APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION


IIUM Engineering Journal, Vol. 22, No. 2, 2021 Mohamed and Inche Ibrahim 
https://doi.org/10.31436/iiumej.v22i2.1559 

INFLUENCE OF LIGHT ABSORPTION PROFILE ON THE 

PERFORMANCE OF ORGANIC PHOTOVOLTAICS 

ABDUL HALIM IKRAM MOHAMED1 AND MOHD LUKMAN INCHE IBRAHIM2* 

1
Department of Electrical and Computer Engineering, 

2
Department of Science in Engineering, 

Kulliyyah of Engineering, International Islamic University Malaysia, 

Jalan Gombak, 53100 Kuala Lumpur, Malaysia. 

*
Corresponding author: lukmanibrahim@iium.edu.my 

         (Received:23rd June 2020; Accepted: 16th January 2021; Published on-line: 4th July 2021) 

ABSTRACT:  We investigate how an enhanced light absorption at a specific position 
inside the active layer affects the performance of organic photovoltaic cells (OPVs), 

namely the short-circuit current density (𝐽𝑠𝑐), the open-circuit voltage (𝑉𝑜𝑐 ), the fill factor
(FF), and the power conversion efficiency (PCE). The performance is calculated using an 

updated version of a previously published analytical current-voltage model for OPVs, 

where the updated model allows the light absorption profile to be described by any 

functions provided that analytical solutions can be produced. We find that the light 

absorption profile affects the performance through the drift current. When the mobility 

imbalance is not very high (when the ratio of the mobility of the faster carrier type to the 

mobility of the slower carrier type is less than about 103), the PCE is maximized when
the light absorption is concentrated at the center of the active layer. When the mobility 

imbalance is very high (when the ratio of the mobility of the faster carrier type to the 

mobility of the slower carrier type is more than approximately 104), the PCE is
maximized when the light absorption is concentrated near the electrode collecting the 

slower carrier type. Therefore, it is important to ensure that the light absorption profile is 

properly tuned so that the performance of OPVs is maximized. Moreover, any efforts 

that we make to improve the performance should not lead to a light absorption profile 

that would actually impair the overall performance.  

ABSTRAK: Kajian ini menilai bagaimana penyerapan cahaya yang tinggi pada bahagian 

tertentu lapisan aktif mempengaruhi prestasi sel fotovoltaik organik (OPV), iaitu 

ketumpatan arus litar pintas (Jsc), voltan litar terbuka (Voc), faktor pengisian (FF), dan 

kecekapan penukaran kuasa (PCE). Prestasi dikira mengguna pakai model terkini yang 

diperbaharui dari model asal analitikal OPV voltan-arus, di mana model ini 

membenarkan mana-mana profil penyerapan cahaya digunakan asalkan penyelesaian 

analitikal terhasil.  Dapatan kajian mendapati profil penyerapan cahaya mempengaruhi 

prestasi berdasarkan arus hanyut. Apabila ketidakseimbangan pergerakan caj tidak begitu 

tinggi (di mana nisbah pergerakan pembawa caj laju kepada perlahan adalah kurang 

daripada 103), PCE menjadi maksimum jika penyerapan cahaya bertumpu pada tengah
lapisan aktif. Apabila ketidakseimbangan pergerakan caj sangat tinggi (di mana nisbah 

pergerakan pembawa caj laju kepada perlahan adalah lebih daripada 104), PCE menjadi
maksimum jika penyerapan cahaya bertumpu pada elektrod yang mengutip pembawa caj 

perlahan. Oleh itu, kedudukan talaan profil penyerapan cahaya yang tepat adalah sangat 

penting bagi menentukan prestasi OPV dimaksimumkan. Tambahan, apa sahaja usaha 

penambahbaikan prestasi seharusnya tidak menyebabkan pengurangan keseluruhan 

prestasi profil penyerapan cahaya. 

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KEYWORDS: carrier mobility; fill factor; open-circuit voltage; power conversion efficiency; 

short-circuit current  

1. INTRODUCTION  

Organic photovoltaic cells (OPVs) have several unique advantages such as 

lightweight, high mechanical flexibility, good transparency, and low fabrication cost [1]. 

Hence, OPVs are viewed as a promising alternative to conventional photovoltaic 

technologies. For example, OPVs have a great potential to be used for powering various 

devices such as wearable devices [1], next generation self-driven biomedical devices [2], 

and off-grid devices for the Internet of Things [3]. The recorded power conversion 

efficiency (PCE) of OPVs has been improving steadily from 11% [4] several years ago to 

17.3% currently [5]. However, the PCE of OPVs is still quite low compared with 

traditional photovoltaics, and even when compared with another emerging photovoltaic 

technology, namely perovskite solar cells [4]. Therefore, improving the performance has 

been a focus of OPV research. 

There are many factors that influence the performance of OPVs such as the carrier 

mobility [6,7], the work function of the electrodes [8,9], the permittivity of the active layer 

[7,10] and the thickness of the active layer [11-13]. The light absorption profile inside the 

active layer has been reported to be another factor that influences the performance [14-16]. 

The light absorption profile is determined by several factors such as the optical properties 

of the OPV components and the thickness of the active layer [13]. 

An improvement in the PCE by optimizing the light absorption profile can be 

achieved in practice since the absorption profile can be controlled to a certain extent. For 

example, optical spacers can be used to alter the absorption profile [14,15]. Another 

method that could be used to control the absorption profile is by employing plasmonic 

nanoparticles, which can be located outside or embedded inside the active layer [17]. Light 

intensity increases near the plasmonic nanoparticles [17], and hence, the light absorption 

should be higher near the nanoparticles. Therefore, one could control the absorption 

profile to a certain extent by choosing the embedment positions of the nanoparticles. 

Although understanding the effect of the light absorption profile is clearly important, 

there is still a gap concerning our understanding on this matter. It has been shown that a 

high light absorption at a specific position inside the active layer improves the 

performance [14-16]. In more detail, Mescher et al. [14] and Islam et al. [16] concluded 

that a higher light absorption at the center of the active layer gives a better performance 

but did not conclude on the possible effect of carrier mobility imbalance. On the other 

hand, Tress et al. [15] concluded that the fill factor (FF) improves if more light is absorbed 

near the contact that collects the slower carrier type even when the mobility imbalance is 

quite low. Hence, there are still uncertainties regarding this matter. For example, from 

previous studies, we still have no clue on how the optimum light absorption profile 

evolves as the carrier mobilities evolve from balanced to highly imbalanced. Furthermore, 

there is a disagreement between the results of previous studies. When the mobility 

imbalance is quite low (when the ratio of the mobility of the faster carrier type to the 

mobility of the slower carrier type is of one order of magnitude), Islam et al. [16] found 

that the best performance is produced when the light absorption is concentrated at the 

center of the active layer whereas Tress et al. [15] found that the best performance is 

produced when the light absorption is concentrated near the electrode collecting the slower 

carrier type. 

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In this paper, we will investigate the effect of the light absorption profile on the 

performance of OPVs with the aim of providing insights on how an enhanced light 

absorption at a specific position inside the active layer affects the OPV performance at 

different levels of mobility imbalance, and enlightening the disagreement between the 

results of previous studies. 

2.   METHODOLOGY 

2.1  Analytical Model 

The current-voltage model for OPVs presented in this section is employed for all 

calculations in this article. The model is essentially the same as a previously published 

model [18]. Therefore, finer details of the model presented here can be obtained in the 

published work [18]. The only difference between the model presented here and the 

previous model is that the charge-transfer (CT) state generation profile here is open to any 

functions provided that the continuity equations can be solved analytically whereas the CT 

state generation profile in the previous model has an exponential function. The device 

structure of the model is illustrated in Fig. 1. 

 

Fig. 1: A schematic showing the energy levels and the device structure. LUMOd and 

HOMOd denote the lowest unoccupied molecular orbital (LUMO) and the highest 

occupied molecular orbital (HOMO) of the donor, respectively. LUMOa and 

HOMOa denote the LUMO and the HOMO of the acceptor, respectively. Eg denotes 

the effective band gap. EFa and EFc denote the Fermi levels of the anode and the 
cathode, respectively. 𝐿 is the thickness of the active layer. φpa and φnc denote the 

hole injection barrier at anode and the electron injection barrier at cathode, 

respectively. In the operation of OPVs, anode is the electrode that collects free holes 

and cathode is the electrode that collects free electrons.  

Light absorption by the active layer generates excitons. When the excitons reach the 

donor-acceptor interface, CT states may be produced. Free charge carriers are generated 

from the dissociation of the CT states. When free electrons and free holes recombine 

bimolecularly, CT states are produced back. Therefore, CT states can be generated due to 

exciton relaxations at the donor-acceptor interface (through photon absorption) and due to 

bimolecular recombination. Since bulk heterojunction design is used to construct the 

active layer, the photogenerated CT states are produced at approximately the same 

location where the excitons are photogenerated (or where the photons are absorbed). 

Therefore, the profile of the light absorption can be assumed to have the same equivalent 

shape as the profile of the CT state photogeneration rate. 

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 According to the previous model, when the non-geminate recombination is not 

considered in the electron (hole) continuity equation, the resulting electron (hole) density 

that arises from the solution of the equation is called the maximum electron (hole) density 

𝑛𝑚𝑎𝑥  (𝑝𝑚𝑎𝑥) [18]. The electron and the hole continuity equations at steady state without 
the non-geminate recombination are given by 

𝐷𝑛
𝜕2𝑛𝑚𝑎𝑥

𝜕𝑥2
+ 𝜇𝑛𝐹

𝜕𝑛𝑚𝑎𝑥

𝜕𝑥
+ 𝑃𝑑 𝐺𝐶𝑇 = 0 (1) 

𝐷𝑝
𝜕2𝑝𝑚𝑎𝑥

𝜕𝑥2
− 𝜇𝑝𝐹

𝜕𝑝𝑚𝑎𝑥

𝜕𝑥
+ 𝑃𝑑 𝐺𝐶𝑇 = 0 (2) 

where 𝐷𝑛 (𝐷𝑝) is the electron (hole) diffusion coefficient, 𝜇𝑛 (𝜇𝑝) is the electron (hole) 

mobility, 𝐹 is the electric field, 𝑃𝑑  is the dissociation probability of CT states, and 𝐺𝐶𝑇  is 
the CT state photogeneration rate per unit volume. The CT state dissociation probability is 

defined as 

𝑃𝑑 =
𝑘𝑑

𝑘𝑑+𝑘𝑓
 (3) 

where 𝑘𝑑  is the CT state dissociation rate coefficient, and 𝑘𝑓  is the CT state decay rate 

coefficient. As in the previous model [18], 𝑘𝑑  given by the work of Inche Ibrahim [19] is 
used in this article, which is an improvement to the Onsager-Braun model [20]. 

Solving Eq. (1) and Eq. (2) are the same as solving non-homogenous second-order 

linear ordinary differential equations with constant coefficients where 𝐷𝑛 , 𝐷𝑝, 𝜇𝑛, 𝜇𝑝, 𝐹, 

and 𝑃𝑑  are independent of 𝑥 [18] with 𝑃𝑑 𝐺𝐶𝑇  being the non-homogenous term. Any 
functions can be assigned to the non-homogenous term (i.e. to 𝐺𝐶𝑇  since 𝑃𝑑  is a constant) 
provided that Eq. (1) and Eq. (2) can be solved analytically. The boundary conditions as 

given in the previous model [18] are used and MATLAB is employed to analytically solve 

Eq. (1) and Eq. (2) to obtain 𝑛𝑚𝑎𝑥  and 𝑝𝑚𝑎𝑥. 

Furthermore, according to the previous model [18], when the non-geminate 

recombination is considered in the electron (hole) continuity equation, the resulting 

electron (hole) density that arises from the solution of the equation is called the net 

electron (hole) density 𝑛𝑛𝑒𝑡 (𝑝𝑛𝑒𝑡). The electron and the hole continuity equations at 
steady state with the non-geminate recombination considered are therefore given by 

𝐷𝑛
𝜕2𝑛𝑛𝑒𝑡

𝜕𝑥2
+ 𝜇𝑛𝐹

𝜕𝑛𝑛𝑒𝑡

𝜕𝑥
+ 𝑃𝑑 𝐺𝐶𝑇 − (1 − 𝑃𝑑 )𝑅𝑏 − 𝑅𝑚𝑛 = 0 (4) 

𝐷𝑝
𝜕2𝑝𝑛𝑒𝑡

𝜕𝑥2
− 𝜇𝑝𝐹

𝜕𝑝𝑛𝑒𝑡

𝜕𝑥
+ 𝑃𝑑 𝐺𝐶𝑇 − (1 − 𝑃𝑑 )𝑅𝑏 − 𝑅𝑚𝑝 = 0 (5) 

where 𝑅𝑏 is the non-geminate bimolecular recombination rate per unit volume, and 𝑅𝑚𝑛 
(𝑅𝑚𝑝) is the non-geminate monomolecular recombination rate per unit volume for 

electrons (holes). 𝑅𝑏 is given by 

𝑅𝑏 = 𝛾𝑘𝐿 𝑛𝑚𝑎𝑥 𝑝𝑚𝑎𝑥  (6) 

where 𝛾 is the bimolecular recombination reduction coefficient, and 𝑘𝐿  is the Langevin 
recombination coefficient [18]. 𝑅𝑚𝑛 and 𝑅𝑚𝑝 are given by 

𝑅𝑚𝑛 = 𝑘𝑚𝑛 𝑛𝑚𝑎𝑥  (7) 

𝑅𝑚𝑝 = 𝑘𝑚𝑝𝑝𝑚𝑎𝑥  (8) 

where 𝑘𝑚𝑛 (𝑘𝑚𝑝) is the monomolecular recombination coefficient for electrons (holes). It 

is important to note that 𝑅𝑚𝑛 = 𝑅𝑚𝑝 as explained in the previous model [18]. Solving Eq. 

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(4) and Eq. (5) are also the same as solving second-order linear ordinary differential 

equations with constant coefficients. Again, the boundary conditions as given in the 

previous model [18] are used and MATLAB is employed to analytically solve Eq. (4) and 

Eq. (5) to obtain 𝑛𝑛𝑒𝑡 and 𝑝𝑛𝑒𝑡. 

The electron current density 𝐽𝑛 and the hole current density 𝐽𝑝 are given by 

𝐽𝑛 = 𝑞𝜇𝑛𝐹𝑛𝑛𝑒𝑡 + 𝑞𝐷𝑛
𝜕𝑛𝑛𝑒𝑡

𝜕𝑥
 (9) 

𝐽𝑝 = 𝑞𝜇𝑝𝐹𝑝𝑛𝑒𝑡 − 𝑞𝐷𝑝
𝜕𝑝𝑛𝑒𝑡

𝜕𝑥
 (10) 

The total current density 𝐽 is the sum of 𝐽𝑛 and 𝐽𝑝. The first terms on the right sides of 

Eq. (9) and Eq. (10) are the electron drift current density 𝐽𝑛,𝑑𝑟 and the hole drift current 
density 𝐽𝑝,𝑑𝑟, respectively, whereas the second terms on the right sides of Eq. (9) and Eq. 

(10) are the electron diffusion current density 𝐽𝑛,𝑑𝑖𝑓𝑓 and the hole diffusion current density 

𝐽𝑝,𝑑𝑖𝑓𝑓, respectively. 

2.2  Light Absorption Profiles and Parameter Values 

To achieve the objective of this paper, we need to use the light absorption profile (and 

hence, the 𝐺𝐶𝑇  profile) that has an enhanced absorption (and hence, an enhanced CT state 
photogeneration rate) at a specific position inside the active layer. To produce such a 

profile, we use 𝐺𝐶𝑇  given by 

𝐺𝐶𝑇 = 𝐺0 + 𝐺1(𝑥 − 𝐶)
2 (11) 

where 𝐺0 and 𝐺1 are the CT state photogeneration rate parameters, and 𝐶 is the location of 
the peak CT state photogeneration rate (or the location of the peak light absorption). The 

CT state photogeneration rate per unit area of the active layer is 𝐺𝐶𝑇,𝑎𝑟𝑒𝑎 = ∫ 𝐺𝐶𝑇
𝐿

0
𝑑𝑥. 

Hence, 

𝐺𝐶𝑇,𝑎𝑟𝑒𝑎 = 𝐺0𝐿 +
1

3
𝐺1[(𝐿 − 𝐶)

3 + 𝐶3] (12) 

To ensure a fair comparison, each of the studied 𝐺𝐶𝑇  profile must have the same the value 
of 𝐺𝐶𝑇,𝑎𝑟𝑒𝑎. When 𝐶 ≤ 𝐿 2⁄ , we impose the condition 𝐺𝐶𝑇 (𝑥 = 𝐿) = 0. Therefore, when 
𝐶 ≤ 𝐿 2⁄ , we have 

𝐺1 =
3𝐺𝐶𝑇,𝑎𝑟𝑒𝑎

𝐶 3+(𝐿−𝐶)3−3𝐿(𝐿−𝐶)2
 (13) 

𝐺0 = −𝐺1(𝐿 − 𝐶)
2 (14) 

When  𝐶 > 𝐿 2⁄ , we impose the condition 𝐺𝐶𝑇 (𝑥 = 0) = 0. Therefore, when 𝐶 >
𝐿 2⁄ , we have 

𝐺1 =
3𝐺𝐶𝑇,𝑎𝑟𝑒𝑎

𝐶 3+(𝐿−𝐶)3−3𝐿𝐶 2
 (15) 

𝐺0 = −𝐺1𝐶
2 (16) 

In this paper, four different values of 𝜇𝑛 𝜇𝑝⁄  (ratio of the electron mobility to the hole 

mobility) are considered. Table 1 shows the values of 𝜇𝑛 𝜇𝑝⁄  together with the 

corresponding values of 𝜇𝑛 and 𝜇𝑝 used in this study. The values of the other parameters 

are shown in Table 2. The parameter values in Table 2 are typical for OPVs [11,18], 

particularly OPVs based on P3HT:PCBM blend [21]. Figure 2 illustrates several 𝐺𝐶𝑇  
profiles used in this study. 

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Table 1: Values of the electron mobility 𝜇𝑛 and the hole mobility 𝜇𝑝, and the 

resulting 𝜇𝑛 𝜇𝑝⁄  used in the calculations 

𝝁𝒏 (m
2V−1s−1) 𝝁𝒑 (m

2V−1s−1) 𝝁𝒏 𝝁𝒑⁄  

1  10−7 1  10−7 1 

9  10−7 9  10−9 102 

9  10−6 9  10−10 104 

3  10−5 3  10−10 105 

Table 2: Parameter values used in the calculations unless otherwise specified. The 

symbols in this table are the same as in the previous works [11,18] 

Symbols Parameter Description Value 

𝑬𝒈 Effective band gap 1.1 eV 

𝑵𝒄, 𝑵𝒗 Density of states 2  10
26 m−3 

𝝁𝒏𝒂 Actual electron mobility 200𝝁𝒏  
𝝁𝒑𝒂 Actual hole mobility 200𝝁𝒑 

𝜺 Effective permittivity of the active layer 3  10−11 F·m−1 
𝝋𝒏𝒂, 𝝋𝒏𝒄 Injection barriers 0.05 eV 

𝒌𝒇 CT state decay rate coefficient 1  10
8 s−1 

𝒂 Electron-hole separation of the CT state 1.8  10−9 m 
𝑻 Temperature 300 K 
𝝀 Donor-acceptor morphology parameter 0.15 
𝑳 Active layer thickness 100 nm 

𝑮𝑪𝑻,𝒂𝒓𝒆𝒂 CT state photogeneration rate per unit area 1  10
21 m−2s−1 

𝜸 Bimolecular recombination reduction coefficient 0.002 

𝒌𝒎𝒏 Monomolecular recombination coefficient for electrons 200 s
−1 

 

 

Fig. 2: Several profiles of the CT state photogeneration rate per unit volume 𝐺𝐶𝑇  that 
are used in this study. The expression for 𝐺𝐶𝑇  is given by Eq. (11). 𝐺𝐶𝑇  profile with 

light absorption peaks 𝐶 at 0 nm, 40 nm, 60 nm and 100 nm are also used but are not 
shown in the figure. 

3.   RESULTS AND DISCUSSION 

In this section, we present and discuss the results of our calculations. Different values 

of the carrier mobilities affect the OPV performance as shown by previous studies 

[6,7,18]. Therefore, it is imperative that we do not compare between performances of an 

OPV with a given 𝜇𝑛 𝜇𝑝⁄  and an OPV with a different 𝜇𝑛 𝜇𝑝⁄ . What we should analyze 

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here is how OPVs with different values of 𝜇𝑛 𝜇𝑝⁄  respond to different 𝐺𝐶𝑇  profiles. 

Furthermore, for clarity and brevity of this study, we only confine our analysis from the 

theoretical perspective. We cannot fully control all device parameters when conducting 

experiments. For example, every time we use a different active layer material in order to 

change the carrier mobilities, we would also inevitably change a few other device 

parameters (e.g. the light absorption properties, and thus the value of  𝐺𝐶𝑇,𝑎𝑟𝑒𝑎). Therefore, 
the inclusion of experimental data into our analysis would complicate our explanation and 

understanding on this matter. 

3.1  Effect on Short-Circuit Current Density 

Figure 3 shows the magnitude of the short-circuit current density (|𝐽𝑠𝑐 |) as a function 
of 𝐶 for each of the studied 𝜇𝑛 𝜇𝑝⁄ . 

 

Fig. 3: Magnitude of the short-circuit current density |𝐽𝑠𝑐 | as a function of the 
position of the light absorption peak 𝐶 when (a) 𝜇𝑛 𝜇𝑝⁄ = 1, (b) 𝜇𝑛 𝜇𝑝⁄ = 10

2, 

(c) 𝜇𝑛 𝜇𝑝⁄ = 10
4, and (d) 𝜇𝑛 𝜇𝑝⁄ = 10

5. If 𝜇𝑛 𝜇𝑝⁄  is increased to a value 

significantly above 105, |𝐽𝑠𝑐 | produced by the light absorption profile with 𝐶 < 𝐿 2⁄  
(e.g. 𝐶 = 20 nm) would even surpass |𝐽𝑠𝑐 | produced by the light absorption profile 

with 𝐶 = 𝐿 2⁄ . 

To explain the trend shown in Fig. 3, let us explore the role of the drift and the 

diffusion currents. We define the average electron drift current density 〈𝐽𝑛,𝑑𝑟 〉 and the 
average electron diffusion current density 〈𝐽𝑛,𝑑𝑖𝑓𝑓 〉 as 

〈𝐽𝑛,𝑑𝑟 〉 = 𝑞𝜇𝑛𝐹 ∫
𝑛𝑛𝑒𝑡

𝐿
𝑑𝑥

𝐿

0
= 𝑞𝜇𝑛𝐹〈𝑛𝑛𝑒𝑡 〉 (17) 

〈𝐽𝑛,𝑑𝑖𝑓𝑓 〉 = 𝑞𝐷𝑛 ∫ (
𝜕𝑛𝑛𝑒𝑡

𝜕𝑥
) 𝑑𝑥

𝐿

0
𝐿⁄ = 𝑞𝐷𝑛 〈

𝜕𝑛𝑛𝑒𝑡

𝜕𝑥
〉 (18) 

where 〈𝑛𝑛𝑒𝑡 〉 is the average 𝑛𝑛𝑒𝑡 and 〈𝜕𝑛𝑛𝑒𝑡 𝜕𝑥⁄ 〉 is the average 𝜕𝑛𝑛𝑒𝑡 𝜕𝑥⁄ . 

Table 3 shows 〈𝐽𝑛,𝑑𝑟 〉’s and 〈𝐽𝑛,𝑑𝑖𝑓𝑓〉’s at short-circuit for the balanced mobility case 

(𝜇𝑛 𝜇𝑝⁄ = 1). As seen in Table 3, as the light absorption is concentrated farther from the 

cathode (i.e. as 𝐶 is lowered), 〈𝐽𝑛,𝑑𝑖𝑓𝑓 〉 remains unchanged because 〈𝜕𝑛𝑛𝑒𝑡 𝜕𝑥⁄ 〉 is 

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unchanged, but the magnitude of 〈𝐽𝑛,𝑑𝑟 〉 increases because 〈𝑛𝑛𝑒𝑡 〉 increases. Even when we 

assume there is no non-geminate recombination in our calculations (by using 𝛾 = 0 and 
𝑘𝑚𝑛 = 0 s

−1), we find that 〈𝑛𝑛𝑒𝑡 〉 still increases if the light absorption is concentrated 
farther from the cathode. Therefore, the non-geminate recombination is not the deciding 

factor why 〈𝑛𝑛𝑒𝑡 〉 increases as 𝐶 is lowered. 

Table 3: The average electron drift current density 〈𝐽𝑛,𝑑𝑟 〉 and the average electron 
diffusion current density 〈𝐽𝑛,𝑑𝑖𝑓𝑓〉 at short-circuit for the balanced mobility case 

(𝜇𝑛 𝜇𝑝⁄ = 1) calculated using 𝐺𝐶𝑇  with different light absorption peak positions C’s. 

C (nm) 〈𝑱𝒏,𝒅𝒓〉 (A·m
−2) 〈𝑱𝒏,𝒅𝒊𝒇𝒇〉 (A·m

−2) 

20 −1.19848  105 1.19762  105 

50 −1.19836  105 1.19762  105 

80 −1.19821  105 1.19762  105 

To rationalize why 〈𝑛𝑛𝑒𝑡 〉 increases as the light absorption is concentrated farther 
from the cathode (the electrode collecting free electrons), we consider the following two 

basic facts. First, the profile of the free electrons per unit volume at steady state (which is 

𝑛𝑛𝑒𝑡 profile) is attained after some of the generated electrons are redistributed within the 
active layer (where some electrons may be (accidentally) extracted from the active layer 

during the redistribution process) and some of the electrons are involved in the non-

geminate recombination. However, we can ignore the effect of recombination since we 

have shown that it is not the deciding factor. Second, the cathode has a higher Fermi level 

than the anode, and thus the resulting boundary conditions [18] mean that the free 

electrons (holes) must have a significantly higher concentration near the cathode (anode) 

than near the anode (cathode). 

Now consider the 𝐶 = 20 nm and 𝐶 = 80 nm cases as shown in Table 3. In both 
cases, the electric field magnitudes are the same since both cases are at the same applied 

voltage 𝑉𝑎 (i.e. at short-circuit). For the 𝐶 = 20 nm case, there should be less free 
electrons extracted from the active layer in attaining 𝑛𝑛𝑒𝑡 than for the 𝐶 = 80 nm case 
since majority of the electrons are generated farther from the cathode, and thus are less 

likely to be extracted during the redistribution process. This means more of the generated 

electrons are retained inside the active layer, thus causing a higher 𝑛𝑛𝑒𝑡 for the 𝐶 = 20 nm 
case than for the 𝐶 = 80 nm case. Therefore, the 𝐶 = 20 nm case would give a higher 
magnitude of 〈𝐽𝑛,𝑑𝑟 〉 than the 𝐶 = 80 nm case due to a higher magnitude of 𝑛𝑛𝑒𝑡 (see Eq. 
(17)).        

From our analysis above, we can conclude that the farther the light absorption is 

concentrated from the cathode (anode), the more the free electrons (holes) are retained 

inside the active layer, and this gives a higher |𝐽𝑛,𝑑𝑟 | (|𝐽𝑝,𝑑𝑟 |), but |𝐽𝑛,𝑑𝑖𝑓𝑓 | (|𝐽𝑝,𝑑𝑖𝑓𝑓 |) is 

effectively unchanged because the gradient of 𝑛𝑛𝑒𝑡 (𝑝𝑛𝑒𝑡) is effectively unchanged. 
Therefore, the farther the light absorption is concentrated from the cathode (anode), the 

higher the |𝐽𝑛| (|𝐽𝑝|).     

Based on the conclusion above, the results in Fig. 3(a) can be rationalized as follows. 

When 𝐶 = 0 nm (i.e. when 𝐶 is the closest to the anode or the farthest from the cathode), 
|𝐽𝑛| is at its highest whereas |𝐽𝑝| is at its lowest. As we start increasing 𝐶 from 𝐶 = 0 nm, 

|𝐽𝑝| increases whereas |𝐽𝑛 | decreases, where the increase in |𝐽𝑝| is stronger than the 

decrease in |𝐽𝑛|, and hence |𝐽𝑠𝑐 | (i.e. |𝐽| at open-circuit) increases. Then, at one point, the 

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decrease in |𝐽𝑛| equals to the increase in |𝐽𝑝|, and at this optimum 𝐶, we have the peak 

|𝐽𝑠𝑐 |. Then, as we further increase 𝐶 beyond the optimum 𝐶, |𝐽𝑠𝑐 | decreases since the 

decrease in |𝐽𝑛| is stronger than the increase in |𝐽𝑝|. For 𝜇𝑛 𝜇𝑝⁄ = 1 case, the contribution 

from electrons (therefore |𝐽𝑛 |) and holes (therefore |𝐽𝑝|) towards |𝐽𝑠𝑐 | are equal (since 

electrons and holes have the same mobilities), and therefore |𝐽𝑠𝑐 | as a function of 𝐶 is 
symmetric about 𝐶 = 𝐿 2⁄  (i.e. the optimum 𝐶). 

When 𝜇𝑛 𝜇𝑝⁄ = 10
2, the contribution of |𝐽𝑛 | on |𝐽𝑠𝑐 | is stronger than the contribution 

of |𝐽𝑝| on |𝐽𝑠𝑐 | due to the fact that the electron mobility is higher now, and since |𝐽𝑛| is 

higher if 𝐶 is closer to the anode, therefore |𝐽𝑠𝑐 | produced by the profile with 𝐶 < 𝐿 2⁄  (i.e. 
𝐶 closer to the anode) is higher than |𝐽𝑠𝑐 | produced by the profile with 𝐶 > 𝐿 2⁄ . This 
explains the results shown in Fig. 3(b).  

As 𝜇𝑛 𝜇𝑝⁄  is increased and becomes more imbalanced, the contribution of |𝐽𝑛 | on |𝐽𝑠𝑐 | 

becomes even more significant, and this widens the difference between |𝐽𝑠𝑐 | produced by 
the profile with 𝐶 < 𝐿 2⁄  and |𝐽𝑠𝑐 | produced by the profile with 𝐶 > 𝐿 2⁄ . This explains 
the trends as we increase 𝜇𝑛 𝜇𝑝⁄  from 1 to 10

5 as shown in Fig. 3. Due to these trends, we 

can expect that when 𝜇𝑛 𝜇𝑝⁄  reaches a threshold value (when 𝜇𝑛 𝜇𝑝⁄ ≫ 10
5), |𝐽𝑠𝑐 | 

produced by the profile with 𝐶 < 𝐿 2⁄  would even surpass |𝐽𝑠𝑐 | produced by the profile 
with 𝐶 = 𝐿 2⁄ . It is worth noting that the explanation presented in section 3.1 is applicable 
at any given applied voltage 𝑉𝑎 in general, and not just at the short-circuit. 

3.2  Effect on Open-Circuit Voltage 

  Figure 4 shows the open-circuit voltage 𝑉𝑜𝑐 as a function of 𝐶 for each of the studied 
𝜇𝑛 𝜇𝑝⁄ . Note that at any given 𝜇𝑛 𝜇𝑝⁄ , the trend of 𝑉𝑜𝑐 as a function of 𝐶 is determined by 

the trend of |𝐽| (the magnitude of the total current density) as a function of 𝐶 at applied 
voltage 𝑉𝑎 near but lower than 𝑉𝑜𝑐. This is because, at a given 𝑉𝑎 that is near but lower than 
𝑉𝑜𝑐, a higher |𝐽| means a higher 𝑉𝑜𝑐  is expected since a higher extra 𝑉𝑎 is required in order 
to push and make |𝐽| = 0. For example, for the 𝜇𝑛 𝜇𝑝⁄ = 1 case at 𝑉𝑎 = 0.7 V as shown in 

Fig. 5(a), the profile with 𝐶 = 50 nm has a higher |𝐽| than the |𝐽| for the profile with 𝐶 =
20 nm, and hence the profile with 𝐶 = 50 nm has a higher 𝑉𝑜𝑐  than the 𝑉𝑜𝑐 for the profile 
with 𝐶 = 20 nm. Therefore, the trend of 𝑉𝑜𝑐 shown in Fig. 4(a) can be explained by the 
trend of |𝐽| shown in Fig. 5(a), where the results shown in Fig. 5(a) can be rationalized in 
the same way as we rationalize the results shown in Fig. 3(a). 

At high 𝑉𝑎 (near but below 𝑉𝑜𝑐), the electric field magnitude is significantly lower and 
has significantly less influence in transporting and extracting the carriers than at short-

circuit. Hence, at high 𝑉𝑎, the light absorption profile (and thus 𝐶) becomes significantly 
more important in determining 𝑛𝑛𝑒𝑡 and 𝑝𝑛𝑒𝑡, and thus in determining |𝐽𝑛|, |𝐽𝑝| and |𝐽|, 

than at short-circuit. Consequently, |𝐽| at high 𝑉𝑎 is more sensitive to 𝐶 compared with |𝐽| 
at short-circuit (i.e. |𝐽𝑠𝑐 |), and therefore the trend of |𝐽| at high 𝑉𝑎 as a function of  𝐶 for a 
given 𝜇𝑛 𝜇𝑝⁄  (where 𝜇𝑛 𝜇𝑝⁄ > 1) is equivalent to the trend of |𝐽𝑠𝑐 | as a function of 𝐶 but at 

a significantly higher 𝜇𝑛 𝜇𝑝⁄ . That is why at a high 𝑉𝑎, the |𝐽| produced by the profile with 

𝐶 < 𝐿 2⁄  already surpasses the |𝐽| produced by the profile with 𝐶 = 𝐿 2⁄  when 𝜇𝑛 𝜇𝑝⁄ =

104 (see Fig. 5(c)) whereas at 𝑉𝑎 = 0 V (i.e. short-circuit), |𝐽𝑠𝑐 | produced by the profile 
with 𝐶 < 𝐿 2⁄  still does not surpass |𝐽𝑠𝑐 | produced by the profile with 𝐶 = 𝐿 2⁄  when 
𝜇𝑛 𝜇𝑝⁄ = 10

4 (see Fig. 3(c)). Figure 5 shows the trends of  |𝐽| at high 𝑉𝑎 as a function of 𝐶 

for each of the studied 𝜇𝑛 𝜇𝑝⁄ , and the results shown in Fig. 5 give rise to the trends of 𝑉𝑜𝑐 

as a function of 𝐶 for each of the studied 𝜇𝑛 𝜇𝑝⁄  which is shown in Fig. 4. As mentioned at 

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the end of section 3.1, the argument that we use to rationalize the results shown in Fig. 3 is 

applicable at any 𝑉𝑎, and therefore, the same argument can be used to rationalize the 
results shown in Fig. 5. 

 

Fig. 4: Open-circuit voltage 𝑉𝑜𝑐 as a function of the position of the light absorption peak 
𝐶 when (a) 𝜇𝑛 𝜇𝑝⁄ = 1, (b) 𝜇𝑛 𝜇𝑝⁄ = 10

2, (c) 𝜇𝑛 𝜇𝑝⁄ = 10
4, and (d) 𝜇𝑛 𝜇𝑝⁄ = 10

5. 

 

Fig. 5: Magnitude of the current density |𝐽| as a function of the position of the light 
absorption peak 𝐶 for (a) 𝜇𝑛 𝜇𝑝⁄ = 1 at 𝑉𝑎 = 0.7 V, (b) 𝜇𝑛 𝜇𝑝⁄ = 10

2 at 𝑉𝑎 = 0.68 

V, (c) 𝜇𝑛 𝜇𝑝⁄ = 10
4 at 𝑉𝑎 = 0.58 V, and (d) 𝜇𝑛 𝜇𝑝⁄ = 10

5 at 𝑉𝑎 = 0.48 V. The 

applied voltage 𝑉𝑎 used for each 𝜇𝑛 𝜇𝑝⁄  are chosen randomly but is close to the 

corresponding 𝑉𝑜𝑐 (e.g. for 𝜇𝑛 𝜇𝑝⁄ = 1 case, 𝑉𝑜𝑐 is around 0.73V as shown in Fig. 

4(a), so we consider 𝑉𝑎 = 0.7 V here).  

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3.3  Effect on Fill Factor 

Figure 6 shows the fill factor (FF) as a function of 𝐶 for each of the studied 𝜇𝑛 𝜇𝑝⁄ . 

To rationalize the trend of the FF as a function of 𝐶, it is important to note that the FF 
occurs somewhere between the short-circuit and the open-circuit. Hence, the trend of the 

FF as a function of 𝐶 at a given 𝜇𝑛 𝜇𝑝⁄  is simply somewhere between the trend of |𝐽𝑠𝑐 | and 

the trend of 𝑉𝑜𝑐 as functions of 𝐶 for the same 𝜇𝑛 𝜇𝑝⁄ . For the device considered in this 

study, the FF occurs at 𝑉𝑎 that is significantly closer to 𝑉𝑜𝑐  than to the short-circuit, and 
hence the trends of the FF as a function of 𝐶 as shown in Fig. 6 is very similar to the 
trends of 𝑉𝑜𝑐 as a function of 𝐶 as shown in Fig. 4. 

 

Fig. 6: Fill factor (FF) as a function of the position of the light absorption peak 𝐶 
when (a) 𝜇𝑛 𝜇𝑝⁄ = 1, (b) 𝜇𝑛 𝜇𝑝⁄ = 10

2, (c) 𝜇𝑛 𝜇𝑝⁄ = 10
4, and (d) 𝜇𝑛 𝜇𝑝⁄ = 10

5. 

Mescher et al. [14] concluded that enhanced light absorption at the two edges of the 

active layer (i.e. near the anode and the cathode) lead to a high unfavorable diffusion 

current which reduces the FF, whereas an enhanced absorption at the center of the active 

layer leads to a high FF. These conclusions partially agree with our results here. First, we 

find the drift current is the one that affects the dependence of |𝐽|, and thus the dependences 
of |𝐽𝑠𝑐 |, 𝑉𝑜𝑐 and FF on the light absorption profile, whereas the diffusion current basically 
has no effect (see section 3.1). Second, we find that an enhanced light absorption at the 

center of the active layer leads to the best FF only if the mobility imbalance (i.e. 𝜇𝑛 𝜇𝑝⁄ ) is 

not very high (see Fig. 6). 

Tress et al. [15] concluded that an enhanced absorption near the electrode collecting 

the slower carrier type would give a higher FF even when the mobility imbalance is very 

low, whereas Islam et al. [16] concluded that when the mobility imbalance is low, an 

enhanced absorption at the center of the active layer gives the best FF. We find that an 

enhanced absorption near the electrode collecting the slower carrier type (hole is the 

slower carrier type in our study here) would give the best FF only when the mobility 

imbalance is very high (see Fig. 6). When the mobility imbalance is not very high, the best 

FF is produced when the light absorption is concentrated at the center of the active layer. 

Therefore, our results here agree with the results from Islam et al. [16], but it is worth 

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noting that they did not investigate the role of mobility imbalance on the optimum light 

absorption profile. 

3.4  Effect on Power Conversion Efficiency 

Figure 7 shows the power conversion efficiency (PCE) as a function of 𝐶 for each of 
the studied 𝜇𝑛 𝜇𝑝⁄ . The PCE is given by 

𝑃𝐶𝐸 = 𝐹𝐹 × |𝐽𝑠𝑐 | × 𝑉𝑜𝑐 𝑃𝑖𝑛⁄  (19) 

where 𝑃𝑖𝑛 is the input power per unit area of the incident light (taken to be under the 1 sun 
condition which is 1000 W/m2). Hence, the trend of the PCE as a function of 𝐶 at a given 
𝜇𝑛 𝜇𝑝⁄  is simply the combination of the trends of |𝐽𝑠𝑐 |, 𝑉𝑜𝑐 and FF as functions of 𝐶 for the 

same 𝜇𝑛 𝜇𝑝⁄ . Therefore, the trends of the PCE shown in Fig. 7 can be understood by 

combining the trends shown in Fig. 3, Fig. 4, and Fig. 6. 

 

Fig. 7: Power conversion efficiency (PCE) as a function of the position of the light 

absorption peak 𝐶 when (a) 𝜇𝑛 𝜇𝑝⁄ = 1, (b) 𝜇𝑛 𝜇𝑝⁄ = 10
2, (c) 𝜇𝑛 𝜇𝑝⁄ = 10

4, and (d) 

𝜇𝑛 𝜇𝑝⁄ = 10
5. 

According to Mescher et al. [14] and Islam et al. [16], the highest PCE is produced 

when the light absorption is concentrated at the center of the active layer. However, the 

conclusion is incomplete according to our results. We find that the highest PCE is 

produced by the light absorption that concentrates at the center of the active layer only 

when the mobility imbalance is not very high (see Fig. 7). However, when the mobility 

imbalance is very high (see Fig. 7), the best PCE is produced when the light absorption is 

concentrated near the electrode collecting the slower carrier type. 

4.   CONCLUSION  

By using our updated current-voltage model for OPVs, we have investigated how the 

light absorption profile with an enhanced absorption at a certain position inside the active 

layer affects the performance of OPVs. It is found that the light absorption profile affects 

the OPV performance through the drift current. The further the light absorption is 

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concentrated from the electrode collecting a given charge carrier type, the higher the drift 

current and the total current for that carrier type, whereas the diffusion current for that 

carrier type is unaffected. When the carrier mobilities are balanced, the best |𝐽𝑠𝑐 |, 𝑉𝑜𝑐, FF
and PCE are produced by the light absorption that concentrates at the center of the active 

layer. When the mobilities become imbalanced, the |𝐽𝑠𝑐 |, 𝑉𝑜𝑐, FF and PCE produced by the
light absorption that concentrates nearer to the electrode collecting the slower carrier type 

improve relative to the ones produced by the light absorption that concentrates at other 

positions inside the active layer. When the mobility imbalance is high enough (i.e. reach a 

threshold value), the best |𝐽𝑠𝑐 |, 𝑉𝑜𝑐, FF and PCE are produced by the light absorption that
concentrates near the electrode collecting the slower carrier type. The mobility imbalance 

threshold values for |𝐽𝑠𝑐 |, 𝑉𝑜𝑐, FF, and PCE are different, where the threshold value for 𝑉𝑜𝑐
is the lowest, whereas the threshold value for |𝐽𝑠𝑐 | is the highest. Therefore, if the mobility
imbalance is not very high (when the ratio of the mobility of the faster carrier to the 

mobility of the slower carrier is less than about 103), it is important to ensure that the light 

absorption is concentrated at the center of the active layer in order to maximize the PCE. 

However, if the mobility imbalance is very high (when the ratio of the mobility of the 

faster carrier to the mobility of the slower carrier is about 104 or more), the light 

absorption should be concentrated near the electrode collecting the slower carrier type in 

order to maximize the PCE. 

ACKNOWLEDGEMENT 

This work was supported by the Ministry of Higher Education Malaysia under the 

Fundamental Research Grant Scheme (grant no. FRGS/1/2017/STG02/UIAM/03/2).  

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