Combined influence of carrier mobility and dielectric constant on the performance of organic bulk heterojunction solar cells

It has been shown that there is an optimum charge carrier mobility that leads to a peak in the efficiency for organic bulk heterojunction solar cells with mobility-dependent recombination rate. Hence, improving the mobility is considered as one of the ways to increase the efficiency. In this study, we investigate the combined influence of charge carrier mobility and dielectric constant on the performance of organic bulk heterojunction solar cells by performing drift-diffusion calculations. We find that a higher dielectric constant leads to a higher peak efficiency together with a lower optimum mobility. We also find that if the dielectric constant of the active material can be increased significantly (to around 8 or higher), it is then possible that the mobility of the active material need not to be improved in order to achieve the maximum efficiency. This study demonstrates the importance of knowing the interplay between the mobility and the dielectric constant with regard to the efficiency.


I. INTRODUCTION
Organic bulk heterojunction (BHJ) solar cell is a promising candidate as a source of energy due to its low cost of fabrication compared to inorganic solar cells.However, their efficiencies are still significantly lower than the efficiencies of inorganic solar cells.This is generally attributed to poor material properties such as low charge carrier mobilities, low dielectric constants, and poor band offsets. 1 Hence, optimization of these properties is crucial in order to achieve a higher efficiency.
Drift-diffusion model has been extensively used and shown to be a powerful tool in modeling organic BHJ solar cells. 2 This general model can help us to describe experimental measurements, to study specific device models, as well as to predict the influences of various parameters on the performance of organic BHJ solar cells, thus guiding us to improve the efficiency.
4][5][6] Mobilities lower than the optimum value lead to lower efficiencies due to significant reductions in the dissociation rate of polaron pairs into free charge carriers, 3 together with less efficient charge extractions.Mobilities above the optimum value lead to decreases in the open circuit voltage and no significant increase in the dissociation rate of polaron pairs. 3This leads to a lower voltage at the maximum power point, thus reducing the efficiency. 3Tress et al. 4 showed that Langevin type recombination leads to an optimum mobility while mobility-independent recombination leads to no optimum value.
Another important factor that influences the performance is the dielectric constant of the active layer.Koster et al. 1 pointed out that increasing the dielectric constant is crucial in order to significantly increase the efficiency of organic solar cells.However, previous studies only concentrated on either a lukman@um.edu.mymobility or dielectric constant alone, while the combined effect of both parameters has been ignored.In order to maximize the efficiency, it is important to know if there is any interplay between the mobility and the dielectric constant with regard to the efficiency.
In this study, we investigate how the mobility and the dielectric constant jointly affect the performance of organic BHJ solar cells.The drift-diffusion model is used for simulations.We adopt the widely used models such as the Onsager-Braun model for polaron pair dissociation [8][9][10] and the Langevin recombination model, which have been shown to well describe the experimental data of organic solar cells.Based on the results, we show that it is important to know the combined influence of mobility and dielectric constant in order to improve the efficiency of organic BHJ solar cells.

II. MODEL
The drift diffusion model essentially solves the continuity and the Poisson equations using numerical schemes. 2,11 he continuity equations are given by where n is the density of electrons, p is the density of holes, q is the elementary charge, J n is the electron current density, J p is the hole current density, G n is the generation rate of electron density, and G p is the generation rate of hole density.J n and J p consist of the drift and the diffusion current components.
The recombination rate follows the bimolecular Langevin recombination, which is given by where γ = (μn+μp)q ε 0 ε r , with μ n is the electron mobility, μ p is the hole mobility, ε 0 is the vacuum permittivity, and ε r is the dielectric constant.The recombination pre-factor ζ is introduced in order to capture the effect of reduced Langevin recombination. 12For the boundary conditions, Boltzmann statistics is used to determine the charge carrier densities at the contacts.The built in potential is taken as the difference between the work functions of the electrodes.
The active layer is considered to be one semiconducting material where the effective band gap E g of the layer is the difference between the LUMO (lowest unoccupied molecular orbital) of the electron accepting material and the HOMO (highest occupied molecular orbital) of the electron donating material.The absorption of light by the active layer first generates excitons, which can then dissociate into polaron pairs (PP) when they reach the donor-acceptor interfaces.The PPs either dissociate into free electrons and holes or decay to the ground state.Furthermore, free electrons and holes can recombine to produce PPs.The continuity equation for PPs is given by where S is the density of polaron pairs, G is the generation rate of PPs per unit volume, k f is the decay rate coefficient of PPs, k d is the dissociation rate coefficient of PPs, and R is the recombination rate [see Eq. ( 2)].
The dissociation rate coefficient k d is taken to follow the Onsager-Braun model 8,9 as proposed by Mihailetchi et al., 10 which has dependencies on both the mobility and the dielectric constant.The dissociation probability of PPs into free carriers is defined as Rearranging Eq. ( 4) by writing k d = P (1 − P) k f , we can write Eq. ( 3) in steady-state condition as Using Eq. ( 5) with k d S = G n = G p , the continuity equations for electrons and holes [Eqs.(1a) and (1b)] in steady-state condition can be alternatively written as In this study, we investigate the combined influence of mobility μ and dielectric constant ε r on the efficiency using two different active layers that represent the properties (obviously except the varied mobility μ and dielectric constant ε r ) of OC 1 C 10 -PPV/PCBM (poly[2-methoxy-5-(3 , 7 -dimethyloctyloxy)-p-phenylene vinylene] and [6,6]-phenyl C 61 -butyric acid methyl ester) and P3HT/PCBM (poly(3-hexylthiophene) and phenyl-C 61 -butyric acid methyl ester) solar cells.We assume that there are no injection barriers at the contacts.The parameters used for the calculations are shown in Table I

A. OC 1 C 10 -PPV/PCBM solar cell 1. The effect of mobility and dielectric constant on short circuit current, open circuit voltage, and fill factor
First, we discuss the effect of mobility μ and dielectric constant ε r on the short circuit current J sc , the open circuit voltage V oc , and the fill factor FF. Integrating Eq. (6a) over x (distance from the anode), we can write where U = [J n (0) − J n (L)] q L is the average net generation rate of charge carriers, R is now the average recombination rate, and P is now the average dissociation probability.It can be seen from Eq. ( 7) that μ and ε r affect U through P and R. From Eq. ( 7), the recombination yield R y is defined as and is used as a measure of the loss in the charge carriers normalized to the generated charge carriers.The open circuit voltage V oc can be described by 13 where k is the Boltzmann constant.From Eq. ( 9), μ and ε r affect V oc through P and the Langevin recombination coefficient γ [see Eq. ( 2)].We can also apply the definition of γ into Eq.( 9) and write Figure 1 shows the short circuit current J sc , the open circuit voltage V oc , and the fill factor FF as functions of charge carrier mobility μ (with equal electron and hole mobilities) and dielectric constant ε r .To help understanding the results in Fig. 1, Fig. 2 shows the average dissociation probability P and the recombination yield R y [see Eq. ( 8)] at short circuit, open circuit, and maximum power point as functions of mobility μ for different dielectric constants ε r .
It can be seen in Fig. 1(a) that J sc basically increases with both mobility μ and dielectric constant ε r .Starting from low mobilities, J sc increases quite rapidly until it is close to a saturated value, where a higher ε r makes J sc to rise and reach the saturated value more quickly.When μ is too low (e.g.μ = 10 −10 m 2 /Vs), J sc remains significantly lower than the saturated value even if ε r is increased to a high value.The reasons behind the results in Fig. 1(a) is clear when considering how the mobility μ and the dielectric constant ε r affect the dissociation probability P at short circuit, as shown in Fig. 2(a).As shown by Eq. ( 7), the recombination rate R also influences U (thus J sc ).However, the influence is small at short circuit because when R is large enough compared to G, the net recombination rate [i.e.(1 − P)R] is small.
From Fig. 1(b), the open circuit voltage V oc increases with dielectric constant ε r but decreases with carrier mobility μ.Furthermore, it can be seen that the increase in V oc becomes lower as ε r is kept increased.These results can be understood by referring to Eq. ( 10) together with the dissociation probability P as a function of μ and ε r at open circuit as shown in Fig. 2(b).It can be seen in Fig. 2(b) that the recombination yield R y is not always 1 at open circuit.This is because the loss of the current density is not only due to recombination, but also due to the extraction of the charge carriers at the "wrong" electrodes.
From Fig. 1(c), it can be seen that the fill factor FF behaves similar to J sc as a function of μ and ε r , except that the fill factor has a peak value instead of a saturated value.The fill factor decreases slowly after it reaches the peak value.Free charge carriers that are not extracted by the electrodes have to recombine.Hence, a high recombination yield R y indicates a low fill factor FF. The results in Fig. 1(c) can be understood by considering the behavior of R y as a function of μ and ε r at maximum power point as shown in Fig. 2(c).The decrease in FF (after the peak value) as the mobility is increased indicates a significant increase in another loss at maximum power point where the charge carriers are collected at the "wrong" electrodes.The increase in the mobility increases the diffusion coefficient, thus helping the charge carriers to diffuse to the wrong electrodes.

B. The effect of mobility and dielectric constant on the efficiency
Figure 3 shows the efficiency η as a function of charge carrier mobility (where μ n = μ p ) for different dielectric constants ε r of the active layer.The results in Fig. 3 originate from the behaviors of the short circuit current J sc , the open circuit voltage V oc , and the fill factor FF as functions of mobility μ and dielectric constant ε r , which are shown in Fig. 1.It can be seen in Fig. 3 that for a given ε r , there is an optimum mobility that leads to a peak efficiency.This can be deduced from Fig. 1 where the optimum mobility occurs when the product of J sc , V oc , and FF is the highest.][5][6] From Fig. 3, a higher dielectric constant ε r leads to a higher efficiency, thus a higher peak efficiency.For example, increasing ε r from 4 to 8 increases the peak efficiency from around 2.45%  to around 3.2%.By looking at Figs. 1(a) and 1(c), it is clear that a higher ε r makes J sc and FF to increase more rapidly with mobility and reach their maximum values at lower mobilities.Since a lower mobility gives a higher V oc [see Fig. 1(b)], this leads to a higher product of J sc , V oc , and FF, hence a higher (peak) efficiency.A higher dielectric constant ε r also leads to a lower optimum mobility.When ε r is increased from 3 to 9 for example, the optimum mobility decreases from around 10 −4 m 2 /Vs to around 10 −7 m 2 /Vs.As just mentioned above, a higher ε r makes J sc and FF to rise more rapidly with mobility and reach their maximum values at lower mobilities, thus leading to a lower optimum mobility.In turn, this is because a higher ε r causes the dissociation probability P to increase more rapidly as the mobility is increased, thus requiring a lower mobility in order to reach the maximum value of P (see Fig. 2).
Figure 3 also indicates that the reduction in the optimum mobility due to the increase in ε r becomes less noticeable at high dielectric constants ε r (particularly starting from ε r = 8).The reason for this is clear when looking at Figs. 1(a) and 1(c).It can be seen that when ε r is increased, the reductions in the mobilities that are required in order to reach the maximum values of J sc and FF are less noticeable at high dielectric constants ε r .
Furthermore, increasing the dielectric constant ε r does not significantly improve the efficiency when the mobility is too low (e.g.μ = 10 −10 m 2 /Vs), and also when the mobility is too high (e.g.μ = 1 m 2 /Vs) if ε r is already relatively high (e.g.ε r = 5).The origin of this can be explained by looking at J sc , V oc , and FF as ε r is increased at very low and very high mobilities, as shown in Fig. 1.
The results in Fig. 3 demonstrate the importance of knowing the relationship between the charge carrier mobility and the dielectric constant ε r when trying to improve and maximize the efficiency of organic solar cells.For example, the average electron and hole mobility for OC 1 C 10 -PPV/PCBM solar cells is 1.4 × 10 −7 m 2 /Vs and the typical ε r ≈ 3.4. 2If the dielectric constant ε r can be increased to ε r ≈ 8 or higher, then we are very close to achieving the peak efficiency without the need to improve the carrier mobility.Furthermore, this would also give a significantly higher efficiency compared to improving to the mobility to the optimum value alone.

C. P3HT/PCBM solar cell
Figure 4 shows the efficiency η of the P3HT/PCBM solar cell as a function of charge carrier mobility (where μ n = μ p ) and dielectric constants ε r .It can be seen that the optimum mobility for the P3HT/PCBM solar cell with typical dielectric constants ε r (between 3-4) is around 10 −7 m 2 /Vs, similar as found by Mandoc et al. 3 With typical mobilities of around 10 −8 m 2 /Vs (Ref.6), the optimum mobility can be obtained by increasing the dielectric constant to ε r ≈ 7 or higher.
The basic features of the efficiency with varying mobility and dielectric constant for OC 1 C 10 -PPV/PCBM and P3HT/PCBM solar cells are the same (see Figs. 3 and 4).The results in Fig. 4 can be derived from the short circuit current J sc , the open circuit voltage V oc , and the fill factor FF as functions of mobility μ and dielectric constant ε r , which are shown in Fig. 5. Figure 6 shows the average dissociation probability P and the recombination yield R y as functions of charge carrier mobility μ for different dielectric constants ε r of the active layer.The results here can be similarly explained as for the OC 1 C 10 -PPV/PCBM solar cell.
Apart from the values of the efficiency and the optimum mobility, the other noticeable difference between the P3HT/PCBM and the OC 1 C 10 -PPV/PCBM solar cells is the reduction in the optimum mobility when the dielectric constant ε r is increased (see Figs. 3 and 4).For example, when ε r is increased from 3 to 9, the optimum mobility for the P3HT/PCBM solar cell decreases from around 5 × 10 −7 m 2 /Vs to around 1 × 10 −8 m 2 /Vs, while for the OC 1 C 10 -PPV/PCBM solar cell, the optimum mobility decreases from around 1 × 10 −4 m 2 /Vs to around 1 × 10 −7 m 2 /Vs.Referring to Figs. 1 and  5, this is because the reductions in the mobilities that are required in order to reach the maximum values of J sc and FF when ε r is increased are lower for the P3HT/PCBM solar cell.In turn, this is because the reduction in the mobility that is required in order to reach the maximum dissociation probability P when ε r is increased is lower for the P3HT/PCBM solar cell, which can be seen by comparing Figs. 2 and 6.Furthermore, it can also be seen from Figs. 3 and 4 that at high dielectric constants (starting from ε r = 8 and ε r = 7 for OC 1 C 10 -PPV/PCBM and P3HT/PCBM solar cells, respectively), the reductions in the optimum mobility for both solar cells when ε r is increased start to become insignificant.

IV. CONCLUSIONS
We have performed simulations using a drift-diffusion model to investigate the combined influence of charge carrier mobility and dielectric constant on the performance of organic BHJ solar cells.We apply the widely used models which have been shown to well describe the experimental data of organic solar cells, such as the Onsager-Braun model for polaron pair dissociation, the Langevin recombination model, and the effective band gap approach.It is well known that there is an optimum mobility that would result in a peak efficiency.Hence, improving the mobility is considered as one of the ways to increase the efficiency.Here, we found that a higher dielectric constant leads to a higher peak efficiency together with a lower optimum mobility.This is because a higher dielectric constant causes the dissociation probability to increase more rapidly as the mobility is increased, thus requiring a lower mobility in order to reach the maximum value of dissociation probability.Using two different active materials that represent the properties of OC 1 C 10 -PPV/PCBM and P3HT/PCBM solar cells, we found that if the dielectric constants ε r can be increased to ε r ≈ 8 or higher, then the optimum mobilities can be achieved without improving the charge carrier mobility.Our results demonstrate the importance of knowing the interplay between the mobility and the dielectric constant with regard to the efficiency.

057133- 5 IncheFIG. 2 .
FIG. 1.(a) Short circuit current density J sc , (b) open circuit voltage V oc , and (c) fill factor FF of the OC 1 C 10 -PPV/PCBM solar cell as functions of charge carrier mobility (with μ n = μ p ) and dielectric constant ε r .

FIG. 3 .
FIG. 3. Efficiency η of the OC 1 C 10 -PPV/PCBM solar cell as a function of charge carrier mobility (where μ n = μ p ) for different dielectric constants ε r of the active layer.

9 FIG. 4 .
FIG. 4. Efficiency η of the P3HT/PCBM solar cell as a function of charge carrier mobility (where μ n = μ p ) for different dielectric constants ε r of the active layer.

9 FIG. 5 .
FIG. 5. (a) Short circuit current density J sc , (b) open circuit voltage V oc , and (c) fill factor FF of the P3HT/PCBM solar cell as functions of charge carrier mobility (with μ n = μ p ) and dielectric constant ε r .

TABLE I .
Values of the parameters used in the simulations.