Sensitivity analysis of current generation in organic solar cells—comparing bilayer, sawtooth, and bulk heterojunction morphologies

Solar Energy Materials & Solar Cells 111 (2013) 66–73

Contents lists available at SciVerse ScienceDirect

Solar Energy Materials & Solar Cells journal homepage: www.elsevier.com/locate/solmat

Sensitivity analysis of current generation in organic solar cells—comparing bilayer, sawtooth, and bulk heterojunction morphologies Hari K. Kodali a, Baskar Ganapathysubramanian a,b,n a b

Department of Mechanical Engineering, Iowa State University, Ames, IA, USA Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 March 2012 Received in revised form 14 June 2012 Accepted 3 December 2012 Available online 20 January 2013

Organic solar cells (OSC) show great potential as a low-cost energy source. In addition, their mechanical flexibility allows the added advantage of use on a wide variety of surfaces. In recent years, progress in experimental strategies and modeling approaches have enabled enhancing the power conversion efficiencies of OSC. In particular, simulation based analysis has played a significant role in improving our understanding of the charge transport phenomena in the active layer of these devices. The excitonic drift-diffusion (EDD) model has been used widely to simulate the generation and transport properties of bulk heterojunction (BHJ) solar cells. The EDD model – which is derived from the Boltzmann transport is dependent on a number of input parameters such as (1) material properties (mobility and permittivity), (2) operating conditions (illumination and device thickness), and (3) active layer morphology. A comprehensive sensitivity analysis of the short-circuit current, Jsc, to the input parameters is performed. This helps in rank ordering the input parameters and operating conditions – by strength and relevance – on their impact on Jsc. We particularly focus our investigations on understanding how the active layer morphology affects the sensitivity of Jsc. To accomplish this we analyze three classes of morphologies: bilayer, BHJ, and sawtooth. The results show significant differences in sensitivities between BHJ, sawtooth, and bilayer morphologies. Short-circuit current in BHJ structure shows higher sensitivity to material properties than either sawtooth and bilayer structure, suggesting that the necessity for finer control of material properties to counteract the increased disorder in the active layer morphology. The electrode current is found to be most sensitive to illumination intensity for all three morphologies. We report some interesting trends that may help choose the most sensitive parameters to vary for designing OSC’s with better performance. & 2012 Elsevier B.V. All rights reserved.

Keywords: Sensitivity analysis Organic solar cells Heterogeneous microstructure Drift-diffusion

1. Introduction Organic solar cells allow the use of low-temperature roll-toroll, and hence low-cost manufacturing [1]. Mechanical flexibility is the other major advantage of these devices which makes them so attractive. Huge progress has been made in the last few years towards improving the problem of low efficiencies associated with organic solar cells. This advancement (highest reported value under laboratory conditions is 9% [2]) is made possible by experimental and numerical investigations of the operating mechanisms and device structure optimization. Numerical simulations not only help to enhance performance through device optimization [3] but also help to improve the fundamental n Corresponding author at: Department of Mechanical Engineering, Iowa State University, Ames, IA, USA. Tel.: þ 1 515 294 7442; fax: þ1 515 294 3261. E-mail address: [email protected] (B. Ganapathysubramanian). URL: http://www3.me.iastate.edu/bglab/ (B. Ganapathysubramanian).

0927-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.solmat.2012.12.004

understanding of the underlying charge transfer processes in the active layer which is not easily accessible to experimental tools [4–8]. Monte-Carlo method based microscopic models [9–11] have been utilized to probe the effect of morphology on the generation, mobility and recombination of charge carriers. But the high computational cost associated with microscopic simulations hampers their use for large scale simulations or in high-throughput analysis of OSC. A computationally efficient alternative in the form of excitonic drift-diffusion model [5,12–17] has been used successfully for simulating the charge transport properties of organic photovoltaics. With wide use of EDD equations it has become increasingly imperative to do a detailed investigation to understand how the material properties and operating conditions (inputs to EDD model) effect the device performance (output from EDD model). When input variables are not known or can be changed, sensitivity analysis informs us about the effect of a small change in input parameters on the output from EDD model. Sensitivity

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

2. Method 2.1. Model In order to determine the short-circuit current density output from the active layer of a BHJ device, we use the EDD model [5] as described below

r  ðErfÞ ¼ qðnpÞ

ð1Þ

r  Jn ¼ qff 0 D½rf,X þ qff 0 R½n,p

ð2Þ

r  Jp ¼ þ qff 0 D½rf,X qff 0 R½n,p

ð3Þ

Jn ¼ qnmn rf þqV t mn rn

ð4Þ

Jp ¼ qpmp rfqV t mp rp

ð5Þ

r  ðV t mX rXÞff 0 D½rf,X R½X ¼ Gff 0 R½n,p

ð6Þ

Eqs. (1)–(6) are solved to obtain the electron (n) and hole (p) densities, and electrostatic potential f. The solution thus obtained is used to determine the electron and hole current densities, Jn and Jp respectively. Here, q is the elementary charge. Morphology dependent parameters: The values of dielectric constant E, electron mobility mn , and hole mobility mp are dependent on morphology, i.e. E, mn , and mp at any location in the active layer depends on the type of material at that location. f 0 and f denote the parameters which are used to track the donor– acceptor (DA) interfaces. f is 1 at interface, 0 elsewhere. To identify the interfaces with pathways to electrodes – for both 0 0 electrons and holes – we use parameter f . f is 0 at interfaces without direct connection to either cathode (for electrons) or anode (for holes). These interfaces are associated with either of 0 the donor and acceptor regions forming a cul-de-sac. f is 1 elsewhere. The exciton generation, diffusion, and dissociation are represented by Eq. (6). G and D½rf,X denote rate of exciton generation and dissociation respectively. Exciton relaxation rate is denoted by R½X . The recombination of charges is given by R½n,p . The expressions for D½rf,X , R½n,p , G, R½X are D½rf,X ¼ kd X

ð7Þ

R½n,p ¼ gr np

ð8Þ

G ¼ aG expðaxÞ

ð9Þ

R½X ¼ X=tX

ð10Þ

where G denotes illumination intensity and x is the distance from the illumination site. Here the exciton generation rate G is approximated using Beer–Lambert Law. The expression for

Current Density (A/m^2)

analysis can also help in ordering the inputs by their effect on performance of organic solar cells. This facilitates decision making process when one is presented with the option of modifying one among a number of competing input parameters, thus providing guidance on process optimization. Also, it would contribute towards increasing the understanding and credibility of the EDD model and making the results more persuasive. In recent years, such sensitivity analysis has been used extensively for a variety of systems like ecological models [18], chemical models [19], structural systems [20], pharmacokinetic models [21], and fluid flows [22]. For OSC devices, sensitivity analysis of 1D homogeneous ¨ model was undertaken by Hausermann et al. [23], wherein the sensitivity to input parameters for a range of device thicknesses, input voltages and time during turn-on was studied. The input parameters selected were electron and hole mobilities mn , mp , recombination efficiency reff, exciton pair separation a, and decay rate kf. The pair separation a was found to have the highest influence on the short-circuit current. However, experimental evidence suggests that the active layer morphology is an important factor determining the performance of organic solar cells [24–26], which indicates significant effect of morphology on the sensitivity of Jsc. This study aims to do an indepth sensitivity analysis of a morphology dependent EDD model for organic solar cells. We analyze the sensitivity of the shortcircuit current to the (a) material properties (electron mobility mn , hole mobility mp , acceptor dielectric constant EA , and donor dielectric constant ED ), (b) device thickness (L), and (c) operating condition (illumination intensity G) under three different morphologies. To investigate the effect of the transition from ordered to disordered morphologies on sensitivity, we study bilayer, sawtooth, and BHJ based devices. As the sensitivity to a specific input parameter may be dependent on the values of other input parameters, we compute the sensitivity to the inputs for a range of values of all input parameters.

67

0

-50

-100

Bilayer BHJ Sawtooth Experimental 0

0.2

0.4

0.6

Voltage (V) Fig. 2. Current–voltage characteristic plots for the bilayer, sawtooth, and BHJ morphologies compared with result from experiment [27].

Fig. 1. Morphologies used for the sensitivity analysis. Dark region represents acceptor and lighter region is donor: (a) bilayer, (b) BHJ, and (c) sawtooth.

68

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

Langevin recombination parameter gr used in this study is gr ¼ ðq=EÞðmn þ mp Þ. The dissociation rate constant kd, absorption coefficient a, average lifetime of exciton tX can be found in Kodali and Ganapathysubramanian [5]. We use ohmic boundary conditions at the electrode boundaries and homogeneous Neumann condition for the rest of the boundaries. The Eqs. (1)–(6) are solved using finite element method [5]. We utilized the National Center for Supercomputing Applications (NCSA)-ABE cluster which is part of XSEDE, to analyze the large number of test cases involved with a detailed sensitivity analysis.

Table 1 Representative values of the input parameters P. Parameter, P

2.2. Morphologies The sensitivity analysis is performed on three OPV devices with structures shown in Fig. 1. The cross-section of the active layer shown in Fig. 1 consists of 1:1 volume ratio of poly(3hexylthiophene):[6,6]-phenyl-C61-butyric acid-methyl ester (P3HT:PCBM). Fig. 1(b) shows a typical BHJ morphology with interpenetrating electron donor (P3HT) and acceptor (PCBM) phases. Fig. 1(a) and (c) shows bilayer and sawtooth structures. The bottom of the active layer is connected to cathode and top to anode. Fig. 2 shows the current–voltage characteristic curves for the representative values of input parameters P shown in Table 1. We note that the values of the relative dielectric constants (Table 1) in the acceptor and donor region correspond to the real part of the complex permittivity.

Symbol Numerical value Units

1. Electron mobility

1.0  10  7

m2 V1 s1

2. Hole mobility

1.0  10  7

5. Active layer thickness 6. Illumination intensity

3.0 3.0 1.0  10  7 4  1021

m2 V1 s1 – – m W/m2

mn mp 3. Donor relative dielectric constant ED 4. Acceptor relative dielectric constant EA L

G

Jsc[A /m2]

150

200

BHJ Sawtooth Bilayer

150

100 50 0 1e-09

The EDD model described in Section 2.1 is used to determine the dependence of short-circuit current Jsc on electron and hole mobilities (mn , mp ), donor and acceptor dielectric constants (EA , ED ), active layer thickness (L), and photon flux (G). We define

Jsc[A /m2]

200

2.3. Sensitivity analysis

100 50

1e-08

1e-07

1e-06

0 1e-09

1e-05

1.13

εArelative

200

BHJ Sawtooth Bilayer

150

100 50 0 10

100

0 0.113

11.3

Jsc[A /m2]

Jsc[A /m2]

150

1e-05

50

50

200

1e-06

BHJ Sawtooth Bilayer

150

100

0 0.113

1e-07

200

BHJ Sawtooth Bilayer Jsc[A /m2]

Jsc[A /m2]

150

1e-08

μp [m 2 /Vs]

μn [m 2 /Vs]

200

BHJ Sawtooth Bilayer

1.13

11.3

εDrelative

BHJ Sawtooth Bilayer

100 50

100 L[nm]

1000

0 0.02

0.2

2

20

SUN

Fig. 3. Jsc plotted against (a) mn , (b) mp , (c) EA , (d) ED , (e) L, and (f) G (in terms of number of Suns) for BHJ, sawtooth, and bilayer morphologies.

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

δ Jsc

100

101

BHJ Sawtooth Bilayer

100 δ Jsc

101

10−1

1e-08

1e-07 μn

10−3 1e-09

1e-05

1e-06

1e-05

[m 2 /Vs]

BHJ Sawtooth Bilayer

100

10−2

10−1 10−2

1.13

εArelative

10−3 0.113

11.3

101

100

100

10−1

δ Jsc

101

BHJ Sawtooth Bilayer

10−2 10

1e-07

101

10−1

10−3

1e-08 μn

δ Jsc

δ Jsc

1e-06

BHJ Sawtooth Bilayer

100

δ Jsc

10−1

[m 2 /Vs]

101

10−3 0.113

BHJ Sawtooth Bilayer

10−2

10−2 10−3 1e-09

69

100

10−1 10−2

1000

L[nm]

10−3 0.02

1.13 εDrelative

11.3

BHJ Sawtooth Bilayer

0.2

SUN

2

20

Fig. 4. Sensitivity of Jsc to (a) mn , (b) mp , (c) EA , (d) ED , (e) L, and (f) G (in terms of number of Suns) for BHJ, sawtooth, and bilayer morphologies.

the sensitivity of Jsc to input parameter P, as Sensitivity½P ðJsc Þ ¼

@J sc @P

ð11Þ

where P can be mn , mp , EA , ED , L or G. Note that the value of G can be  30 orders of magnitude larger than other parameters (see Table 1). This results in  30 order of magnitude difference in the values of Sensitivity½G ðJ sc Þ and sensitivity of Jsc to mn , mp , EA , ED , L. Hence we define a normalized form of sensitivity as

dd P J sc ¼

@Jsc =J sc @P=P

ð12Þ

This enables comparison of sensitivity to all the parameters irrespective of the magnitude of the physical values. All subsequent mention of sensitivity refer to normalized sensitivity.

3. Results and discussion We study the response of the three devices as light is incident from the top through the transparent anode onto the active layer. The absorption of photons and consequent exciton generation is

restricted to the donor regions, as it is comparatively negligible in the acceptor region for P3HT:PCBM based devices [28]. The absorption depth is taken as 50 nm [29]. The electron and hole mobilities are assumed to be independent of the electric field in this study [30,31]. Fig. 3 shows the comparison of the effect of varying parameters P on Jsc for the three device morphologies under consideration on a lin-log plot. The mobilities (Fig. 3(a) and (b)) and dielectric constants (Fig. 3(c) and (d)) have a comparatively small effect on Jsc. Photon flux G and active layer thickness L in Fig. 3(e) and (f) show a relatively higher impact. Among the three structures considered, BHJ structures show greatest variation in Jsc due to change in the permittivity (Fig. 3(c) and (d)). This may be due to increased bending of the electric field lines around the donor–acceptor interfaces, thereby assisting charges to bypass interface recombination while moving towards the electrodes [32]. BHJ and sawtooth structures show maximum Jsc when the active layer thickness L is equal to the absorption depth (50 nm) and any further increase in thickness leads to negligible increase in charge generation while significant reduction in charge transport and collection (Fig. 3(e)). Bilayer devices show a monotonic decrease in Jsc with increasing L due to change in the distance of

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

101

101

100

100

10−1

10−1

δ Jsc

δ Jsc

70

10−2 10−3 10−4 10−5 1e-09

μn μp εA εD 1e-08

10−2 10−3 10−4

1e-07

1e-06

10−5 0.113

1e-05

μn μp εA εD 1.13

11.3

εDrelative

μp [m 2 /Vs] 101 100

δ Jsc

10−1 10−2

μn μp εA εD

10−3 10−4 10−5

10

100 L[nm]

1000

Fig. 5. Sensitivity of J sc to different parameters over a range of (a) mp , (b) ED , and (c) L for BHJ morphology.

101

100

100

−1

10−1

δ Jsc

10

δ Jsc

101

10−2 10

−3

10−4 10−5 1e-09

μn μp εA 1e-08

10−3

εD L Γ 1e-07

1e-06

10−2 10−4 10−5 1e-09

1e-05

μn [m 2 /Vs]

100

100

−1

10−1

δ Jsc

10−2 10

10−4 10−5 0.113

1.13

10−4 10−5 0.113

11.3

εArelative

101

101

100

100

10−1

10−1

10−2 10−3

μn μp εA

10−4 10−5

10

1e-05

10−4 1000

εD L Γ

μn μp εA 1.13 εDrelative

11.3

10−2 10−3

εD L Γ 100 L[nm]

1e-06

10−2 10−3

εD L Γ

μn μp εA

δ Jsc

δ Jsc

δ Jsc

101

−3

1e-07

μp [m 2 /Vs]

101 10

εD L Γ

μn μp εA 1e-08

10−5 0.02

εD L Γ

μn μp εA 0.2

SUN

2

20

Fig. A1. Sensitivity of J sc to different parameters over a range of (a) mn , (b) EA , (c) L, and (d) G (in terms of number of Suns) for bilayer morphology.

101

101

100

100

10−1

10−1 δ Jsc

δ Jsc

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

10−2 10−3 10−4 10−5 1e-09

1e-08

1e-07

1e-05

10−5 1e-09

101

100

100

10−1

10−1

10−2

10−2

10−3 10−5

εD L Γ

μn μp εA

0.113

1.13

10−4 10−5 0.113

11.3

εArelative

10−3

101

100

100

10−1

10−1 δ Jsc

101

10−2 10−3 10−5

10

100 L[nm]

1e-07

1e-06

1e-05

10−4 1000

εD L Γ

μn μp εA

1.13 εDrelative

11.3

10−2 10−3

εD L Γ

μn μp εA

10−4

εD L Γ

μn μp εA 1e-08

μp [m 2 /Vs]

[m 2 /Vs]

101

10−4

δ Jsc

10−4

δ Jsc

δ Jsc

μn

1e-06

10−2 10−3

εD L Γ

μn μp εA

71

10−5

0.02

εD L Γ

μn μp εA

0.2

2

20

SUN

Fig. A2. Sensitivity of J sc to different parameters over a range of (a) mn , (b) EA , (c) L, and (d) G (in terms of number of Suns) for BHJ morphology.

the donor–acceptor interface from the location of greatest exciton generation, i.e. the P3HT region closest to anode. Fig. 3(f) shows a rapid increase in Jsc with increase in G. As given in Table 1, the ratio of representative values of electron–hole mobilities as well as donor–acceptor dielectric constant is 1:1. The sensitivity of Jsc (Fig. 4) to different parameters is calculated for a range of each of the parameter values, while keeping all the other parameters at their representative values shown in Table 1. dd P J sc is determined by calculating the relative change in Jsc for a 10% variation in each of the input parameter P. From Fig. 4, it can be inferred that for the representative values of parameters, the sensitivity for BHJ morphology is highest to G. As shown in Fig. 4 (a)–(d), sensitivity of Jsc to mobilities and permittivities is higher for BHJ structure compared to sawtooth or bilayer structure. On the other hand, sensitivity of Jsc to device thickness L is lowest for BHJ structure (Fig. 4(e)). This suggests greater performance benefit by optimizing thickness of devices with regular structure compared to those with BHJ structure. The value of dd P J sc for BHJ and sawtooth is lowest (Fig. 4(e)) when the device thickness L is equal to the absorption depth of 50 nm. All three structures show the same sensitivity to G, suggesting that all devices – independent of the morphology – benefit equally from increase in illumination intensity.

After comparing the values of dd P J sc to each of the parameters P separately for the three morphologies, we analyze the sensitivity of Jsc to multiple parameters simultaneously for a range of values of each d d d parameter (Fig. 5). dd mn J sc , dmp J sc , dEA J sc , and dED J sc are plotted for a range of values of mp , ED , and L (Fig. 5) for a device with BHJ morphology. Plots with sensitivity to all the input parameters for bilayer, BHJ, and sawtooth morphologies are shown in Figs. A1–A3 in Appendix. Fig. 5(a) and (b) shows that as the sensitivity of Jsc to EA and ED increases with increase in mobilities. On the other hand, sensitivity of Jsc to mn and mp decreases with increase in the values of permittivities. Hence it is more difficult to increase the Jsc by increasing mobilities of materials with higher dielectric constants. Also, Jsc can be increased by making relatively small changes to the permittivities of a material with high carrier mobilities for BHJ devices. dd mn J sc increases significantly for larger thicknesses. In addid tion, dd mn J sc is greater than dmp J sc for larger thicknesses, suggesting that the electron mobility, mn , becomes a critical factor for thicker devices. Among mn , mp , EA , ED , L, G the short-circuit current is found to be most sensitive to illumination intensity G. The values of dd G J sc are closely followed by the sensitivity of short-circuit current to device thickness. dd L J sc is also invariant to changes in most of the input parameters, as dd G J sc , but shows change when the thickness itself is changed. Shortcircuit current plotted against device thickness shows a maximum

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

101

101

100

100

10−1

10−1 δ Jsc

δ Jsc

72

10−2 10−3 10−4 10−5 1e-09

10−3

εD L Γ

μn μp εA 1e-08

1e-07

1e-06

10−2 10−4 10−5 1e-09

1e-05

1e-08

101

100

100

10−1

10−1

10−2

10−5 0.113

δ Jsc

δ Jsc

101

10−4

μn μp εA

10−3

εD L Γ

1.13

10−4 10−5 0.113

11.3

εArelative

101

100

100

10−1

10−1

10−2 10−4 10−5 10

100 L[nm]

1e-05

10−4 1000

εD L Γ

μn μp εA

1.13 εDrelative

11.3

10−2 10−3

εD L Γ

μn μp εA

1e-06

10−2

101

10−3

1e-07 μp [m 2 /Vs]

δ Jsc

δ Jsc

μn [m 2 /Vs]

10−3

εD L Γ

μn μp εA

10−5 0.02

εD L Γ

μn μp εA

0.2

2

20

SUN

Fig. A3. Sensitivity of Jsc to different parameters over a range of (a) mn , (b) EA , (c) L, and (d) G (in terms of number of Suns) for sawtooth morphology.

when L is equal to absorption depth for BHJ and sawtooth structures. At the same thickness, sensitivity to thickness is lowest for these structures. Hence, optimizing the device thickness (to be close to the absorption depth) is one order more effective in increasing Jsc than changing any of the material properties. Short-circuit current sensitivity to material properties dmd J and n9p sc dEd J is higher for BHJ structure compared to sawtooth or bilayer A9D sc structures. Sensitivity to electron mobility increases significantly for thicker devices and higher illumination intensities. Jsc is found to be least sensitive to hole mobility. Sensitivity to electron mobility is shown to be an order of magnitude higher than that to hole mobility for most of the values of the input parameters considered.

2. For materials with higher mobilities, Jsc can be changed by making relatively small changes to the permittivities of devices with BHJ structures (Fig. 5(a)). 3. Greater performance benefit can be achieved by optimizing the thickness of devices with regular (sawtooth and bilayer) structure compared to devices with disordered (BHJ) structure (Fig. 4(e)). 4. The sensitivity to permittivities is highest for BHJ devices (Fig. 4(c) and 4(d)), which may be due to charge transport assisted by the electric field lines, bent around the donor– acceptor interfaces [5]. 5. For relatively thicker devices, electron mobility mn is the most critical material property effecting the short-circuit current (Fig. 5(c)).

4. Conclusions We have performed a morphology dependent sensitivity analysis of the short-circuit current in organic solar cells using the exitonic drift-diffusion equations. Following is an enumeration of some of the insights gained is this study:

We envision this analysis and the trends reported to be of value to our experimental colleagues for choosing the most sensitive parameters to vary for designing OSC’s with better performance.

Acknowledgments 1. For materials with higher permittivities, it is difficult to increase Jsc by increasing carrier mobilities of devices with BHJ structures (Fig. 5(b)).

B.G. and H.K.K. were supported in part by NSF PHY-0941576, NSF CCF-0917202 and an NSF CAREER Award. We thank Olga

H.K. Kodali, B. Ganapathysubramanian / Solar Energy Materials & Solar Cells 111 (2013) 66–73

Wodo for the images of BHJ structures. We also thank Prof. Evan Reed (Department of Materials Science and Engineering, Stanford) for motivating the notion of sensitivity analysis. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant number OCI-1053575.

Appendix A. Sensitivity of Jsc to parameters P for bilayer, BHJ and sawtooth structures Figs. A1–A3 show the sensitivity of Jsc to P for bilayer, sawtooth, and BHJ morphologies respectively. For regular structures – bilayer and sawtooth – the Jsc is most sensitive to device thickness for most of the parameter values considered. Whereas in case of BHJ structure, Jsc is more to illumination intensity than device thickness. This is expected as the distribution of the donor phase in the active layer region within the absorption depth governs the exciton generation rate. This donor phase distribution does not change for regular structured devices. dmn J sc is observed to be larger than dmp Jsc for all three device structures, but the difference between dmn J sc and dmp J sc is larger of BHJ compared to sawtooth, which is in-turn larger than bilayer structure. A.1. Detailed procedure for sensitivity calculation

[8]

[9]

[10]

[11]

[12]

[13] [14]

[15]

[16] [17]

Initially representative morphologies (bilayer, sawtooth, and BHJ) are selected. Then we choose the input parameters mn , mp , EA , ED , L and G on which the sensitivity of the selected output parameter Jsc is calculated. A set of representative values are selected for the input parameters. Each of the input six parameters is investigated over a range at eight discrete values. The first value is from the representative values shown in Table 1. Hence, in stage we have one case with values from Table 1 and seven cases (for seven discrete values) for each of the six input parameters, with a total of 43 cases. This is the phase space of parameter values at which sensitivity is calculated. To calculate the sensitivity each of the six parameters is varied by 10% resulting in 258 simulations for each morphology. The sensitivity of the short circuit current is calculated by taking the difference between the values of short circuit current obtained at the 43 cases and the ones obtained with the input parameters varied by 10%. Thus, the sensitivities to each input parameter are determined for a range of parameter values.

[18] [19]

[20] [21]

[22]

[23]

[24]

[25]

References [26] [1] F.C. Krebs, Fabrication and processing of polymer solar cells: a review of printing and coating techniques, Solar Energy Materials and Solar Cells (93) (2009) 394–412. [2] Konarka technologies advances award winning power plastic solar cell efficiency with 9% certification, (February (28)) (2012) /http://www. konarka.com/index.php/site/pressreleasedetail/konarka_technologies_advan ces_award_winning_power_plastic_solar_cell_efficiS. Accessed on the Jan 04 2013. [3] Patrick Boland, Keejoo Lee, Gon Namkoong, Device optimization in PCPDTBT: PCBM plastic solar cells, Solar Energy Materials and Solar Cells 94 (May (5)) (2010) 915–920. [4] G.A. Buxton, N. Clarke, Computer simulation of polymer solar cells, Modelling and Simulation in Materials Science and Engineering 13 (2007). [5] Hari K. Kodali, Baskar Ganapathysubramanian, Computer simulation of heterogeneous polymer photovoltaic devices, Modelling and Simulation in Materials Science and Engineering 20 (3) (2012) 035015. [6] O. Wodo, S. Tirthapura, S. Chaudhary, B. Ganapathysubramanian, A novel graph based formulation for characterizing morphology with application to organic solar cells, Organic Electronics, 13 (2012) 1105–1113. [7] Kanwar S. Nalwa, John A. Carr, Rakesh C. Mahadevapuram, Hari K. Kodali, Sayantan Bose, Yuqing Chen, Jacob W. Petrich, Baskar Ganapathysubramanian,

[27]

[28]

[29] [30]

[31]

[32]

73

and Sumit Chaudhary, Enhanced charge separation in organic photovoltaic films doped with ferroelectric dipoles, Energy & Environmental Science 5 (2012) 7042–7049. Kanwar S. Nalwa, Hari K. Kodali, Baskar Ganapathysubramanian, Sumit Chaudhary, Dependence of recombination mechanisms and strength on processing conditions in polymer solar cells, Applied Physics Letters 99 (26) (2011) 263–301. P.K. Watkins, A.B. Walker, G.L.B. Verschoor, Dynamical monte carlo modelling of organic solar cells: the dependence of internal quantum efficiency on morphology, Nano Letters 5 (9) (2005) 1814–1818. R.A. Marsh, C. Groves, N.C. Greenham, A microscopic model for the behaviour of nanostructured organic photovoltaic devices, Journal of Applied Physics 101 (2007) 083509. C.R. McNeill, S. Westenhoff, C. Groves, R.H. Friend, N.C. Greenham, Influence of nanoscale phase separation on the charge generation dynamics and photovoltaic performance of conjugated polymer blends: balancing charge generation and separation, Journal of Physical Chemistry C 111 (51) (2007) 19153–19160. J. Barker, C. Ramsdale, N. Greenham, Modeling the current–voltage characteristics of bilayer polymer photovoltaic devices, Physical Review B 67 (February (7)) (2003) 1–9. G.A. Buxton, N. Clarke, Predicting structure and property relations in polymeric photovoltaic devices, Physical Review B 74 (August (8)) (2006) 1–5. Biswajit Ray, Muhammad A. Alam, Random vs regularized OPV: limits of performance gain of organic bulk heterojunction solar cells by morphology engineering, Solar Energy Materials and Solar Cells 99 (January) (2012) 204–212. C.M. Martin, V.M. Burlakov, H.E. Assender, D.A.R. Barkhouse, A numerical model for explaining the role of the interface morphology in composite solar cells, Journal of Applied Physics 102 (10) (2007) 104506. J. Williams, A.B. Walker, Two-dimensional simulations of bulk heterojunction solar cell characteristics, Nanotechnology 19 (October (42)) (2008) 424011. M. Shah, V. Ganesan, Correlations between morphologies and photovoltaic properties of rod-coil block copolymers, Macromolecules 43 (January (1)) (2010) 543–552. J. Cariboni, D. Gatelli, R. Liska, A. Saltelli, The role of sensitivity analysis in ecological modelling, Ecological Modelling 203 (12) (2007) 167–182. Andrea Saltelli, Marco Ratto, Stefano Tarantola, Francesca Campolongo, Sensitivity analysis for chemical models, Chemical Reviews 105 (July (7)) (2005) 2811–2828. E.J. Haug, K.K. Choi, V. Komkov, Design Sensitivity Analysis of Structural Systems, 1986. Ivan A. Nestorov, Sensitivity analysis of pharmacokinetic and pharmacodynamic systems: I. A structural approach to sensitivity analysis of physiologically based pharmacokinetic models, Journal of Pharmacokinetics and Pharmacodynamics 27 (1999) 577–596, http://dx.doi.org/10.1023/ A:1020926525495. I.Y. Gejadze, G.J.M. Copeland, Adjoint sensitivity analysis for fluid flow with free surface, International Journal for Numerical Methods in Fluids 47 (8–9) (2005) 1027–1034. R. Hausermann, E. Knapp, M. Moos, N.A. Reinke, T. Flatz, B. Ruhstaller, Coupled optoelectronic simulation of organic bulk-heterojunction solar cells: parameter extraction and sensitivity analysis, Journal of Applied Physics 106 (10) (2009) 104507. L. Chen, Z. Hong, G. Li, Y. Yang, Recent progress in polymer solar cells: manipulation of polymer:fullerene morphology and the formation of efficient inverted polymer solar cells, Advanced Materials 21 (2009) 1434–1449. J. Peet, A.J. Heeger, G.C. Bazan, ‘‘Plastic’’ solar cells: self-assembly of bulk heterojunction nanomaterials by spontaneous phase separation, Accounts of Chemical Research 42 (November (11)) (2009) 1700–1708. C. Groves, O.G. Reid, D.S. Ginger, Heterogeneity in polymer solar cells: local morphology and performance in organic photovoltaics studied with scanning probe microscopy, Accounts of Chemical Research 43 (5) (2010) 615–620, PMID: 20143815. Veronique S. Gevaerts, L. Jan Anton Koster, Martijn M. Wienk, Rene A. J. Janssen, Discriminating between bilayer and bulk heterojunction polymer:fullerene solar cells using the external quantum efficiency. ACS Applied Materials & Interfaces, August 2011, pp. 12–15. C. Deibel, V. Dyakonov, C.J. Brabec, Organic bulk-heterojunction solar cells, IEEE Journal of Selected Topics in Quantum Electronics 16 (6) (2010) 1517–1527. A.L. Fahrenbruch, R.H. Bube, Fundamentals of Solar Cells, 1983. L.B. Schein, R.S. Narang, R.W. Anderson, K.E. Meyer, A.R. McGhie, Electricfield-independent electron mobilities in anthracene, Chemical Physics Letters 100 (1) (1983) 37–40. Veaceslav Coropceanu, Jrme Cornil, Demetrio A. da Silva Filho, Yoann Olivier, Robert Silbey, Jean-Luc Brdas, Charge transport in organic semiconductors, Chemical Reviews 107 (4) (2007) 926–952. Hari K. Kodali, Baskar Ganapathysubramanian, A computational framework to investigate charge transport in heterogeneous organic photovoltaic devices, Computer Methods in Applied Mechanics and Engineering 247–248 (2012) 113–129.