**3. Results and discussions**

#### **3.1. Two-level systems with local interaction**

Relying on the new formalism described in the previous section, here we are going to investigate the effects of short-range electron-hole interaction on the performance of a molecular photocell. The short-range interaction term implies that interaction between the electron and the hole occurs only when they both are inside the absorber molecule. To simplify the discussion, only mono-channel configuration, where there is just one possible evacuation channel for each charge carrier (i.e., *C*<sup>1</sup> & *<sup>C</sup>*<sup>2</sup> <sup>≠</sup> 0), is considered. The discussion later could be generalized to cover the multi-channel configuration; interested readers are referred to [32].

#### *3.1.1. Local density of states*

For the numerical simulations, we use *J* <sup>1</sup> = *J* <sup>2</sup> <sup>=</sup> *<sup>J</sup>* <sup>=</sup> 0.2*eV* and Δ <sup>=</sup> 0.2*eV*; therefore, the energy continuum lies between 1.2 and 2.8 ev. The energy continuum is the possible energy states of the independent electron-hole pairs far from the two-level system.

The total energy continuum is simply the sum of the electron and hole energies. In **Figure 3**, the LDOS is plotted as a function of the absorbed photon energy. In these plots, the dependence of LDOS on short-range interaction energy (*U*), strength of coupling parameters (*C*), and recombination rate (Γ*<sup>R</sup>* ) is examined. As can be seen from panels (a) and (b), for a given set of coupling parameters and in the absence of recombination, the number of LDOS peaks is dependent on the interaction strength. For small values of |*U*|, there is a single peak which tends to become

**Figure 3.** Local density of states as a function of incident photon energy in a mono-channel system under different conditions. ((a) and (b)) for different values of interaction energy (*U*); (c) for different coupling parameters (*C*<sup>1</sup> & *C*<sup>2</sup> ); and (d) for different interaction energies and recombination rates (*U* & Γ*R*).

narrower for larger |*U*|. This peak appears at an energy *Eres* <sup>≃</sup> *<sup>ε</sup><sup>e</sup>* (0) <sup>+</sup> *<sup>ε</sup><sup>h</sup>* (0) <sup>−</sup> *<sup>U</sup>*. Indeed, *Eres* is the energy at which photons are most easily absorbed. The peak eventually splits into two for growing values of |*U*| and the resulting two peaks separate further with increasing |*U*| as depicted in panel (b). The narrow peak outside the continuum is called excitonic state, which blocks the charge carrier injection to the energy continuum. Next, we study the effects of the coupling parameters. The corresponding LDOS is shown in panel (c). As discussed, increasing *C* enhances charge carrier transfer from HOMO and LOMO to the respective evacuation channels; it can be detected through the extended width of the LDOS line shape. In both cases, the effect of Γ*<sup>R</sup>* is to slightly shift the LDOS peak to the left and slightly broaden the line shape width.

#### *3.1.2. Charge separation yield*

the number of absorbed photons per unit time, (2) the fluxes of electron-hole pairs that

to a detailed analysis of the photovoltaic cell performance. The yield *Y*(*E*) of the photocell at a given photon energy *E* is proportional to the ratio of photo-generated electrons or holes that arrive at the electrodes and the total number of absorbed photons at this given energy.

The average yield or in other words the charge separation yield, *Y*, which is the proportion of all electron-hole pairs, generated by different photons and giving rise to the photovoltaic

where *n*(*E*) is the local density of states which is related to the flux of absorbed photons through

Relying on the new formalism described in the previous section, here we are going to investigate the effects of short-range electron-hole interaction on the performance of a molecular photocell. The short-range interaction term implies that interaction between the electron and the hole occurs only when they both are inside the absorber molecule. To simplify the discussion, only mono-channel configuration, where there is just one possible evacuation channel for each charge carrier (i.e., *C*<sup>1</sup> & *<sup>C</sup>*<sup>2</sup> <sup>≠</sup> 0), is considered. The discussion later could be generalized

to cover the multi-channel configuration; interested readers are referred to [32].

<sup>1</sup> = *J*

independent electron-hole pairs far from the two-level system.

tinuum lies between 1.2 and 2.8 ev. The energy continuum is the possible energy states of the

The total energy continuum is simply the sum of the electron and hole energies. In **Figure 3**, the LDOS is plotted as a function of the absorbed photon energy. In these plots, the dependence of LDOS on short-range interaction energy (*U*), strength of coupling parameters (*C*), and recombi-

parameters and in the absence of recombination, the number of LDOS peaks is dependent on the interaction strength. For small values of |*U*|, there is a single peak which tends to become

) is examined. As can be seen from panels (a) and (b), for a given set of coupling

Φ*C* Φ*Ph* (*E*) \_\_\_\_\_

(*E*), and (3) the flux of pairs whose escape from the molecule

(*E*). The determination of these quantities gives access

<sup>Φ</sup>*Ph*(*E*) (5)

= ∫*n*(*E*)*Y*(*E*)*dE* (6)

<sup>2</sup> <sup>=</sup> *<sup>J</sup>* <sup>=</sup> 0.2*eV* and Δ <sup>=</sup> 0.2*eV*; therefore, the energy con-

recombine in the molecule Φ*<sup>R</sup>*

34 Solar Panels and Photovoltaic Materials

current, can be defined as:

Fermi's golden rule [31].

*3.1.1. Local density of states*

nation rate (Γ*<sup>R</sup>*

For the numerical simulations, we use *J*

**3. Results and discussions**

results in the photovoltaic current Φ*<sup>C</sup>*

*<sup>Y</sup>*(*E*) <sup>=</sup> <sup>Φ</sup>*<sup>C</sup>*

*Y* = \_\_\_

**3.1. Two-level systems with local interaction**

The other important quantity that can be investigated is the charge separation yield, *Y*, which is computed as an average over all the absorbed photon energies.

The dependence of the charge separation yield of the interacting electron-hole pair is examined as a function of short-range interaction strength *U* and recombination rate Γ*<sup>R</sup>* , for different coupling parameters *C*. As shown in **Figure 4**, in all cases, for small values of interaction energy, the yield remains 1 for Γ*<sup>R</sup>* <sup>=</sup> 0. The effect of Γ*<sup>R</sup>* and *U* is to reduce the yield. The behavior can be understood based on the spectral information provided in **Figure 3**. For larger values

and quantum yield of the molecular photocell by considering the effects of long-range electron-

For the numerical simulation, the same parameters as in the previous section are used. In all the calculations, *C* stands for the first coupling parameters and we treat the symmetric condi-

In **Figure 5**, the LDOS is plotted as a function of the absorbed photon energy. In these plots, the dependence of LDOS on short- and long-range interaction energy (*U* and *V*) and strength of coupling parameters (*C*) is examined. Here, the coupling parameter related to the panels of top and bottom rows is *C* = 0.1 *eV* and *C* = 0.2 *eV*, respectively. As shown, under the influence of the long-range interaction, a series of excitonic peaks appears outside the energy continuum, below the lower band edge. This was expected, as it is known that the long-range Coulomb interaction creates localized states. It should be noted that upon increasing the interaction strength, the total weight of excitonic states increases. Furthermore, in all cases, as the coupling parameter

increases, the width of the LDOS peak inside the energy continuum increases as well.

The dependence of the charge separation yield of the interacting electron-hole pair is exam-

**Figure 5.** LDOS as a function of the energy of the absorbed photon. The impact of the coupling parameter *C* and the

strength of the electron-hole interaction *U* & *V* are illustrated in the various panels.

for different coupling parameters *C* and long-range electron-hole

Quantum Two-Level Model for Excitonic Solar Cells http://dx.doi.org/10.5772/intechopen.74996 37

hole interaction and non-radiative recombination.

.

*3.2.1.1. Local density of states*

*3.2.1.2. Charge separation yield*

ined as a function of *U* and Γ*<sup>R</sup>*

tion, that is, *<sup>C</sup>* <sup>=</sup> *<sup>C</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup>*<sup>2</sup>

**Figure 4.** Photovoltaic yield as a function of interaction energy (*U*) and recombination rate (Γ*R*) in a mono-channel system for different values of coupling parameters (*C*<sup>1</sup> and *C*<sup>2</sup> ).

of |*U*|, the charge carriers will stay on the molecule to form a localized state because their energy does not lie in the energy continuum of the contacts. Besides, for large values of the coupling parameters (*C*<sup>1</sup> and *C*<sup>2</sup> ), more charge carriers will transfer to the evacuation channels and hence the cell remains efficient over a wider range of the recombination parameter.

#### **3.2. Two-level systems with non-local interaction**

This section is intended to investigate the effects of non-local interaction on the performance of photovoltaic cells. This means that in contrast to the results presented in the previous section there are interactions even if the charge carriers are outside the molecule. An important case of non-local interaction is the long-range Coulomb interaction between the photo-generated electron and hole. This means that the electron and the hole do interact even if they are not both inside the molecule. Here of course the Coulomb interaction is not the bare interaction but is screened by all the charges of the materials around the electron-hole pair. This screening effect is well represented by considering an effective dielectric constant of the medium. The other case of non-local interaction is the coupling between the electron (or the hole) with the lattice distortion around it when the electron (or the hole) moves out of the initial two-level system. The first part of this section is devoted to the long-range Coulomb interaction case and the second part deals with the coupling to the optical phonon modes.

#### *3.2.1. Long-range electron: hole Coulomb interaction*

Similar to the previous section, we consider the mono-channel case where there is only one evacuation channel for each charge carrier. We analyze photon absorption, energy conversion and quantum yield of the molecular photocell by considering the effects of long-range electronhole interaction and non-radiative recombination.

#### *3.2.1.1. Local density of states*

For the numerical simulation, the same parameters as in the previous section are used. In all the calculations, *C* stands for the first coupling parameters and we treat the symmetric condition, that is, *<sup>C</sup>* <sup>=</sup> *<sup>C</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup>*<sup>2</sup> .

In **Figure 5**, the LDOS is plotted as a function of the absorbed photon energy. In these plots, the dependence of LDOS on short- and long-range interaction energy (*U* and *V*) and strength of coupling parameters (*C*) is examined. Here, the coupling parameter related to the panels of top and bottom rows is *C* = 0.1 *eV* and *C* = 0.2 *eV*, respectively. As shown, under the influence of the long-range interaction, a series of excitonic peaks appears outside the energy continuum, below the lower band edge. This was expected, as it is known that the long-range Coulomb interaction creates localized states. It should be noted that upon increasing the interaction strength, the total weight of excitonic states increases. Furthermore, in all cases, as the coupling parameter increases, the width of the LDOS peak inside the energy continuum increases as well.

### *3.2.1.2. Charge separation yield*

of |*U*|, the charge carriers will stay on the molecule to form a localized state because their energy does not lie in the energy continuum of the contacts. Besides, for large values of the

**Figure 4.** Photovoltaic yield as a function of interaction energy (*U*) and recombination rate (Γ*R*) in a mono-channel

 and *C*<sup>2</sup> ).

This section is intended to investigate the effects of non-local interaction on the performance of photovoltaic cells. This means that in contrast to the results presented in the previous section there are interactions even if the charge carriers are outside the molecule. An important case of non-local interaction is the long-range Coulomb interaction between the photo-generated electron and hole. This means that the electron and the hole do interact even if they are not both inside the molecule. Here of course the Coulomb interaction is not the bare interaction but is screened by all the charges of the materials around the electron-hole pair. This screening effect is well represented by considering an effective dielectric constant of the medium. The other case of non-local interaction is the coupling between the electron (or the hole) with the lattice distortion around it when the electron (or the hole) moves out of the initial two-level system. The first part of this section is devoted to the long-range Coulomb interaction case and the

Similar to the previous section, we consider the mono-channel case where there is only one evacuation channel for each charge carrier. We analyze photon absorption, energy conversion

and hence the cell remains efficient over a wider range of the recombination parameter.

), more charge carriers will transfer to the evacuation channels

coupling parameters (*C*<sup>1</sup>

36 Solar Panels and Photovoltaic Materials

and *C*<sup>2</sup>

second part deals with the coupling to the optical phonon modes.

*3.2.1. Long-range electron: hole Coulomb interaction*

**3.2. Two-level systems with non-local interaction**

system for different values of coupling parameters (*C*<sup>1</sup>

The dependence of the charge separation yield of the interacting electron-hole pair is examined as a function of *U* and Γ*<sup>R</sup>* for different coupling parameters *C* and long-range electron-hole

**Figure 5.** LDOS as a function of the energy of the absorbed photon. The impact of the coupling parameter *C* and the strength of the electron-hole interaction *U* & *V* are illustrated in the various panels.

interaction strength *V* (see **Figure 6**). The coupling parameters in the panels of left and right columns are *C* = 0.1 *eV* and *C* = 0.2 *eV*, respectively. As shown, upon increasing interaction strength inside the molecule (*U*), less charge carriers exit through the contacts because of localized state formation and hence the yield decreases. The interesting point is that in the case of *U* = *V*, the effect of the Coulomb interaction is to diminish the global potential nearly uniformly, which has a small effect on the localization of the electron-hole pair. Therefore, the maximum of the charge separation yield is at *U* ≃ *V*. In panels (a) and (b) where the effect of the long-range interaction has been neglected, the maximum in the charge separation yield is at *U* = 0 *eV*; in the other panels, as a consequence of the electron-hole long-range interaction (*V* ≠ 0 *eV*), the maximum of the yield is at lower values of the interaction energy.

states increases and consequently the possibility of recombination and annihilating the charge carriers enhances. For a given electron-hole interaction strength, the yield improves with increasing values of the coupling parameter. Since the strong coupling extends the width of the DOS line shape and consequently improves the escaping rate, this behavior is understandable. Furthermore, the effect of non-radiative recombination is to diminish the yield, and its impact is more important under the influence of the strong long-range interaction condition.

Quantum Two-Level Model for Excitonic Solar Cells http://dx.doi.org/10.5772/intechopen.74996 39

In the performance of excitonic solar cells, coupling to the phonon modes can play a major role as it may lead to the occurrence of polarons, where a polaron is a moving charge surrounded by a cloud of virtual phonons. To address how the electron-phonon coupling (in addition to the electron-hole interaction) can affect the charge separation process, here we propose a simple tight-binding-based model. We analyze the spectrum of polaronic bands and focus on their effects on the charge separation yield, which is defined as a proportion of emitted electrons that arrive at the cathode electrode. We start the discussion by the model description.

We present a mathematical model to investigate the influence polaron formation has on the charge separation process of excitonic solar cells. This model can be applied to any type of excitonic solar cells. We suppose that an electron emitted at first site (*l* = 0) of a chain that represents the acceptor material and hole is fixed at the interface. We construct a simple vibration

**Figure 7.** Schematic depiction of the chain model to describe the electron-vibration coupling which can occur on every single acceptor site when the electron arrives at that site. *C* represents the coupling parameter between interface state and initial acceptor site. The coupling between adjacent sites on the acceptor chain is shown by *J*. *g* is the strength of

*3.2.2. Charge injection in polaronic bands and quantum yield of excitonic solar cells*

*3.2.2.1. Coupling to the phonon modes: theoretical model in the small polaron limit*

chain model which is schematically illustrated in **Figure 7**.

coupling to the phonon chains.

Additionally, for a given coupling parameter, the yield is higher for weakly interacting electron-hole pairs. This behavior can be understood based on the spectral information provided in **Figure 5**; explicitly under the influence of the strong interaction, the weight of localized

**Figure 6.** Charge separation yield as a function of the interaction energy (*U*) and recombination rate (Γ*R*). The various plots are obtained upon the variation of the electron-hole interaction and for different strengths of the coupling parameters.

states increases and consequently the possibility of recombination and annihilating the charge carriers enhances. For a given electron-hole interaction strength, the yield improves with increasing values of the coupling parameter. Since the strong coupling extends the width of the DOS line shape and consequently improves the escaping rate, this behavior is understandable. Furthermore, the effect of non-radiative recombination is to diminish the yield, and its impact is more important under the influence of the strong long-range interaction condition.
