# A Theoretical Study on the Operation Principle of Hybrid Solar Cells

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Operation Mechanism in Hybrid Solar Cells

#### 2.1. Excitation of Singlet and Triplet Excitons

_{LUMO}), leaving a hole in the highest occupied molecular orbital, (E

_{HOMO}), which instantly form an exciton due to strong Coulomb interaction between e and h caused by the low dielectric constant (ε = 3–4) of organic solids. As the electronic intermolecular interaction is weak in organics, the formation of such excitons is usually of Frenkel type [4,14,15,16]. The rate of a photon absorption for a singlet excitation due to exciton-photon interaction, ${R}_{a}^{S}$, and triplet excitation due to exciton-spin-orbit-photon interaction, ${R}_{a}^{T}$, are derived using the transition matrix element and Fermi’s Golden rule and are given as [6]:

_{o})

^{−}

^{1}= 9 × 10

^{9}, e = electronic charge, c = speed of light, ε = dielectric constant, ℏ = reduced Planck’s constant, Z = atomic number of heavy metal atom, ε

_{o}is the vacuum permittivity and μ

_{x}is the reduced mass of exciton. The excitonic Bohr radius for singlet and triplet excitons a

_{xs}and a

_{xt}, respectively, are given as [4]:

_{x}is the reduced mass of an exciton, a

_{o}= 5.29 × 10

^{−}

^{11}m is the Bohr radius and μ is the reduced mass of electron in a hydrogen atom.

#### 2.2. Exciton Diffusion to Organic-Inorganic Interface

_{f}= Förster radius that determines the actual distance between the donor and acceptor, R

_{da}= donor-acceptor separation distance and τ is the lifetime, L is the average length of a molecular orbital and R

_{d}is the Dexter radius between donor and acceptor at which the efficiency of such transfer remains at 50% [17]. The diffusion length, ${L}_{D}=\sqrt{D\tau}$, for singlet, and triplet excitons are obtained as [4]:

#### 2.3. Exciton Dissociation at the Organic-Inorganic Interface

_{d}, for singlet and triplet excitons in hybrid OSCs is derived using the interaction operator between CT exciton and molecular vibration energy, transition matrix element and Fermi’s Golden rule and can be written as [7]:

_{v}is the phonon frequency and ${E}_{B}^{S}$ and ${E}_{B}^{T}$ are the singlet and triplet exciton binding energies, respectively, given as [4]:

## 3. Results and Discussion

_{xs}= 4.35 nm and a

_{xt}= 0.32 nm, Z = 77 for Iridium, ε = 7, μ

_{x}= 1.99 × 10

^{−26}kg, ${E}_{B}^{S}$ = 0.06 eV, ${E}_{B}^{T}$ = 0.76 eV [4], R

_{da}= 0.05 nm, R

_{f}= 8 nm, τ = 2230 ps [28], L = 0.11 nm, R

_{d}= 1 nm [29] ${E}_{LUMO}^{P3HT}$ = −3.03 eV, ${E}_{CB}^{SiNW}$= −4.00 eV [30]. Using the above design and parameters, the various rates are calculated as shown in Table 1. These rates depend on the above input parameters and therefore some uncertainty may be expected due to the uncertainties in the parameters themselves. It is not possible exactly to estimate variations in the input parameters however according to Equations (3), (4), (11) and (12), singlet and triplet Bohr radii and corresponding binding energies depend on the parameters α and μ

_{x}which are estimated to be equal to 1.37 and 0.5 m

_{e}, respectively [4]. The estimate of the effective mass of charge carriers in organic materials to be equal to the electronic rest mass m

_{e}is regarded to be a good approximation as the HOMO and LUMO levels are electronic molecular states instead of energy bands. The parameter α is estimated from the measured value of the exchange energy ΔE = 0.7 eV [31] between singlet and triplet excitons, which is also a reasonable approximation. However, according to Equations (9) and (10), the rates of dissociation of excitons depend on the squires of both the exciton binding energy and exciton Bohr radius. This means that any uncertainty in these quantities may be expected to get squared in the rates.

**Table 1.**Rates of excitation, diffusion and dissociation in PET/PEDOT:PSS/TFB/P3HT:SiNW/Ca hybrid solar cell.

Mechanism | Parameter | Value |
---|---|---|

Excitation | ${R}_{a}^{S}$ | 1.27 × 10^{1}^{0} s^{−}^{1} |

${R}_{a}^{T}$ | 3.09 × 10^{5} s^{−}^{1} | |

Diffusion | D_{S} | 1.99 cm^{2} s^{−}^{1} |

D_{T} | 2.71 × 10^{−}^{6} cm^{2}·s^{−}^{1} | |

${L}_{D}^{S}$ | 2.82 × 10^{2} nm | |

${L}_{D}^{T}$ | 1.17 nm | |

Dissociation | ${R}_{d}^{S}$ | 2.60 × 10^{17} s^{−}^{1} |

${R}_{d}^{T}$ | 2.98 × 10^{12} s^{−}^{1} | |

Charge transport | F | 3.36 × 10^{−}^{12} N |

_{LUMO}− E

_{HOMO})

^{3}while ${R}_{a}^{T}$ is only linearly dependent on (E

_{LUMO}− E

_{HOMO}). The triplet excitation rate is facilitated by the incorporation of heavy metal atom, Iridium, due to enhanced exciton-spin-orbit-photon interaction which also flips the spin to singlet configuration to facilitate the absorption by spin conservation. At the same time, flipped spins to singlet configuration makes it easier to dissociate the exciton into free charge carriers due to their low binding energy [4].

^{n}. As 4π ~ 10, the further the exciton is from the interface, much less is the probability that it will reach the interface to form a CT exciton. This basically means that every single step away from the interface will reduce the probability by at least one order of magnitude [7].

^{2}, which is nearly three orders of magnitude less than that obtained from Equations (6) and (7) and hence the dissociation time will be longer by three orders of magnitude.

_{e}> μ

_{h}i.e., electron will reach the Ca electrode much faster than the hole can reach PEDOT:PSS. This leads to hole accumulation within the bulk which may lead to non-radiative quenching of holes and a low free carrier collection efficiency is obtained. However, the results presented in this study may be expected to assist in understanding the operation and loss mechanisms in hybrid solar cells.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

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Narayan, M.; Singh, J.
A Theoretical Study on the Operation Principle of Hybrid Solar Cells. *Electronics* **2015**, *4*, 303-310.
https://doi.org/10.3390/electronics4020303

**AMA Style**

Narayan M, Singh J.
A Theoretical Study on the Operation Principle of Hybrid Solar Cells. *Electronics*. 2015; 4(2):303-310.
https://doi.org/10.3390/electronics4020303

**Chicago/Turabian Style**

Narayan, Monishka, and Jai Singh.
2015. "A Theoretical Study on the Operation Principle of Hybrid Solar Cells" *Electronics* 4, no. 2: 303-310.
https://doi.org/10.3390/electronics4020303