Porous Silicon Antireflective Coatings for Silicon Solar Cells

-This study presents a numerical investigation of the reflectivity of a Single Anti-Reflective Layer (SARL) and a stack of antireflective layers made of porous silicon. The stack consists of a certain number of periods, and each period contains two layers with different porosity. The simulations were conducted using the well-known Stratified Medium Theory (SMT) framework and the effect of porosity was studied. The optimal value was determined at 60% for the SARL and 65/55% for the stack of 12 periods and 6 layers. The angle of incidence was found to have more influence on the stack reflection than on the SARL reflection. The results of this investigation show that porous silicon can be used as an effective anti-reflective coating for silicon solar cells.


INTRODUCTION
The photovoltaic effect is defined as the direct transformation of electromagnetic energy (solar radiation) into directly usable continuous electrical energy, and it was discovered in 1839 [1]. The photovoltaic effect has been exploited for the design of photovoltaic solar cells. Since the 1990s, photovoltaic energy has attracted growing interest and many technological advances have been made through several technological sectors. Photovoltaic electricity production has increased exponentially since 2001, and projections are quite optimistic for the coming years [2]. Cell efficiency will certainly increase and, according to the Community Research and Development Information Service (CORDIS), the lifespan of cells in 2030 will be 40 years with an energy payback time of only one year and very satisfactory efficiency [3]. The efficiency of a solar cell is mainly limited by the various losses that occur in the cell. Among these losses, the reflection of incident photons at the surface of the cell (emitter, or window layer) contributes to the degradation of the collection efficiency of the cell. To remedy this problem, antireflection layers are often used on the front face of the cell to reduce the reflected fraction of incident radiation and improve the transport of carriers via the passivation of defects inside the cell [4][5][6]. The simplest way to create an anti-reflective coating is to deposit a quarter-wave layer of a dielectric with an intermediate refractive index between that of the emitter (or the window) and the air (TiO 2 , Si 3 N 4 , Ta 2 O 5 , etc). For silicon, for example, a layer of Si 3 N 4 quarter wavelength centered on the maximum of the solar spectrum AM1.5 reduces the reflection by 12% on average in the wavelength range 400-1100nm, increasing the photocurrent by 45% [7]. It should be noted that the reflection of silicon is between 35-40%, depending on the synthesis process [7][8][9][10]. In addition, porous silicon can be used as an alternative antireflective coating for silicon solar cells since the basis material is the same.
Porous silicon was accidentally discovered in the 1950s in an attempt to develop an electrochemical method to fabricate silicon substrates for microelectronic devices. Under the appropriate electrochemical conditions, the silicon substrate did not dissolve uniformly as expected. However, fine holes appeared, propagating primarily within the material in the [100] direction. Since this did not provide the desired smooth polish, the material was somewhat abandoned. During the 1970s and 1980s, significant interest arose because the high surface area of porous silicon was found to be useful as a model of the crystalline silicon surface in spectroscopic studies [11][12][13][14], as a precursor to generate thick oxide layers on silicon, and as a dielectric layer in capacitance-based chemical sensors [15]. Interest in porous silicon, and in particular its nanostructure, exploded in the early 1990s when Ulrich Goesele at Duke University identified quantum confinement effects in the absorption spectrum of porous silicon, and almost simultaneously Leigh Canham in England reported efficient bright red-orange photoluminescence from the material [16,17]. The quantum confinement effects arise when the pores become extensive enough to overlap with each other, generating nanometer-scale silicon filaments. As expected from the quantum confinement relationship [18], the red to green color of photoluminescence occurs at energies significantly higher than the bandgap energy of bulk silicon (1.1eV, in the near-infrared). A little after, enormous work focused on creating silicon-based optoelectronic switches, displays, and lasers. At the same time, the unique properties of the material, such as large surface area, controllable pore sizes, convenient surface chemistry, and compatibility with conventional silicon microfabrication technologies, prompted research into novel applications: various biomedical sensors, optics, and electronics applications emerged [19].
This study examines the use of porous silicon with different porosity values as an antireflection coating to reduce light reflection and improve the conversion efficiency of silicon solar cells. Two different cases were considered: single antireflection layer and multilayered stack antireflection coating. The effect of the incidence angle was examined along with the determination of the reflectivity in each case.
II. COMPUTATIONAL METHODOLOGY The Stratified Medium Theory (SMT) was used for the calculation of the reflectivity of the studied ARCs. This approach is known to combine a simple formalism and great flexibility of use. More insights and details about SMT are given in [20][21][22]. Within SMT, the inhomogeneous medium is subdivided into N inhomogeneous strata with refractive index ܰ ෩ . Each stratum is represented by a complex characteristic matrix ‫ܯ‬ . Matrices ‫ܯ‬ and ‫ܯ‬ ௦ correspond to ambient and substrate semi-infinite media, respectively, as detailed in [21,22]. The simplified relation: can be written more explicitly: where ߜ is the phase shift due to the j th stratum: and ߠ is determined from the incidence angle ߠ according to the Snell-Descartes law: For Transverse Electric (TE) polarization, ‫‬ ൌ ܰ ෩ ‫ߠݏܿ‬ , and for Transverse Magnetic (TM) polarization, ‫‬ ൌ ܰ ෩ / ‫ߠݏܿ‬ . Fresnel coefficients are calculated for both polarizations and each wavelength using: Reflectivity can then be expressed as [21,22]:

III. RESULTS AND DISCUSSION
A. Single Antireflection Layer Figure 1 represents the reflectivity as a function of the incident wavelength for a single antireflection layer made of porous silicon with different values of porosity ranging between 55 and 80%. As can be noted, the reflectivity at first increases, reaching a maximum value for wavelengths below 400nm and then decreases until the wavelengths approach 630nm. Beyond 630nm, the reflectivity increases again. Furthermore, the lowest reflectivity is obtained in the vicinity of 630nm and corresponds to porosity values of 55, 60, and 65%. The best-retained result corresponds to a porosity of 60% and therefore to a refractive index of 1.96. This is in agreement with the findings reported in [23]. However, the reflectivity in the vicinity of 630nm in this study is better than the one obtained in [23]. In general, good agreement is recorded for other wavelengths.

B. Effect of the Number of Layers in the Stack
The anti-reflective coating is obtained by stacking a certain number of periods, each of them consisting of two layers of porous silicon with different porosity values. The effect of the number of layers of the stack on the reflectivity is illustrated in Figure 2. As can be noted, the stack of layers with minimum reflectivity consists of alternating layers of respective porosities 65/60% with n L =6 periods. However, the corresponding wavelength range is relatively narrow.  [24][25][26][27][28]. These results are qualitatively similar in trend to the requirements reported in [29][30][31][32][33][34][35].

C. Effect of the Angle of Incidence for a Single ARL
The influence of the angle of incidence on the reflectivity of a simple porous silicon antireflection layer of 60% porosity with refractive index 1.96 is shown in Figure 3. Reflectivity as a function of the angle of incidence for a single porous silicon anti-reflection coating with a porosity of 60%.
As it can be observed, the reflectivity is almost zero for angles of incidence less than 30°. This reflectivity increases up to 0.05 for angles between 30° and 53°, then it increases more for higher values of the angle of incidence. Therefore, it can be concluded that the angle of incidence affects the reflectivity only slightly until the angle of incidence exceeds 55°. Qualitatively similar behavior was reported in [22]. This result is confirmed in Figure 4 which represents the dependence on the wavelength of the reflectivity of a single layer of porous silicon used as an antireflection coating on a silicon-based photovoltaic cell, for different angles of incidence ranging from 0 to 80°. It is noted that the lowest reflectivity is recorded for normal incidence, and increasing the angles to more than 30° deteriorates the reflectivity. Reflectivity vs. wavelength for a single porous silicon antireflection layer with 60% porosity for different incidence angles.

D. Effect of the Angle of Incidence for a Stack of Anti-Reflection Layers
The chosen stack of layers was the optimal one consisting of stacking of porosity 65/55% with 1.78/2.14 refractive indices and n L =6 (n L being the number of layers in the stack). The corresponding reflectivity is plotted as a function of the angle of incidence in Figure 5.  Reflectivity vs. wavelength for a stack of porous silicon antireflection layers with a porosity of 65%/55% for different incidence angles.
As can be observed, the reflectivity is between 1.66% and 18% for angles of incidence less than 20°, and between 18 and 22.7% for angles between 20°-30°, and increases rapidly for angles above 30°. This confirms that the lowest reflectivity corresponds to the normal incidence, while the reflectivity of the stack increases when moving away from normal incidence. Figure 6 shows the reflectivity as a function of wavelength for angles of incidence ranging from 0 to 80°, for a stack of porous silicon anti-reflection layers with a porosity of 65/55%. It can be noted that the lowest reflectivity is recorded for the case of normal incidence. Furthermore, as the angle of incidence exceeds 20°, the reflectivity increases and shifts towards the lowest wavelengths.
The conversion efficiency of a solar cell can be improved by making a judicious choice of the number of junctions in the cell and the materials whose energy gap can be controlled. This control can be easily obtained by alloying binary compounds to obtain ternary semiconductor alloys such as GaAs 1-x N x , GaSb 1x N x , or InAs 1-x N x [36][37][38][39]. Other alloys that contain low concentration magnetic elements, called dilute magnetic semiconductors, can also be considered [40][41][42][43]. Their gaps are controllable by varying the concentration of the magnetic element and also by taking advantage of the spin channels.

IV. CONCLUSION
The reflectivity of a single antireflective layer made of porous silicon along with that of a multilayered stack of the same material was examined to reduce the reflection losses occurring in silicon solar cells. The effect of porosity, number of layers, and incidence angle was addressed and analyzed. The optimal porosity value was found to be 60% for the single antireflective layer and 65/55% for the multilayered antireflective stack with 6 periods. The incidence angle effect was less for the single antireflective layer than for the stack. These results show that porous silicon can be used efficiently as an anti-reflective coating and can lead to improvements in the photovoltaic solar cell conversion efficiency.