INTRODUCTION
Photovoltaic materials (Grasso et al., 2005;
Moller et al., 2005; Shah et
al., 2003) 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 and hence making
the penetrating rays rate to reach 90%. With the prototype presented in this
model three reflection can be minimized to less than 10%, leading to the improvement
of the spectral response, the absorption coefficient, the generation rate and
the photovoltaic efficiency.
DEVELOPMENT OF THE SUGGESTED MODEL
In this model, the textured surface is obtained by an anisotropic chemical attack on semiconductor material. The pyramids obtained by this mechanism present regular orientations and having a height comprised between 5 and 10 μm.
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.
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 II 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 I and α the angle between 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
I 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 I and II, permits
to obtain the refraction angles θ, θ’, θ”. The angles correspond respectively
to the incidences angles i, i^{′} and i”= (Table 1).

Fig. 1: 
Textured surface with three successive reflections (suggested
model) 
Table 1: 
Incidence and refraction angles 

i: 1^{er} Incidence angle, i′: 2^{ ėme}
incidence angle, i′′=: 3^{ėme} incidence angle,
θ: 1^{er} refraction angle, θ′: 2^{θme}
refraction angle, θ′′=: 3^{θme} 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 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.
SPECTRAL RESPONSES
The spectral response is an essential parameter in the characterisation 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 relation 6 becomes:
where, N(1r^{2}) represents the absorbed rays. By applying the model
that uses three successive incidences, the relations 7 becomes:
where, N(1r^{3}) represents the absorbed rays.
If we write x = J_{ph}/qN we get

Fig. 2: 
Spectral response vs. reflection coefficient (a) Ideal case,
(b) Suggested model, (c) Textured surface and (d) Normal surface 
In the case of this model the variation of the spectral response as a function of the reflection coefficient r is shown in Fig. 2.
This variation is compared on the same figure with the ideal case (complete absorption of incident rays), with the case of normal plane and with that of the textured plane. The results show that spectral response of the suggested model approaches more and more the ideal case.
ABSORPTION COEFFICIENT AND GENERATION RATE
The absorption coefficient varies linearly with the incident rays; this coefficient
is given by the relation:
where, d is the cell thickness.
Thus:
and
The generation rate is given by the relation:

Fig. 3: 
Absorption coefficient vs. reflection coefficient, (a) Ideal
case, (b) Suggested model, (c) Textured surface and (d) Normal surface 

Fig. 4: 
Generation rate vs. reflection coefficient, (a) Ideal case,
(b) Suggested model, (c) Textured surface and (d) Normal surface 
Figure 3 and 4, show the variation of the
absorption coefficient and the generation rate against the reflection coefficient
for d= 100 μm.
These results show that these two parameters approach the theoretical ideal case.
CONCLUSION
This study is based on the use of successive reflections 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. 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 materials.