# Analytical Model for Voltage-Dependent Photo and Dark Currents in Bulk Heterojunction Organic Solar Cells

^{*}

## Abstract

**:**

## 1. Introduction

_{r}) of organic materials reduces the number of photogenerated free carriers [3]. The photoionized electron and its twin hole (geminate pair) cannot immediately escape from their mutual columbic attraction and the geminate pair dissociates to free charge carriers with probability M. Then the free carriers drift across the photoconductor layer by the built-in electric field and some of the carriers are lost by deep trapping/recombination. At the optimum operating output voltage, the built-in electric field is decreased, which reduces the charge collection efficiency of the photogenerated carriers. Moreover, at the same time, the forward diode-like current (commonly known as the dark current) increases considerably. Both the photo and dark currents critically depend on the carrier transport properties of the blend (active layer) and cell structure. Thus the overall cell efficiency is mainly dominated by the photon absorption, dissociation efficiency of bound EHPs, charge collection efficiency and dark current. Therefore, an explicit physics-based model for the voltage-dependent photo and dark currents is highly desirable for enhancing the efficiency and optimizing the design.

## 2. Theoretical Model

#### 2.1. Dark Current

_{bi}is the built-in potential.

_{n}is the recombination rate (=r

_{m}n(x) = n(x)/τ

_{n}; where r

_{m}is the monomolecular recombination coefficient and τ

_{n}is the electron lifetime), F is the electric field, μ

_{n}and D

_{n}are the electron mobility and electron diffusion coefficient, respectively. The diffusion coefficient is assumed to be independent of n and can be determined using Einstein’s relation, D

_{n}/μ

_{n}= kT/e = V

_{t}. The electric field can be written as:

_{bi}can be calculated from Equation (1).

_{n}(= $\sqrt{{D}_{n}}{\mathsf{\tau}}_{\mathrm{n}}$) is the diffusion length of electrons.

_{1}and A

_{2}can be determined by the following two boundary conditions:

_{c}is the effective density of states in the conduction band.

_{1}and A

_{2}are:

_{v}is the effective density of states in the valence band.

_{1}and B

_{2}are,

_{Fn}) and holes (E

_{Fp}) coincide with each other at the contacts, yet in most part of the blend thickness the difference remains the same, i.e., E

_{Fn}– E

_{Fp}≈ eV. For simplicity, if we consider E

_{Fn}– E

_{Fp}≈ eV throughout the active layer, then the electron and hole profiles can be simplified as:

_{i}is the intrinsic carrier concentration in the blend.

_{dark}. However, as shown in Figure 1, holes are the majority carrier in the first half of the active layer and electrons are the majority carriers in the other half. Thus, R ≈ (n - n

_{0})/τ

_{n}for x = 0 to L/2 provided ${\mathsf{\phi}}_{1}$ ≈ ${\mathsf{\phi}}_{2}$. Therefore, an analytical expression of the dark current due to the SHR recombination can be determined as:

#### 2.2. Dissociation Efficiency

_{d}is the separation rate, K

_{f}(= S/r

_{0}) is the recombination rate of bound EHPs, r

_{0}is the initial separation between a bound EHP, S is the reactivity parameter. Reactivity parameter is the relative velocity between bound electron and hole at the reaction radius. Wojcik et al. [11] have showed that Modified Braun model agrees well with the exact extension of Onsager theory except at extremely high fields. According to the Modified Braun’s model [11]:

_{c}(= ${e}^{2}/4{\mathsf{\pi}\mathsf{\epsilon}}_{o}{\mathsf{\epsilon}}_{r}kT)$ is the Onsager radius, J

_{1}is the first order Bessel function, ${\mathsf{\epsilon}}_{o}{\mathsf{\epsilon}}_{r}$ is the effective dielectric constant of the blend, e is the elementary charge, k is the Boltzmann constant, T is the absolute temperature, and the reduced field, $b={e}^{3}F/8{\mathsf{\pi}\mathsf{\epsilon}}_{o}{\mathsf{\epsilon}}_{r}{k}^{2}{T}^{2}$.

#### 2.3. Photocurrent

_{0}is the intensity of the solar spectra (W/cm

^{2}-nm), c is the speed of light, h is the Plank constant, R is the reflectance or the loss factors, α(λ) is the absorption coefficient of the blend and λ is the incident photon wavelength.

_{1}and C

_{2}are:

#### 2.4. Net External Current

_{s}and R

_{p}are the series and shunt area resistances, respectively. Therefore, the expression of the electric field (Equation (3)) has to be modified to $F=\frac{\left(V-J{R}_{s}\right)-{V}_{bi}}{L}$and V has to be replaced by (V − JR

_{s}) in all expressions above for the calculation of the external current.

## 3. Results and Discussion

_{61}BM BHJ organic solar cells are given in Table 1. The effective bandgap is the difference between acceptor LUMO (lowest unoccupied molecular orbital) level and donor HOMO (highest occupied molecular orbital) level. Unless otherwise stated, the parameters shown in Table 1 are the fixed parameters used in all model calculations.

#### 3.1. Dark Current Density

_{61}BM solar cell. The experimental data are extracted from References [19,20]. The active layer thickness, L = 200 nm [19]. The symbols, dashed, and solid lines represent experimental results, drift-diffusion model of Kumar el al. [10], present model fit to experimental data, respectively. As evident from Figure 2, the dark current models considering the SRH recombination provide better fittings. The dark current calculations using Equations (19) and (21) are almost identical because of symmetrical carrier profile across the active layer (${\mathsf{\varphi}}_{1}$ = ${\mathsf{\varphi}}_{2}$). The best fitted parameters in Figure 2 are; µ

_{p}= 2 × 10

^{−4}cm

^{2}/Vs, µ

_{n}= 2 × 10

^{−3}cm

^{2}/Vs, R

_{s}= 1 Ω·cm

^{2}. The carrier lifetimes, τ

_{n}= τ

_{p}= 3 and 6 μs in Figure 2a,b, respectively. The drift-diffusion model of Kumar el al. [10] shows much higher dark current than the experimental results.

_{71}BM (poly[[4,8-bis[(2-ethylhexyl)oxy]benzo[1,2-b:4,5-b′]dithiophene-2,6-diyl][3-fluoro-2-[(2-ethyl-hexyl)carbonyl]thieno [3,4-b]thiophenediyl]]) solar cell is shown in Figure 3. The experimental data are extracted from Reference [19]. Similar to P3HT:PC

_{61}BM solar cell, the SRH recombination is the main source of dark current in PTB7:PC

_{71}BM solar cells. The best fitted values of carrier lifetimes are τ

_{n}= τ

_{p}= 45 μs. All other parameters in Figure 3 are the same as in Figure 2. Since the results using Equations (19) and (21) are almost identical and show the best fit to the experimental data, Equation (21) is used for calculating the dark current in the rest of this paper.

^{−4}cm

^{2}/Vs whereas it decreases abruptly by reducing the mobility below 10

^{−4}cm

^{2}/Vs.

#### 3.2. Net External Current

_{70}BM and P3HT:PC

_{61}BM) at different wavelengths are taken into account [21,22]. The carrier lifetimes are kept within an acceptable range while fitting the experimental results [5,23]. Figure 5 shows the J-V curves of P3HT:PCBM solar cells at different sun intensities (i.e., 0.5, 0.75, 1 and 1.4 sun) for L = 230 nm. The symbols represent experimental data, and the solid lines represent the model fit to the experimental results. The experimental data for different sun intensities are extracted from Figure 4a of Reference [13]. The exciton dissociation efficiency at the operating voltage is about 87% for r

_{0}= 1.5 nm. In order to ensure the best fit to the experimental results, the electron and hole lifetimes are kept fixed at 2 μs and 18 μs, respectively. The values of other fitting parameters in Figure 2 are; µ

_{p}= 5 × 10

^{−4}cm

^{2}/Vs, µ

_{n}= 5 × 10

^{−3}cm

^{2}/Vs, R

_{s}= 0.3 Ω·cm

^{2}, and R = 0.11. The power conversion efficiency for 1 sun intensity is 2.87%. The analytical model agrees well with the experimental data.

_{71}BM solar cells. The symbols and solid lines represent the experimental results [4] and model fit, respectively. The cell performance, particularly the fill factor, deteriorates with increasing the active layer thickness from 70 to 150 nm. Low carrier mobility in PCDTBT are responsible for lower charge collection efficiency in thicker devices. Therefore, the active layer thickness is usually kept around 70–80 nm. The bandgap and dielectric constant (ε

_{r}) of PCDTBT:PCBM blend are 1.2 eV and 3.8. The dissociation efficiencies at maximum power points are 99% and 98.5% for W = 70 nm and W = 150 nm, respectively, which indicates that the dissociation efficiency in PCDTBT:PCBM blend is much higher as compared to P3HT:PCBM blend. The other fitting parameters in Figure 6 are; μ

_{n}= 5 × 10

^{−5}cm

^{2}/Vs, μ

_{p}= 6 ×10

^{−5}cm

^{2}/Vs, τ

_{n}= 13 μs, τ

_{p}= 25 μs, r

_{0}= 1.8 nm, and R

_{s}= 0.3 and 1 Ω·cm

^{2}for W = 70 and 150 nm, respectively. The fill factor decreases from 67.8% to 55% by increasing the active layer thickness from 70 nm to 150 nm.

_{sc}) for a P3HT: PCBM based BHJ solar cell is shown in Figure 7. The symbols and solid lines represent the experimental data and the model fit to the experimental results, respectively. The experimental data are extracted from Figure 6 of Reference [24]. The short circuit current should increase with increasing the blend thickness because the thicker layer absorbs more photons (i.e., higher quantum efficiency). On the other hand, the charge collection efficiency decreases with increasing the thickness, which results in lower short circuit current. Thus, there exists an optimum thickness that maximizes the short circuit current as shown in Figure 7. The fitting parameters in Figure 7 are; μ

_{n}= 10

^{−3}cm

^{2}/Vs, μ

_{p}= 2 × 10

^{−4}cm

^{2}/Vs, τ

_{n}= 0.1 μs, τ

_{p}= 0.2 μs, R = 0.02, and R

_{s}= 0.5 Ω·cm

^{2}.

_{n}= 2 × 10

^{−3}cm

^{2}/Vs, μ

_{p}= 2 × 10

^{−4}cm

^{2}/Vs, τ

_{n}= τ

_{p}= 20 μs, L = 180 nm, R = 0, and R

_{s}= 0.3 Ω·cm

^{2}.

_{n}and τ

_{p}on the J-V characteristics of a 230 nm thick P3HT:PCBM solar cell are shown in Figure 10a,b, respectively. All other parameters in Figure 10 are the same as in Figure 8. The open circuit voltage and short circuit current decrease with decreasing both the electron and hole lifetimes. However, the open circuit voltage is more prone to the electron lifetime whereas the short circuit current is more prone to the hole lifetime. The enhancement of the dark current with decreasing carrier lifetimes reduces the open circuit voltage.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A typical energy band diagram of a bulk heterojunction (BHJ) solar cell under applied bias (V). Here x is the distance from the anode (radiation-receiving electrode).

**Figure 4.**Theoretical dark current-voltage characteristics of P3HT:PC

_{61}BM solar cells for (

**a**) varying μ

_{n}with μ

_{p}= 2 × 10

^{−4}cm

^{2}/Vs and (

**b**) varying μ

_{p}with μ

_{n}= 2 × 10

^{−3}cm

^{2}/Vs.

**Figure 5.**Current-voltage characteristics of a P3HT:PCBM solar cell at different sun intensities. The symbols represent experimental data and solid lines represent the model fit to the experimental data.

**Figure 6.**Current-voltage characteristics of PCDTBT solar cells for different active layer thicknesses. The symbols and solid lines represent experimental data and model fit to the experimental data, respectively.

**Figure 7.**Short circuit current density (J

_{sc}) versus active layer thickness (L). Symbols: experimental data and solid line: model fit to the experimental results.

**Figure 8.**Theoretical net current density versus voltage curves of P3HT:PCBM solar cells for varying (

**a**) electron injection barrier (${\mathsf{\varphi}}_{1}$), and (

**b**) hole injection barrier (${\mathsf{\varphi}}_{2}$).

**Figure 9.**Theoretical current-voltage characteristics of P3HT:PCBM solar cells for (

**a**) varying electron mobility with µ

_{p}= 2 × 10

^{−4}cm

^{2}/Vs and (

**b**) varying hole mobility with µ

_{n}= 2 × 10

^{−3}cm

^{2}/Vs. Carrier lifetimes are: τ

_{n}= τ

_{p}= 20 μs.

**Figure 10.**Theoretical current-voltage characteristics of P3HT:PCBM solar cells for (

**a**) varying electron lifetimes with τ

_{p}= 20 μs and (

**b**) varying hole lifetime with τ

_{n}= 20 μs. Carrier mobilities are: µ

_{p}= 2 × 10

^{−4}cm

^{2}/Vs and µ

_{n}= 2 × 10

^{−3}cm

^{2}/Vs.

Parameters | Value |
---|---|

Effective Bandgap, E_{g} | 1 eV |

Electron (Hole) injection barrier, ${\mathsf{\varphi}}_{1}$ (${\mathsf{\varphi}}_{2}$) | 0.1 eV |

Effective density of states in conduction (valence) band | 2 × 10^{20} cm^{−3} |

Relative dielectric constant ε_{r} | 3.5 |

Parallel area resistance, R_{p} | 10^{6} ohm·cm^{2} |

Initial separation, r_{0} | 1.5 nm |

Reactivity parameter, S | 0.05 cm/s |

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**MDPI and ACS Style**

Saleheen, M.; Arnab, S.M.; Kabir, M.Z. Analytical Model for Voltage-Dependent Photo and Dark Currents in Bulk Heterojunction Organic Solar Cells. *Energies* **2016**, *9*, 412.
https://doi.org/10.3390/en9060412

**AMA Style**

Saleheen M, Arnab SM, Kabir MZ. Analytical Model for Voltage-Dependent Photo and Dark Currents in Bulk Heterojunction Organic Solar Cells. *Energies*. 2016; 9(6):412.
https://doi.org/10.3390/en9060412

**Chicago/Turabian Style**

Saleheen, Mesbahus, Salman M. Arnab, and M. Z. Kabir. 2016. "Analytical Model for Voltage-Dependent Photo and Dark Currents in Bulk Heterojunction Organic Solar Cells" *Energies* 9, no. 6: 412.
https://doi.org/10.3390/en9060412