INTRODUCTION
Photovoltaic materials (Grasso et al., 2005;
Moller et al., 2005; Shah et
al., 2003; Metzdorf et al., 2000; Lawrence
et al., 2000; Mashanovich et al., 2008;
Usuda et al., 2007) are distinguished by an index
of refraction greater then 3 and a high reflection coefficient in the visible
spectrum.
The reflection can be more reduced by covering the cell surface with an antireflective
layer in order to bring the percentage of the reflected rays to a reasonable
value. In effect a normal plane of silicon can reflect up 35% of the received
rays (Goetzberger et al., 2003). This rate can
be reduced to 10% if the plane is covered with an antireflective layer (Caneau
et al., 1993) and hence, making the penetrating rays rate to reach
90%. With the model of three successive reflection (Hamel
et al., 2009) can be minimized to less than 10%, leading to the improvement
of the spectral response, the absorption coefficient and the generation rate.
TEXTURED SURFACE WITH THREE SUCCESSIVE REFLECTIONS
The model suggested in this study allows the material to have three successive absorptions of the incident rayon, by varying the incident angle i, the aperture between the pyramids f and their height h (Height varying between 5 and 10 μm).
This model is based on reflection and refraction laws of incident rays on the surfaces of two neighbouring pyramids. By considering N the number of rays that are incident on the surface of the pyramid I and r the reflection coefficient, the proportion of the absorbed rays by the material is given by N(1r) whereas, that of the reflected rays is Nr. These tatters fall on the surface of the pyramid 2 where they are absorbed with Nr(1r) proportion, while Nr^{2} proportion is reflected. A change of the aperture between the summits the neighbouring pyramids will allow the Nr^{2} rays to fall a second time on pyramid I (Fig. 1).
This mechanism will permit to recuperate an third proportion of the incident rays Nr^{2 }(1r), that will participate to the improvement of the photovoltaic properties such as the spectral response, the absorption coefficient and generation rate. The total amount of the absorbed rays in the sum of the three successive incidences and is given by N (1 r^{3}).
Let φ to be the angle between the incident ray Nr^{2} and the face of pyramid 1 and α the angle between the two neighbouring pyramids:

Fig. 1:  Textured surface with three successive reflections 
Table 1:  Incidence
and refraction angles 

i: 1^{er} incidence angle, i’: 2^{ème}
incidence angle, i’=: 3^{ème} incidence angle, θ:
1^{er} refraction angle, θè’: 2^{ème}
refraction angle, θ’=: 3^{ème} refraction angle,
φ: angle between the incident ray Nr^{2} and the face of pyramid
I, α: angle between the two neighbouring pyramids 
Table 2:  Distance
between the summits of two neighbouring pyramids 

i:
incident angle, f : aperture between the summits of the two neighbouring
pyramids 
The sum of a triangle angles is π, Thus:
And as φ>0 then:
If i represents the angle of the first projection on the surface of pyramids 1 and i’ the angle the second projection on the surface of the second pyramid then:
In the case of crystalline silicon and for wave length λ = 590 nm; the
application of Snell‘s law between the surfaces of the pyramids 1 and 2,
permits to obtain the refraction angles θ, θ’, θ’=.
The angles correspond respectively to the incidences angles i, i^{’}
and i’= (Table 1).
The aperture f between the summits of the two neighbouring pyramids is given by the relation:
The calculated values of f for different heights h and different incidence angles i are assembled in Table 2.
ANTIREFLECTIVE LAYER
One way to mix angles is bay photon reabsorption and reemission, but this requires extremely good material quality and low parasitic absorption.

Fig. 2: 
Typical photon trajectories. (a) In the planer structure,
photons’ trajectories are randomized by selfabsorption/reemissions
events represented by dots (photons recycling) and (b) in the textured surface,
angular randomization is achieved by strong surface 
Photon recycling implemented with high quality material (Fig.
2a).
However, it was also demonstrated that this high external efficiency is very
susceptible to parasitic loss mechanisms (freecarrier absorption in doped layers,
for example) or slight degradation in the internal quantum efficiency. Angular
randomization by photon recycling is not practical since, it is vulnerable to
the slightest deterioration of material or mirror quality. A more practical
approach and the one which we adopt here, is angular randomization by scattering
of the photons from textured semiconductor surfaces. Thus, the textured surface
geometry, schematically shown in Fig. 2b, can boost the external
efficiency to 50% or more.
SPECTRAL RESPONSE, ABSORPTION COEFFICIENT AND GENERATION RATE
The spectral response is an essential parameter in the characterization of solar cells for a silicon normal plane this parameter in given by the relation:
where, N(1r) represents the proposition of absorbed rays.
For a textured plane the Eq. 6 becomes:
where, N(1r^{2}) represents the absorbed rays. By applying the model that uses three successive incidences, the Eq. 7 becomes:
where, N(1r^{3}) represents the absorbed rays. If we write x = P_{ph}/qN we get:
The absorption coefficient varies linearly with the incident rays; this coefficient is given by the equation:
where, d is the cell thickness.
Thus:
and
The generation rate is given by the equation:
If we cover the surface of the cell with an antireflective layer we will get good results especially for the treated plan surface where reflection coefficient r is near zero and consequently the spectral response, the absorption coefficient and the generation rate increases almost to the ideal values, which helps improve In Fig. 3 sums up what has been mentioned above for the various wavelengths located in the visible spectrum.

Fig. 3:  Spectral
response vs. wavelength. (a) ideal case, (b) with an antireflective layer
(d = 0, 12 μm) and (c) without an antireflective layer 

Fig. 4:  Absorption
Coefficient Vs. wavelength. (a) ideal case, (b) with an antireflective
layer (d = 0, 12 μm) (c) without an antireflective layer 

Fig. 5:  Generation
rate Vs. wavelength. (a) ideal case, (b) with an antireflective layer
(d = 0, 12 μm) and (c) without an antireflective layer 
Figure 4 and 5, show the variation of the
absorption coefficient and the generation rate against the various wavelengths
located in the visible spectrum.
These results show that these two parameters approach also the theoretical ideal case.
CONCLUSION
This study is based on the use of the antireflective layer on the surface
of textured planes of solar cells, in order to improve the photovoltaic efficiency.
For achieving this goal we developed a model that can recuperate a second reflection
instead of one currently, by varying the incidence angle and the aperture between
the neighbouring pyramids. This model permits the solar incident rays to have
three successive absorptions by the material. The calculations of incidence
angles on the pyramids surfaces and the aperture f between the neighbouring
pyramids were carried out for different pyramid heights. The application of
the suggested model shows a significant improvement of the photovoltaic parameters
such as the spectral response, the absorption coefficient and the generation
rate.
The representative curves of these parameters in the case of this model approach
those representing the ideal case and if we cover the surface of the cell with
an antireflective layer we will get good results. In conclusion we can say
that this model can contribute to a significant improvement of the photovoltaic
efficiency and can be applied to other photovoltaic material.