Antireflection coatings (ARC) are used in processing solar cells to reduce reflection and to offer better passivation properties  . This results in a significant increase in the current generation, leading to an improvement in the cell efficiency. Modern ARCs are usually fabricated using single or multilayer thin films. An ideal antireflection structure should lead to zero reflection loss on solar cell surfaces over an extended solar spectral range for all incident angles. It is well known that normal-incidence reflection at a specific wavelength (usually 600 nm) can be minimized using a single layer coating with quarter wavelength optical thickness. However, the materials used to deposit an antireflection coating must have a refractive index n = (nSi × n0)1/2, where n0 is the refractive index of the surrounding medium. Single-layer thin film antireflective coatings are limited by the availability of materials with required refractive indices  . But a single-layer antireflection coating is also known to be unable to cover a broad range of the solar spectrum   , and using double-layer antireflection coating is considered. The refractive indices and thickness of each of the upper and bottom layers forming the antireflection stack must satisfy a number of conditions.
In this study, a numerical optimization of the antireflection proprieties of simple layer and double layers are performed by modeling the refractive index and thickness of adequate materials. Thus, to better evaluate the reflectivity on the surface of a cell coated with a single or double antireflection layer, two new concepts will be introduced: the refractive index difference (RID) and the thickness difference (TD). RID and TD values will help to predict how the performance of a solar cell panels will be affected from reflectivity losses.
2. Models Description and Optimization Procedure
The matrix method for calculating spectral coefficients of the first layered media was suggested by F. Abeles (1950) and has been ever since widely employed  . For each configuration, a modified transfer matrix method    is used to calculate the reflectance from the silicon surface. It is used for the reflectance simulation. It is essential to calculate the average reflectivity Rw due to the solar spectrum’s long range of wavelength from 400 to 1100 nm.
For a PV cell, it is important to have a minimum of reflection over a whole spectral range. The average residual reflection factor is defined by   :
where and are the maximum and minimum values of the wavelength range respectively. is the reflection factor. In order to compare the effectiveness of an AR coating on a solar cell, it is important to take the AM 1.5 solar spectrum into account. For this investigation, we have chosen the wavelength range from 400 nm to 1100 nm of the solar spectrum. The reason is that for wavelengths shorter than 400 nm, the spectral power density in the AM 1.5 spectrum is almost zero, while photons with wavelengths longer than 1100 nm are hardly absorbed by the Silicon.
The reflection can be calculated by the transmission matrices of the ARC layers. It depends on both and . For each configuration, a transfer matrix method     is used to calculate the reflectance from the silicon surface. The ARC structures represented in Figure 1(b) can be
Figure 1. (a) Structure of silicon solar cell with a single layer antireflection coating; (b) Structure of silicon solar cell with a double stack antireflection coatings.
considered as composed of two layers (therefore, three interfaces) on a Si substrate (Figure 1(b)).
First, we describe the characteristic matrix of a single layer. The relationship of matrix defining the problem of single antireflection layer is given by the following relation
M is a matrix given by:
The characteristic matrix of a multilayer is a product of corresponding single layer matrices. For a double stack antireflection coating, relation 2 become:
with , represent respectively the refractive index of the top and the bottom antireflection layers. and are respectively dephasing between the reflected waves of layers k and k + 1.
The detailed derivation of amplitude reflection (r) and transmission (t) coefficients is given in   . The resulting expressions are shown in Equation (6) and Equation (7).
Mij are the elements of the characteristic matrix of the multilayer. The energy coefficients (reflectivity, transmissivity, and absorptance) are given by:
with nSi and no is refractive index of the silicon and vacuum respectively.
2.1. Simple Layer Antireflection Coating
In order to reduce reflection, a film with intermediate index of refraction can be applied according to Figure 2. If the optical film thickness (d × nSi) is 1/4 of the wavelength (l), the phase difference becomes π and the two reflected waves cancel out. For complete annihilation, the amplitude of the interfering radiation also has to be identical. Under normal incidence, this requirement is fulfilled when the refractive index of the film is equal to the square root of the refractive index of the substrate    . Then, optimal thickness and optimal refractive index of the antireflection layer are given respectively by following relations:
l is wavelength (m), n0 is the air refractive index and it taken equal to 1. ncar and nSi represent refractive indices of AR layer and silicon respectively. Equation (12) and Equation (13) permit us to obtain the following figure.
Figure 2. Optimal refractive index (curve green) and optimal thickness (blue) of simple layer antireflective coating for silicon, for annihilation of reflectance at wavelength l(nm).
Figure 3 shows the result refractive index and thickness of optimal simple ARC as a function of wavelength. According to Figures 2-5, the optimal thickness of an ARC for silicon (blue curve) to cancel the reflection at the wavelength λ, can be approximated as the following equation:
By replacing this thickness in Equation (2), the optimal refractive index is then given by:
Relation which can be written as follows:
with A = 1.736, B = 134.12 and C = 77.26
We find the Cornu equation  giving the refractive index of a transparent materials.
The refractive index can also be found by extrapolation of the curve 2-5. We obtain the Cauchy equation  . These equations are completely empirical and were first proposed by Cauchy (1789-1827).
Following the Cauchy equation, the optimal refractive index of a single ARC is given by the following expression:
Figure 3. Comparison of different models (Cornu, Cauchy) giving the optimal refractive index (left) and the optimal thickness (right) of single layer ARC with those given by the conditions of phase and amplitude.
Figure 4. Optimal refractive indices (left) and optimal thicknesses (right) of upper (narc1, earc1) and bottom (narc2, earc2) antireflection layers for a double stack on the silicon surface.
Figure 5. Comparison of different models (Cauchy, Cornu and exact values) giving the optimal refractive indices (left curve) and the optimal thicknesses (right curve) of each of the upper and bottom layers ARC as a function of the wavelength.
These results are summarized in Table 1.
It can be seen from Figure 3 that there is no general trend in the variation of reflectance with the models. However, when the wavelength is inferior to 480 nm, the curve corresponding to Cornu model hasn’t the same allure with other. Table 2 shows the influence of the thickness and the refractive index of an antireflection layer for the annihilation of reflectance at a wavelength λ. As can be seen, the reflectivity is almost zero for each of wavelengths. The choice of an antireflection layer material is accordingly guided by its refractive index (we consider that its absorbance is negligible); then we look for the optimal thickness.
2.2. Double Layer Antireflection Coating
For a double layer coating for a zero reflection minimum in a large wavelength
Table 1. models of optimum thickness and refractive index of a single layer ARC for silicon.
Table 2. Characteristics of an optimal antireflection layer (for silicon solar cells) to eliminate reflection at the wavelength λ.
range, the refractive index of the two layers have to fulfill these following relations  :
And the optimal thicknesses of the top and bottom layers are respectively equal to:
According to Equation (5) and Equation (6), the refractive index and thicknesses of an optimal double antireflection layer are given respectively by Figure 5 and Figure 6. These figures show the optimal values of the refractive index (left) and the thicknesses (right) of each of the upper and bottom layers forming the stack.
According to the curves (3) on the right, these thicknesses are governed by the following equations:
The thicknesses and wavelength are in nanometer (nm). From relation (6), the optimal refractive index is given by the following expression:
These relations can be written in the form of the dispersion equation giving the Cornu refractive index:
Figure 6. Reflectance spectra for a stack of four-layer ARC with the order of their deposition on silicon (A, B, C and D) under normal incidence.
Besides by extrapolation of the left Figure 3, we obtain the refractive index according to the Cauchy equation. We can then deduce the optimal thicknesses.
These results are summarized in Table 3.
Figure 4 on the left shows the comparison of different models giving the refractive index of the optimal antireflection layers (upper and bottom). As can be seen, curves of refractive index of the inner AR layer are same allure. This means that the models used reflect the behavior (refractive index) of the optimal antireflection layer. The same observation is noted for the upper layer. But the Cauchy model better reflects the behavior of the optimal refractive index. Indeed for wavelengths less than 480 nm, the curve giving the refractive index obtained with the model of Cornu is detached from those of Cauchy and exact values. Curve 6 on the right compares the models giving the optimal thicknesses of the upper and inner antireflection layers for a minimization of the average reflectance. As can be seen, these curves have same allure.
3. Refractive Index Difference (RID)
In order to better search for the ideal materials for an antireflection layer on the silicon surface, we introduced the notions of refractive index difference (RID) and thickness difference (TD). Thus, the refractive index difference of a material at the wavelength λ is called the spectral refractive index difference (SRID). Over the whole range of the spectrum considered, we speak of difference of total refractive index difference, which is the sum of all the difference of spectral refractive index on the spectrum [400 - 1100 nm].
3.1. Refractive Index Difference for Simple Layer Antireflection Coating
Table 3. Modeling of refractive index and optimal thicknesses for double layer antireflection coatings ARC1/ARC2 with reflective indices narc1/narc2 and thicknesses earc1/earc2.
layer n(λ) and the optimal one, given by the phase condition at the wavelength λ.
can be given by the following relation:
Relation (15) is the spectral refractive index difference (SRID).
To obtain the total refractive index difference, we sum on all the wavelengths of the spectrum considered. The expression of the total refractive index difference (or refractive index difference) is then given by the following relation:
3.2. Refractive Index Difference for Double Layer Antireflection Coatings
Let be the refractive index difference between the upper layer of refractive index and that of optimal refractive index for a layer of the same position and the difference between the refractive index of the bottom layer ARC and that optimal for a layer of the same position (described by relation 13). Then, the difference in refractive index for a double AR layer at a wavelength is given by the average of and .
and represent respectively the optimal refractive indices of the top and the bottom antireflective layers. And represent respectively refractive indices of the chosen antireflective coatings of top and bottom layers
Taking into account relations 20 and 21, it comes:
Relation (19) is the spectral refractive index difference, which depending only on the wavelength and the refractive index of the bottom and top layers. Over the entire solar spectrum, the refractive index difference (RID) is defined by the sum of all differences of spectral refractive index.
3.3. Case of Antireflective Multi Layers
If the number of layers is greater than 3, the refractive index difference will be defined in another way. Consider a stack of N layers deposited follow that order with refractive indices , respectively. The spectral refractive index difference for a multilayer coating is defined by the following equation:
In the case where the study is done on all the wavelengths, refractive index difference (RID) is defined by the following relation:
4. Thickness Difference (TD)
4.1. Thickness Difference for Single AR Layer
It is defined as the difference between the optimal thickness ( ) of an antireflection layer for silicon and that given by the amplitude condition.
Following relations (14) and (38), thickness difference (TD), is related to the wavelength l by the following expression:
4.2. Case of Double AR Layers
Let e1 and e2 be the thicknesses of the respective lower and upper layers forming the AR double layer stack. Noting earc1 and earc2 those optimal for a double-layer antireflection coating, the thickness difference (TD) of a double antireflection layer is defined by the following relations:
ncar1 and ncar2 respectively represent the optimal refractive index of the inner and the upper antireflection layers. According to relations (19) and (20), these expressions can be written as following relations:
Thickness difference (TD) of a double layers is presented as follows: , l = 600nm is chosen as wavelength of reference in this paper.
5. Results and Discussion
Average Reflectance (Rav) Depending on Refractive Index Difference (RID)
1) Case of Simple Layer Antireflection Coating.
Table 4 shows the average reflectance values and the corresponding refractive indices difference for each of the AR layers. As can be seen, there is a dependency relationship between the refractive index difference and the average reflectance. Reflectivity is as lower than this difference is low. For example, a low reflectivity of 9.1% is obtained with the SiNx corresponding to a low refractive index difference of 61; and a refractive index difference of 445 corresponds to a large reflectivity of 15%. These results are explained by the fact that this refractive index difference is even bottom; the refractive index of the AR layer is close to that of the optimal antireflection layer (given by the phase condition). Using Table 4, the variation of the average reflectance Rav on the silicon coated with a single antireflection layer can be modeled as a function of the total refractive index difference (Dn) by the following relation:
Table 4. Reflectance on silicon surface coated a single antireflection layer, depending of refractive index difference. Optimal thickness (Δe = 0) were taken for each of the AR layer.
2) Case of Double Layer Antireflection Coatings.
Reflectance losses values and refractive index difference (RID) corresponding for each of double stack ARCs are presented in Table 5.
Table 5 shows the dependence between refractive index difference and average reflectance. As can be seen, the reflection is weaker as this difference is small. The average reflectance of the SiO2/TiO2 AR double layer is lower, corresponding to an index difference of 84. This reflection is greater for the double SiO2 /Al2O3 antireflection layer (10.8%) corresponding to a refractive index difference of 278. However, the refraction index difference for a single antireflection layer is not comparable to that coated with a double antireflection layer. In other words, the reflection on a cell coated with a double AR layer may be greater than that coated with a single AR layer while the latter has a difference in refractive index greater than the first. The average reflectance (Rav) on the silicon surface coated with a double antireflection layer is a function of the refractive index difference (Δn) and can be modeled as follows:
3) Case of multilayer antireflection coatings
Consider these different configurations of deposition of the AR layers on silicon with the same materials:
➢ A. TiO2/Al2O3/ZrO2/SiO2 ARC on silicon (degrowth of refractive indices).
➢ B. SiO2/Al2O3/ZrO2/TiO2 ARC on silicon (growth of the refractive indices).
➢ C. ZrO2/Al2O3/TiO2/SiO2 ARC on silicon (refractive indices neither growth nor degrowth).
➢ D. Al2O3/ZrO2/SiO2/TiO2 ARC on silicon (refractive indices neither growth nor degrowth).
Table 6 shows the reflectivity of a stack multilayer depends greatly on deposition order of these layers on silicon surface. In fact that deposition order governs the refractive indices difference values those impacts directly on solar cell reflectivity. Figure 6 confirms theses results.
Table 5. Dependences of the average reflectance on the refractive index difference (RID) for a silicon surface coated double layer antireflection coatings.
Table 6. Importance of deposition configuration of the multi stack antireflection coating on silicon surface for minimizing reflectance losses.
With Refractive index difference (RID) between layer number j and layer number k: then represents RID between the n0 (air refractive index) and n1 (refractive index of upper layer).
Figure 6 shows the variation of the reflectivity of a four-layer AR stack according to their deposition orders (A, B, C and D) on the silicon surface. Among these different curves, that A representing the order of deposition where the refractive index increase from nair to that of silicon (nSi) leads to a lower reflectance (less than 4%) over the entire spectrum wavelength range of solar. These results were expected. Indeed, the high reflectivity on the surface of silicon solar cells is due to the large discontinuity between the refractive index that exists at the interface (air-cell). The deposition of a stack where the refractive indices grow, makes a sudden change in this index is replaced by a continuous transition from a low refractive index material to a high refractive index material. Thus, for a minimization of the reflectivity with a multilayer stack, the refractive index difference must be minimal. Consequently, the refractive indices of the different layers composing this AR stack must grow from the refractive index of air to that of silicon.
6. Average Reflectance Depending on Thickness Difference (TD)
6.1. Case of Simple Layer Antireflection Coating
As shown in Figure 7, the average reflectance increases rapidly with this thickness difference. In fact, for antireflection layers of thicknesses less than or greater
Figure 7. Thickness difference (TD) dependence of the reflectance spectra of different single antireflection layer.
than optimal, in other words, as one moves away from the zero thickness difference (Δe = 0), the reflectivity grow rapidly. ZrO2 ARC is an example where a zero refractive index difference (Δe = 0 then e = 72 nm) corresponds to an average reflectance of 9.4% while thickness difference of Δe = 10 (e = 82 nm) and Δe = −10 (e = 62) correspond to mean reflectivity of 10% and 10.4%, respectively. That is an increase in reflection of more than 6% in each case.
6.2. Case of Double Layers Anti-Reflection Coatings
At the reference wavelength λref = 600 nm, the thickness value of each layer given by the phase condition, (relation 8) is compared with that optimal (making the minimum reflectance) for different types of antireflection layer. The thickness difference (TD) and average reflectance values then obtained with each of antireflection layer) are summarized in Table 7.
In Table 7, for each antireflection layer, optimal thickness is used for calculating average reflectance. As can we see, there is little difference between optimal thickness and that given by phase condition. Consequently, the thickness difference (TD) values are low.
Figures 8(a)-(f) show variation of reflectance losses with thickness difference (TD) for each of stack double layer ARC on silicon.
As can be seen, the reflectivity on a double layer AR strongly depending on the respective thicknesses of the different stacks constituting it. For each type of AR layer, there is a couple of optimal thickness minimizing the average reflectance of the light rays arriving on the surface. As can be seen, this pair of thickness differs little from that given by the relationships (25) which would lead to a
(a) (b) (c) (d) (e) (f)
Figure 8. (a) Reflectance according to the thickness difference of the SiO2/TiO2 double layer antireflection coatings; (b) Reflectance according to the thickness difference of the SiO2/ZrO2 double layer antireflection coatings; (c) Reflectance according to the thickness difference of the SiO2/Al2O3 double layer antireflection coatings; (d) Reflectance according to the thickness difference of the SiO2/SiNx (n = 2.2) double layer antireflection coatings; (e) Reflectance according to the thickness difference of the SiO2/SiNX (n = 2) double layer antireflection coatings; (f) Reflectance according to the thickness difference of the SiO2/SiNX (n = 1.8) double layer antireflection coatings.
zero thickness difference ( ) of the curves 8, only the one representing the AR double-layer stack SiO2/SiNx (n = 1.8) leads to a minimal reflectance outside the zone where the thickness differences are zero. Indeed, in
Table 7. Influence of the thickness difference of each antireflection layer on average reflectance.
this case the reflectivity is minimal (2% on average) if Δe (SiNx) is between −15 and 0; and Δe (SiO2) is between −20 and −10.
We have modeled the optimal refractive index for single layer and multilayer antireflection coatings for minimization of reflectivity losses at the silicon surface. Silicon nitride (SiNx, n = 1.8) single layer ARC reduces consequently reflectance but It was found that the antireflection effect of SiO2/ARC2 double-layer ARC is better than that of single layer. For SiO2/SiNx double-layer ARC, the optimal antireflection effect is obtained with refractive indices of 1.46 and 2 for the top and the bottom layer, respectively. Refraction Index Difference (RID) and thickness Difference (TD) have allowed us to better understand the mechanisms of photon losses at the surface of silicon solar cells coated antireflection layer: Reflectivity is even lower than these refractive index difference and thickness difference are low. The results reported in this study can be used as a significant tool for efficiency improvement in thin film silicon solar cells. However, for an ideal antireflection layer, the absorption loss especially in the wavelength range [400 - 1100 nm], must be low.