Liang 2014

Modeling and simulation of bulk heterojunction polymer solar cells

Chunjun Liang a,n, Yongsheng Wang a, Dan Li b, Xingchen Ji b, Fujun Zhang a, Zhiqun He a

a Key Laboratory of Luminescence and Optical Information, Ministry of Education, Institute of Optoelectronic Technology, Beijing Jiaotong University,Beijing 100044, Chinab School of Science, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e i n f o

Article history:Received 16 November 2013Received in revised form4 April 2014Accepted 8 April 2014

Keywords:Numerical simulationPolymer solar cellsBulk heterojunctionP3HT

a b s t r a c t

This review summarizes the optical and electrical models of bulk heterojunction (BHJ) polymer solarcells (PSCs) and numerically simulates and analyzes the performance of the PSCs. A complete simulationof a conventional BHJ device based on the polymer P3HT is presented and results are compared with theexperimental data. Key factors affecting the device performance, including the photo absorption,quantum efficiency, short-circuit current, fill factor, and open-circuit voltage of the device, are analyzedand summarized. Simulations on inverted, semitransparent, and large-area PSCs are performed andfindings are compared with experimental results. Simulations reveal the effects of optical spacer layers,different thicknesses, carrier mobilities, light intensities, contact barriers, effective bandgaps, recombi-nation coefficients, and energy-level bending on the quantum efficiency, short-circuit current, fill factor,and open-circuit voltage of the PSCs. Differences between conventional and inverted geometry, opacityand semitransparency, and small and large-area PSCs are discussed based on the simulations. A powerconversion efficiency of 11.0% is predicted for the PSC based on P3HT. Results suggest the need to furtherreduce the series resistance in large-area PSCs.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Polymer solar cells (PSCs) serve as renewable sources ofelectrical energy because of their many advantages, which includelow cost of fabrication [1–4] and easy processing on flexiblesubstrates [5–9]. The performance of PSCs has been greatlyimproved by introduction of the bulk-heterojunction (BHJ) con-cept [10–15] as an active layer where electron donor and acceptormaterials are mixed in a solution and cast into a thin filmsandwiched between two electrodes. The power-conversion effi-ciency of state-of-the-art PSCs has exceeded 9% for single cells and10% for tandem cells in recent published research [16,17].

Materials innovation is one of the major forces driving theperformance of PSCs. MEH-PPV, P3HT, PCPDTBT and PCDTBT aresome important electron-donor polymers in the history of PSCsresearch [18]. The most impressive high-performance polymersare those composed of thieno[3,4-b]-thiophene (TT) and benzo-dithiophene (BDT) alternating units [16,19,20], which are thefirst donor system capable of reaching power conversion efficiencyof 7–9%. The C60 derivative PCBM is the most widely used

electron-acceptor material in PSCs. However C70 derivativesexhibit better absorption than those of C60. Replacing C60derivatives with C70 derivatives often enhances photocurrent byaround 10%. Another important approach for improving PSCperformance is by replacing the conventional acceptor PCBM witha new soluble C60 derivative (ICBA), the lowest unoccupiedmolecular orbital (LUMO) energy level of which is 0.17 eV higherthan that of PCBM; this approach leads to higher open-circuitvoltages and enhanced power conversion efficiencies [21,22].

Besides great progress on donor and acceptor materials [18,23–27],device processing techniques and their corresponding influence onthe performance of PSCs have been investigated intensively. The mostprominent processing steps for PSC devices include growth-ratecontrol [28], solvent annealing [29], solvent additive [30,31], andthermal annealing [32–34]; these approaches are effective approachesfor increasing optical absorption and carrier mobilities in the activeblends [35]. Another approach involves the use of a relatively thin(100 nm) layer of the active blend to yield satisfactory deviceperformance [36–38]. Small thickness, however, leads to insufficientoptical absorption in the active blend layer. Kim et al. [39] introducedthe concept of an optical spacer, a thin layer inserted between theactive layer and the reflective electrode, to increase photo absorptionin the active layer through the optical interference effect. The proper-ties of the electrodes [40–43], the buffer layer [44–46], and the devicearea [47–49] have great impacts on the performance of PSCs.

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Solar Energy Materials & Solar Cells

http://dx.doi.org/10.1016/j.solmat.2014.04.0090927-0248/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.:þ86 10 51688675.E-mail address: [email protected] (C. Liang).

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Device architectures for PSCs have been diversified. ConventionalPSCs refer to architectures consisting of a transparent conductingmetal oxide coated with a PEDOT:PSS hole-transporting layer fol-lowed by the active BHJ layer. A low work function metal electrode(Al, Ca/Al) is evaporated on top as an electron-collecting electrode tocomplete the device [50–55]. An inverted PSC refers to an architec-ture wherein the nature of charge collection is reversed [6,56–81].The device ITO/ZnO/P3HT:PCBM/MoO3 (or PEDOT:PSS)/Ag is a typicalinverted PSC [82,83]. This architecture exhibits stability [84–86] andprocessing [87,88] advantages over its conventional counterpart.Semitransparent or transparent PSCs [43,89–98] and solution pro-cessed large-area PSCs [3,99,100] are some important ongoingdevelopments in the field of PSCs.

Experimental results show that the performance of solar cells isa function of multiple variables, such as the absorption coefficient,layer thickness, carrier mobility, contact barrier, energy bandgap,recombination coefficient [101–104], and device architecture [16].These factors impart a remarkable influence on the performance ofa BHJ PSC. A comprehensive understanding of the physics of thedevice, including the relationships between processing techniqueand device performance as well as differences between normaland inverted geometries, opaque and transparent devices, andsmall- and large area devices, is necessary for device optimizationand further improvement. Modeling and numerical simulation arepowerful tools for exploring the physics of the device and provideskey insights into the internal mechanisms of PSCs. These tools alsoallow the extrapolation of possible but undiscovered deviceconcepts.

We review the progress on the modeling and simulation of BHJPSCs. A comprehensive and intuitive discussion of device perfor-mance is provided to help further endeavors toward deviceoptimization. In the second section, a brief review on the devel-opment optical and electrical models is given. The simulation of aconventional PSC is presented with a basic model in the thirdsection, and key factors that affect the device performance,specifically, the photo absorption, quantum efficiency, short circuitcurrent (Isc), fill factor (FF), and open circuit voltage (Voc) of thedevice, are analyzed and discussed. Simulations of inverted,semitransparent PSCs, and large-area PSCs are also performedand discussed. Results are compared with the experimental datafrom related works and discussed accordingly. Finally, a summaryis provided in the fourth section.

2. Device models of BHJ PSCs

A typical PSC is shown in Fig. 1(a), where the incident lightpasses through a transparent substrate and its successive layers;the light reflects back at the metal electrode. Excitons are formedand undergo dissociation into free electrons and holes in the BHJlayer after light absorption. The charge carriers are driven by aninternal electric field and collected by the electrodes beforerecombination. The optical model is applied to calculate thenumbers of the absorbed photon in the multilayer structure, andthe electrical model is used to address the carrier generation andcollection process of the photovoltaic structure.

2.1. Optical model

Because the layer thicknesses in organic solar cells are compar-able with the wavelength of sunlight, the optical interferenceeffect should not be ignored in the multilayer structure, especiallywhen the highly reflective metal electrode is used. The mathema-tical treatment for optical interference in multiple-layer thin-filmstacks has long been established [105]. The optical transfer-matrix

theory introduced by Heavens [106] was applied to organic hetero-junction solar cells by Pettersson et al. [107] and others [108,109].The basic concepts of the method are briefly described here.A complete description of the method is presented in Ref. [107]and a free Matlab program is presented by the authors of Ref. [109].

Consider a plane wave incident from the left on a multilayerstructure having m layers between a semi-infinite transparentambient and a semi-infinite substrate as schematically describedin Fig. 1(b). Each layer j (j¼1, 2,…,m) has a thickness dj and opticalproperties described by a complex index of refraction. The opticalelectric field at any point in the system can be resolved into twocomponents: one propagating in the positive x-direction and onein the negative x-direction, which, at a position x in layer j, aredenoted Ejþ(x) and Ej�(x), respectively. The total electric field inan arbitrary plane in layer j at a distance x to the right of boundary(j�1)|j is given by

EjðxÞ ¼ Eþj ðxÞþE�

j ðxÞ ð1Þ

The energy absorbed at position x in the layered structure isproportional to the product of the square of the modulus of theelectric field, the refractive index, and the absorption coefficient atthe actual position x. Thus, jEj2 versus position x in the filmdirectly represents the number of absorbed photons at each point.The absorbed optical power in layer j at position x at normalincidence is finally given by

QjðxÞ ¼ αjT jI0 e�αjxþρ″2j e�αjð2dj �xÞ þ2ρ″je�αjdj cos

4πηjλ

ðdj�xÞþδ″j� ��

ð2Þwhere I0 is the intensity of the incident light, Tj is the internalintensity transmittance, and ρ″j and δ″j are the absolute value andthe argument of the complex reflection coefficient for the secondsubsystem (see Ref. [107]), and αj is the absorption coefficient.

Fig. 1. (a) Device configuration of a typical bulk-heterojunction polymer solar cell.(b) A multilayer structure having m layers between a semi-infinite transparentsubstrate, an ambient substrate, and a semi-infinite substrate. Each layer j has athickness of dj and its optical property is described by its complex index ofrefraction. The optical electric field at any point in layer j is represented by twocomponents: one propagating in the positive and one in the negative x direction,Ejþ and Ej

� , respectively.

C. Liang et al. / Solar Energy Materials & Solar Cells 127 (2014) 67–8668

Here, the energy dissipation in a layered structure at each positionx in layer j is described by three terms. The first originates from theoptical electric field propagating in the positive x-direction, thesame direction as the incident propagating electromagnetic field;the second comes from the field propagating in the negativex-direction; and the third is brought about by the interference oftwo waves. This interference term is especially important foroptically thin layers and when the layered structure has a highlyreflective interface in the structure.

The optical electrical-field and photon absorption rate in themultilayer structure can be calculated if the complex indices ofrefraction and the layer thickness of the materials are known. Theabsorptivity of the incident light for each layer at any spectrumcan be obtained by integration.

2.2. Electrical model

The electrical model considers the generation, recombination,drift, diffusion, and collection process of the electron and hole inthe active BHJ layer.

2.2.1. The effective medium modelThe BHJ layer is a mixture of donors and acceptors exhibiting a

complex morphology. Fig. 2(a) shows the energy levels of bothmaterials and the electronic processes in the device. The photo-generated excitons in the donor and the acceptor dissociate intofree charge carriers at the interface after diffusion. The photo-generated electrons and holes are transported in the acceptor anddonor phases, respectively. In the effective medium model the BHJlayer is considered a homogeneous semiconductor. The energydifference between the LUMO of the acceptor (PCBM) and the

HOMO of the donor (P3HT) functions is considered the effectiveband gap (Egap) of the semiconductor. Fig. 2(b) is a schematicshowing the transport of electrons and holes in the semiconductorunder operation condition.

2.2.2. The semiconductor equationsThe one-dimensional equations that describe the behaviors of

electrons and holes in the BHJ layer are identical to those used ininorganic semiconductor devices, such as the Poisson equation

∂2

∂x2ψ ðxÞ ¼ q

ε½NðxÞþnðxÞ�pðxÞ� ð3Þ

where q is the elementary charge, ε is the dielectric constant, N isthe doping density, ψ represents the potential, and n and p are theelectron and hole densities, respectively. The relevant currentequations are

Jn ¼ Jdiffusionþ Jdrift ¼ qDn∂n∂x

þμnqn∂ψ∂x

ð4Þ

Jp ¼ �qDp∂p∂x

þμpqp∂ψ∂x

ð5Þ

where Jdiffusion represents the diffusion current density due todifferent charge carrier concentrations and Jdrift represents thedrift current density due to internal electric field. μnðμpÞ is theelectron (hole) mobility and Dn (Dp) is the electron (hole) diffusioncoefficient given by the Einstein relation

Dn

μn¼Dp

μp¼ kBT

qð6Þ

where kB is the Boltzmann constant and T is the temperature. Thecurrent continuity equations at steady-state conditions are

�1qdJndx

¼ G�R ð7Þ

1qdJpdx

¼ G�R ð8Þ

where Jn and Jp are the current densities of the electron and hole,respectively, G is the free charge generation rate, and R is therecombination rate of electrons and holes.

Appropriate specification of the boundary conditions is neces-sary to obtain a unique solution. One assumption is that thesemiconductor is always in thermodynamic equilibrium at thecontact interface. Assuming the energy barrier for electrons andholes are Be and Bh (see Fig. 2(b)), respectively, using Boltzmannstatistics

nð0Þ ¼Nceð�Be=kBTÞ ð9Þ

pð0Þ ¼Nve�ðEgap �Be=kBTÞ ð10Þ

nðLÞ ¼Nce�ðEgap �Bh=kBTÞ ð11Þ

pðLÞ ¼Nveð�Bh=kBTÞ ð12Þwhere Nc is the effective density of states of the conduction bandand Nv is the effective density of states of the valence band. If thecontact for electron (hole) is ohmic, there is no energy barrier forelectrons (holes). The boundary condition for the potential is

ψ ðLÞ�ψ ð0Þ ¼ Egap�Be�Bh

q�Va ð13Þ

where Va is the applied voltage.A more general condition is that the extraction of the carriers at

the contacts is not particularly efficient; thus, a finite surfacerecombination (or extraction) velocity [110,111] must be consid-ered. At the left side (x¼0) of the device, the surface electron

Fig. 2. (a) Schematic diagram showing the energy level of the materials and theelectronic processes in a typical BHJ PSC. (b) The electrical model with positiveapplied bias Va under operating conditions; here, Be and Bh represent the energybarrier for electron and hole, respectively.

C. Liang et al. / Solar Energy Materials & Solar Cells 127 (2014) 67–86 69

current Je(0) can be written as

Jnð0Þ ¼ ðnð0Þ�n0ÞSe0 ¼ ½nð0Þ�Nceð�Be=kBTÞ�Se0 ð14Þwhere Se0 is the electron surface recombination velocity at left sideand n0 is the equilibrium charge carrier density defined by theeffective density of states Nc and the barrier Be. Similarly, thesurface hole current at left side Jh(0) is

Jpð0Þ ¼ ½pð0Þ�Nve�ðEgap �Be=kBTÞ�Sh0 ð15ÞAt the right side (x¼L), the surface electron current Je(L) is

JnðLÞ ¼ ½nðLÞ�Nce�ðEgap �Bh=kBTÞ�SeL ð16Þand the surface hole current Jh(L) is

JpðLÞ ¼ ½pðLÞ�Nveð�Be=kBTÞ�ShL ð17Þwhere Sh0 is the hole surface recombination velocity at the leftcontact and SeL and ShL are the electron and hole surface recombi-nation velocities at the right side, respectively. The value of thesurface extraction velocities defines the extraction abilities of thecontacts. The boundary conditions described by Eqs. (9)–(12)mean very sufficient charge carrier extractions, i.e., S-1. Theconditions state that the contacts are in equilibrium and indepen-dent of the working point of the device because all excess chargecarriers are instantaneously extracted.

At this point, we need to specify the charge generation rate Gand the recombination rate R in Eqs. (7) and (8) to complete thedevice model.

2.2.3. The charge carrier generation and recombination in organicBHJ layer

The unique features that define the difference between inor-ganic semiconductors and the organic BHJ materials lie in theprocesses of carrier generation and recombination.

2.2.3.1. The charge carrier generation. The dielectric constant islower (ε� 3–4) in organic semiconductors than in theirinorganic counterparts (ε410). Such a phenomenon exerts adirect consequence on the primary photo excitation in thesematerials. In inorganic semiconductors, photon absorption leadsto the excitation of an electron from the valence band to theconduction band, further resulting in a free electron in theconduction band and a free hole in the valence band. These twocharge carriers do not usually feel mutual electrostatic attractionbecause of the high dielectric constant. In organic semiconductors,the electrostatic attraction between the charge carriers is notefficiently screened because of their low dielectric constant;thus, the charge carriers experience stronger attraction. Theabsorption of a photon primarily leads to the formation of astrongly bound exciton in organic semiconductors. To obtain freecharge carriers, the exciton must be dissociated, which can beachieved by blending two organic semiconductors with differentenergy levels so that an electron can easily undergo a charge-transfer process from the bound exciton state to transform into aless tightly bound charge-transfer exciton [112], leading toefficient generation of free electrons and holes.

Onsager [113] developed an equation for the relative effect ofan applied electric field on the dissociation of a weak electrolyte. Itis a solution to a steady-state diffusion equation which describesthe dissociation and recombination kinetics of ion pairs in anapplied electric field. The ions in solutions are infinitely long lived.The relative increase of the dissociation rate constant K(E) to thedissociation equilibrium constant K(0) is

KðEÞ=Kð0Þ ¼ J1ð2ffiffiffiffiffiffiffiffiffiffiffi�2b

pÞ=

ffiffiffiffiffiffiffiffiffiffiffi�2b

p¼ 1þbþb2=3þb3=18þ…… ð18Þ

where J1 is the Bessel function of order one, b¼ q3E=ð8πεrε0k2T2Þ

and E is the electric field. Braun [114] applied this result todescribe the electric field assisted dissociation of charge transferstates (CT) in donor–acceptor system. He completed the form ofthe dissociation rate to be

kdiss ¼3kR4πa3

e�EB=kT J1ð2ffiffiffiffiffiffiffiffiffiffiffi�2b

pÞ=

ffiffiffiffiffiffiffiffiffiffiffi�2b

p

¼ 3kR4πa3

e�EB=kT ð1þbþb2=3þb3=18þ ::::::Þ ð19Þ

where EB¼q2/(4πεrε0a) is the electron–hole pair binding energy, ais the pair distance, and kR¼q(mnþmp)/εrε0. Braun also noticed thatthe CT states should have a finite lifetime. The bound electron–hole pair may decay to the ground state with a decay rate kf ordissociate into free carriers with a dissociation rate kdiss; thus theprobability of electron–hole pair dissociation is given by P¼kdiss/(kdissþkf).

Blom et al. [115,116] introduced Braun's theory to explain theobserved field and temperature dependence of the photo-currentin PPV:PCBM blends. In this theory, the charge transfer (CT) statewith binding energy EB is considered an intermediate state,leading to modification of the free charge carrier generation(G-PG0), where G0 is the exciton generation rate and P is theprobability of the dissociation of the CT state. The significance of aCT state can be seen in experiments with a strongly pronouncedelectric field and a characteristic temperature-dependent photo-current, such as that observed by Mihailetchi et al. in Ref. [114].

Ultrafast spectroscopy reveals that photo absorption in annealedRR-P3HT:PCBM blends immediately produces free charges, even in theabsence of a field. The yield of free charges is lower in unannealedblends than in annealed ones and very small in disordered P3HT:PCBM blends; these yields approach unity in annealed blends[117,118]. Monte Carlo models [119,120] have also been used to explainthe high dissociation efficiency in BHJ layers. The charge separation isreportedly affected by small organic additives [121].

The free carrier generation rate G can be described as:

G¼ PG0 ð20Þwhere G0 is the photo-absorption induced exciton generation ratecalculated by the optical model and P is the dissociation probabilityof the exciton. The dissociation probability P depends on thetemperature, field, and morphology; in some ideal cases, e.g., inannealed RR-P3HT:PCBM blends, P can approach unity.

2.2.3.2. The charge carrier recombination2.2.3.2.1. Langevin recombination in low mobility materials.

Langevin recombination is expected to occur in low-mobility(μ o1 cm2/V s) materials, that is, when the carrier mean free pathis smaller than the Coulomb capture radius [122]. The recombina-tion rate is determined by the probability of charge carriersmeeting in space. Langevin recombination is described as abimolecular process, with a rate

R¼ γðnp�ni2Þ ð21Þ

where the recombination coefficient γ is given by

γ ¼ qεðμnþμpÞ ð22Þ

where μn and μp the electron and hole mobilities, respectively2.2.3.2.2. Reduced Langevin recombination in BHJ layers. Recom-

bination occurs only at the interface of the donor and acceptor in BHJblends; thus, Koster et al. argued that bimolecular recombinationshould be governed by a slower charge carrier. The recombinationcoefficient afforded by the minimum mobility is given as [123124]

γ ¼ qεminðμn;μpÞ ð23Þ


Pivrikas et al. [125] and others [50,126] have demonstrated thatbimolecular recombination coefficient in a PSC may be three tofour orders lower than the Langevin recombination coefficient.Deibel et al. [127] explained reduction in terms of the carrierconcentration gradient in an ambipolar device.

Koster et al. [115] found that reduced recombination is broughtabout by the CT state. In this model, free carriers recombine intoCT states and these CT states may again dissociate during theirlifetime. Here, the overall recombination rate is reduced by a factorof (1�P)

R¼ ð1�PÞγðnp�ni2Þ ð24Þ

where P is the dissociation probability of the CT state defined inEq. (18). If P approaches unity, the factor (1�P) approaches zero.High dissociation probabilities therefore imply low bimolecularrecombination rates.

A prefactor ξ can be used to address the reduced recombina-tion rate. The recombination rate is still given by

R¼ γðnp�ni2Þ ð25Þ

but the recombination coefficient is

γ ¼ ξqεminðμn;μpÞ ð26Þ

2.2.3.2.3. Trap-assisted recombination. In the case of stronglyreduced bimolecular recombination, trap-assisted recombinationmay become a measurable loss mechanism in organic solar cells[128]. A simple trap-assisted Shockley–Read–Hall (SRH) expres-sion for midgap traps with density Nt and the same capturecoefficient ct for electrons and holes is given by [110]

RSRH ¼ CtNtnp�n2

i

nþpþn1þp1ð27Þ

where n1 and p1 are characteristic charge carrier densities. In thecase of midgap traps, the values are very low compared with theamount of photogenerated charge carriers. Charge carriers recom-bining via traps are directly lost.

2.2.4. The multi-dimensional and Monte Carlo models of PSCsPSCs could be simulated by solving one-dimensional drift-

diffusion equations with proper generation, recombination, andboundary conditions [129–131], assuming a homogeneous semi-conductor. To take full account of the effect of a multi-dimensionalmorphology on the key processes of charge transport, generation,and recombination, two-dimensional [132,133] and three-dimensional [134,135] models have been solved for device simula-tion. The kinetic Monte Carlo model, introduced in Refs. [136,137]and explored in Refs. [138–143], is another important approach forstudying the influence of morphology on the performance of PSCs.These approaches have provided novel insights into PSCs at themicroscopic level.

3. Simulation results and discussion

Here, we simulate the performance the PSCs using the one-dimensional effective medium model. Results from the simulationare compared with experimental findings in previous works. Thesimulation is divided into two parts. First, we simulate a typicalpolymer solar cell with normal geometry and discuss the mainfactors that affect its quantum efficiency, Isc, FF, and Voc. Second,we extend our simulation to the inverted geometry and transpar-ent and large-area devices with/without grids.

We choose the equilibrium boundary condition (Eqs. (9)–(13)),the general generation rate (Eq. (20)), and the recombinationcoefficient (Eq. (26)). More general cases from other works arecarefully reviewed. Using the finite difference method, the

differential equations are discretized on a finite number of points,thereby yielding a finite number of algebraic equations. TheScharfetter–Gummel discretization scheme for drift-diffusionequations [144,145] is used to avoid instability of the solution.Newton's iteration method is applied to solve the discretized non-linear equations, and the simulation is performed using Matlab.

3.1. PSCs with normal geometry

3.1.1. A complete numerical simulation of a typical BHJ PSCZhao et al. [146] presented one of the best performances reported

thus far for a BHJ PSC device based on the polymer P3HT. Thus, wecarry out a simulation and compare our findings with the experi-mental data given in this reference. We take the absorption coeffi-cient data of P3HT:PCBM from this work. Other optical parameters ofthe materials are selected from Refs. [108,147]. Fig. 3 shows thedependence of extinction coefficient kj, which is calculated by therelationship

kj ¼ αjλ=4π ð28Þ

and the index of refraction. We note that the extinction coefficientobtained is inconsistent in the literature. The peak of the extinctioncoefficient in Refs. [146,148] was about �0.2, approximately �0.6 inRef. [108], and �0.3 in Ref. [147]. The absorption ability of the activelayer is important for the device performance as shown in thefollowing context. For the simulation of power conversion efficiency(PCE), data of the AM1.5 solar spectrum [149] are used.

Table 1 displays an overview of the parameters used in thesimulation of a typical P3HT:PCMB PSC. In this simulation, thethickness of the active layer is set to 80 nm because our simulationshows that this value is an optimum one. As shown in Fig. 2(a), thevalue of the effective bandgap is 1.0 eV, which is the differencebetween the LUMO of PCBM and the HOMO of P3HT [150–152] asindicated in Fig. 2. The energy barriers for electrons and holes arezero since the contacts are considered ohmic.

The optical electrical field of an incident electromagnetic wavewith a representative wavelength of 510 nm may be calculatedfrom the optical model, as shown in Fig. 4. Because of opticalinterference between the incident (from the glass side) and back-reflected light (from the reflective electrode), the strength of theelectrical field is at its lowest point at the metallic (Al) electrode.The fluctuant distribution of the optical electrical-field shows thefeature of standing waves. To maximize photo absorption at theP3HT:PCBM blend layer, the active layer must cover the peakregion of the distribution, as shown in the figure.

The energy absorbed at any position in the layer is proportionalto the product of the square of the modulus of the electric field,

1.6

1.7

1.8

1.9

2.0

2.1

2.2

400 500 600 700 800

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Ref

ract

ive

inde

x

Ext

inct

ion

coef

ficie

nt

Wavelength (nm)

Fig. 3. Extinction coefficients (dashed line) and refractive indices (solid line) of thepolymer blend used in the simulation.


the refractive index, and the absorption coefficient at the actualposition. Fig. 5 shows the absorptivity, calculated as the fraction ofradiation absorbed at a given wavelength of the incident light foreach layer at various wavelengths as well as the reflectivity of themultilayer structure. Absorption from the ITO layer is prominentunder a wavelength of 400 nm, and the absorption from the P3HT:PCBM layer declines continuously as the wavelength decreases.The active layer shows major absorption in the range of 400–650 nm with the highest absorptivity of 68% obtained at about510 nm. At wavelengths above 650 nm, absorption from the P3HT:

PCBM layer is negligible, and most of the incident light is reflectedout of the device.

Fig. 6 shows the photo absorption profile in the layered struc-ture calculated using the standard AM 1.5 spectrum. A peak absorp-tion rate of 1.0�1028 /m3s is located at the middle of the active layer.In our simulation the photo absorption profile in the P3HT:PCBMlayer is also the carrier generation rate G0, which is ready for transferto the electric model for calculating the photo current.

The current–voltage characteristics (I–V curve) and the externalquantum efficiency (EQE) of the PSC are computed using theelectrical model based on the photon absorption profile. Fig. 7shows the calculated EQE as a function of wavelength. The deviceshows an EQE of around 63% at the range of 480–610 nm; the EQEquickly drops below 10% at wavelengths beyond 650 nm. Thecurve shows a lower plateau of EQE of �40% at the short-wavelength region and drops continuously at wavelengths lowerthan 370 nm. We note that our EQE curve is very similar in shapeand magnitude to the measured EQE response obtained by Zhaoet al. [146]. For comparison the photo absorptivity of the activelayer is also shown in (Fig. 7); a similar shape that is slightly higherthan the EQE curve is observed. The internal quantum efficiency(IQE), which is simply the quotient of EQE and absorptivity (alsoshown in Fig. 7), is above 95% throughout the whole region. Wuet al. [36] reported that a short circuit current of 11.33 mA/cm2 canbe achieved in an 60-nm thin P3HT:PCBM PSC, and Li et al. [153]

Table 1An overview of the parameters in the simulation of a typical PSC.

Parameter Symbol Numerical value

P3HT:PCBM Thickness D 80 nmBandgap Egap 1.0 eVElectron mobility un 1.0�10�7 m2/V sHole mobility up 1.5�10�8 m2/V sEffective density of states Nc, Nv 3.9�1025 m�3

Dielectric constant ε 2.7�10�11 F/mEnergy barrier for electron Bn 0 eVEnergy barrier for hole Bp 0 eVSeries resistance Rs 3 Ω cm2

Recombination prefactor ξ 1.0Dissociation probability P 1.0Temperature T 300 K

0

5

10

15

20

25

30

35

0

5

10

15

20

25

30

35

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

Ele

ctric

Fie

ld (V

/m)

Optical distance from glass (wavelength)

Glass ITO PEDOT:PSS P3HT:PCBM Reflectiv

electrode

Fig. 4. Calculated profile of the optical electrical field of the incident light at510 nm.

400 500 600 700 8000.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Ref

lect

ivity

Wavelength (nm)

Abs

orpt

ivity

Wavelength (nm)

P3HT:PCBM ITO PEDOT:PSS

Absorption:

Fig. 5. The calculated absorptivity of the incident light for each layer at variouswavelengths and the reflectivity of the multilayer structure.

0 50 100 150 2000

2

4

6

8

10

12

Pho

ton

abso

rptio

n R

ate

(1027

m-3s-1

)

Distance from glass (nm)

ITOPEDOT:PSS

P3HT:PCBM

Fig. 6. Calculated photon absorption rate in the multilayer structure.

400 500 600 700 8000

20

40

60

80

100

Abs

orpt

ivity

(%)

Qua

ntum

Effi

cien

cy (%

)

Wavelength (nm)

EQE IQE Absorptivity

Fig. 7. Calculated EQE, absorptivity, and IQE of a typical PSC.


reported that a short circuit current of 8.08 mA/cm2 was achievedin an ultra-thin 43-nm P3HT:PCBM device. These results suggestthat very high IQE can be achieved in the thin P3HT:PCBM PSC.Park et al. [154] observed almost 100% IQE in a PCDTBT:PCBM PSC.

Fig. 8 shows the simulated I–V characteristics (line) of thedevice in dark and illuminated conditions. For comparison, theexperimental P3HT:PCBM results determined by Zhao et al. [146]are also presented in the figure. Using the parameters specified inTable 1, our simulation results agree very well with the experi-mental data on both dark and illuminated I–V conditions, whichindicates that the combined optical and electrical model in thisstudy covers the essential part of the underlying physics of thedevice.

The inset in Fig. 8 shows the semilog plot of the simulated darkand illuminated I–V curve. The dark semilog I–V curve showsstraight line at the voltage range of 0–0.5 V, which implies that thecurve obeys the ideal diode equation at this region:

I ¼ I0ðeðqv=nkBTÞ �1Þ ð29Þwhere I0 denotes the saturation current density, n is the idealityfactor, kB is the Boltzmann constant, and T is the temperature. Theideality factor is a measure of the slope of the I–V characteristics ona semilogarithmic plot. In our simulation, the fitted ideal factorn¼1.0, which agrees with the finding that Langevin recombinationleads the ideality factor to unity. Wetzelaer et al. [128,155] demon-strated that the ideality factor of the electroluminescence from theCT state of several polymer:fullerene solar cells (polymers:PCPDTBT, MEH-PPV, PF10TBT) are close to unity and concludedthat the main recombination mechanism in the devices is bimole-cular Langevin recombination. In the case of greatly reducedbimolecular recombination, trap-assisted recombination maybecome a measurable loss mechanism, as demonstrated in P3HT:PCBM PSCs [128]. Our simulations indicate that inclusion of trap-assisted recombination increases the ideality factor by up to 2-foldif such recombination plays a dominant role (data not shown).

Here, we discuss the main factors affecting the performance ofthe PSC. For the purpose of discussion, we label the values inTable 1 as “typical” conditions, and we tune each parameteraround typical conditions to show the influence of a specificparameter. Unless otherwise noted, when discussing a certainparameter, the others remain the same as in Table 1.

3.1.2. Quantum efficiency and short-circuit current of the PSCAn efficient PSC device should have a high short-circuit current

(Isc) under solar illumination, which means that it should present a

high EQE under the short-circuit condition. The EQE is the productof photon absorptivity of the absorption layer and the IQE of thedevice. Here, we discuss the factors that have a major impact onthe light absorption and IQE.

3.1.2.1. The thickness of the BHJ blend layer. Fig. 9 shows thecalculated photon absorptivity of a P3HT:PCBM layer at thicknessesof 80, 140, and 220 nm, respectively. Because of the opticalinterference effect, the photon absorptivity of the 140-nm layer issignificantly lower than that of the 80 nm layer for most wavelengthranges. At a thickness of 220 nm, the optical absorptivity of the activelayer has much higher values, as shown in the figure, which agreeswith the common notion that thicker layers absorb more photonsthan thinner ones. Fig. 9 also shows the computed EQE and IQE of thePSC device (the illumination intensity is 8�1020 /m2s) as a functionof the wavelength. The PSC with a thinner active layer of 80 nmshows a high IQE of 95% at most wavelength ranges, and the 140-nmdevice shows a relatively lower IQE of about 92% in its mainabsorptive region. Meanwhile, the IQE of the 220-nm device dropsconsiderably to �80%, which indicates that the IQE is sensitive to thethickness of the absorption layer. This result is understandablebecause the IQE denotes the collection efficiency of the dissociatedcarrier by electrodes, a process that is in competition with the carrierrecombination. The thinner the active layer, the higher the possibilityof the carrier being swept out of the device by the internal fieldbefore recombination. Results suggest that the mobility of the chargecarrier is another key factor affecting the IQE because highermobilities imply a greater possibility that the internal electricalfield will cause the photo-generated carrier to drift and becollected before recombination.

Under solar illumination, the number of absorbed photons(Nph) of the P3HT:PCBM layer at difference thicknesses is calcu-lated and shown in Fig. 10. Photon absorption increases with theactive-layer thickness but shows apparent oscillation [156–158]brought about by the optical interference effect. Absorption peaksat thicknesses of 80, 220 and 350 nm, while the troughs of thecurve are located at 140 and 270 nm. The Isc of the PSC device withdifferent P3HT:PCBM thicknesses is simulated and results are alsoshown in Fig. 10. Fluctuation features similar to that of the Nph

curve are observed; however, unlike the rising trend of the Nph

curve, the Isc decreases with increasing P3HT:PCBM thickness,consistent with a previous conclusion stating that the thicker PSCpossesses a lower IQE. Fig. 10 shows the Isc peaks at 80 and220 nm. The 80 nm PSC shows a photon absorptivity of �68% andan IQE of �95%, while the 220 nm device shows a higher photonabsorptivity of �83% and a lower IQE of �79%. An EQE of 63%

-15

-10

-5

0

5

10

15

20

25

0.750.500.250.00

-15

-10

-5

0

5

10

15

20

25

Cur

rent

Den

sity

(mA

/cm

2 )

0. 0 0. 2 0. 4 0. 6 0. 8 1.01E-9

1E-6

1E-3

1C

urre

nt D

ensi

ty (m

A/c

m)

Voltage (V)

Voltage (V)

Fig. 8. Experimental (scatter, from Ref. [146]) and simulated (line) current–voltagecharacteristics of the device under dark and illuminated conditions. Inset: Semilogplot of the simulated curves.

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

400 500 600 700 8000

20

40

60

80

100 80 nm

140 nm

220 nm

Abs

orpt

ivity

(%)

Qua

ntum

Effi

cien

cy (%

)

Wavelength (nm)

EQE Photo Absorptivity IQE

140 nm

80 nm

Fig. 9. EQE, IQE, and absorptivity of active layers with different thicknesses of 80,140, and 220 nm.


implies that the EQE could still be improved, particularly in termsof absorption enhancement in thinner devices and IQE improve-ment in thicker ones.

3.1.2.2. The carrier mobility of the active layer. Fig. 11 shows thecombined effect of carrier mobility and active-layer thickness onthe IQE and EQE of the PSC device. As mentioned above, the typicaldevice has a high IQE of �95% and EQE of �63% in its mainabsorptive region. The IQE of the device whose mobility is lowerby a factor of 10�2 (as in the case of un-annealed P3HT:PCBM film[35]) drops to �70%, and the EQE drops to �45% in the mainabsorptive region. Combining the low mobility with a higherP3HT:PCBM thickness of 160 nm leads to drastic drops in IQE to�25% and EQE to �15, which indicates that the active-layerthickness and mobility are crucial elements in determining thequantum efficiency of the PSC device.

3.1.2.3. The concept of optical spacer. Kim et al. [39] introduced theidea of adding an optical spacer between the active layer and theAl electrode to spatially change the distribution of the lightintensity inside the device and improve the absorption in thethin active layer. Here, we discuss the effect of the optical spacerlayer on the absorption of the active layer. Fig. 12 shows thecalculated number of absorbed photons in the P3HT:PCBM layer

with different thicknesses of the optical spacer, the TiO2 layer. Theeffect of the spacer layer depends on the thickness of P3HT:PCBMlayer. For the device with a P3HT:PCBM thickness of 80 nm, whichis an optimized number in the conventional device (without theoptical spacer layer), the effect of the TiO2 spacer layer is negativebecause insertion of the spacer layer reduces the absorption oflight in the active layer, as shown in the figure. However, for thedevice with an active layer of 140 nm, in which the absorption iscompressed for the conventional device, as shown in Fig. 10,adoption of the spacer layer enhances the photon absorption inthe active layer. The Nph of the active layer increases from4.3�1020 to 6.0�1020 /m2s, a 40% increase, upon insertion ofthe 50 nm TiO2 spacer layer. Fig. 13 shows the change in photonabsorptivity of the active layer after application of the spacer layer.The absorptivity increases from 50% to 80% at its main absorptionwavelength. Note, however, that although the TiO2 layer canenhance the absorption at a thickness of 140 nm, the overallabsorption of 6.0�1020 /m2s is only slightly (o2%) higher thanthe total absorption of the 80 nm active layer without the spacerlayer, which means the enhancement effect of the optical spacer islimited and the ultimate absorptivity is still determined by theabsorption coefficient. To achieve higher absorptivity in a thin

0

2

4

6

8

10

0 100 200 300 4000

2

4

6

8

10

12

14

16

Nph

(m-2

s-1)

Nph

Jsc

Jsc

(mA

/cm

2 )

Thickness (nm)

Fig. 10. Calculated number of absorbed photons (Nph) of P3HT:PCBM active layersof different thickness and short-circuit current densities (Jsc) of PSCs with differentactive layer thicknesses.

0

20

40

60

80

100

0

20

40

60

80

100

400 500 600 700 8000

20

40

60

80

100

Qua

ntum

Effi

cien

cy (%

)

Wavelength (nm)

M2

Thickness: typical x 2Mobility: typical x 0.01

EQE IQE

Typical

Mobility: typical x 0.01

Fig. 11. EQE and IQE of PSCs with different thicknesses of the active layer anddifferent mobilities.

0 20 40 60 80 1002

3

4

5

6

Nph

(1020

m-2s-1

)

TiO2 Thickness (nm)

P3HT:PCBM Thickness 140 nm 80 nm

Fig. 12. Effects of different thicknesses of the optical spacer, a TiO2 layer, on thelight absorption of the P3HT:PCBM layer with thicknesses of 80 and 140 nm.

400 500 600 700 8000.0

0.2

0.4

0.6

0.8

1.0

Abs

orpt

ivity

(%)

Wavelength (nm)

TiO2 thickness 0 nm 50 nm

Fig. 13. Calculated photon absorptivity of the 140 nm active layer with and withoutthe 50 nm TiO2 spacer layer.


layer, a polymer blend with higher absorption coefficient isneeded.

3.1.3. Fill factor and open-circuit voltageIn this section, we discuss the major factors influencing the

shape of the I–V curve, with emphasis on the parameter fill factor(FF) and the open-circuit voltage (Voc).

3.1.3.1. The thickness of the blend layer. The active layer thicknesshas a great impact on the Isc of the PSC. Here, we check theinfluence of thickness on FF and Voc. Fig. 14 shows the I–V curve ofthe PSC under one sun illumination at different P3HT:PCBMthicknesses of 80, 220, and 350 nm. The curves become flatterwith increasing thickness, which means the FF decreases withincreasing thickness, as confirmed in Fig. 15. Fig. 15 also shows thedependence of Voc and power conversion efficiency (PCE) on thethickness of the active layer. The FF decreases continuously from�70% to o40% when the thickness is raised from tens ofnanometers to 200 nm. A thicker active layer means the photo-generated carrier has to travel a longer average distance out of thedevice. As well, the internal electrical field is weaker in this case,given that the built-in potential is constant. Thus, the thicker theactive layer, the lower the possibility of the photo-generatedcarrier being swept out of the device by the internal field beforerecombination. The Voc curve fluctuates slightly in the range of

0.52–0.58 V in the same manner as the Isc curve, as shown inFig. 15. While the PCE curve shows peaks at 80 and 220 nm, thelatter is significantly lower, which indicates that 80 nm is anoptimized thickness for the active layer in our simulation.

3.1.3.2. The light intensity. The simulation indicates that Isc islinearly dependent on the light intensity (data not shown). Voc

increases monotonically with the light intensity, as shown inFig. 16. Further, Voc exhibits a straight line when plotted as afunction of the logarithm of light intensity. The fitted slope is0.026 V, which is the exact value of kBT/q when T¼300 K. Thisresult agrees well with the analysis of the light-intensity-dependent Voc of PSCs by Koster et al. [159]. Note that oursimulations only consider bimolecular Langevin recombination;if strong trap-assisted recombination is involved, the illumination-intensity dependence of the open-circuit voltage becomesstronger, i.e., the slope will become higher than kBT/q [128].

FF decreases continuously with increasing light intensitybecause of intensified recombination according to the character-istics of bimolecular recombination described in Eq. (21), whichimplies that recombination is proportional to the square of lightintensity rather than linearly correlated with the intensity.

3.1.3.3. The carrier mobility. Figs. 17 and 18 show the impact ofcarrier mobilities on the performance of the PSC. The magnitude ofthe mobility affects the shape of the I–V curve significantly. Fig. 18shows that low mobility leads to low Isc and FF values. Forexample, the Isc and FF are as low as 3 mA/cm2 and 0.30, respec-tively, when the mobility is three orders lower than that of atypical device. The Isc and FF increase monotonically with incre-asing mobility. The Isc shows sign of saturation when the mobilityis one order lower than that of typical device, which suggeststhat the IQE approaches unity at this range. However the Voc

decreases continuously with increasing mobility because (1) ahigher mobility also leads to higher recombination, as describedby Eq. (26)), and (2) a higher mobility also means more surfacerecombination losses because of the infinite surface recombinationvelocity assumed. Enhanced recombination losses in the bulk andsurface lead to negative effects on Voc. The model predicts a PCEpeak at an optimum charge carrier mobility and indicates thatfurther enhancements in mobility will not lead to better PCE[160,161]. Simulations by Tress et al. [110] and Kirchartz et al. [162]showed that selective contacts and a constant recombinationcoefficient will not lead to decreases in Voc in the high mobility

0.0 0.2 0.4 0.6

-10

-5

0

5

10

Cur

rent

(mA

/cm

2 )

Voltage (V)

80 nm 220 nm 350 nm

Fig. 14. I–V curve of the PSC under one sun illumination at different P3HT:PCBMthicknesses of 80, 220, and 350 nm.

0.5

0.6

0.4

0.6

0.8

0 10 200 300 4000

1

2

3

Voc

(V)

FFP

CE

(%)

Thickness (nm)

Fig. 15. FF, Voc, and PCE of the PSC under one sun illumination as a function oftheP3HT:PCBM thickness.

0.50

0.55

0.60

10.10.60

0.65

0.70

Voc

(V)

FF

Light Intensity (Sun)

Fig. 16. Voc and FF of the PSC as a function of the incident light intensity.


case. A simulation with temperature- and field-dependentmobility is given by Zhang and Lacic et al. [163,164].

3.1.3.4. The contact barrier. Ohmic contacts for both electrodesare important for PSCs. Anode layers with high work function arerequired for hole collection, while low-work function cathodesare required for electron collection. Non-ohmic contact wouldgreatly change the shape of the I–V characteristics of the device.Fig. 19 shows the simulated I–V characteristics of the PSCs withvarious contact barriers. An energy barrier of 0.1 eV for bothelectron and hole does not change the curve significantly, but asthe barrier increases, the change becomes more and more obvious.The Isc, FF, and Voc decrease continuously with increasing barrierheight. The current becomes saturated at higher voltages when thebarrier height reaches 0.3 eV or higher. A simple device made ofITO/P3HT:PCBM/Al on ITO with no special treatment (e.g., ozone orplasma) is a good example of a non-ohmic device. The workfunctions of Al and ITO are about 4.4 and 4.6 eV, respectively,leading to an energy barrier of 0.4 eV for both electron and hole atthe contact in this case. We fabricated such a device and show inFig. 20 the measured and simulated I–V characteristics of the PSC.Good matches between the simulation and experimental results areshown in the figure. Note that the interface between the active layerand the contact is important in determining the performance ofPSCs. Reduced majority surface recombination velocities [111,165]

or improper dipoles [166] may lead to “S” shaped I–V charac-teristics, which degrades device performance. Great efforts havebeen made to improve the contact property of PSCs through theinterface layer [167–172].

3.1.3.5. The effective bandgap. For PSCs, the effective bandgap ofthe absorption layer, which is defined as the energy differencebetween the LUMO of the accepter and the HOMO of the donor inthe blends, plays a crucial role in the determination of Voc and FF.Fig. 21 shows the variation in the I–V characteristics obtainedunder different effective bandgaps. Fig. 22 shows that the changein Voc is linearly correlated with the variation of the bandgap,

ΔVoc ¼ΔEgap

qð30Þ

The Voc of a “typical” device is 0.58 eV. An increase of 0.17 eV inthe bandgap will lead to a 0.17 V improvement in Voc, resulting in avalue of 0.75 V, as shown in Fig. 21. A higher bandgap also leads tosignificant FF improvement, which indicates that higher bandgapsare greatly beneficial for improving PSC performance. However, a0.17 eV increase in the LUMO level of ICBA cannot fully explain thejump in Voc from 0.58 to 0.84 V, which is a 0.26 V increase, afterthe adoption of ICBA in experiments [21,146].

3.1.3.6. The recombination rate. Pivrikas et al. [125] demonstratedthat the experimental bimolecular recombination rate constantin the PSC can be several orders lower than that of Langevin recombi-

0.0 0.2 0.4 0.6 0.8

-10

-8

-6

-4

-2

0

2

4

Cur

rent

Den

sity

(mA

/cm

2 )

Voltage (V)

Mobility: Typical Typical x 0.1 Typical x 0.01

Fig. 17. I–V characteristics of PSCs with different carrier mobilities of the active layer.

2468

10

0.5

0.6

0.7

0.4

0.6

1E-3 0.01 0.1 1 10

1

2

3

Isc

(mA/

cm2 )

Voc

(V)

FFP

CE

(%)

Mobility (10-7 m2/Vs for electron, 1.5x10-8 m2/Vs for hole)

Fig. 18. Isc, Voc, FF, and PCE of the PSC as a function of the mobility of theactive layer.

0.0 0.2 0.4 0.6 0.8

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

Cur

rent

Den

sity

(mA

/cm

2 )

Voltage (V)

Energy Barrier 0 eV 0.1 eV 0.2 eV 0.3 eV 0.4 eV

Fig. 19. Simulated I–V characteristics of PSCs with various contact barriers.

5.00.05.0-

-10

-8

-6

-4

-2

0

2

4

6

8

10

Cur

rent

Den

sity

(mA

/cm

2 )

Voltage (V)

Simulation withB

n = 0.4 eV

Bp = 0.4 eV

Experimental

Fig. 20. Measured and simulated I–V characteristics of a PSC with a non-ohmiccontact.


nation. Thus, we change the prefactor ξ in Eq. (26) to examinethe influence of the recombination rate constant on the I–V charac-teristics, as shown in Fig. 23. Reduction of the recombinationstrength clearly leads to continuous improvements in all of theparameters determined, especially the Voc, as indicated in Fig. 23.However, even in the case of zero recombination (prefactor¼0),the Voc of 0.67 V is still much lower than 1.0 V, the built-in volt-age. This result implies that there exists some other loss mechanismfor Voc.

3.1.3.7. The energy-level bending. The inset in Fig. 24 shows thedecline in energy-level with the built-in field at equilibriumconditions. The simulation shows that if the active layer has a loweffective density of states Nc (3.9�1021 m�3), the energy level showsa straight line, indicating a uniform built-in field in the active layer.At high Nc values (3.9�1025 m�3), however, the energy-level showsobvious bending (Ebnd) near the contacts. This bending leads to lossof Voc [40,110] because it reduces the field in the device to a value of(Vbi�Ebnd/q)/d. Bending at the electrodes occurs because of the highmajority charge carrier density in the area. The value of Nc deter-mines the majority charge carrier density near the contacts and thusaffects the bending of energy-level there. Fig. 24 shows that the Vocincreases continuously and even approaches the built-in voltage atlow Nc and low recombination conditions.

3.1.4. Simulation of the P3HT:ICBA PSCsFinally, a numerical simulation of the P3HT:ICBA PSC is per-

formed and results are compared with the experimental I–V curve[146], as shown in Fig. 25. As mentioned above, increases of thebandgap and reductions in recombination and effective density ofstates could lead to increases in the Voc. However, Guerrero et al.observed that the difference in the recombination coefficients inthe P3HT:PCBM and P3HT:ICBA system is very limited [173]. Thus,

0.0 0.2 0.4 0.6 0.8 1.0

-10

-5

0

5

10C

urre

nt D

ensi

ty (m

A/c

m2 )

Voltage (V)

Egap

1.0 eV 1.17 eV 1.26 eV

0.58 0.75 0.84

Fig. 21. Simulated I–V characteristics of PSCs with different effective bandgaps.

0.6

0.7

0.8

0.9

1.0 1.1 1.2 1.30.64

0.66

0.68

0.70

Voc

(V)

A

Fill

fact

or

Effective bandgap (eV)

Fig. 22. Voc and FF of the PSC as a function of the effective bandgap.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-8

-4

0

4

Cur

rent

den

sity

(mA

/cm

2 )

Voltage (V)

Recombination prefactor: 1.0 0.2 0

Fig. 23. Simulated I–V characteristics of PSCs with different recombinationprefactors.

1021 1022 1023 1024 1025

0.6

0.7

0.8

0.9

1.0

E /2

0 20 40 60 80

-1.0

-0.5

0.0

0.5

1.0

Ene

rgy

leve

l (eV

)

X (nm)

Nc=3.9x10 m Nc=3.9x10 m

Voc

(V)

Nc (m-3)

Fig. 24. Simulated Voc at different Nc values; here, the recombination prefactor waskept at 0.2. Inset: Energy-level of the active layer at equilibrium conditions (dark,open circuit.).

00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-15

-10

-5

0

5

10

15

-15

-10

-5

0

5

10

15

Cur

rent

den

sity

(mA

/cm

2 )

Voltage (V)

Fig. 25. Measured (scattered line) and simulated (solid line) I–V characteristics ofthe P3HT:ICBA PSC and simulated I–V curve (dashed line) with improved lightabsorption (absorption coefficient is twice that in Fig. 3) and a higher effectivebandgap of 1.75 eV.


we only adjust the value of the bandgap and the effective densityof states. Good matches are achieved when the following para-meters are used

Egap ¼ 1:17 eV

Nc ¼Nv ¼ 0:7� 1025 m�3

The simulation suggests that the performance improvement,especially the 0.26 V increase in Voc, is a combined result of theincreased bandgap and reduced effective density of states in theP3HT:ICBA system.

3.2. Inverted, semitransparent, and large-area PSCs

Kang et al. reported the performance of inverted, semitran-sparent, and large-area PSCs in one paper [47]. We simulate thesedevices and compare findings with the experimental data.

3.2.1. Inverted PSCsFig. 26 shows the simulated I–V curve of the inverted PSC with a

device structure of glass/ITO (100 nm)/ZnO (40 nm)/P3HT:PCBM(250 nm)/PEDOT:PSS (40 nm)/Ag film (120 nm), as in the reference.Simulation results match experimental ones when the parameters inTable 2 are used. The electron mobility of the P3HT:PCBM blends is inthe range of 10�8–10�6 m2/V s and the hole mobility is in the rangeof 10�11–10�7 m2/V s [35,174]. In the simulation the order of the car-rier mobilities are chosen within these ranges, but the actualnumbers are selected based on the fitting of the simulated I–Vcurves and the experimental ones. Keeping all of the electricalparameters the same, only when changes in the device structure toa normal geometry of glass/ITO/PEDOT:PSS (40 nm)/P3HT:PCBM(250 nm)/LiF/Al are made does the PSC show a slightly (6%) lowerphotocurrent than the inverted device. Fig. 27 shows the differentlight absorptivities of the active layer in the normal and invertedconfigurations. The inverted PSC shows slightly higher peak absorp-tivity. Fig. 28 shows the different absorption profiles of the devicesunder solar illumination. Absorption from the PEDOT:PSS buffer layerin the normal structure is significantly higher than that from the ZnObuffer layer in the inverted structure. The low absorption in the ZnObuffer layer is believed to be one of the reasons for the higherphotocurrent in the inverted PSC. He et al. [175] observed a10% absorption increase in inverted PSCs over the conventionaldevice. Albrecht et al. predicted that inverted devices basedon PCPDTBT should be able to deliver high power conversion

efficiencies of more than 7% compared with the regular devices[176]. Zou et al. [177] showed that because the peak of excitongeneration is located at a different position, a different blending ratioof fullerene is required to provide suitable electron mobility indifferent devices.

0.0 0.2 0.4 0.6

-10

-8

-6

-4

-2

0

2

4

6

Cur

rent

den

sity

(mA

/cm

2 )

Voltage (V)

Inverted Normal Experimental

Fig. 26. Comparison of the simulated I–V characteristics of a normal PSC (dottedline), an inverted PSC (solid line), and the experimental data (scattered line) in Ref.[47]. The normal structure is ITO/PEDOT:PSS (40 nm)/P3HT:PCBM (250 nm)/LiF/Alfilm (120 nm), and the inverted structure is ITO (100 nm)/ZnO (40 nm)/P3HT:PCBM(250 nm)/PEDOT:PSS (40 nm)/Ag film (120 nm).

Table 2The parameters in the simulation of the inverted and transparent PSCs.

Parameter Symbol Numerical value

P3HT:PCBM Thickness D 250 nmBandgap Egap 1.0 eVElectron mobility un 2.0�10�7 m2/V sHole mobility up 2.0�10�7 m2/V sEffective density of states Nc, Nv 6.5�1024 m�3

Dielectric constant ε 2.7�10�11 F/mEnergy barrier for electron Bn 0 eVEnergy barrier for hole Bp 0 eVSeries resistance Rs 3 Ω cm2

Recombination prefactor ξ 0.3Dissociation probability P 0.7Temperature T 300 K

400 500 600 700 800 9000.0

0.2

0.4

0.6

0.8

1.0

Abs

orpt

ivity

Wavelength (nm)

Inverted Nomal

Fig. 27. Comparison of the light absorptivity in the P3HT:PCBM layer of normal andinverted PSCs.

100 200 300-1

0

1

2

3

4

5

6

100 200 300

Pho

ton

abso

rptio

n ra

te (1

027 m

-3S-1

)

Distance from the substrate (nm)

Nomal

Buffer P3HT:PCBM

Inverted


ZnO

PEDOT:PSS

Fig. 28. Photon absorption profiles in the buffer and P3HT:PCBM layers in normaland inverted PSCs.


3.2.2. Semitransparent PSCsUsing highly conductive PEDOT:PSS as the anode layer, a PSC

with a structure of ITO/ZnO/P3HT:PCBM/PEDOT:PSS is used as aninverted semitransparent device. The transmittance T and thesheet resistance Rsheet of the PEDOT:PSS film are functions of thefilm thickness x and can be estimates using the relationships

T ¼ e�ax ð31Þ

Rsheet ¼1cx

ð32Þ

where a is the absorption coefficient and c is the conductivity ofthe PEDOT:PSS film. The inset in Fig. 29 shows the fitting of thetransmittance and sheet resistance data from Ref. [47] using theseequations. Fitting gives an absorption coefficient of 0.73/μm, and aconductivity of 450 S/cm. Thus, for transmittances of 50, 38, 21and 3%, as described by the reference, the PEDOT:PSS thicknesses Tare 0.95, 1.33, 2.14 and 4.8 μm and the sheet resistances Rsheet are23.4, 16.8, 10.4, and 4.6 Ω/sqr, respectively.

3.2.2.1. Bottom illumination. No peak is observed in the absorptionprofile of the (semi)transparent devices because of the absence ofa reflective metal layer. The absorption profile in the active layerdecreases monotonically (exponential decay could be a first-orderapproximation if the reflection from the interface is ignored) withthe distance from the substrate when illumination is performedfrom the substrate (bottom) side. Fig. 29 shows the absorptionprofile of the active layer in a device of ITO/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS (0.9 μm or higher). When thethickness of the PEDOT:PSS layer is higher than 0.9 μm, theabsorption profile remains nearly constant despite increases inthickness. The absorption profile in a device of ITO/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS (2100 nm)/Ag film (120 nm) isalso the same, which indicates that a thick PEDOT:PSS layersignificantly weakens the interference effect from the reflectivemetal layer.

Fig. 30 shows the simulated absorptivities of the P3HT:PCBMand PEDOT:PSS layers as well as the transmittances of ITO/ZnO(40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS devices where thethicknesses of the PEDOT:PSS layer are 0.95, 1.33, and 2.14 μm,corresponding to transmittances of 50, 38, and 21%, respectively.The absorptivity of the P3HT:PCBM layer is constant among thedevices. Light absorption from the PEDOT:PSS layer increases withthe layer thickness, leading to decreases in the transmittance of

the PSC devices. The shapes and trends of the simulated transmit-tance curves are very similar to the measured curves in Ref. [47].

The absence of change in the absorption profiles of semitran-sparent PSCs with different PEDOT:PSS thicknesses implies thatsharp changes in the I–V curves observed under different PEDOT:PSS transmittances in Ref. [47] originate from the series resistanceof the devices. Here, we consider that the series resistance Rs ismainly caused by the high sheet resistance of the PEDOT:PSS layerRsheet

Rs ¼ BRsheet ð33Þwhere B is a shape factor related to the average distance photo-generated carriers must travel to the resistance-free electrode.Large cell areas mean higher shape factors (see the relationshipbetween B and the geometry of the cell area in Appendix A).

As shown in Fig. 31, we simulated the bottom-illuminated I–Vcharacteristics of PSCs with a structure of ITO (100 nm)/ZnO(40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS and PEDOT:PSS sheetresistances of 23.4, 16.8, 10.4, and 4.6 Ω/sqr, corresponding totransmittances of 50, 38, 21, and 3%, respectively. If the shapefactor B is 2.85 (and other parameters are held as in Table 2), the

100 150 200 250 300 350 4000

1

2

3

4

5

6

0

20

40

60

80

100

0 1 2 3 4 5 60

40

80

120

Pho

ton

abso

rptio

n ra

te (1

027m

-3S-1

)


Tra

nsm

ittan

ce(%

)

Transmittance The fitting curve

Thickness ( m)

She

et re

sist

ance

(Ohm

/sqr

.)

Sheet resistance

Fig. 29. Photon absorption profile of the active layer in a PSC with a structure of ITO/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS (0.9 μm or higher). The absorptionprofile in a device ITO/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS (2100 nm)/Agfilm (120 nm) is identical to that of the first device. Inset: Fitting of the transmittanceand sheet resistance of the PEDOT:PSS film using Eqs. (31) and (32).

0.0

0.3

0.6

0.9

0.0

0.3

0.6

0.9

0.0

0.3

0.6

0.9

400 500 600 700 800 900 1000

0.0

0.3

0.6

0.9 T=21%T=38%

T=21%

T=50%

T=38%

Wavelength (nm)

P3HT:PCBM absorption PEDOT:PSS absorption

T=50%

T=21%T=38%T=50%

Abs

orpt

ivity

Tran

smitt

ance

(%)

Device transmittance

Fig. 30. Simulated absorptivity of the P3HT:PCBM and PEDOT:PSS layers and thetransmittance of glass/ITO/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS deviceswhen illuminated from the glass side. The thicknesses of the PEDOT:PSS layers are0.95, 1.33, and 2.14 μm, corresponding to transmittances of 50, 38, and 21%,respectively.

-10

-8

-6

-4

-2

0

2

0.0 0.2 0.4 0.6 0.80.1 0.3 0.5 0.7 0.9-10

-8

-6

-4

-2

0

2

Cur

rent

den

sity

(mA

/cm

2 )

Voltage (V)

T=50% (Bottom) T=38% (Bottom) T=21% (Bottom) T=3% (Bottom) T=3% (Experiment) Rs=0 (Bottom)

T=50% (Top) T=38% (Top) T=21% (Top)

Fig. 31. Simulated I–V characteristics of PSCs (bottom- and top-illuminated) with astructure of ITO (100 nm)/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS. Thethicknesses of the PEDOT:PSS layers are 0.95, 1.33, 2.14, and 4.8 μm, correspondingto transmittances of 50, 38, 21, and 3%, respectively, and sheet resistances of 23.4,16.8, 10.4, and 4.6 Ω/sqr, respectively. The shape factor B is 2.85. The experimentaldata (scattered line) are from Ref. [47].


simulated I–V curves are highly similar to the measured ones (cellarea, 0.36 cm2). Fig. 31 also shows the I–V curve with Rs¼0 Ω cm2;which yields a much better FF than the other curves. This findingindicates that the main factor limiting semitransparent PSCs is thehigh surface resistance of the polymer anode layer.

3.2.2.2. Top illumination. When illuminated from the PEDOT:PSSside (top illumination), the absorptivities of both P3HT:PCBM andPEDOT:PSS are sensitive to the thickness of PEDOT:PSS layer, asshown in Fig. 32. A lower PEDOT:PSS thickness leads to a highertransmittance, higher absorptivity in the P3HT:PCBM layer, and ahigher photocurrent, as indicated in Fig. 31. The overall photocurrentis lower in top illumination mode than in bottom illumination modebecause of strong light absorption in the polymer anode layer in theformer case. The trends of the simulated I–V curves obtained fromtop illumination are similar to the measured ones in Ref. [47].

In semi-transparent devices, the carrier transport distance canbe controlled via the local light absorption profile by appropriateselection of the illumination side (top or bottom) and incidentwavelength. Tumbleston et al. [178] demonstrated the utility ofthe local absorption profile in probing recombination mechanisms.From light intensity measurements, they determined which carriercauses the onset of recombination.

3.2.3. Large-area PSCs3.2.3.1. PSCs without grids. An increase in cell area result in highershape factors B. Fig. 33 shows the simulated I–V characteristics of alarge-area PSC (15.25 cm2) with a structure of ITO (100 nm)/ZnO(40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS; here, the sheetresistance of the PEDOT:PSS layer is 10.4 Ω/sqr, corresponding toa transmittance of 21%. If we chose a shape factor B of 9.5 for thedevices without grids, results match the measured I–V curves inboth bottom- and top-illumination conditions. In this case, theseries resistance of the PSC is as high as �100 Ω cm2, which is themain obstacle for solar cells.

3.2.3.2. The PSC with grids. A simple way to reduce the overallresistance of the PEDOT:PSS anode is to deposit metal grids on top ofthe anode. Fig. 29 shows that deposition of the metal layer will notchange the light absorption of the active layer when illuminatedfrom the bottom; therefore, the major function of metal griddeposition is to reduce the series resistance of the PSCs. Assuminga series resistance of 82 Ω cm2 yields matching of the simulated andexperimental I–V curves in both bottom- and top-illumination

conditions (the grid cover ratio is 50% for the top-illuminationcase), as shown in Fig. 33. The I–V curve of an ideal device withRs¼0 Ω cm2 is also shown in the figure. The ideal device shows amuch better FF and Isc than other devices, including the controldevice, in which the anode is fully covered by an Ag layer. Thisfinding suggests that the high surface resistance of the electrodelayer (including ITO) remains the main limiting factor for large-areaPSCs, especially at high illumination intensity [179]. Some cleverdesign of the electrode geometry [180] or integration of metal gridsinto the transparent electrode can reduce the series resistance of thePSCs [48,181–183].

4. Summary and outlook

4.1. Summary

We reviewed optical and electrical models of polymer BHJ PSCsand numerically simulated and analyzed the performance of con-ventional, inverted, semitransparent, and large-area PSCs based onP3HT. The agreement between the simulated and experimental dataof the PSCs shows the validity of the models used.

Sufficient optical absorption is the first step toward achievinghigh quantum efficiency and Isc. Our simulations suggest that theEQE of �63% in typical P3HT PSCs is mainly limited by the opticalabsorption of the active layer. The simulations also show that anoptical spacer layer can enhance the optical absorption of theactive layer but only when the active-layer thickness is unfavor-able in a conventional device. The photon absorptivity is ulti-mately determined by the absorption property and thickness ofthe active layer. High dissociation probability for forming freecarriers is another necessary condition; this possibility may varywith temperature, field, and morphology [124]. The carrier-collection process competes with the recombination process inthe active layer. High carrier mobilities, low recombination coeffi-cients, and thin active layers are favorable conditions for achievinghigher collection efficiencies. The simulations show that improperthicknesses and low mobilities could easily lead to very poor EQEperformance. High levels of P-type doping [111,184–187] orreduced generation zones [188] may also be important factorsaffecting EQE and Isc.

The simulations also intuitively show the effects of differentthicknesses, carrier mobilities, light intensities, contact barriers,

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

300 400 500 600 70 800 9000.0

0.2

0.4

0.6

0.8

1.0

PEDOT:PSS P3HT:PCBM

T=21%

T=21%

T=38%

T=38%

T=50%T=50%

Abs

orpt

ivity

Wavelength (nm)

Fig. 32. Simulated absorptivities of P3HT:PCBM and PEDOT:PSS layers in a deviceof glass/ITO/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS when illuminated fromthe PEDOT:PSS side. The thicknesses of the PEDOT:PSS layers are 0.95, 1.33, and2.14 μm, corresponding to transmittance of 50, 38, and 21%, respectively.

-10

-8

-6

-4

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0

2

4

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0

2

4

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-8

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0

2

4

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-8

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-2

0

2

4

0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0 0.2 0.4 0.6 0.8 1.0-10

-8

-6

-4

-2

0

2

4

Cur

rent

den

sity

(mA

/cm

2 )

Without grids (Bottom) Rs=0 (Bottom) Without grids (Experiment)

Voltage (V)

Without grids (Top) With grids (Top)

Control device (Experiment) with grids (Experiment)

With grids (Bottom)

Fig. 33. Simulated I–V characteristics of a large-area PSC (15.25 cm2) with astructure of ITO (100 nm)/ZnO (40 nm)/P3HT:PCBM (250 nm)/PEDOT:PSS; here,the sheet resistance of the PEDOT:PSS layer is 10.4 Ω/sqr, corresponding to atransmittance of 21%. The shape factor B is 9.5 for devices without grids. Theexperimental data (scattered line) are from Ref. [47].


effective bandgaps, and recombination coefficients on the FF andVoc of the PSCs. High carrier mobilities, ohmic contact, highbandgaps, and low recombination coefficients are the most favor-able conditions for achieving high FFs. Ohmic contact, highbandgaps, low recombination coefficients, and low effective den-sities of states are required for higher Voc values. Finally, oursimulations suggest that performance improvement, especially the0.26 V increase in Voc, in the P3HT:ICBA PSC is a combined result ofthe increased bandgap and reduced effective density of states inthis system.

The PSCs in the inverted architecture show stability andprocessing advantages over conventional devices. The simulationspredicted slightly higher (6–10%) photo absorption in the inverteddevice, and a peak-position shift in the absorption profiles ofthe two types of devices may be observed. No peak is observed inthe absorption profile of (semi)transparent devices because of theabsence of a reflective metal layer, and the absorption profile inthe active layer decreases monotonically with the distance fromthe bottom electrode (bottom illumination) or the top electrode(top illumination). The performance of semi-transparent PSCs isclosely related to the sheet resistance of the transparent electrodeused. High series resistance, signaling a high shape factor B, is amajor obstacle for large-area PSCs.

4.2. Outlook

Despite many low-absorption-band-gap polymers emerging asalternative absorption materials [30,151,189–194], PSCs based onP3HT remain at the frontier of PSC technology in terms ofindustrialization [195–200]; these PSCs are also suitable sub-cells[201,202] in tandem PSC configurations [203–215]. Researchefforts should focus on absorptivity enhancements in thinnerdevices and IQE improvements in thicker ones. For example, theplasmonic effects of metal nanostructures were recently demon-strated to be effective in enhancing the light absorption of thinactive layers [74,216–224].

Considering that the absorption bandgap of P3HT is as high as1.9 eV, another area for improvement in P3HT-based PSCs is theVoc. Raising the LUMO level of the acceptor and increasing theeffective bandgap of the PSC are effective methods for achievinghigher Voc. As such, research on this path should continue. Kanget al. showed that controlling the number of indene-solubilizinggroups in multi-adduct fullerenes is capable of lifting LUMO levelsand increasing the Voc to 0.92 V in P3HT-based PSCs [225]. Theeffective bandgap of P3HT:F8TBT blends can reach 1.75 eV becauseof the high LUMO of F8TBT. A Voc of 1.15 V was achieved and avalue of 1.35 V was predicted in the P3HT:F8TBT blends [226,227].As an optimistic estimation, in Fig. 25, we show the simulated I–Vcharacteristics of a PSC with an effective bandgap of 1.75 eV and ahigh absorption coefficient (twice the value in Fig. 3). This I–Vcurve shows that a Voc of 1.34 V, an Isc of 11.7 mA/cm2 (EQE, 79%),and a PCE of 11.0% can be expected in P3HT-based small-area PSCsif both the EQE and Voc approach their full potential.

To meet demands for industry-related large-area devices,research on low cost, highly conductive, and transparent filmsfor electrodes [43,66,89,98,228–246] continues to be a necessity.Another important approach for reducing series resistance is tointegrate metal grids into the transparent electrode of PSCs. Therole of these highly conductive grids is to “divide” large-area PSCsinto multiple small-area devices. Using these means, if the seriesresistance of large-area PSCs can be kept to �3 Ω cm2 or lower, thesimulations suggest that the device efficiency obtained willapproach that of laboratory-based small-area PSCs.

Acknowledgements

This work is supported in part by FRFCU under 2013JBZ004 and2009JBZ019, by NSFC under 60825407 and 21174016, by ISTCPunder 2008DFA61420, by RFDP under 20120009110031.

Appendix A

Fig. A1 is a schematics showing the current flow through theconducting film (such as PEDOT:PSS or ITO ) and collected with aresistance-free electrode (such as a silver grid). A deduction of theseries resistance of the conducting film is as following.

Suppose J is the photogenerated current density in the activearea of the cell, the current increase linearly with the position inthis region as

I¼ JWx ðA:1ÞIn the range of dx, the potential increase is

dV ¼ IRsheetdxW

¼ JRsheetxdx ðA:2Þ

Set the electric potential zero at x¼0, thus in the cell area

V ¼ 12x2JRsheet ðA:3Þ

At position x1,

Iðx1Þ ¼ JWL1 ðA:4Þ

Vðx1Þ ¼12L1

2JRsheet ðA:5Þ

At position x2

Iðx2Þ ¼ Iðx1Þ ¼ JWL1 ðA:6Þ

Vðx2Þ ¼ Vðx1Þþ Iðx1ÞRsheetL2W

¼ 12L1

2JRsheetþL1L2JRsheet ðA:7Þ

Thus the series resistance for the solar cell with the area ofA¼WL1 is

RsðAÞ ¼Vðx2ÞIðx2Þ

¼ 12L1Rsheet

WþL2Rsheet

WðA:8Þ

The equivalent series resistance for the device with unityarea is

Rs ¼ RsðAÞA¼ 12L1Rsheet

WþL2Rsheet

W

� �WL1 ¼ Rsheet

12L1

2þL1L2

� �

ðA:9Þ

Fig. A1. Schematics showing the current flow through the conducting film to theresistance-free electrode. Suppose one part of the film is on the active area of thesolar cell with a length L1, the other part is the conducting-film-only region with alength L2, the width of the film is W.


Compared with Eq. (33), we get

B¼ 12L1

2þL1L2

� �ðA:10Þ

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