In Table 1 the characterization results of CuIn₃Se₅ materials by EDS are presented. The samples show a nearly perfect stoechiometry since the magnitude of deviation from stoechiometry, Dy ( Δ y = ( 3 × Cu ) / In − 1 ), is very small [8] . The results of the chemical compositions of a CuIn₃Se₅ sample have a large value of Dy (0.3). A detailed analysis is done in order to study the secondary phases that appear in this sample which will be discussed later on. The CuIn₃Se₅ samples present a n-type conductivity.

The spectra of different CuIn₃Se₅ samples obtained by X-ray diffraction are shown in Figure 1. These samples are well-crystallized and all existent peaks are similar to those found in previous publications [9] [10] [11] . Thus, our XR spectra show the presence of several preferential orientations according to planes (112), (220) and (312) for all samples. Since the characteristic peaks concerning the chalcopyrite structure were not observed, the conclusion was made that unlike the CuInSe₂ compounds, the CuIn₃Se₅ compounds do not have a chalcopyrite structure [1] [12] . They either have a Stanite structure [6] , an Ordered Defect Chalcopyrite structure (ODC), or an Ordered Vacancy Chalcopyrite structure (OVC).

The a and c lattice parameters of CuIn₃Se₅ have been calculated from the spectra where a = 5.76, c = 11.52. These values of a and c are in agreement with those reported in G. Marin et al. and T. Negami et al. [13] [14] .

Figure 2 presents the photoluminescence spectrum of CuIn₃Se₅ obtained at 4.2 K. The spectrum shows the exciton position which is required to determine the gap value of 1.23 eV [15] . The activation energy level implicated in the optical transition is found by subtracting the emission peak located at 1.09 eV from the gap value. In other words the activation energy is equal to 1.23 − 1.09 = 0.140 eV (140 meV).

We have determined the band gap energy value by analyzing our samples using spectral Photoconductivity [16] . Figure 3 illustrates the Photoconductivity spectrum ((αhν)² as a function of hν) of CuIn₃Se₅. This spectrum denotes high speeds of surface recombination. A saturation level at high energy was not observed. In these cases, the gap value is given by an approximate value which was found by taking the abscissa of each curve at PC_max/2 [17] . The gap value at room temperature is 1.22 eV, which match those found by Photoluminescence and in literature [18] .

a) Method

Spectroscopic Ellipsometry (SE) is a characterization technique which allows the determination of energy levels above the gap in the band scheme. The real part of the dielectric pseudo-function 〈 ε r 〉 is related to the substrate densities of state D n . The calculation of the 3^rd order derivative of the quantity E 2 ε r ( E ) [18] , is given by:

D n ( E ) = 1 E 2 d 3 d E 3 [ E 2 ε r ( E ) ]

where E = ℏ ω is the photon energy, allows the determination of the energy levels of optical transitions that correspond to the peaks of D_n. In order to illustrate this fact, a harmonic oscillator model, centred at the frequency ω 0 is used. Its’ Lorentz nature is characterized by an elastic amplitude α e and an absorption one α a :

α e = ω 0 2 − ω 2 ( ω 0 2 − ω 2 ) 2 + r 2 ω 2

α a = r 2 ω 2 ( ω 0 2 − ω 2 ) 2 + r 2 ω 2

where r is the damping constant.

α e and α a functions are shown in Figure 4. The frequency ω 0 is determined by the maximum of α a and the zero of α e . The 3^rd and 4^th order derivatives of functions α e and α a are respectively calculated. The numerical calculation of the n^th derivative is made by repeating n times the standard algorithm of the derivative calculation using three points, which is expressed as:

f ′ ( x i ) = f ( x i + 1 ) − f ( x i − 1 ) 2 Δ x

Figure 5 illustrates the spectra of α a ( ω ) and of its 4^th order derivative. Figure 6 shows the spectra of α e ( ω ) and of its 3^rd order derivative. In the 2 cases, ω 0 value was determinate. The function α a ( ω ) presents a narrow peak which permit to determine ω 0 with precision. The peaks fineness in the α a ( ω ) 4^th derivative and in the α e ( ω ) 3^rd derivative spectra, respectively (Full Width Half Maximum (FWHM) lower than that of the initial function), shows that this method is a great way to determine the ω 0 position.

To verify the accuracy of the determination of E₁, E₂... levels, this method was applied to the case of Si. ε i and ε r represent α e and α a type for each harmonic oscillator, where the energy level is an optical transition described in the band scheme. The spectra of ε_i and ε_r functions are shown in Figure 7.

The maximum 3^rd and 4^th order derivatives of ε r and ε i are shown in Figure 8 representing the transitions of energy levels in the band scheme [19] . The three energy levels observed E₁ (3.6 eV), E₂ (4.2 eV), and E₃ (5 eV) are identical to those found by Aspnes et al. [19] , which shows the efficiency of our method.

b) Case of CuIn₃Se₅

CuIn₃Se₅ samples were characterized by Spectroscopic Ellipsometry, and the observed transitions were determined by the method described above. The different spectra of the imaginary part (ε_i) and the real part (ε_r) respectively of the dielectric function, for different CuIn₃Se₅ samples are shown in Figure 9. Figure 10 illustrates the results of spectra at 3^rd and 4^th order derivatives of ε r and ε i respectively, of one sample of CuIn₃Se₅.

Zeaiter et al. [20] were studied the stœchiometry effect on the dielectric functions. They found that the imaginary dielectric function (ε_i) of CuInSe₂ evolves for different values of Cu/In ratio. In fact the peaks widen when the percentage of indium is increased, because the dielectric function is directly related to the crystal structure. Since the crystal is rich in In, involves a disorder in the crystal lattice and a loss of crystallinity which proved more significantly for high energies. This effect was observed in the case of silicon [21] [22] . So the excess of In has the effect of expanding and decreasing the intensity peaks of the imaginary dielectric function (ε_i). We noticed the same effects on our spectra of CuIn₃Se₅.

Table 2 shows the observed transitions (E₁, E₂ and E₃) for different samples A, B and C. The transitions E₁ and E₃ can be attributed to the d levels ionization of Cu. In contrary, the transition E₂ is related to the hybridization s-s between the s-levels of In and that of Cu. The highest levels of the valence band are essentially constituted by the 4p levels of Se and the d-levels of high density localized of Cu.