Follow us on :


Take a look at the Recent articles

Charge-carrier-mobility-dependent-open-circuit-voltage-in-organic-and-hybrid-solar-cells

David Ompong

School of Engineering and IT, Faculty of EHSE, Charles Darwin University, Darwin, NT 0909, Australia

Jai Singh

School of Engineering and IT, Faculty of EHSE, Charles Darwin University, Darwin, NT 0909, Australia

E-mail : Jai.Singh@cdu.edu.au

DOI: 10.15761/FNN.1000108

Article
Article Info
Author Info
Figures & Data

Abstract

A better understanding of the open-circuit 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 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.

Key words

 charge carrier mobility, donor-acceptor, open-circuit voltage, organic solar cells, quasi-fermi 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 [1-4]. 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 short-circuit current densityand open-circuit voltage, and hence, reduction in the power conversion efficiency of OSCs and OHSCs. 
The current densityin the drift-diffusion model is a function of both the electrical and chemical potentials gradients, denoted byand, 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 optimizingwith respect to one can determine the optimal value of  corresponding to.
It is established that the 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.
In this work, we have derived an analytical expression for  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 (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 Open-Circuit Voltage  ()

The open-circuit voltage is given by the energy difference between the electron and hole quasi-Fermi levels (Gregg, 2003)
,                                                                                                                 (1)
In OSCs and OHSCs the open-circuit 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 drift-diffusion 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) quasi-Fermi level.
The charge-carrier 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 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. 

Entry

Active Layer          

 (cm2V-1s-1)

 (cm2V-1s-1) 

 

Ref.

OSC

     PTB7:PCBM            

1.0x10-3            

2.0x10-4           

5.0                 

[20]   

OSC

PCDTBT:PCBM         

2.9x10-3            

3.0x10-5           

96.7 

[21]  

OSC

P3HT:PCBM                    

x10-3                

x10-4            

10.0                            

[19]   

OSC

MDMOPPV: PCBM     

  x10-3                

x10-4            

10.0                

[19]   

OSC

PBDTBDD:Bis-PCBM  

9.6x10-5            

1.3x10-4           

0.7                 

[10]   

OSC

PBDTBDD:PCBM       

8.8x10-4            

1.4x10-3           

0.6                

[10]   

OSC

   P3HT: Bis-PCBM         

   
9.6x10-5

              1.0x10-4           

1.0                

[10]   

OSC

MEHPPV: PCBM             

   x10-3                  

x10-6          

1000.0           

[25]   

OSC

Si-PCPDTBT:PCBM   

2.5x10-4            

3.0x10-5           

8.3                

[22]   

H

MDMOPPV:nc-ZnO  

2.8x10-5            

5.5x10-6           

5.1                

[23]   

H

P3HT: Si-NCs              

x10-3                  

x10-3              

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 current-voltage characteristics normalized to the short-circuit 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 current-voltage characteristics normalized to the short-circuit 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 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 (Koster et al., 2006). The measured mobility P3HT:PCBM of electrons and holes is found to be(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.

Abbrevations

H: Hybrid
PTB7:(poly[[4,8-bis[(2-ethylhexyl)oxy]benzo[1,2-b:4,5-b']dithiophene-2,6-diyl][3-fluoro-2-(2- ethylhexy)carbonyl]thieno[3,4-b]thiophenediyl]])
PCBM: 1-(3-methoxycarbonyl)-propyl-1-phenyl-(6,6)C
PCDTBT:poly[N-9''-hepta-decanyl-2,7-carbazole-alt-5,5-(4',7'-di-2-thienyl-2',1',3'-benzothiadiazole)]
P3HT: poly(3-hexylthiophene)
MDMOPPV:poly[2-methoxy-5-(3’,7’-dimethyloctyloxy)-1-4-phenylene vinylene]
PBDTBDD:poly(((4,8-Bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b:4,5-b′]dithiophene-2,6-diyl) bis(trimethyl))-co-(5,7-bis(2-ethylhexyl)benzo[1,2-c:4,5-c′]dithiophene-4,8-dione))

Bis-PCBM: bisadduct of phenyl-C61-butyric acid methyl ester)
MEHPPV :poly[2-methoxy-5-(2-ethylhexyloxy)-1,4-phenylenevinylene]  
Si-PCPDTBT:poly[2,1,3-benzothiadiazole-4,7-diyl[4,4-bis(2-ethylhexyl)-4H-cyclopenta2,1-b:3,4-b′]dithiophene-siloe 2,6-diyl]]
nc-ZnO: Zinc oxide nanoparticles  
Si NCs: Silicon nanocrystals    

Discussions

According to equation (10) the open-circuit voltage becomes equal to the effective band gap energy and hence independent of the charge carrier mobilities when the hole quasi-Fermi 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:Bis-PCBM 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: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.

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     

Bis-PCBM 

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    

Bis-PCBM 

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]   

Si-PCPDTBT

4.86      

PCBM

3.88        

0.98          

0.05        

0.63          

0.59 

[22]   

MDMOPPV

5.20      

nc-ZnO     

4.20        

1.00          

0.04        

0.64         

0.74          

[23]  

P3HT

5.10       

Si-NCs

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.

References

  1. Rita Narayan M, Singh J (2013) Study of the Mechanism and Rate of Exciton Dissociation at the Donor-Acceptor Interface in Bulk-Heterojunction Organic Solar Cells. Journal of Applied Physics 114: 735101-735107.
  2. Ompong D, Singh J (2015) Diffusion length and Langevin recombination of singlet and triplet excitons in organic heterojunction solar cells. Chemphyschem 16: 1281-1285. [Crossref]
  3. Narayan MR, Singh J (2012) Roles of Binding Energy and Diffusion Length of Singlet and Triplet Excitons In Organic Heterojunction Solar Cells. Physica Status Solidi 9: 2386-2389.
  4. Narayan MR, Singh J (2015) Excitonic and Photonic Processes In Materials, Singapore, Springer.
  5. Würfel U, Neher D, Spies A, Albrecht S (2015) Impact of charge transport on current-voltage characteristics and power-conversion efficiency of organic solar cells. Nat Commun 6: 6951. [Crossref]
  6. Deibel C, Wagenpfahl A, Dyakonov V (2008) Influence of Charge Carrier Mobility on the Performance of Organic Solar Cells. Physica Status Solidi (Rrl)- Rapid Research Letters 2: 175-177.
  7. Gregg BA (2003) Excitonic Solar Cells. The Journal of Physical Chemistry B 107: 4688-4698.
  8. Brabec CJ, Cravino A, Meissner D, Sariciftci NS, Fromherz T, et al. (2001) Origin of the Open Circuit Voltage of Plastic Solar Cells. Advanced Functional Materials 11: 374-380.
  9. Scharber MC, Mühlbacher D, Koppe M, Denk P, Waldauf C, et al. (2006) Design Rules For Donors In Bulk-Heterojunction Solar Cells-Towards 10% Energy-Conversion Efficiency. Advanced Materials 18: 789-794.
  10. Ye L, Zhang S, Qian D, Wang Q, Hou J (2013) Application of Bis-Pcbm In Polymer Solar Cells With Improved Voltage. The Journal of Physical Chemistry C 117: 25360-25366
  11. Cravino A (2007) Origin of the Open Circuit Voltage Of Donor-Acceptor Solar Cells: Do Polaronic Energy Levels Play A Role? Applied Physics Letters 91: 243502.
  12. Koster LJA, Van Strien WJ, Beek WJE, Blom PWM (2007) Device Operation Of Conjugated Polymer/Zinc Oxide Bulk Heterojunction Solar Cells. Advanced Functional Materials 17: 1297-1302.
  13. Dastoor PC, Mcneill CR, Frohne H, Foster CJ, Dean B, et al. (2007) Understanding And Improving Solid-State Polymer/C60-Fullerene Bulk-Heterojunction Solar Cells Using Ternary Porphyrin Blends. The Journal of Physical Chemistry C 111: 15415-15426.
  14. Liu Y, Zhao J, Li Z, Mu C, Ma W, et al. (2014) Aggregation and morphology control enables multiple cases of high-efficiency polymer solar cells. Scharber [Crossref]
  15. Proctor CM, Love JA, Nguyen TQ (2014) Mobility guidelines for high fill factor solution-processed small molecule solar cells. Adv Mater 26: 5957-5961. [Crossref]
  16. Servaites JD, Ratner MA, Marks TJ (2009) Practical Efficiency Limits In Organic Photovoltaic Cells: Functional Dependence of Fill Factor And External Quantum Efficiency. Applied Physics Letters 95: 163302.
  17. Nelson J (2003) The Physics of Solar Cells, London, Imperial College Press
  18. Wagenpfahl A, Rauh D, Binder M, Deibel C, Dyakonov V (2010) S-Shaped Current-Voltage Characteristics Of Organic Solar Devices. Physical Review B 82: 115306.
  19. Mandoc MM, Koster LJA, Blom PWM (2007) Optimum Charge Carrier Mobility in Organic Solar Cells. Applied Physics Letters 90: 133504.
  20. Ebenhoch B, Thomson SAJ, Genevicius K, Juška G, Samuel IDW (2015) Charge Carrier Mobility of the Organic Photovoltaic Materials Ptb7 And Pc71bm And Its Influence On Device Performance. Organic Electronics 22: 62-68.
  21. Philippa B, Stolterfoht M, Burn PL, Juška G, Meredith P, et al. (2014) The impact of hot charge carrier mobility on photocurrent losses in polymer-based solar cells. Sci Rep 4: 5695. [Crossref]
  22. Albrecht S, Vandewal K, Tumbleston JR, Fischer FS, Douglas JD, et al. (2014) On the efficiency of charge transfer state splitting in polymer: fullerene solar cells. Adv Mater 26: 2533-2539. [Crossref]
  23. Koster LJA, Mihailetchi VD, Blom PWM (2006) Bimolecular Recombination in Polymer/Fullerene Bulk Heterojunction Solar Cells. Applied Physics Letters 88: 052104.
  24. Liu CY, Holman ZC, Kortshagen UR (2009) Hybrid solar cells from P3HT and silicon nanocrystals. Nano Lett 9: 449-452. [Crossref]
  25. Liu XD, Xu Z, Zhang FJ, Zhao SL, Zhang TH, et al. (2010) Influence of Small-Molecule Material on Performance of Polymer Solar Cells Based On Meh-Ppv:Pcbm Blend. Chinese Physics B 19: 118601.
  26. Sun Y, Takacs CJ, Cowan SR, Seo JH, Gong X, et al. (2011) Efficient, air-stable bulk heterojunction polymer solar cells using MoO(x) as the anode interfacial layer. Adv Mater 23: 2226-2230. [Crossref]

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

Corresponding author

Jai Singh

School of Engineering and IT, Faculty of EHSE, Charles Darwin University, Darwin, NT 0909, Australia, Tel: 414-805-9160, Fax: 414-805-9170

E-mail : Jai.Singh@cdu.edu.au

Entry

Active Layer          

 (cm2V-1s-1)

 (cm2V-1s-1) 

 

Ref.

OSC

     PTB7:PCBM            

1.0x10-3            

2.0x10-4           

5.0                 

[20]   

OSC

PCDTBT:PCBM         

2.9x10-3            

3.0x10-5           

96.7 

[21]  

OSC

P3HT:PCBM                    

x10-3                

x10-4            

10.0                            

[19]   

OSC

MDMOPPV: PCBM     

  x10-3                

x10-4            

10.0                

[19]   

OSC

PBDTBDD:Bis-PCBM  

9.6x10-5            

1.3x10-4           

0.7                 

[10]   

OSC

PBDTBDD:PCBM       

8.8x10-4            

1.4x10-3           

0.6                

[10]   

OSC

   P3HT: Bis-PCBM         

   
9.6x10-5

              1.0x10-4           

1.0                

[10]   

OSC

MEHPPV: PCBM             

   x10-3                  

x10-6          

1000.0           

[25]   

OSC

Si-PCPDTBT:PCBM   

2.5x10-4            

3.0x10-5           

8.3                

[22]   

H

MDMOPPV:nc-ZnO  

2.8x10-5            

5.5x10-6           

5.1                

[23]   

H

P3HT: Si-NCs              

x10-3                  

x10-3              

1.0              

[24]   

Table 1. Input values for calculating  with donor –accepter materials forming the active layer, electron mobility , hole mobility, mobility ratio P

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     

Bis-PCBM 

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    

Bis-PCBM 

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]   

Si-PCPDTBT

4.86      

PCBM

3.88        

0.98          

0.05        

0.63          

0.59 

[22]   

MDMOPPV

5.20      

nc-ZnO     

4.20        

1.00          

0.04        

0.64         

0.74          

[23]  

P3HT

5.10       

Si-NCs

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) .                                     


Figure 1. , in equation (10) plotted as a function of electron: hole mobility ratio, .

Figure 2. Measured current-voltage characteristics normalized to the short-circuit 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.