Abstract
A better understanding of the opencircuit voltage () related losses in organic solar cells (OSCs) is desirable in order to assess their photovoltaic performance. We have derived as a function of charge carrier mobilities (and) for organic and hybrid solar cells by optimizing the driftdiffusion current density. The thus obtained depends on the energy difference between the highest occupied molecular orbital (HOMO) level and the quasiFermi level of holes of the donor material and on the ratio of the electron () and hole () mobilities in the blend. It is found that the increases with the increase of the mobility ratio. The most loss in is contributed by the energetics of the donor and acceptor materials.
charge carrier mobility, donoracceptor, opencircuit voltage, organic solar cells, quasifermi levels
Introduction
Research interest in organic solar cells (OSCs) is currently on the increase mainly because of their cost effectiveness, flexibility, easy fabrication techniques, large scale production and the potential integration of OSCs into a wide variety of devices [14]. The development of new materials for photovoltaic applications coupled with device optimization has led to a dramatic increase in OSCs’ performance in recent years [5]. A major research focus now lies in finding ways for further optimization of the power conversion efficiency (PCE), guided by a deeper understanding of the fundamental processes that influence the photovoltaic properties of OSCs [6]. The following four processes of OSCs and organic hybrid solar cells (OHSCs) make them remarkably different from their inorganic counterparts: i) photon absorption and exciton generation; ii) diffusion of excitons to the donor acceptor (DA) interface; iii) dissociation and charge separation at the interface; and iv) carrier collection by the electrodes [1,2]. These four processes have to be sufficiently efficient to reduce or eliminate energy losses leading to reduction in the shortcircuit current densityand opencircuit voltage, and hence, reduction in the power conversion efficiency of OSCs and OHSCs.
The current densityin the driftdiffusion model is a function of both the electrical and chemical potentials gradients, denoted byand, respectively. In OSCs, is negligible because there is no builtin electric field like the one in inorganic solar cells due to the property of pn junction [7]. Therefore, in OSCs and OHSCs depends mainly on the gradient of the chemical potential which is a function of as shown below. Thus, becomes a function of and by optimizingwith respect to one can determine the optimal value of corresponding to.
It is established that the of OSCs [811] depends on the energy difference between the highest occupied molecular orbital (HOMO) of the donor material and lowest unoccupied molecular orbital (LUMO) of the acceptor material or the conduction band of the inorganic nanoparticle in the case of OHSCs [12]. In addition, simulation [5,6] and experimental [1315] works show that charge transport have effect on PCE of OSCs and a detailed analysis of bulk heterojunction organic solar cells reveals that low is the main factor limiting this efficiency [9]. This implies that the of an OSC depends on the transport properties of the charge carrier in the material, which has not yet been studied adequately.
In this work, we have derived an analytical expression for by optimizing the driftdiffusion current density. The thus obtained depends explicitly on the electron and hole mobilities and donor and acceptor HOMO and LUMO energy levels. In a previous study (Wurfel et al., 2015), the effective carrier mobility is used to define the external voltage applied across an OSC, however in our approach the concept of the effective mobility is not used. Instead, it is found that the depends on the ratio of the electron () to hole () mobility such that if the ratio increases the also increases.
Derivation of OpenCircuit Voltage ()
The opencircuit voltage is given by the energy difference between the electron and hole quasiFermi levels (Gregg, 2003)
, (1)
In OSCs and OHSCs the opencircuit voltage is also related to the HOMO energy level of the donorand the LUMO energy level of the acceptoras [16]:
, (2)
where is an empirical value representing energy losses in transporting charge carriers to the electrodes.
According to the driftdiffusion model the total current density J in a semiconductor under bias can be written as the sum of the electron and hole current densities, given by [17]:
, (3)
where is the electron current density and is the hole current density . Here is the electron (hole) density, is the electron (hole) mobility, and is the gradient of the electron (hole) quasiFermi level.
The chargecarrier densities and of electrons and holes inside the active layer are, respectively, given as [18]
, (4)
and
, (5)
whereis the effective density of states for the LUMO (HOMO) of acceptor (donor) material and is the energy of the corresponding Fermi levels. Using equations (1)(5), the total current density in (3) can be written as a function of as:
, (6)
The total current density in equation (6) can be optimized with respect to as , which gives:
, (7)
In OSCs the chemical potential energy gradient drives the electrons and holes in the opposite direction (Gregg, 2003), this explains the significance of the minus sign on the left hand side of equation (7) the minus sign is dropped from here onwards for convenience.
Multiplying both sides of equation (7) by we get:, (8)
where is the effective band gap or the DA interface energy gap. Rearranging equation (8) we obtain as:
, (9)
Following earlier works [5, 18] {Wagenpfahl, 2010 #121;Wurfel, 2015 #110} we assume and which gives;
where, (10)
Here is the energy loss contributed by the energetic (first term) and charge transport (second term)
Results
We have used equation (10) to calculate in several donoracceptor (DA) materials listed in Table 1. The input parameters required for each DA in the calculations are also listed in Table 1. In addition, for calculating from equation (10) we need the values of the energy of the donor HOMO () and acceptor LUMO () which are listed in Table 2. It may be noted that following [18] we have used eV in equation (10) for all DA materials used in Table 1 and 2. Using these input parameters the calculated values of are listed in Table 2 along with their experimental values obtained for these materials. According to Table 2, the calculated values are in reasonable agreement with those obtained experimentally.
Entry 
Active Layer 
(cm2V1s1) 
(cm2V1s1) 

Ref. 
OSC 
PTB7:PCBM

1.0x103 
2.0x104 
5.0 
[20] 
OSC 
PCDTBT:PCBM 
2.9x103 
3.0x105 
96.7 
[21]

OSC 
P3HT:PCBM 
x103 
x104 
10.0 
[19] 
OSC 
MDMOPPV: PCBM

x103 
x104 
10.0 
[19] 
OSC 
PBDTBDD:BisPCBM 
9.6x105 
1.3x104 
0.7 
[10] 
OSC 
PBDTBDD:PCBM 
8.8x104 
1.4x103 
0.6 
[10] 
OSC 
P3HT: BisPCBM 
9.6x105 
1.0x104 
1.0 
[10] 
OSC 
MEHPPV: PCBM 
x103 
x106 
1000.0 
[25] 
OSC 
SiPCPDTBT:PCBM

2.5x104 
3.0x105 
8.3 
[22] 
H 
MDMOPPV:ncZnO

2.8x105 
5.5x106 
5.1 
[23] 
H 
P3HT: SiNCs

x103 
x103 
1.0 
[24] 
Table 1. Input values for calculating with donor –accepter materials forming the active layer, electron mobility , hole mobility, mobility ratio P
According to equation (10) the
increases if the ratio
, that means, when the electron mobility is higher than the hole mobility as shown in Figure 1. In a material with equal mobility of electrons and holes, the contribution of the transport term to the
vanishes.
Figure 1. , in equation (10) plotted as a function of electron: hole mobility ratio, .
Figure 2. Measured currentvoltage characteristics normalized to the shortcircuit current (open circles) of two P3HT/PCBM solar cells annealed at 52°C (a) and 70°C (b). The solid lines denote simulations using slowest carrier recombination constant, while the dashed lines correspond to simulations using average carrier recombination constant . is the dielectric constant (Reproduced with permission from (Koster et al. [23]. Copyright 2006, AIP Publishing LLC.
Figure 2 Measured currentvoltage characteristics normalized to the shortcircuit current (open circles) of two P3HT/PCBM solar cells annealed at 52 °C (a) and 70°C (b). The solid lines denote simulations using slowest carrier recombination constant, while the dashed lines correspond to simulations using average carrier recombination constant . is the dielectric constant (Reproduced with permission from [12].
The analytical results of the dependence of on the charge carrier mobilities derived in equation (10) agree with the experimental observation as well as with the numerical simulation [12]. In Figure 2(a) and (b) we have reproduced the JV characteristics measured on P3HT:PCBM bulk heterojunction organic solar cells (BHJ OSCs) annealed at two different temperatures, 52°C and 70°C, respectively (Koster et al., 2006). The measured mobility P3HT:PCBM of electrons and holes is found to be(m2V1s1),(m2V1s1) at 52°C (Figure:2a) and (m2V1s1), (m2V1s1) at 70°C (Figure 2b) [12]. Using these values, we find that the mobility ratio P decreases from 8.3 x 103 to 1.0 x 103 when one anneals the sample at 52°C and 70°C. According to equation (10), this means that one should get a higher value of at the annealing temperature of 52°C than at 70°C. This result is quite consistent with that shown in Figures 2(a) and (b), where the measured and simulated at 52°C is about 0.04 V higher than that at 70°C. Mobility dependent JV characteristics have also been simulated by assuming [5]. The is found to be independent of the charge carrier mobility in the range from 1 to 106 cm2/Vs. According to equation (10) also, the mobility dependent term vanishes for and hence Voc becomes constant which is consistent with this result.
Abbrevations
H: Hybrid
PTB7:(poly[[4,8bis[(2ethylhexyl)oxy]benzo[1,2b:4,5b']dithiophene2,6diyl][3fluoro2(2 ethylhexy)carbonyl]thieno[3,4b]thiophenediyl]])
PCBM: 1(3methoxycarbonyl)propyl1phenyl(6,6)C
PCDTBT:poly[N9''heptadecanyl2,7carbazolealt5,5(4',7'di2thienyl2',1',3'benzothiadiazole)]
P3HT: poly(3hexylthiophene)
MDMOPPV:poly[2methoxy5(3’,7’dimethyloctyloxy)14phenylene vinylene]
PBDTBDD:poly(((4,8Bis(5(2ethylhexyl)thiophen2yl)benzo[1,2b:4,5b′]dithiophene2,6diyl) bis(trimethyl))co(5,7bis(2ethylhexyl)benzo[1,2c:4,5c′]dithiophene4,8dione))
BisPCBM: bisadduct of phenylC61butyric acid methyl ester)
MEHPPV :poly[2methoxy5(2ethylhexyloxy)1,4phenylenevinylene]
SiPCPDTBT:poly[2,1,3benzothiadiazole4,7diyl[4,4bis(2ethylhexyl)4Hcyclopenta2,1b:3,4b′]dithiophenesiloe 2,6diyl]]
ncZnO: Zinc oxide nanoparticles
Si NCs: Silicon nanocrystals
Discussions
According to equation (10) the opencircuit voltage becomes equal to the effective band gap energy and hence independent of the charge carrier mobilities when the hole quasiFermi level is equal to the HOMO level of the donor molecule and the electron and hole mobilities are equal. It is to be noted that the derived in equation (10), depends on the electron and hole mobilities directly. The material withwill have greater energy lossand hence lower in comparison with materials with, which will have lesser and hence higher . From this point of view, one may prefer to use materials with for obtaining higherin OSCs.
As stated above, in the calculation of from equation (10), we have assumed a constant value for eV, which is valid only if the charge carrier concentration remains constant and that means the mobilities of charge carriers are not very high or very low. For example, in OSCs based on P3HT:PCBM where a mobility ratio=10 is considered [19], it is found that if both charge carrier mobilities at this ratio are high, then this will lead to the efficient extraction of charge carriers which reduces the charge carrier concentration. This reduction in carrier concentration is expected to draw the hole quasi Fermi level away from the HOMO level of the donor material, which according to equation (10) will reduce the . This will eventually reduce the PCE as found in [19]. Likewise, at low charge carrier mobilities at the same ratio, the recombination will be enhanced which will reduce the short circuit current [6, 19], leading to reduction in PCE. In this view, the derived in equation (10) may be regarded to be valid only at moderate electron and hole mobilities leading to high PCE.
Table 2 Donor Acceptor materials, Donor HOMO level , Acceptor LUMO level , Effective band gap , transport loss term and from equation (10) .
For highlighting the role of the charge carrier mobility, it may be desirable to consider the two DA combination materials MDMOPPV:PCBM and P3HT:BisPCBM in Table 2. These two combinations have the same effective gap of 1.30 eV but the second term of in equation (10) is 0.06 eV for the first combination and zero for the second (Table 2). As a result the value of is less in the first combination than that in the second, producing higher (0.96 eV) in MDMOPPV:PCBM in comparison with that of 0.89 eV in P3HT:BisPCBM. It may be interesting to note that, using eV in equation (10), we get, which shows that the loss of 0.4 eV due to the energy difference is much bigger than the second term due to the charge transport whose calculated values are listed in column 6 of Table 2.
Donor
material 
(eV) 
Acceptor
material 
(eV) 
(eV) 
(eV) 
(V) 
(V) 
Ref. 
PTB7 
5.15 
PCBM 
4.06 
1.09 
0.04 
0.73 
0.75 
[20] 
PCDTBT 
5.50 
PCBM 
4.30 [11] 
1.20 
0.12 
0.92 
0.85 
[21] 
P3HT 
5.10 
PCBM 
4.06 
1.04 
0.06 
0.69 
0.63 
[26] 
MDMOPPV 
5.36 
PCBM 
4.06 
1.30 
0.06 
0.96 
0.83 
[11] 
PBDTBDD 
5.23 
BisPCBM 
3.80 
1.43 
0.01 
0.97 
1.00 
[10] 
PBDTBDD 
5.23 
PCBM 
3.94 
1.29 
0.01 
0.88 
0.86

[10] 
P3HT 
5.10 
BisPCBM 
3.80 
1.30 
0.00 
0.89 
0.74 
[10] 
MEHPPV 
5.20 
PCBM 
3.95 
1.00 
0.18 
0.88 
0.74 
[13] 
SiPCPDTBT 
4.86 
PCBM 
3.88 
0.98 
0.05 
0.63 
0.59

[22] 
MDMOPPV 
5.20 
ncZnO 
4.20 
1.00 
0.04 
0.64 
0.74 
[23]

P3HT 
5.10 
SiNCs 
3.95 
1.15 
0.00 
0.75 
0.75 
[24]

Table 2. Donor Acceptor materials, Donor HOMO level , Acceptor LUMO level , Effective band gap , transport loss term and from equation (10) .
Conclusion
We have derived a mobility dependant expression for of OSCs and OHSCs. We have shown that if the difference between the electron and hole mobilities is small, the derived here does not depend on the charge carrier mobilities significantly. According to our model, the of a DA material depends on two terms; the first depends on the energetics and the second on the electron and hole mobility ratio. This may be expected to be useful in predicting the PCE of OSCs and OHSCs prior to their fabrications from a combination of DA materials.
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