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In this work, the excitons distribution function in organic bulk hetero junction solar cells, at a depth z has been determined from solving the charge continuity equation, by exploiting the Laplace transform with appropriate boundary conditions. Next, the influence of the electron-hole pair separation distance on the excitons dissociation probability, the internal quantum efficiency and the binding energy, has been studied. The simulated results show that the probability density of the carriers photo generated depends on the generation rate, excitons dissociation and the charge carriers in the cells. The potential improvement of the internal quantum efficiency of charge generation depends on electron-hole pair separation distance, the excitons dissociation probability into free charges and depends strongly on the optical absorption of the photons in the active layer.

The active layer of organic solar cells is based on the concept of donor-acceptor heterojunction introduced by C. W. Tang to create a better charge separation [_{SC} and a net increasing in the internal quantum efficiency [

This energy conversion efficiency of about 10.7% obtained by Heliatek and M. A. Green et al., reached the high power conversion efficiency of 12% recently, and obtained on a standard size of 1.1 cm^{2} combining two different absorbing materials [

The electronic and optical properties such as the efficiency of photon absorption, the exciton dissociation rate and geminate recombination, exciton diffusion length and nongeminate bimolecular recombination can play important roles in the performance of organic bulk heterojunction solar cells [_{OC}, the short circuit current density J_{SC} and therefore the efficiency of the device [

Considerable interest for an improvement to the quantum efficiency conversion by incorporating plasmonic nanoparticles (NPs) in active layer of organic solar cells has been showed experimentally [

To determine the optical propagation in the organic solar cells, the transfer matrix method based on Maxwell’s equations is used by considering optical absorption of plasmonic active layers [_{ij}_{ }and transmission t_{ij} coefficients for a propagating plane wave along the surface normal between two adjacent layers j and k have been determined from Fresnel’s theory by considering the effect of optical interference [

Recently, in order to understand the origin of the phenomena of surface losses, W. Yang et al. have defined quantitatively and calculated the loss probability of free charge carriers at the metal/organic (M/O) interfaces from numerical simulations [

C. De Falco et al., for highlighting the role of the exciton dynamics in determining the transient activation time in organic photovoltaic devices, they carried out a temporal semi-discretization using an implicit adaptive method for the numerical treatment [

However, there is very little theoretical works on the distribution theory of the phenomenon of the charge photogeneration and the excitons dynamic in organic bulk heterojunction solar cells. Moreover, charge photogeneration phenomenon, transport phenomenon, recombination and excitons dissociation into-free-charge, the surface loss of free charge carriers at the metal/organic (M/O) interface are important processes in the organic solar cells.

The objective of this paper is to study theoretically the photogeneration and excitons distribution function in the organic bulk heterojunction solar cells in order to understand the excitons dynamics in the cells. The charge continuity equation has been solved using Laplace transform and the residue theorem. Next, the influence of the Coulomb interaction between the electron and hole on the dissociation probability, the internal quantum efficiency, the binding energy distribution and efficiency of the excitons absorption have been analyzed and interpreted. The simulated results have been compared with those existing in the literature [

The paper is organized as following: in Section 2, the architecture of the organic bulk heterojunction solar cell considered is described. Subsequently, a theoretical approach of the generation mechanism, recombination and dissociation of excitons in free charge carriers and excitons transport are presented. An overview of relevant equations and the solving of the charge continuity equation are shown. The simulated results and discussion are presented in Sections 3. Finally, the conclusion and outlook are listed in Section 4.

The structure of the solar cell considered, is blend of ITO/PEDOT: PSS/rrP3HT: PC70BM (1:0.7 weight ratio)/Yb/Al. The layer Yb/Al is cathode and ITO/PEDOT: PSS is anode. The ITO (85 nm) layer on glass is a transparent semi-conductor and serves of anode in the structure. It collects the holes after the separation of excitons and the charges transport. It is composed a blend of 90% indium oxide (In_{2}O_{3}) and 10% of tin oxide (SnO_{2}). The ITO electrodes are covered with a film of PEDOT: PSS (40 nm) deposited by spin-coating from an aqueous solution. The PEDOT: PSS layer collects the holes and limits the electrochemistry between the active layer and the anode. It decreases the roughness of the ITO surface. PEDOT favours the holes transport but presents the drawback of being hydrophilic. rrP3HT: PC70BM (1:0.7 weight ratio) (100 nm) is the active layer; it is the seat of the charge photogeneration. PC70BM is a good electron acceptor; it presents a great chemical stability and good charge mobility. RrP3HT is used as the electron donor, it presents good electrical properties (good mobility of the charge carriers), good optical properties and an appreciable chemical stability for manufacturing of the organic photovoltaic devices. The Al/Yb layer collects the electrons and the aluminum (Al (200 nm)) layer presents a great surface roughness. Ytterbium (20 nm) layer limits electrochemistry between aluminum (Al) and sulfur of rrP3HT in film and allows improvement of open circuit voltage, shunt resistance and fill factor (FF). Ytterbium (Yb) forms a barrier layer between the active layer and aluminum (Al) [

The BHJ blend is represented by the HOMO and LUMO levels of the donor and acceptor. The band offset between the LUMO levels of donor and acceptor guarantees the charge separation at the interface if the lifetime of the exciton is sufficiently long to meet a split site as shown in _{OC} of the solar cell while the energy difference between the LUMO of the two materials ensures exciton dissociation [

The quantity energy of incident light absorbed by active layer of an organic solar cell depends on the complex index of refraction

where c: is the vacuum speed of light,

the absorption coefficient, and E(z): the electrical optical field at point z. ^{3} [

[

Assuming that every photon generates one exciton, the exciton generation rate at position z in the material is:

where h: is Planck constant, v: is the frequency of incident light.

We have modeled

where

In organic bulk heterojunction semiconductors, the mechanism of recombination is Langevin-type controlled by the mobility of the charge carriers [

with q: the elementary charge, J_{n}(z) et J_{p}(z): the electron and hole current density at a depth z in the active layer, G(z): the photon absorption rate, P: the dissociation probability of charge carrier pair and R(z): the bi-molecular recombination rate from the Langevin model is:

where n (z): electron density, p (z): hole density,

Incorporating both the drift and diffusion of charge carriers, we obtain the current density equations for electrons and holes, respectively:

where

The electric field distribution E(z) is determined by the Poisson equation

The built-in voltage

where

To find the exciton distribution, the continuity equation can be applied with a one dimensional diffusion model for exciton movement, the generation rate from (Equation (1)) and recombination in the bulk becomes:

where P: the excitons distribution, D: excitons diffusion constant, L: diffusion length.

Boundary conditions must be chosen to solve Equation (10) in each layer.

where

The solution of Equation (10) is obtained by determining its image using the residue theorem.

1) First case: let us assume that

The Laplace transform gives:

The image of Equation (10) is obtained by posing X = P and

The original of Equation (12) solution of Equation (10) gives the excitons distribution function in the cell:

2) Second case:

The Laplace transform gives

The image of Equation (10) is obtained:

The original of Equation (14), solution of Equation (10) gives the excitons distribution function in the organic cell:

3) Third case: we took into account, the excitons field dependent mobility parameter then we have:

The Laplace transform is expressed as followed:

The image of Equation (10) is given by:

The original of Equation (16), solution of Equation (10) gives the excitons distribution function in the organic solar cell according to the residue theorem.

The photon absorption by an organic material generates in the majority of cases an exciton of Frenkel-type. These excitons photogenerated, not all of them can be dissociated into-free carriers. In our numerical simulation, the field dependency of charges generation is processed by the Onsager-Braun model which gives the probability of electron-hole pair dissociation [

where a: the electron-hole separation distance,

Taking into account the disorder in the blend, a distribution of the excitons probability having different separation distances have been incorporated into the numerical simulations [

where f(a, z) is a normalized distribution function given by:

Charge transport in the organic solar cells device is governed by the set of continuity equations [

where n and p: electrons and holes density,

The charge carriers’ flux density is given by:

where E: electric field,

We denote by X(z) the volume density of geminate pairs and we express its rate of charge as:

where

The Equation (22) and Equation (25) become:

where

After substituting Equation (27) into Equation (10) and imposes to Equation (23), the condition:

we have:

The Laplace transform gives the images of Equation (28) for the electrons and holes:

The originals of Equation (29) and Equation (30) solution of Equation (28) gives the distribution function of the electrons and holes:

The excitons distribution function in the cell is given by the relation:

Taking into account the absorption coefficients

where

The excitons photogenerated distribution function in the active layer is given by:

Parameters used in the device model simulations: a = 1.8 nm: electron-hole separation distance in the CT state,_{irr}_{ }= I_{o}_{ }= 100 mW・cm^{−}^{2}:

We ﬁrst note that the distribution remains constant whatever the variation of the organic thickness layer. Next, it decreases linearly with an abrupt variation of the organic layer thickness.

This linear decreasing is due to the excitons loss mobility in the cell; there are drift of charges. This factor can be attributing to the loss of free charge carriers at the electrodes/organic (M/O) interfaces. It causes an increasing of the interfacial dissociation rate and a diffusivity of the electron-hole pair at D/A interface. This result has been obtained by W. Yang and al [

Considering the organic layer as an homogeneous material, the simulated results show that the excitons generation rate decreases with the increasing of the active layer thickness as described in Equation (3), which makes the corresponding average exciton generation rate (total exciton generation rate divided by the thickness) become smaller. However, when we took into account the excitons mobility, on remarked that, at the initial stage, the average exciton generation rate increases with the increasing thickness of the active layer as shown in

cause the absolute values for the peaks become small, which leads to the corresponding decrease of average exciton generation rate.

Moreover, when the active layer thickness increases from 20 to 50 nm, the total excitons generation rate increases from

_{SC}, both the optical and the electric properties should be considered at the same time.

Moreover, when the active layer thickness increases,

into free charge carriers with the increasing of the active layer thickness. These results have been obtained experimentally by [

work, a = 0.92 nm as shown in

We determined the distribution function of the excitons binding energy (U_{B}) and external quantum efficiency (EQE) as a function of the photons absorption energy by varying the electron-hole pair separation distance “a”

We determined the excitons distribution function in an organic bulk heterojunction solar cell ITO/PEDOT: PSS/rrP3HT: PC70BM (1:0.7 weight ratio)/Yb/Al using the Laplace transforms with the residue theorem. The influence of the electron-hole pair separation distance on the excitons dissociation probability, the quantum efficiency of charge carriers generation, the current density photo-generated on the excitons binding energy have been studied.

The simulated results indicate that the linear decreasing of the excitons generation function is attributed to the loss of free charge carriers at the electrodes/organic (M/O) interfaces. It causes the increasing of the interfacial dissociation rate and a diffusivity of the electron-hole pair at D/A interface. The result is consistent with the finding that there is a loss of open circuit voltage when Yb/Al is employed as the cathode which can form Ohmic contact with the PCBM active layer. We remarked that the total exciton generation rate does not monotonously increase with the increasing of the active layer thickness, but behaves wavelike which induces the corresponding variation of

Our results are in agreement with those predicted by the literature. Moreover, the phenomena of charge generation and the excitons dissociation into-free-charge carriers are important processes in organic bulk heterojunction solar cells. The potential improvement of the internal quantum density of charge generation depends on the exciton dissociation probability into-free-charge carriers and strongly depends on the photons optical absorption.

Future work can be about:

1) extensions to the model; 2) improvement of the analytical results and 3) the model study as a stochastic processes using Fokker Planck equations.

We thank Professor Jean Chabi Orou, Dr. Amoussa S. Hounkpatin, MC. Basile Kounouhéwa, Dr. M. Ossénath, Dr. K. N’GOBI Gabin and Dr. Luis Adébayo Essoun for reading the manuscript. We acknowledge the financial supports by the Trading Company SOCOMA-Benin, “Ecole Doctorale Sciences des Matériaux (EDSM)” and “Laboratoire de Physique du Rayonnement LPR, FAST-UAC 01BP 526 Cotonou, Benin”.

V. I.Madogni,W.Yang,B.Kounouhéwa,M.Agbomahéna,S. A.Hounkpatin,C. N.Awanou,11, (2015) Dynamic, Charge Photogeneration and Excitons Distribution Function in Organic Bulk Heterojunction Solar Cells. Open Journal of Applied Sciences,05,509-525. doi: 10.4236/ojapps.2015.58050