) 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 drift-diffusion current density. The 
thus obtained depends on the energy difference between the highest occupied molecular orbital (HOMO) level and the quasi-Fermi 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.
and open-circuit voltage
, and hence, reduction in the power conversion efficiency of OSCs and OHSCs. 
in the drift-diffusion model is a function of both the electrical and chemical potentials gradients, denoted by
and
, respectively. In OSCs, 
is negligible because there is no built-in electric field like the one in inorganic solar cells due to the property of p-n 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 optimizing
with respect to 
one can determine the optimal value of 
corresponding to
. 
of OSCs [8-11] 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 [13-15] 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.
by optimizing the drift-diffusion 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 (
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.
)
, (1)
and the LUMO energy level of the acceptor
as [16]:
, (2)
is an empirical value representing energy losses in transporting charge carriers to the electrodes.
, (3)
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) quasi-Fermi level. 
and
of electrons and holes inside the active layer are, respectively, given as [18] 
, (4)
, (5)
is 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)
in equation (6) can be optimized with respect to 
as 
, which gives: 
, (7)
drives the electrons and holes in the opposite direction (
we get:
, (8)
is the effective band gap or the DA interface energy gap. Rearranging equation (8) we obtain 
as:
, (9)
and 
which gives; 
where
, (10)
is the energy loss contributed by the energetic (first term) and charge transport (second term) 
in several donor-acceptor (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. 
(cm2V-1s-1) 
(cm2V-1s-1) 
with donor –accepter materials forming the active layer, electron mobility 
, hole 
mobility, mobility ratio P 
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. 
, in equation (10) plotted as a function of electron: hole mobility ratio, 
.
, while the dashed lines correspond to simulations using average carrier recombination constant
. 
is the dielectric constant (Reproduced with permission from (
, while the dashed lines correspond to simulations using average carrier recombination constant
. 
is the dielectric constant (Reproduced with permission from [12]. 
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 J-V characteristics measured on P3HT:PCBM bulk heterojunction organic solar cells (BHJ OSCs) annealed at two different temperatures, 52°C and 70°C, respectively (
(m2V-1s-1),
(m2V-1s-1) at 52°C (Figure:2a) and 
(m2V-1s-1), 
(m2V-1s-1) 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 J-V 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 10-6 cm2/Vs. According to equation (10) also, the mobility dependent term vanishes for 
and hence Voc becomes constant which is consistent with this result. 
derived in equation (10), depends on the electron and hole mobilities directly. The material with
will have greater energy loss
and 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 higher
in OSCs. 
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. 
, Acceptor LUMO level 
, Effective band gap 
, transport loss term 
and 
from equation (10) . 
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:Bis-PCBM. 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. 
(eV) 
(eV)
(eV)
(eV)
(V)
(V)
, Acceptor LUMO level 
, Effective band gap 
, transport loss term 
and 
from equation (10) . 
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. Editorial Information
Editor-in-Chief
Harry E. Ruda
University of Toronto
Article Type
Research Article
Publication history
Received: October 29, 2015
Accepted: January 11, 2016
Published: Janaury14, 2016
Copyright
©2016 Jai Singh, This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation
Ompong D, Singh J (2016) Charge carrier mobility dependent open-circuit voltage in organic and hybrid solar cells. Front Nanosci Nanotech 2: DOI: 10.15761/FNN.1000108
