**3.2 Opto-thermal analysis**

The Amlouk-Boubaker optothermal expansivity is defined by:

$$\mathcal{W}\_{\text{AB}} = \frac{D}{\hat{\alpha}} \tag{1}$$

Where *D* is the thermal diffusivity and ˆ is the effective absorptivity, defined in the next section.

A New Guide to Thermally Optimized Doped Oxides Monolayer

**3.2.2 The Optothermal expansivity** 

**3.2.3 The optimizing-scale 3-D Abacus** 

material.

Fig. 2. The 3D abacus

in initially hard binary Zn-S material.

material.

Spray-Grown Solar Cells: The Amlouk-Boubaker Optothermal Expansivity

**AB** 

The Amlouk-Boubaker optothermal expansivity unit is m3s-1. This parameter, as calculated in Eq. (1) can be considered either as the total volume that contains a fixed amount of heat per unit time, or a 3D expansion velocity of the transmitted heat inside the

According to precedent analyses, along with the definitions presented in § 3.2, it was obvious that any judicious material choice must take into account simultaneously and conjointly the three defined parameters: the band gap E , Vickers Microhardness H <sup>g</sup> υ and The Optothermal Expansivity ψAB . The new 3D abacus (Fig. 2) gathers all these parameters

For particular applications, on had to ignore one of the three physical parameters gathered

The projection in Hυ - Eg plane, which is interesting in the case of a thermally neutral

It is the case, i.e. of the ZnS1-xSex compounds, it is obvious that the consideration of Band gap-Haredness features is mor important than thermal proprieties. The E - H <sup>g</sup> υ projection (Fig. 3) gives relevant information: the selenization process causes drastical loss of hardness

in the abacus. The following 2D projections have been exploited:

and results in a global scaling tool as a guide to material performance evaluation.

AB 31

#### **3.2.1 The effective absorptivity**

The effective absorptivity ˆ is defined as the mean normalized absorbance weighted by AM1.5 *I*( ) , the solar standard irradiance, with : the normalised solar spectrum wavelength:

$$\begin{cases} \tilde{\mathcal{A}} = \frac{\mathcal{\lambda} - \mathcal{\lambda}\_{\text{min}}}{\mathcal{\lambda}\_{\text{max}} - \mathcal{\lambda}\_{\text{min}}}\\ \mathcal{\lambda}\_{\text{min}} = 200.0 \text{ nm} \quad \mathcal{\lambda}\_{\text{max}} = 1800.0 \text{ nm}. \end{cases} \tag{2}$$

and :

$$\alpha = \frac{\int I(\tilde{\mathcal{A}})\_{\text{AM1.5}} \times \alpha(\tilde{\mathcal{A}}) d\tilde{\mathcal{A}}}{\int I(\tilde{\mathcal{A}})\_{\text{AM1.5}} d\tilde{\mathcal{A}}} \tag{3}$$

where: AM1.5 *I*( ) is the Reference Solar Spectral Irradiance.

The normalized absorbance spectrum ( ) is deduced from the Boubaker polynomials Expansion Scheme *BPES* (Oyedum et al., 2009; Zhang et al., 2009, 2010a, 2010b; Ghrib et al., 2007; Slama et al., 2008; Zhao et al., 2008; Awojoyogbe and Boubaker, 2009; Ghanouchi et al.,2008; Fridjine et al., 2009 ; Tabatabaei et al., 2009; Belhadj et al., 2009; Lazzez et al., 2009; Guezmir et al., 2009; Yldrm et al., 2010; Dubey et al., 2010; Kumar, 2010; Agida and Kumar, 2010). According to this protocol, a set of *m* experimental measured values of the transmittance-reflectance vector: 1.. ( ); ( ) *ii ii i m T R* 

versus the normalized wavelength 1.. *<sup>i</sup> i m* is established. Then the system (4) is set:

$$\begin{cases} R(\tilde{\mathcal{A}}) = \left[ \frac{1}{2N\_0} \sum\_{n=1}^{N\_0} \xi\_n \times B\_{4n}(\tilde{\mathcal{A}} \times \mathcal{B}\_n) \right] \\ T(\tilde{\mathcal{A}}) = \left[ \frac{1}{2N\_0} \sum\_{n=1}^{N\_0} \xi\_n^{\prime} \times B\_{4n}(\tilde{\mathcal{A}} \times \mathcal{B}\_n) \right] \end{cases} \tag{4}$$

where *<sup>n</sup>* are the 4n-Boubaker polynomials *B*4n minimal positive roots (*N*0 is a given integer and *<sup>n</sup>* and ' *n* are coefficients determined through Boubaker Polynomials Expansion Scheme BPES.

Finally, the normalized absorbance spectrum ( ) is calculated using the relation (5) :

$$\alpha(\tilde{\lambda}) = \frac{1}{d\sqrt{2}}\sqrt{\left(\ln \frac{1 - R(\tilde{\lambda})}{T(\tilde{\lambda})}\right)^2 + \left(\ln \frac{\left(1 - R(\tilde{\lambda})\right)^2}{T(\tilde{\lambda})}\right)^2} \tag{5}$$

where *d* is the layer thickness.

The effective absorptivity ˆ is calculated using (Eq. 3) and (Eq. 5).

#### **3.2.2 The Optothermal expansivity AB**

30 Solar Cells – New Aspects and Solutions

min max 200.0 nm ; 1800.0 nm.

 

AM1.5

( )

Expansion Scheme *BPES* (Oyedum et al., 2009; Zhang et al., 2009, 2010a, 2010b; Ghrib et al., 2007; Slama et al., 2008; Zhao et al., 2008; Awojoyogbe and Boubaker, 2009; Ghanouchi et al.,2008; Fridjine et al., 2009 ; Tabatabaei et al., 2009; Belhadj et al., 2009; Lazzez et al., 2009; Guezmir et al., 2009; Yldrm et al., 2010; Dubey et al., 2010; Kumar, 2010; Agida and Kumar, 2010). According to this protocol, a set of *m* experimental measured values of the

() ()

*I d*

 

> 

AM1.5

*I d*

min max min

 

1

0 1

ˆ

 

is the Reference Solar Spectral Irradiance.

0

 

*T R* 

> 

 

0

*N*

<sup>1</sup> ( ) ( ) <sup>2</sup>

 

0

 

'

<sup>1</sup> ( ) ( ) <sup>2</sup>

 

 

<sup>2</sup> <sup>2</sup> <sup>2</sup> 1 1 ( ) (1 ( )) ( ) ln ln 2 ( ) ( )

ˆ is calculated using (Eq. 3) and (Eq. 5).

*d T T* 

0 1

*n N*

*R B N*

*T B N*

0 1

*n*

4

*nn n*

 

4

*R R*

*<sup>n</sup>* are the 4n-Boubaker polynomials *B*4n minimal positive roots (*N*0 is a given integer

*nn n*

 

are coefficients determined through Boubaker Polynomials Expansion

is established. Then the system (4) is set:

 

 

ˆ is defined as the mean normalized absorbance weighted by

: the normalised solar spectrum wavelength:

is deduced from the Boubaker polynomials

is calculated using the relation (5) :

 

 

(5)

(2)

(3)

(4)

**3.2.1 The effective absorptivity**  The effective absorptivity

AM1.5 *I*( ) 

and :

where: AM1.5 *I*( ) 

where

and *<sup>n</sup>* 

Scheme BPES.

 and ' *n* 

where *d* is the layer thickness. The effective absorptivity

, the solar standard irradiance, with

The normalized absorbance spectrum ( )

transmittance-reflectance vector: 1.. ( ); ( ) *ii ii i m*

Finally, the normalized absorbance spectrum ( )

 

versus the normalized wavelength 1.. *<sup>i</sup> i m*

The Amlouk-Boubaker optothermal expansivity unit is m3s-1. This parameter, as calculated in Eq. (1) can be considered either as the total volume that contains a fixed amount of heat per unit time, or a 3D expansion velocity of the transmitted heat inside the material.
