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

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

**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 coupling to the phonon chains.

**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.

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 maxi-

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

mum of the yield is at lower values of the interaction energy.

38 Solar Panels and Photovoltaic Materials

In this model, the charge separation process follows an interesting scenario: after the exciton dissociation at the interface, the electron either recombines with the hole which is fixed at the interface or moves through a set of acceptor sites where it can be coupled to one single phonon mode. The physical interpretation of the model is that the charge-transfer process is viewed as a hopping process when the electron interacts sufficiently strongly with intramolecular vibrations. The Hamiltonian of the considered system can be written based on the Holstein model [35]:

Let us recall that the yield is the proportion of the electrons that arrive at the cathode electrode, the other electrons recombining at the interface with the hole. We assume that the hole is localized at the two-level system (called site *l* = 0). The effects of hole propagation can be described by extending Hamiltonian Eq. (7) and normally it is expected that the hole propagation decreases the effective Coulomb potential. To study the effects of lattice distortion, we consider a coupling between electron and single intramolecular vibration mode. Besides, a Coulomb interaction between the photo-generated electron-hole pair is considered but in principle other types of interaction potentials should not significantly affect the results [38]. Since both electronic and vibrational energy

ing we examine the charge separation yield in the presence of electron-phonon and long-range electron-hole interaction. The charge separation yield under the influence of long-range Coulomb

As can be seen, first, the yield keeps the periodic resonance structure as a consequence of polaronic band formation. Second, the combined effect of Coulomb interaction and polaronic dressing

**Figure 9.** Local density of states in the bulk part and close to the interface for different injection energies. The electron-

phonon coupling constant is *α* = 0.4 and the coulomb interaction energy is *V* = −1 *eV*.

interaction and a given strength of electron-phonon coupling is represented in **Figure 8**.

*<sup>B</sup> <sup>T</sup>*, the investigation is done at zero temperature [39]. In the follow-

Quantum Two-Level Model for Excitonic Solar Cells http://dx.doi.org/10.5772/intechopen.74996 41

scales are much larger than *k*

$$H = \left. \varepsilon\_{o} c\_{o}^{\*} c\_{o} + \sum\_{l=1}^{\underline{N}} \frac{V}{l} c\_{l}^{\*} c\_{l} + \mathbb{C} \{ c\_{o}^{\*} c\_{1} + c\_{1}^{\*} c\_{o} \} + \bar{f} \sum\_{l=1}^{\underline{N} - 1} \{ c\_{l}^{\*} c\_{l+1} + c\_{l+1}^{\*} c\_{l} \} + \hbar \, \omega\_{o} \sum\_{l=1}^{\underline{N}} a\_{l}^{\*} a\_{l} + \underline{g} \sum\_{l=1}^{\underline{N}} c\_{l}^{\*} c\_{l} \{ a\_{l}^{\*} + a\_{l} \} \quad \text{(7)}$$

where *al* + (*al* ) and *<sup>c</sup> l* + (*c l* ) are, respectively, the phonon and electron creation (destruction) operators on-site *l*, <sup>ℏ</sup> *<sup>ω</sup>*<sup>0</sup> is the energy of the relevant molecular vibration, *g* is the electron-phonon coupling constant, *C* is the coupling parameter between the initial site (*l* = 0) and acceptor chain, *ε*<sup>0</sup> is the energy of the LUMO orbital of the electron at the interface, *V* sets the typical Coulomb potential binding the electron-hole pair, and *J* is the hopping amplitude within the chain. In the following section, we use the dimensionless Huang-Rhys parameter *α*<sup>2</sup> <sup>=</sup> (*g*/<sup>ℏ</sup> *<sup>ω</sup>*0) 2 to characterize the strength of the electron-phonon interaction [36]. In the calculations, we consider only the nearest neighbor's tight-binding interaction and study the model at zero temperature. This assumption is justified as the electronic and vibrational energies are much larger than *k <sup>B</sup> <sup>T</sup>*. Furthermore, we examine the model in the small polaron limit, which means that there are excited phonons only on the site where the electron arrives. The small polaron effect is present in a variety of materials, including many polymers (trans-polyacetylene, etc.) and most transition metals (MnO, NiO, etc.) [37]. Now we intend to examine the charge separation yield in the presence of electron-hole and electron-phonon interactions.

**Figure 8.** Yield as a function of incoming electron energy, *ε*<sup>0</sup> , for various values of the electron-phonon coupling constant *α* in the presence of long-range electron-hole binding *V* = −1 *eV*.

Let us recall that the yield is the proportion of the electrons that arrive at the cathode electrode, the other electrons recombining at the interface with the hole. We assume that the hole is localized at the two-level system (called site *l* = 0). The effects of hole propagation can be described by extending Hamiltonian Eq. (7) and normally it is expected that the hole propagation decreases the effective Coulomb potential. To study the effects of lattice distortion, we consider a coupling between electron and single intramolecular vibration mode. Besides, a Coulomb interaction between the photo-generated electron-hole pair is considered but in principle other types of interaction potentials should not significantly affect the results [38]. Since both electronic and vibrational energy scales are much larger than *k <sup>B</sup> <sup>T</sup>*, the investigation is done at zero temperature [39]. In the following we examine the charge separation yield in the presence of electron-phonon and long-range electron-hole interaction. The charge separation yield under the influence of long-range Coulomb interaction and a given strength of electron-phonon coupling is represented in **Figure 8**.

In this model, the charge separation process follows an interesting scenario: after the exciton dissociation at the interface, the electron either recombines with the hole which is fixed at the interface or moves through a set of acceptor sites where it can be coupled to one single phonon mode. The physical interpretation of the model is that the charge-transfer process is viewed as a hopping process when the electron interacts sufficiently strongly with intramolecular vibrations. The Hamiltonian of the considered system can be written based on the Holstein model [35]:

coupling constant, *C* is the coupling parameter between the initial site (*l* = 0) and acceptor

Coulomb potential binding the electron-hole pair, and *J* is the hopping amplitude within the chain. In the following section, we use the dimensionless Huang-Rhys parameter *α*<sup>2</sup> <sup>=</sup> (*g*/<sup>ℏ</sup> *<sup>ω</sup>*0)

to characterize the strength of the electron-phonon interaction [36]. In the calculations, we consider only the nearest neighbor's tight-binding interaction and study the model at zero temperature. This assumption is justified as the electronic and vibrational energies are much

that there are excited phonons only on the site where the electron arrives. The small polaron effect is present in a variety of materials, including many polymers (trans-polyacetylene, etc.) and most transition metals (MnO, NiO, etc.) [37]. Now we intend to examine the charge

separation yield in the presence of electron-hole and electron-phonon interactions.

is the energy of the LUMO orbital of the electron at the interface, *V* sets the typical

*<sup>B</sup> <sup>T</sup>*. Furthermore, we examine the model in the small polaron limit, which means

<sup>+</sup> *cl*+1 + *cl*+1 <sup>+</sup> *cl*

) are, respectively, the phonon and electron creation (destruction) opera-

is the energy of the relevant molecular vibration, *g* is the electron-phonon

) + ℏ *ω*<sup>0</sup> ∑ *l*=1 *N al* <sup>+</sup> *al* + *g*∑ *l*=1 *N cl* + *cl* (*al* <sup>+</sup> + *al*

, for various values of the electron-phonon coupling constant

) (7)

2

*H* = *ε*<sup>0</sup> *c*<sup>0</sup>

tors on-site *l*, <sup>ℏ</sup> *<sup>ω</sup>*<sup>0</sup>

where *al* + (*al* ) and *<sup>c</sup> l* + (*c l*

chain, *ε*<sup>0</sup>

larger than *k*

<sup>+</sup> *c*<sup>0</sup> + ∑ *l*=1 *<sup>N</sup>* \_\_ *V l cl*

40 Solar Panels and Photovoltaic Materials

<sup>+</sup> *cl* + *C*(*c*<sup>0</sup>

**Figure 8.** Yield as a function of incoming electron energy, *ε*<sup>0</sup>

*α* in the presence of long-range electron-hole binding *V* = −1 *eV*.

<sup>+</sup> *c*<sup>1</sup> + *c*<sup>1</sup>

<sup>+</sup> *c*0) + *J*∑ *l*=1 *N*−1 (*cl*

> As can be seen, first, the yield keeps the periodic resonance structure as a consequence of polaronic band formation. Second, the combined effect of Coulomb interaction and polaronic dressing

**Figure 9.** Local density of states in the bulk part and close to the interface for different injection energies. The electronphonon coupling constant is *α* = 0.4 and the coulomb interaction energy is *V* = −1 *eV*.

demonstrated that this new methodology provides a quantitative picture of the fundamental processes underlying solar energy conversion, including photon absorption, exciton dissociation and charge separation as well as an understanding of their consequences on the cell performance. Interestingly, this theory could successfully analyze excitonic solar cell in the presence of strong Coulomb interaction between the electron and the hole. Here we highlight

**I.** We showed that there is a competition between injection of charge carriers in the leads and recombination in the two-level system. This competition depends sensitively on the parameters of the model such as the local electron-hole interaction, the recombination

**II.** We found that the electron-hole Coulomb interaction and non-radiative recombination reduce the photocell yield, especially under the weak coupling condition where the

**III.** Finally, we provided microscopic evidence that the efficiency of charge transfer is subtly controlled by interplay of electrostatic confinement and coherent coupling of charge

rate, the coupling to the leads, and the band structure of the leads.

1 Néel Institute, CNRS and University of Grenoble Alpes, Grenoble, France

Solar Cells. 2008;**92**:1628-1633. DOI: 10.1016/j.solmat.2008.07.012

2 Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran

[1] Lee KM, Hu CW, Chen HW, Ho KC. Incorporating carbon nanotube in a low-tempera-

[2] Yu C, Choi K, Yin L, Grunlan JC. Light-weight flexible carbon nanotube based organic composites with large thermoelectric power factors. ACS Nano. 2011;**5**:7885-7892. DOI:

[3] Wu J, Wang ZM. Quantum dot solar cells. Spring. 2014. 387p. DOI: 10.1007/978-1-4614-

[4] Kalyanasundaram K. Dye-Sensitized Solar Cells. Lausanne: EPFL press; 2010. 609p

solar cells. Solar Energy Materials and

Quantum Two-Level Model for Excitonic Solar Cells http://dx.doi.org/10.5772/intechopen.74996 43

charge carriers cannot readily escape into the contacts.

carrier(s) to high-energy quantized vibrational modes.

Tahereh Nemati Aram1,2\* and Didier Mayou<sup>1</sup>

\*Address all correspondence to: th.nemati@gmail.com

ture fabrication process for dye-sensitized TiO<sup>2</sup>

**Author details**

**References**

10.1021/nn202868a

8148-5

some of the important achievements of this study.

**Figure 10.** Yield as a function of incoming electron energy *ε*<sup>0</sup> in the presence of long-range electron-hole interaction *V* = −1*eV* for the given electron-phonon coupling constant *α* and different electron-hole recombination rate Γ*R*. The legend presented in the first panel is valid for all the other panels.

of the carriers leads to a strong overall suppression of the yield such that it never reaches one. To have clear understanding, again we refer to the spectral information. **Figure 9** represents the yield and corresponding LDOS. The LDOS is represented in the bulk part (far from the interface) and also at the interface for different injection energies. The electronic structure in the bulk part gives a view of all possible polaronic bands and the energy gap regimes. For a given injection energy *ε*<sup>0</sup> , the electronic structure may contain the energy states on the allowed polaronic bands and also localized states in the energy gap (compared to the bulk DOS).

The charge carriers lying in a polaronic band can evacuate and arrive at the electrodes. On the other hand, the charge carriers localized in the bound state in the gap recombine quickly and cannot lead to photovoltaic current which diminishes the yield. Through this physical interpretation, yield values around the red marked points are in good agreement with the corresponding electronic structure. As shown, the long-range Coulomb interaction leads to an intricate spectrum with many localized and nearly localized states. This tendency to localizing the spectrum induces a lowering of the efficiency of the cell. **Figure 10** represents the effect of recombination (Γ*<sup>R</sup>* ) on the yield. The effects of recombination can be detected through the global reduction of yield.
