*3.2.3 Current generation*

These are the Poisson equation in the presence of an electric potential φ, and the continuity equations of electrons and holes with well-specified boundary conditions [23].

The Poisson equation for semiconductors is:

$$\frac{d}{d\mathbf{x}}\left(\varepsilon \frac{d\rho}{d\mathbf{x}}\right) = -q\left(p - n + N\_D^+ - N\_A^-\right) \tag{12}$$

The continuity equations for electrons and holes are:

$$\frac{d}{d\mathbf{x}}(J\_n) = q(\mathbf{R} - \mathbf{G}) + q\frac{\partial \mathbf{n}}{\partial t};\tag{13}$$

$$\frac{d}{d\mathbf{x}}\left(f\_p\right) = -q(\mathbf{R} - \mathbf{G}) + q\frac{\partial p}{\partial t} \tag{14}$$

*ε* is the dielectric constant of the material, *φ* is the electrostatic potential, *n* and *p* are respectively the concentration of free carriers for electrons and holes, *N*<sup>þ</sup> *<sup>D</sup>* and *N*� *<sup>A</sup>*- are the densities of ionized donors and acceptors, *Jn* and *Jp* are the current densities due to electrons and holes. *R* and *G* are the rates of recombination and generation of electron-hole pairs, respectively.

$$J\_n = q\mu\_\epsilon n \nabla \rho + qD\_\epsilon \nabla n;\tag{15}$$

$$J\_p = q\mu\_p p \nabla \rho + qD\_p \nabla p \tag{16}$$

The current density in the cell can be written in the following form: (17)

$$J = J\_{\text{Geferation}} - J\_{\text{Recombination}} = -q \int\_{-d}^{W+L} G\_L(\lambda, \mathbf{x}) d\mathbf{x} - J\_{ir} + q \int\_0^{W+L} R(\mathbf{x}) d\mathbf{x} \tag{17}$$

Where *GL* is the generation function, *Jir* is the recombination current at the interface, R is the recombination function in the volume of the absorbent layer. *d*, *W* and *L* are the widths of the buffer layer, ZCE and ZQN respectively [23].

The rate of generation of electron-hole pairs at one dimension of the surface of the semiconductor is given by:

$$G\_L(\lambda, \mathfrak{x}) = a\_1(\lambda) F(\lambda) (\mathbf{1} - R(\lambda)) \exp\left(-a\_1 \mathfrak{x}\right) \tag{18}$$

*<sup>α</sup>*<sup>1</sup> (λ) is the number of incident photons per *cm*<sup>2</sup> per *<sup>s</sup>* per unit wavelength. *<sup>R</sup>*ð Þ*<sup>λ</sup>* is the fraction of photons reflected from the surface, *α*1is the absorption coefficient in the semiconductor.
