**2. Formalism and numerical methods**

#### **2.1. Two-level photovoltaic system**

The basic idea of our methodology is described through the example of the two-level molecular photocell where the energy conversion process takes place in a single molecular donoracceptor complex attached to electrodes. The two-level system is characterized by the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Initially, the whole system is in the ground state with filled valence bands and empty conduction bands. Following the photon absorption by the molecule, one electron and one hole are created in LUMO and HOMO, respectively. Both charge carriers interact via the Coulomb potential and can be recombined in the molecule or can be transferred to their respective channels where they produce photovoltaic current (see **Figure 2**).

Here, the first coupling energies to the charge evacuation channels are denoted by *C*. The hopping matrix element inside each evacuation channel is considered uniform (i.e., independent of the electron-hole positions) and denoted by *J*. The on-site energies of the electron at site (*x*) and the hole at site (*y*) are assumed to be *ε<sup>e</sup>* (*x*) and *εh*(*y*), respectively. Additionally, their Coulomb-type interaction *I*(*x*, *y*) is modeled by

$$\begin{array}{rcl} \text{(1)} & \text{(10)} & \text{(1)}\\\\ \text{I} \{\mathbf{x}, y\} = \begin{cases} \text{I} & \text{if } \mathbf{x} = \mathbf{0} \text{ and } y = \mathbf{0} \\\\ \frac{\text{V}}{\text{(x} + y)} & \text{if } \mathbf{x} \neq \mathbf{0} \text{ or } y \neq \mathbf{0} \end{cases} \end{array} \tag{1}$$

where both charge carriers are in the same place, that is, the absorber molecule. Furthermore, *V* is the strength of long-range electron-hole interaction, that is, the situation where at least one of the charge carriers is in its respective lead. It has to be noted here that this form of interaction is suitable for the mono-channel configuration where photo-generated electron

> |*i*〉〈*i*| + ∑ *i*,*j J i*,*j* |*i*〉〈*j*

Here, the first term indicates the total on-site energy of each electron-hole basis state which is defined as a summation over the electron on-site energy, the hole's on-site energy, and the

The second term in Eq. (2) represents the coupling energy between two adjacent basis states. In the other words, coupling represents either the hoping of a hole or of an electron from a given initial site of the electron-hole pair to a neighboring site. As pointed earlier, the coupling energies between molecular states and their first neighbors are taken to be different from the

In this formalism, we consider a photovoltaic cell as a system submitted to an incident flux of photons and assume that the whole system (PV cell plus electromagnetic field) is in a stationary state that obeys the fundamental Lippmann-Schwinger Equation [31–34]. By applying quantum scattering theory, in particular the Lippmann-Schwinger equation, the photovoltaic system is described by a wave function. The incoming state of the theory |Φ*inc*〉 represents the photon field with the PV cell in its ground state. By the dipolar interaction between the photovoltaic system and the electromagnetic field, this incident state |Φ*inc*〉 is coupled to a state where one photon is absorbed and one electron-hole pair is created. Based on the Lippmann-Schwinger equation, the total wave function of the system with incident photons


The second term in the right-hand side of the above equation is called the scattered wave

tion of a photon with energy *E* and plays an important role in this formalism. Knowing

The main three fluxes are the following: (1) the flux of absorbed photons Φ*Ph*

〉 enables one to compute all the essential fluxes to describe the photocell operation.

〉 <sup>=</sup> *<sup>G</sup>*<sup>0</sup> *<sup>V</sup>*|*ψ*(*E*)〉, which represents the charge carriers photo-generated by absorp-


Quantum Two-Level Model for Excitonic Solar Cells http://dx.doi.org/10.5772/intechopen.74996 33

(*E*), which is

(*x*) + *εh*(*y*) + *I*(*x*, *y*) (3)

and hole are directed toward different evacuation channels.

The effective Hamiltonian of the system is of the tight-binding type

*i εi*

**2.2. The electron-hole pair Hamiltonian**

*H* = ∑

Coulomb interaction energy between them:

*ε*(*x*, *y*) = *ε<sup>e</sup>*

other coupling energies.

of energy *E* is:

function, |*ψ<sup>P</sup>*

<sup>|</sup>Ψ*<sup>P</sup>* (*E*) (*E*)

**2.3. Fluxes & quantum yield**

Since *I*(*x*, *y*) is an attractive Coulomb potential, *U* and *V* have negative values. In the above equation, *U* represents the strength of short-range electron-hole interaction, that is, the situation

**Figure 2.** A molecular photocell with one HOMO and one LUMO orbital attached to the electrodes in materials I (right) and II (left). The red wiggly line represents the electron-hole interaction and recombination and the hopping integrals of electron and hole are denoted by *C* and *J*.

where both charge carriers are in the same place, that is, the absorber molecule. Furthermore, *V* is the strength of long-range electron-hole interaction, that is, the situation where at least one of the charge carriers is in its respective lead. It has to be noted here that this form of interaction is suitable for the mono-channel configuration where photo-generated electron and hole are directed toward different evacuation channels.

#### **2.2. The electron-hole pair Hamiltonian**

**2. Formalism and numerical methods**

channels where they produce photovoltaic current (see **Figure 2**).

site (*x*) and the hole at site (*y*) are assumed to be *ε<sup>e</sup>*

Coulomb-type interaction *I*(*x*, *y*) is modeled by

*<sup>I</sup>*(*x*, *<sup>y</sup>*) <sup>=</sup> {

electron and hole are denoted by *C* and *J*.

The basic idea of our methodology is described through the example of the two-level molecular photocell where the energy conversion process takes place in a single molecular donoracceptor complex attached to electrodes. The two-level system is characterized by the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Initially, the whole system is in the ground state with filled valence bands and empty conduction bands. Following the photon absorption by the molecule, one electron and one hole are created in LUMO and HOMO, respectively. Both charge carriers interact via the Coulomb potential and can be recombined in the molecule or can be transferred to their respective

Here, the first coupling energies to the charge evacuation channels are denoted by *C*. The hopping matrix element inside each evacuation channel is considered uniform (i.e., independent of the electron-hole positions) and denoted by *J*. The on-site energies of the electron at

Since *I*(*x*, *y*) is an attractive Coulomb potential, *U* and *V* have negative values. In the above equation, *U* represents the strength of short-range electron-hole interaction, that is, the situation

**Figure 2.** A molecular photocell with one HOMO and one LUMO orbital attached to the electrodes in materials I (right) and II (left). The red wiggly line represents the electron-hole interaction and recombination and the hopping integrals of

*U if x* = 0 *and y* = 0 \_\_\_\_\_ *<sup>V</sup>*

(*x*) and *εh*(*y*), respectively. Additionally, their

(*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*) *if <sup>x</sup>* <sup>≠</sup> <sup>0</sup> *or <sup>y</sup>* <sup>≠</sup> <sup>0</sup> (1)

**2.1. Two-level photovoltaic system**

32 Solar Panels and Photovoltaic Materials

The effective Hamiltonian of the system is of the tight-binding type

$$H = \sum\_{i} \varepsilon\_{i} |i\rangle\langle i| + \sum\_{\forall} I\_{\downarrow i} |i\rangle\langle j| \tag{2}$$

Here, the first term indicates the total on-site energy of each electron-hole basis state which is defined as a summation over the electron on-site energy, the hole's on-site energy, and the Coulomb interaction energy between them:

$$
\varepsilon(\mathbf{x}, \mathbf{y}) = \varepsilon\_c(\mathbf{x}) + \varepsilon\_h(\mathbf{y}) + I(\mathbf{x}, \mathbf{y}) \tag{3}
$$

The second term in Eq. (2) represents the coupling energy between two adjacent basis states. In the other words, coupling represents either the hoping of a hole or of an electron from a given initial site of the electron-hole pair to a neighboring site. As pointed earlier, the coupling energies between molecular states and their first neighbors are taken to be different from the other coupling energies.

#### **2.3. Fluxes & quantum yield**

In this formalism, we consider a photovoltaic cell as a system submitted to an incident flux of photons and assume that the whole system (PV cell plus electromagnetic field) is in a stationary state that obeys the fundamental Lippmann-Schwinger Equation [31–34]. By applying quantum scattering theory, in particular the Lippmann-Schwinger equation, the photovoltaic system is described by a wave function. The incoming state of the theory |Φ*inc*〉 represents the photon field with the PV cell in its ground state. By the dipolar interaction between the photovoltaic system and the electromagnetic field, this incident state |Φ*inc*〉 is coupled to a state where one photon is absorbed and one electron-hole pair is created. Based on the Lippmann-Schwinger equation, the total wave function of the system with incident photons of energy *E* is:

$$\left|\psi(E)\right> = \left|\mathfrak{O}\_{\rm loc}\right> + G\_0 V \left|\psi(E)\right>\tag{4}$$

The second term in the right-hand side of the above equation is called the scattered wave function, |*ψ<sup>P</sup>* (*E*) 〉 <sup>=</sup> *<sup>G</sup>*<sup>0</sup> *<sup>V</sup>*|*ψ*(*E*)〉, which represents the charge carriers photo-generated by absorption of a photon with energy *E* and plays an important role in this formalism. Knowing <sup>|</sup>Ψ*<sup>P</sup>* (*E*) 〉 enables one to compute all the essential fluxes to describe the photocell operation. The main three fluxes are the following: (1) the flux of absorbed photons Φ*Ph* (*E*), which is the number of absorbed photons per unit time, (2) the fluxes of electron-hole pairs that recombine in the molecule Φ*<sup>R</sup>* (*E*), and (3) the flux of pairs whose escape from the molecule results in the photovoltaic current Φ*<sup>C</sup>* (*E*). The determination of these quantities gives access 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.

$$\mathcal{Y}(\mathcal{E}) = \frac{\Phi\_c(\mathcal{E})}{\Phi\_{\mathcal{W}}(\mathcal{E})} \tag{5}$$

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 current, can be defined as:

$$Y = \frac{\Phi\_c}{\Phi\_{p\_h}} = \int n(E)Y(E)dE\tag{6}$$

narrower for larger |*U*|. This peak appears at an energy *Eres* <sup>≃</sup> *<sup>ε</sup><sup>e</sup>*

(d) for different interaction energies and recombination rates (*U* & Γ*R*).

*3.1.2. Charge separation yield*

(0) <sup>+</sup> *<sup>ε</sup><sup>h</sup>*

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

**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>

The other important quantity that can be investigated is the charge separation yield, *Y*, which

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

ent coupling parameters *C*. As shown in **Figure 4**, in all cases, for small values of interaction

can be understood based on the spectral information provided in **Figure 3**. For larger values

through the extended width of the LDOS line shape. In both cases, the effect of Γ*<sup>R</sup>*

ined as a function of short-range interaction strength *U* and recombination rate Γ*<sup>R</sup>*

shift the LDOS peak to the left and slightly broaden the line shape width.

is computed as an average over all the absorbed photon energies.

energy, the yield remains 1 for Γ*<sup>R</sup>* <sup>=</sup> 0. The effect of Γ*<sup>R</sup>*

(0) <sup>−</sup> *<sup>U</sup>*. Indeed, *Eres* is the energy

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

is to slightly

); and

35

, for differ-

and *U* is to reduce the yield. The behavior

where *n*(*E*) is the local density of states which is related to the flux of absorbed photons through Fermi's golden rule [31].
