationship, Figure 3 portraits the relation between the absorption coefficient as a function of wavelength for nanoPS layers of different porosities. In all cases, the graphs show a reduction in with increasing wavelength. In addition, α has higher values for low porosity samples (e.g. nanoPS 10 mA/cm 2 ). This behavior is attributed to the reduction in the optical absorption at low porosity.
In addition, the values of the optical band gap for the nanoPS self-standing layers were extracted using Tauc plots 20]. Si is an indirect bandgap material in which the band gap energy equals 1.12 eV [21]. As has been previously demonstrated, the changes in the crystal structure and the morphology after etching the Si wafers to fabricate nanoPS lead to widening the optical bandgap and to a direct bandgap character [22]. Accordingly, the relation between ( h ) 2 and photon energy ( h ) was represented and is shown in Fig. 4a, b and c for three self-standing layers with different porosities. The band gap energy value ( E g ) can be calculated by extrapolating the straight line of the relation between ( h ) 2 and ( h ) . From the values of E g it is concluded that an increase in the porosity of the nanoPS layers leads to an increase in the value of the optical E g . This behavior is attributed to the change in the composition from bulk Si to a matrix of air (pores), amorphous silicon and Si nanocrystals [10,15,23]. Accordingly, the optical properties of nanostructured porous silicon can be adjusted depending on the intended application by the change in the morphology and porosity of the bulk material.

Conclusions
The values of the optical constants, n and k, and bandgap energy ( E g ) of self-standing nanostructured porous silicon layers as a function of the fabrication parameters were determined using a simulation program based on the matrix method. The employed model in this work avoids the usual approximations which lead to the loss of solution during the numerical inversion process. These include homogeneous layers, semi-infinite substrate, normal incidence angle for reflectance measurement, as well as smoothness of the optical constants and the use of the criteria of continuity. From the experimental results, it was found that increasing etching current density leads to a reduction in both the effective index of refraction and the extinction coefficient. However, from the experimentally determined values of E g it is concluded that an increase in the porosity of the layers leads to an increase in the value of the optical bandgap, which is associated to a reduction in the size of the Si nanocrystals which compose nanoPS.
The accurate determination of the values of the optical constants of self-standing nanostructured porous silicon layers allows us to be specifically involve it in different photonic devices with the appropriate parameters and optimal performance.