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Device physics of donor/acceptor-blend solar cellsKoster, Lambert

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Device physics of donor/acceptor-blendsolar cells

Lambert Jan Anton Koster

Device physics of donor-acceptor/blend solar cellsLambert Jan Anton KosterPhD thesisUniversity of Groningen, The Netherlands

MSC PhD thesis series 2007-04ISSN 1570-1530ISBN-13: 9789036729413

The research described in this thesis forms part of the research program of the DutchPolymer Institute (DPI), project #323.

RIJKSUNIVERSITEIT GRONINGEN

Device physics of donor/acceptor-blendsolar cells

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, dr. F. Zwarts,in het openbaar te verdedigen op

vrijdag 23 februariom 16.15 uur

door

Lambert Jan Anton Kostergeboren op 24 mei 1975

te Hoogeveen

Promotor : Prof. dr. ir. P. W. M. Blom

Beoordelingscommissie : Dr. N. C. GreenhamProf. dr. J. C. HummelenProf. dr. L. D. A. Siebbeles

CONTENTS

1 Introduction to organic solar cells 11.1 Solar energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Conjugated polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Transport of charges in conjugated polymers . . . . . . . . . . . . . . . . . 3

1.3.1 Hopping transport in disordered systems . . . . . . . . . . . . . . . 41.3.2 Transport in conjugated polymers . . . . . . . . . . . . . . . . . . . 51.3.3 Measuring the charge carrier mobility . . . . . . . . . . . . . . . . . 61.3.4 Conjugated polymers used in this thesis . . . . . . . . . . . . . . . . 6

1.4 Organic photovoltaics in a nutshell . . . . . . . . . . . . . . . . . . . . . . . 71.5 Device fabrication and characterization . . . . . . . . . . . . . . . . . . . . 101.6 Objective and outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . 11

2 Metal-insulator-metal model for bulk heterojunction solar cells 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Generation of free charge carriers . . . . . . . . . . . . . . . . . . . 242.2.3 Bimolecular recombination in bulk heterojunctions . . . . . . . . . 27

2.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Simulation results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Bimolecular recombination: Experimental results . . . . . . . . . . 302.4.2 Modeling MDMO-PPV/PCBM bulk heterojunction solar cells . . . 34

2.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Open-circuit voltage of bulk heterojunction solar cells 453.1 Open-circuit voltage in p-n junction models . . . . . . . . . . . . . . . . . . 463.2 Open-circuit voltage in the MIM model . . . . . . . . . . . . . . . . . . . . 49

3.2.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 49

v

3.2.2 Formula for the open-circuit voltage . . . . . . . . . . . . . . . . . . 493.3 Comparison with other solar cells . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.1 Influence of non-homogeneity . . . . . . . . . . . . . . . . . . . . . 533.3.2 Comparison with (in)organic low mobility solar cells . . . . . . . . 55

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Short-circuit current of bulk heterojunction solar cells 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Uniform field approximation . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Space-charge-limited photocurrents . . . . . . . . . . . . . . . . . . 63

4.2 Intensity dependence of the short-circuit current . . . . . . . . . . . . . . . 644.2.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Hybrid organic/inorganic solar cells 715.1 Introduction to hybrid organic/inorganic solar cells . . . . . . . . . . . . . 725.2 Hybrid solar cells with acceptors from a precursor . . . . . . . . . . . . . . 72

5.2.1 Using a precursor for titanium dioxide . . . . . . . . . . . . . . . . 725.2.2 Using a precursor for zinc oxide . . . . . . . . . . . . . . . . . . . . 73

5.3 Polymer solar cells with zinc oxide nanoparticles . . . . . . . . . . . . . . . 775.3.1 Charge transport in MDMO-PPV/nc-ZnO blends . . . . . . . . . . 795.3.2 Improving the efficiency of MDMO-PPV/nc-ZnO solar cells . . . . 80

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.5 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Improving the efficiency of bulk heterojunction solar cells 896.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Improving polymer/fullerene solar cells . . . . . . . . . . . . . . . . . . . . 916.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Publications 99

Summary 101

Samenvatting 105

Dankwoord 109

CHAPTER

ONE

Introduction to organic solar cells

Summary

As the need for renewable energy sources becomes more urgent, photovoltaic en-ergy conversion is attracting more and more attention. In this introductory chapterseveral aspects of polymer solar cells will be introduced. After discussing the transportof charge in conjugated polymers, the electro-optical processes in bulk heterojunctionsolar cells are discussed. Finally, an overview of this thesis is given.

1

Chapter 1. Introduction to organic solar cells

1.1 Solar energy

What can be a more attractive way of producing energy than harvesting it directly fromsunlight? The amount of energy that the Earth receives from the sun is enormous: 1.75× 1017 W. As the world energy consumption in 2003 amounted to 4.4 × 1020 J, Earth re-ceives enough energy to fulfill the yearly world demand of energy in less than an hour.Not all of that energy reaches the Earth’s surface due to absorption and scattering, how-ever, and the photovoltaic conversion of solar energy remains an important challenge.State-of-the-art inorganic solar cells have a record power conversion efficiency of close to39%, [1] while commerically available solar panels, have a significantly lower efficiencyof around 15–20%.

Another approach to making solar cells is to use organic materials, such as conju-gated polymers. Solar cells based on thin polymer films are particularly attractive be-cause of their ease of processing, mechanical flexibility, and potential for low cost fab-rication of large areas. Additionally, their material properties can be tailored by mod-ifying their chemical makeup, resulting in greater customization than traditional solarcells allow. Although significant progress has been made, the efficiency of convertingsolar energy into electrical power obtained with plastic solar cells still does not warrantcommercialization: the most efficient devices have an efficiency of 4-5%. [2] To improvethe efficiency of plastic solar cells it is, therefore, crucial to understand what limits theirperformance.

1.2 Conjugated polymers

Since Shirakawa, MacDiarmid, and Heeger demonstrated in 1977 that the conductivityof conjugated polymers can be controlled by doping, [3] a new field has emerged. Theywere rewarded for their discovery with the Nobel prize in chemistry in 2000. These con-jugated polymers have been used successfully in, e.g., light-emitting diodes (LEDs) [4,5]

and solar cells. [6–8]

The insulating properties of most of the industrial plastics available stem from theformation of σ bonds between the constituent carbon atoms. In conjugated polymers,e.g., polyacetylene, the situation is different: In these polymers, the bonds between thecarbon atoms that make up the backbone are alternatingly single or double (see Fig. 1.1);this property is called conjugation. In the backbone of a conjugated polymer, each car-bon atom binds to only three adjacent atoms, leaving one electron per carbon atom ina pz orbital. The mutual overlap between these pz orbitals results in the formation ofπ bonds along the conjugated backbone, thereby delocalizing the π electrons along theentire conjugation path. The delocalized π electrons fill up to whole band and, therefore,conjugated polymers are intrinsic semiconductors. The filled π band is called the highestoccupied molecular orbital (HOMO) and the empty π* band is called the lowest unoc-cupied molecular orbital (LUMO). This π system can be excited without the chain, held

2

1.3. Transport of charges in conjugated polymers

Figure 1.1: In polyacetylene, the bonds between adjacent carbon atoms are alternatingly single ordouble.

together by the σ bonds, falling apart. Therefore, it is possible to promote an electronfrom the HOMO to the LUMO level upon, for example, light absorption.

As the band gap (energy difference between the HOMO and LUMO) of a conjugatedsystem depends on its size, [9] any disturbance of the conjugation along the polymer’sbackbone will change the local HOMO and LUMO positions. Real conjugated polymersare therefore subject to energetic disorder. The density of states of these systems is oftenapproximated by a Gaussian distribution. [10]

1.3 Transport of charges in conjugated polymers

How are charges transported in conjugated polymer films? Since polymers do not havea three dimensional periodical lattice structure, charge transport in polymers cannot bedescribed by standard semiconductor models. As these systems show energetic andspatial disorder, the concept of band conduction of free charge carriers does not apply. Inthis section, a summary is given of how charge carrier transport in conjugated polymersand akin materials is described theoretically and how it is characterized experimentally.

The field of molecularly doped polymers is much older than that of conjugated poly-mers and valuable insights can be gained from studying this field. As early as in the1970s the charge transport in molecularly doped polymers was studied by performingtime-of-flight (TOF) measurements. In this type of experiment, a sample is sandwichedbetween two non-injecting electrodes.∗ A short light pulse is used to illuminate one sideof the sample through an transparent electrode. Under the action of an applied field,charge carriers of the same electrical polarity as the illuminated electrode will traversethe sample. By monitoring the current flow in the external circuit, the charge carrier mo-bility can be determined as a function of the applied voltage. In these TOF experiments,the mobility µ of carriers in molecularly doped polymers, can empirically be describedby [11–15]

µ = µ0 exp(γ√

F), (1.1)

where µ0 is the zero-field mobility, F is the field strength, and γ is the field activationparameter.

∗Note, that no direct physical contact between the electrodes and the sample is necessary.

3


1.3.1 Hopping transport in disordered systems

How can the results summarized in Eq. (1.1) be rationalized? As these materials aredisordered, the concept of band conduction does not apply. Instead, localized states areformed and charge carriers proceed from one such a state to another (hopping), therebyabsorbing or emitting phonons to overcome the energy difference between those states.

Conwell [16] and Mott [17] proposed the concept of hopping conduction in 1956 to de-scribe impurity conduction in inorganic semiconductors. Miller and Abrahams calcu-lated that the transition rate Wij for phonon-assisted hopping from an occupied state i

with an energy ǫi to an unoccupied state j with energy ǫj is described by [18]

Wij = ν0 exp(−2γRij)

exp(

− ǫj−ǫi

kBT

)

ǫi < ǫj

1 ǫi ≥ ǫj,(1.2)

where ν0 is the attempt-to-jump frequency, Rij is the distance between the states i andj, γ is the inverse localization length, kB is Boltzmann’s constant, and T is temperature.The wave function overlap of states i and j is described by the first exponential term inEq. (1.2), while the second exponential term accounts for the temperature dependence ofthe phonon density.

In his pioneering work, Bassler described the transport in disordered organic systemsas a hopping process in a system with both positional and energetic disorder. [10] Thehopping rates between sites were assumed to obey Eq. (1.2) and the site energies variedaccording to a Gaussian distribution with a standard deviation σ. Such a system cannotbe solved analytically. By performing Monte Carlo simulations, the following expressionfor the charge carrier mobility µ was proposed [10]

µ = µ∞e−(

2σ3kBT

)2

exp(

C[

(σ/kBT)2 − Σ2]√

F)

Σ ≥ 1.5

exp(

C[

(σ/kBT)2 − 2.25]√

F)

Σ < 1.5,(1.3)

where µ∞ is the mobility in the limit T → ∞,∗ C is a constant that is related to the latticespacing, and Σ describes the positional disorder.

Although Eq. (1.3) predicts a functional dependence on field strength similar toEq. (1.1), the agreement with experiments is limited to high fields. [13] Gartstein and Con-well found that the agreement with experiments could be improved by taking spatialcorrelations between site energies into account. [19] In this model, the mobility takes theform [20,21]

µ = µ∞ exp

[

−(

3σ

5kBT

)2

+ 0.78

(

(

σ

kBT

)3/2

− Γ

)

√

qaF

σ

]

, (1.4)

∗Since, Eq. (1.3) is an expression which describes the outcome of Monte Carlo simulations, this is a purelymathematical definition of µ∞ and does not mean that it has the physical meaning of the mobility at infinitetemperature. At best, it may be interpreted as the mobility if there would be no barriers to hopping at all.

4

1.3. Transport of charges in conjugated polymers

where q is the elementary charge, a is the intersite spacing, and Γ is the positional dis-order of transport sites. This model was successfully used to describe the transport ofcharges in molecularly doped polymers. [20]

1.3.2 Transport in conjugated polymers

The stretched exponential dependence on field strength as described by Eq. (1.1) was alsoobserved for conjugated polymers. [22] Subsequently, Eq. (1.4) was also applied success-ful to explain the charge transport in conjugated polymers [23,24] as well as other organicsystems. [25]

In the foregoing discussion, only the dependence of the mobility on temperature andfield strength was taken into account. When the applied voltage is increased in a TOFexperiment, only the field across the sample changes. However, in organic solar cells,as well as organic LEDs, changing the applied voltage does not merely change the field.Due to the nature of the contacts, it influences the charge carrier density as well. Recently,it has been shown that the mobility of charge carriers in conjugated polymers also has animportant dependence on charge carrier density. [26–29] Moreover, it was shown that theincrease of the mobility with increasing bias voltage (and concomitant increase in carrierdensity) observed in polymer diodes is, at least for some systems and temperatures,completely due to an increase in charge carrier density. [26]

Throughout this thesis, the increase of the mobility with increasing bias voltage isinterpreted as an effect of the field only. It should be noted, however, that the polymersused in this thesis show only a rather small dependence of the mobility on bias, suggest-ing that the influence of either field strength or carrier density for the system describedhere is quite weak. Additionally, as we will see in chapter 2, the carrier density in solarcells is fairly modest.

Several alternative models exist for explaining charge transport; one of them is theso-called polaron model which was first applied to inorganic crystals [30] and later toconjugated polymers. [31] An excess charge carrier in a solid causes a displacement of theatoms in its vicinity thus lowering the total energy of the system. This displacement ofatoms results in a potential well for the charge carrier, thereby localizing it. The chargecarrier and its concomitant atomic deformation is called a polaron.

The transition rate for polaron hopping from site i to site j is given by [32]

Wij ∝1√ErT

exp

(

−(Ej − Ei + Er)2

4ErkBT

)

, (1.5)

where Er is the intramolecular reorganization energy. The resulting charge carrier mo-bility is of the form [33]

µ = µ0 exp

[

− Er

4kBT− (aF)2

4ErkBT

]

sinh(aF/2kBt)

aF/2kBT. (1.6)

5


The polaron contribution to the activation of the mobility is, as predicted by this model,rather low; it amounts to 25–75 meV, [33] which is much smaller than the activation dueto disorder.

1.3.3 Measuring the charge carrier mobility

When an insulator is contacted by an electrode that can readily inject a sufficiently largenumber of charge carriers — a so-called Ohmic contact — and another electrode thatcan extract these charges, the current flow will be limited by a buildup of space charge.These space-charge-limited (SCL) currents can be used as a simple, yet reliable, tool todetermine the mobility in an experimental configuration that is relevant for solar cells.Considering only one charge carrier (either electrons or holes), the SCL current densityJSCL flowing across a layer with thickness L is given by [34]

JSCL =9

8εµ

V2int

L3, (1.7)

where ε is the dielectric constant of the material and Vint is the internal voltage dropacross the active layer. When the mobility is of the form as given in Eq. (1.1), one canapproximate JSCL by [35]

JSCL =9

8εµ0e0.891γ

√Vint/L V2

int

L3. (1.8)

The internal voltage in an actual device is related to the applied voltage Va by

Vint = Va − Vbi − VRs, (1.9)

where Vbi is the built-in voltage which arises from the difference in work function ofthe bottom and top electrode and VRs is the voltage drop across the series resistance ofthe substrate (typically 30–40 Ω). The built-in voltage is determined from the current-voltage characteristics as the voltage at which the current-voltage characteristic becomesquadratic, corresponding to the SCL regime.

By judiciously choosing the electrode materials, the injection of either carrier type canbe suppressed or enhanced, thereby enabling one to selectively assess either the hole orelectron mobility. The way to do this, is to make sure that the work function of one of theelectrodes is close to the energy level of the transport band under investigation, whilethere exists a large barrier for injection of the other carrier type into the material. Thus,in order to study the hole transport in conjugated polymers, high work function metals,such as gold and palladium, are used. Conversely, low work function metals can be usedas Ohmic contacts for electron injection.

1.3.4 Conjugated polymers used in this thesis

Up to now the photoactive polymers used in this research have not been specified. Thepolymer poly(2-methoxy-5-(3’,7’-dimethyl octyloxy)-p-phenylene vinylene) (MDMO-PPV) had for a long time been the workhorse in polymer photovoltaics. Consequently, its

6

1.4. Organic photovoltaics in a nutshell

Figure 1.2: The chemical structures of the BEH-PPV, MDMO-PPV, and P3HT.

charge transport properties are well documented, making this polymer well suited formodeling purposes. Recently, another polymer has emerged: poly(3-hexylthiophene)(P3HT),∗ which is used in the most efficient polymer solar cells to date. [2] The finalpolymer considered in this thesis is poly(2,5-bis(2’-ethylhexyloxy)-p-phenylene viny-lene) (BEH-PPV). The chemical structure of these polymers is shown in Fig. 1.2.

The charge transport in MDMO-PPV has been extensively studied: Typically, thezero field mobility amounts to 5 × 10−11 m2/V s. [36] Surprisingly, the hole mobility ofMDMO-PPV is enhanced when mixed with 6,6-phenyl C61-butyric acid methyl ester

(PCBM), as reported by several researchers: [37,38] When 80% (by weight) of this blendconsists of PCBM, the hole mobility of the polymer phase is equal to 2 × 10−8 m2/V s,an encrease of more than two orders of magnitude as compared to pristine MDMO-PPV.This spectacular behavior of the hole mobility in MDMO-PPV is the main reason for itssucces as a donor in BHJ solar cells with PCBM.

P3HT is unique in its own right: Padinger et al. observed that solar cells made fromP3HT and PCBM showed a great increase in the efficiency upon thermal annealing. [39]

Mihailetchi et al. have shown that this enhancement is in part due to an increase in themobility: [40] In its pristine form the hole mobility amounts to 10−8 m2/Vs, see Fig. 1.3.For comparison, Fig. 1.3 also shows the electron mobility of the PCBM phase in theseblends. When blended with PCBM, the hole mobility initially decreases, however, uponannealing the hole mobility in the P3HT phase of the blend with PCBM is restored to itspristine value, as depicted in Fig. 1.3. [40]

1.4 Organic photovoltaics in a nutshell

The field of organic photovoltaics dates back to 1959 when Kallman and Pope discov-ered that anthracene can be used to make a solar cell. [41] Their device produced a pho-tovoltage of only 0.2 V and had an extremely low efficiency. Attempts to improve theefficiency solar cells based on a single organic material (a so-called homojunction) wereunsuccessful, mainly because of the low dielectric constant of organic materials (typ-

∗In this research, only regio-regular P3HT is used

7


20 40 60 80 100 120 140 160

10-11

10-9

10-7

pristine P3HT holes

P3HT:PCBM electrons holes

as-cast

[m2 /V

s]

Annealing Temperature [oC]

Figure 1.3: Electron and hole mobility in P3HT/PCBM blends as a function of annealing temper-ature, as well as the hole mobility in pristine P3HT.

ically, the relative dielectric constant is 2–4). Due to this low dielectric constant, theprobability of forming free charge carriers upon light absorption is very low. Instead,strongly bound excitons are formed, with a binding energy of around 0.4 eV in the caseof PPV. [42–44] Since these excitons are so strongly bound, the field in a photovoltaic de-vice, which arises from the work function difference between the electrodes, is much tooweak to dissociate the excitons.

A major advancement was realized by Tang who used two different materials,stacked in layers, to dissociate the excitons. [45] In this so-called heterojunction, an elec-tron donor material (D) and an electron acceptor material (A) are brought together. Bycarefully matching these materials, electron transfer from the donor to the acceptor, orhole transfer from the acceptor to the donor, is energetically favored. In 1992 Sariciftciet al. demonstrated that ultrafast electron transfer takes place from a conjugated poly-mer to C60, showing the great potential of fullerenes as acceptor materials. [46] In order tobe dissociated the excitons must be generated in close proximity to the donor/acceptorinterface, since the diffusion length is typically 5–7 nm. [47–49] This need limits the partof the active layer that contributes to the photocurrent to a very thin region near thedonor/acceptor interface; excitons generated in the remainder of the device are lost.

How can the problem of not all excitons reaching the donor/acceptor interface beovercome? In 1995 Yu et al. devised a solution: [7] By intimately mixing both componentsthe interfacial area is greatly increased and the distance excitons have to travel in order toreach the interface is reduced. This device structure is called a bulk heterojunction (BHJ)and has been used extensively since its introduction in 1995. An important breakthroughin terms of power conversion efficiency was reached by Shaheen et al. who showed thatthe solvent used has a profound effect on the morphology and performance of BHJ solarcells. [50] By optimizing the device processing, an efficiency of 2.5% was obtained. State-of-the-art polymer/fullerene BHJ solar cells have an efficiency of more than 4%. [2]

8

1.4. Organic photovoltaics in a nutshell

Figure 1.4: Organic photovoltaics in a nutshell: Part (a) shows the process of light absorptionby the polymer yielding an exciton which has to diffuse to the donor/acceptor interface. If theexciton reaches this interface, electron transfer to the acceptor phase is energetically favored, asshown in part (b), yielding a Coulombically bound electron-hole pair. The dissociation of theelectron-hole pair, either phonon- or field assisted, produces free charge carriers, as depicted in (c).Finally, the free carriers have to be transported through their respective phases to the electrodesin order to be extracted (d). Exciton decay is one possible loss mechanism, see part (e), whilegeminate recombination of the bound electron-hole pair and bimolecular recombination of freecharge carriers (f) are two other possibilities.

9


Figure 1.5: Schematic layout of a BHJ solar cell. A part of the active layer is enlarged to illustratethe processes of light absorption and charge transport.

The main steps in photovoltaic energy conversion by organic solar cells are depictedin Fig. 1.4. The foremost process is light absorption by the polymer, yielding an excitonwhich has to diffuse to the donor/acceptor interface. If the exciton reaches this interface,electron transfer to the acceptor phase is energetically favored, resulting in a Coulombi-cally bound electron-hole pair. The dissociation of this electron-hole pair, either phonon-or field assisted, produces free charge carriers. Finally, the free carriers have to be trans-ported through their respective phases to the electrodes in order to be extracted. Possi-ble loss mechanisms are exciton decay, geminate recombination of bound electron-holepairs, and bimolecular recombination of free charge carriers.

1.5 Device fabrication and characterization

A typical BHJ solar cell has a structure as shown in Fig. 1.5. The active layer is sand-wiched between two electrodes, one transparent and one reflecting. The glass substrateis coated with indium-tin-oxide (ITO) which is a transparent conductive electrode witha high work function, suitable to act as an anode. To reduce the roughness of this ITOlayer and increase the work function even further, a layer of poly(3,4-ethylene dioxythio-phene):poly(styrene sulfonate) (PEDOT:PSS) is spin cast, followed by the active layer.The top electrode usually consists of a low work function metal or lithium fluoride (LiF),topped with a layer of aluminum, all of which are deposited by thermal deposition invacuum through a shadow mask.

In order to determine the performance and electrical characteristics of the photo-voltaic devices, current-voltage measurements are performed (positive Va correspondsto positive biasing of the anode), both in dark and under illumination. A typical current-voltage characteristic of a solar cell under illumination is shown in Fig. 1.6. The currentdensity under illumination at zero applied voltage Va is called the short-circuit currentdensity Jsc.∗ The maximum voltage that the cell can supply, i.e., the voltage where the

∗ Jsc is taken positive throughout this thesis, as is customary.

10

1.6. Objective and outline of this thesis

0.0 0.3 0.6 0.9

-60

-30

0

30

JscVoc

Voc

Jsc

J L [A

/m2 ]

Va [V]

FF = |JL Va|max

Figure 1.6: Typical current-voltage characteristics of a BHJ solar cell showing the Voc, Jsc, and FF.The shaded area corresponds to the maximum power that the solar cell can supply.

current density under illumination JL is zero is designated as the open-circuit voltageVoc. The fill factor FF is defined as

FF =|JLVa|max

Voc Jsc, (1.10)

relating the maximum power that can be drawn from the device to the open-circuit volt-age and short-circuit current. The power conversion efficiency χ is related to these threequantities by

χ =JscVocFF

I, (1.11)

where I is the incident light intensity. Because of the wavelength and light intensitydependence of the photovoltaic response, the efficiency should be measured under stan-dard test conditions. The conditions include the temperature of the cell (25°C), the lightintensity (1000 W/m2) and the spectral distribution of light (air mass 1.5 or AM1.5,which is the spectrum of sunlight after passing through 1.5 times the thickness of theatmosphere). [51]

1.6 Objective and outline of this thesis

Although significant progress has been made, the efficiency of current BHJ solar cellsstill does not warrant commercialization. A lack of understanding makes targeted im-provement troublesome. The main theme of this thesis is to introduce a simple model forthe electrical characteristics of BHJ solar cells, relating their performance to basic physicsand material properties such as charge carrier mobilities.

11


The basis of this research is laid down in chapter 2, which describes the MIM modelused throughout this thesis. This numerical model describes the generation and trans-port processes in the BHJ as if occurring in one virtual semiconductor. Drift and diffusionof charge carriers, the effect of charge density on the electric field, bimolecular recom-bination and a temperature- and field-dependent generation mechanism of free chargesare incorporated. From the modeling of current-voltage characteristics, it is found thatthe bimolecular recombination strength is significantly reduced, and is governed by theslowest charge carrier. Subsequently, the numerical model is successfully applied to ex-perimental data on MDMO-PPV/PCBM solar cells, showing field and carrier densityprofiles.

In chapter 3, two competing models for the open-circuit voltage are introduced: First,a model valid for p-n junctions is examined. By studying the dependency of the open-circuit voltage on light intensity, it is demonstrated that this model does not correctlydescribe the open-circuit voltage of BHJ solar cells. Within the framework of the MIMmodel an alternative explanation for the open-circuit voltage is presented. Based on thenotion that the quasi-Fermi potentials are constant throughout the device, a formula forVoc is derived that consistently describes the open-circuit voltage. Next, the predictionsof the MIM model and its relation to other types of solar cells are discussed.

One other key parameter of solar cells, the short-circuit current, is the subject of chap-ter 4. Following the description of some simple analytical expressions for the short-circuit density, the dependence of the short-circuit current density on incident light in-tensity is discussed in more detail. A typical feature of polymer/fullerene based solarcells is that the short-circuit current density does not scale exactly linearly with light in-tensity. Instead, a power law relationship is found given by Jsc ∝ Iα, where α rangesfrom 0.85 to 1. In this chapter, it is shown that this behavior does not originate frombimolecular recombination but is a consequence of space charge effects.

Hybrid organic/inorganic solar cells, as discussed in chapter 5, are an auspicious al-ternative to polymer/fullerene devices. In this case, an inorganic semiconductor, eithertitanium dioxide or zinc oxide, is used as the electron acceptor. One way of makingthese cells is the precursor route: A precursor for the inorganic semiconductor is mixedwith the solution of the polymer. Upon spin casting of the active layer in ambient condi-tions, the precursor reacts with moisture from the air and the inorganic semiconductor isformed. Although promising, this method seems to harm the transport of charge carri-ers through the active layer. Alternatively, the inorganic semiconductor, in this case zincoxide, can be formed ex situ. This enables one to better control the reaction conditionsand purity of the material. The transport of charge carriers as well as limitations to theefficiency are investigated in detail.

In chapter 6, various ways to improve the efficiency of bulk heterojunction solar cellsare identified by using the MIM model as outlined in chapter 2. A much pursued way toincrease the performance is to increase the amount of photons absorbed by the film bydecreasing the band gap of the polymer. Calculations based on the MIM model confirmthat this would indeed enhance the performance. However, it is demonstrated that theeffect of minimizing the energy loss in the electron transfer from the polymer to the

12

1.6. Objective and outline of this thesis

fullerene derivative is even more beneficial. By combining these two effects, it turns outthat the optimal band gap of the polymer would be 1.9 eV. Ultimately, with balancedcharge transport, polymer/fullerene solar cells can reach power conversion efficienciesof 10.8%.

Table 1.1: List of symbols and abbreviations used in this thesis.

Symbol description

A acceptora electron-hole pair distanceα exponent in Jsc ∝ Iα

AM1.5 air mass 1.5BEH-PPV poly(2,5-bis(2’-ethylhexyloxy)-p-phenylene vinylene)BHJ bulk heterojunctionD donorDn(p) electron (hole) diffusion coefficient

DSSC dye-sensitized solar cell

Eeffgap effective band gap

ε dielectric constantη Poole-Frenkel detrapping parameterF field strengthFF fill factorφn(p) electron (hole) quasi-Fermi potential

G generation rate of free charge carriersGe−h generation rate of bound electron-hole pairsγ field activation parameter of mobilityhi grid spacingHOMO highest occupied molecular orbitalI incident light intensityITO indium tin oxideJD current density in darkJL current density under illuminationJn(p) electron (hole) current density

Jph photogenerated current densityJsc short-circuit current densitykB Boltzmann’s constantkdiss electron-hole pair dissociation ratek f electron-hole pair decay ratekr bimolecular recombination rateL active layer thicknessLUMO lowest unoccupied molecular orbitalMDMO-PPV poly(2-methoxy-5-(3’,7’-dimethyl octyloxy)-p-phenylene vinylene)MIM model metal-insulator-metal modelµn(p) electron (hole) mobility

n electron densityNcv effective density of states of valance and conduction bandsnc-ZnO nanocrystalline zinc oxidenint intrinsic carrier densityp hole densityP electron-hole pair dissociation probabilityψ potential

Continued on next page

13


Symbol descriptionP3HT poly(3-hexylthiophene)PCBM 6,6-phenyl C61-butyric acid methyl esterPEDOT:PSS poly(3,4-ethylene dioxythiophene):poly(styrene sulfonate)Photo-CELIV photoinduced charge carrier extraction in a linearly increasing voltagePPV poly(phenylene vinylene)prec-ZnO zinc oxide by precursor routeq elementary chargeR recombination rate of charge carriersS slope of Voc vs. ln(I)SCL space-charge-limitedSRH Shockley-Read-Hallσ width of Gaussian distribution energy distributionT absolute temperatureTOF time-of-flightU net generation rate of free carriersV0 compensation voltageVa applied voltageVbi built-in voltageVint internal voltage across active layerVoc open-circuit voltageVt thermal voltagewn(p) electron (hole) drift length

x positionX density of bound electron-hole pairsχ power conversion efficiency〈. . .〉 spatial average

14

References chapter 1

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[21] S. V. Novikov, D. H. Dunlap, V. M. Kenkre, P. E. Parris, and A. V. Vannikov, Phys. Rev. Lett. 81,4472 (1998).

15


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16


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17

CHAPTER

TWO

Metal-insulator-metal model for bulkheterojunction solar cells

Summary

In this chapter, a numerical device model is introduced that consistently describesthe current-voltage characteristics of polymer/fullerene BHJ solar cells. This numericalmodel hinges on the description of the active blend layer as one effective medium (theso-called metal-insulator-metal model), described by basic semiconductor equations.Drift and diffusion of charge carriers, the effect of charge density on the electricfield, bimolecular recombination and a temperature- and field-dependent generationmechanism of free charges are incorporated. From the modeling of current-voltagecharacteristics, it is found that the bimolecular recombination strength is significantlyreduced, and is governed by the slowest charge carrier.

Subsequently, the numerical model is successfully applied to experimental data onMDMO-PPV/PCBM (1:4 by weight) solar cells. As a result, it is demonstrated that inthese devices space charge effects only play a minor role, leading to a relatively constantelectric field in the active layer. Furthermore, at short-circuit conditions only a smallfraction of free carriers is lost due to bimolecular recombination.

19

Chapter 2. Metal-insulator-metal model for bulk heterojunction solar cells

2.1 Introduction

An accurate and reliable numerical description of BHJ solar cells is highly desirablewhen searching for ways to optimize their performance. There is, however, still consid-erable controversy on the most suited basic physical description of BHJ devices. In a re-cent article, Mihailetchi et al. [1] have demonstrated that the separation of bound electron-hole pairs into free charges is an important process in solar cells based on MDMO-PPVand PCBM as donor and acceptor, respectively. At high reverse bias saturation of thephotocurrent is observed, indicating that all electron-hole pairs are separated. From thisobservation it follows directly that at short-circuit conditions only 61% of the electron-hole pairs are dissociated, which is a major loss mechanism in these devices. The impor-tance of other processes such as bimolecular recombination is not known. It has beensuggested [2,3] that bimolecular recombination can be excluded because of the (nearly)linear dependence of the short-circuit current on light intensity. It remains to be seen,however, whether this holds only at short-circuit conditions, or at all biases and espe-cially at maximum power output (typically around 0.6 V applied bias).

Goodman and Rose [4] have presented a model for extraction of uniformly photogen-erated charges from a photoconductor with noninjecting contacts. In the case of a largedifference in mean-free path for electrons and holes, caused by, for example, a large dif-ference in electron and hole mobility, the electric field in the device adjusts itself in sucha way that the transport of the slowest carrier is enhanced. This results in a nonuniformfield, since the charges of photogenerated electrons and holes do not cancel. Conse-quently, the slowest charge carrier will dominate the device because the faster carriercan leave the device much easier. In MDMO-PPV/PCBM bulk heterojunctions the elec-tron mobility in the PCBM is one order of magnitude higher than the hole mobility inMDMO-PPV. [5,6] This raises the question of whether the photocurrent will be dominatedby a nonuniform electric field and resulting space charge. So far, no detailed descriptionof bulk heterojunction solar cells clarifying field distribution and carrier densities hasbeen given and interpretation of current-voltage curves is often done by using mod-els developed for inorganic p-n junctions. [7–10] Recently, Barker et al. [11] have presenteda numerical model describing the current-voltage characteristics of bilayer conjugatedpolymer photovoltaic devices. However, since the electronic structures of bilayers andbulk heterojunctions are distinct, their operational principles are fundamentally differ-ent.

In this chapter a device model is presented that quantitatively addresses the role ofcontacts, drift and diffusion of charge carriers, charge carrier generation, and recombi-nation. First a description of the model is given, followed by an overview of the relevantequations. Because of the particular morphology of BHJ solar cells, a new expressionfor the bimolecular recombination strength is proposed. Subsequently, the generationmechanism of free charge carriers is described, which completes the description of themodel. The second part of this chapter contains details on the numerical scheme, i.e.,on the iteration procedure and discretization of the equations. Finally, the results of thesimulations are presented, showing field and carrier density distributions for this type of

20

2.2. Description of the model

Figure 2.1: (a) Schematic representation of the energy levels (energies given in eV) of the elec-tron donating and electron accepting materials. After charge separation, the electron and holeare transported through the respective materials and collected by the electrodes. As the anode alayer of poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) deposited on topof indium-tin-oxide (ITO) was used, while on top of the active layer lithium fluoride and alu-minum were deposited by thermal evaporation under vacuum. (b) The resulting device model inthe metal-insulator-metal representation with positive applied bias Va, under operating conditions(Va smaller than open-circuit voltage).

devices. An important loss process in solar cells is the bimolecular recombination of freecharge carriers. The numerical simulations show that the recombination losses in BHJsolar cells strongly depend on the bias conditions; at short-circuit only a small fractionof free charge carriers is lost due to bimolecular recombination. At the maximum poweroutput, however, the losses increase due to the decrease of the internal electric field.

2.2 Description of the model

How can one model these devices, considering their complicated morphology? Themetal-insulator-metal (MIM) model used in this thesis, is based on an effective mediumapproach, treating the blend of both components as one intrinsic semiconducting mate-rial. The lowest unoccupied molecular orbital (LUMO) of the acceptor and the highestoccupied molecular orbital (HOMO) of the donor act as valence and conduction bandof this virtual semiconductor, respectively (see Fig. 2.1 for the relevant energy levels ofboth materials). [8,12] The energy difference between the LUMO of the acceptor and theHOMO of the donor functions as the effective band gap (Eeff

gap) of the semiconductor.Note, that in a disordered system, like an organic solar cell, the band gap will not be arigorously defined quantity due to the Gaussian density of states of both the acceptorand the donor material. [13]

The model used in this thesis contains drift and diffusion of charge carriers, and theeffect of space charge on the electric field in the device. It should be mentioned, thatthe influence of disorder on the transport of carriers is only taken into account throughthe magnitude and field/temperature dependence of the mobility, leaving the equationsthemselves unaltered. The resulting basic equations describing transport through semi-conductors are solved self-consistently. [14] Recombination is described as a bimolecular

21


process, with the rate given by Langevin. [15] The rate of generation of bound electron-hole pairs is assumed to be homogeneous throughout the device. Although this isknown not to be strictly correct, the incorporation of an exponential dependence of thegeneration rate on distance, resulting from absorption of light by the active layer, doesnot significantly influence our results. As the devices considered in this study are verythin (120 nm), the assumption of uniform generation of electron-hole pairs does not giverise to serious inconsistencies. [16]

The generation of free charge carriers is a two-step process: exciton dissociationacross the donor-acceptor interface, which yields a bound electron-hole pair, and sub-sequent dissociation of this electron-hole pair. [1] The ultrafast (within 100 fs) excitondissociation, driven by the difference in LUMO levels of MDMO-PPV and PCBM, hasa quantum efficiency of almost unity [12] and is assumed to be field independent. Theresulting electron-hole pair is metastable (up to milliseconds at 80 K) and its dissociationis strongly field and temperature dependent.

In solar cells based on amorphous silicon traps play a dominant role in the descrip-tion of the solar cell characteristics. [17] In contrast, the current in MDMO-PPV basedhole-only diodes has been shown to have a quadratic dependence on voltage and ex-hibits a third power dependence on sample thickness. [18] This behavior is characteristicfor a space-charge-limited (SCL) current. The occurrence of SCL current enables one todirectly determine the hole mobility from the current-voltage characteristics. It shouldbe noted that a material with shallow traps would also exhibit an identical voltage andthickness dependence, and the observed mobility would be an effective mobility in thatcase, including trapping effects. However, transient measurements demonstrated thatthe measured mobility does not show any evidence of trapping effects. [19] The sameholds for the electron transport in bulk PCBM. [5] Additionally, the electron- and holemobility in the blend of both materials have been addressed, both showing trap-freeSCL current-voltage characteristics. [6,20] Therefore, it can be safely concluded that trap-ping effects do not play a role in polymer/fullerene devices, and hence can be neglectedin the model.

2.2.1 Basic equations

The equations [14] used to describe the transport through the virtual semiconductor arethe Poisson equation∗

∂2

∂x2ψ(x) =

q

ε[n(x)− p(x)], (2.1)

where q is the elementary charge and ε is the dielectric constant, relating the potentialψ(x) to the electron and hole densities n(x) and p(x), respectively. The current continu-

∗Through the use of the Poisson equation and the fact that we ignore any morphological effects in n(x) andp(x), it is implicitly assumed that space charge caused by electrons in one phase can be neutralized by holes inthe other phase, which is not obvious considering the actual morphology of a BHJ. However, this assumptionis supported by the consistent description of space charge effects given in chapter 4.

22


ity equations∂

∂xJn(x) = −qU(x), (2.2a)

∂

∂xJp(x) = qU(x), (2.2b)

where Jn(p)(x) is the electron (hole) current density and U(x) is the net generation rate,i.e., the difference between generation of free carriers and recombination of free carriers.In the remainder of this thesis, the position (x) dependence of variables is dropped fornotational convenience, unless stated otherwise. Only one spatial dimension is consid-ered, since the device has a planar structure with a very small thickness (typically 100nm) compared to the lateral dimensions (typically several mm).

In order to solve the basic equations, a set of equations is needed relating the currentdensities to the carrier densities and the potential. Incorporating both drift and diffusionof charge carriers, one has∗

Jn = −qnµn∂

∂xψ + qDn

∂

∂xn, (2.3a)

Jp = −qpµp∂

∂xψ − qDp

∂

∂xp, (2.3b)

where Dn,p are the carrier diffusion coefficients, which are assumed to obey the Einstein

relation [14]

Dn,p = µn,pVt, (2.4)

with Vt the thermal voltage, i.e., Vt = kBT/q, where kB is Boltzmann’s constant and T isthe absolute temperature. However, it should be noted that at high carrier densities thediffusion coefficient may be increased. [21]

To obtain a unique solution of the system of equations formed by Eqs. (2.1)–(2.4) it isnecessary to specify the carrier densities and potential at both contacts. The contact atx = 0 will be called the top contact and the contact at x = L, where L is the device thick-ness, the bottom contact. The work function of the top contact is assumed to match theconduction band of the semiconductor and is therefore Ohmic as far as electron injectionis considered. Using Boltzmann statistics

n(0) = Ncv, (2.5a)

p(0) = Ncv exp(

−Eeff

gap

qVt

)

, (2.5b)

where Ncv is the effective density of states of both the conduction and valence band. Thisassumption implies that the semiconductor is in thermodynamic equilibrium with the

∗These relations imply, in conjunction with the convention of having the cathode at x = 0, a positive short-current density under illumination at zero bias. In the graphs in this thesis, however, the sign of the current isreversed, as is customary in this field.

23


contacts. Since the exact values of the effective densities of states of valence and con-duction band are not known and are of little importance, one value (2.5 × 1025 m−3) [22]

for both bands is used. Similarly, the bottom contact is assumed to be an Ohmic holecontact, thus

n(L) = Ncv exp(

−Eeff

gap

qVt

)

, (2.6a)

p(L) = Ncv. (2.6b)

The Ohmic nature of both contacts is supported by current-voltage measurements onboth materials, which clearly show space-charge-limited behavior. [5,6] The boundarycondition for the potential reads

ψ(L) − ψ(0) =1

qEeff

gap − Va, (2.7)

where Va is the applied voltage.

2.2.2 Generation of free charge carriers

As outlined in chapter 1, exciton dissociation at the donor/acceptor interface does notdirectly yield free charge carriers, but rather bound electron-hole pairs. The dissociationof these bound pairs into free carriers has been explained in terms of the Onsager the-ory on geminate recombination. [23] Braun [24] has made an important refinement to thistheory by pointing out that the bound electron-hole pair, which acts as a precursor forfree charge carriers, has a finite lifetime, see Fig. 2.2. The bound electron-hole pair maydecay to the ground state with a decay rate k f or dissociate into free carriers. The sep-aration into free carriers is a competition between dissociation (rate constant kdiss) andrecombination (rate constant kr), which revives the charge transfer state. Furthermore,Braun uses the 1934 theory by Onsager on the dissociation of weak electrolytes, [25] in-stead of the 1938 theory of initial recombination of ions. The difference between thesetwo models is in the boundary conditions: Whereas Onsager’s 1934 theory has a sourceat the origin and a sink at infinity, the 1938 theory places the source at a finite distance r0

and sinks at the origin and at infinity.In Braun’s model the probability of electron-hole pair dissociation, for a given

electron-hole pair distance y, is given by

p(y, T, F) =kdiss(y, T, F)

kdiss(y, T, F) + k f (T), (2.8)

depending on both temperature T and field strength F. The ratio of kdiss(y, T, F) tokdiss(y, T, 0) is given by Onsager’s theory for field-dependent dissociation of weak elec-trolytes as [25]

kdiss(y, T, F)

kdiss(y, T, 0)= J1(2

√−2b)/

√−2b = (1 + b + b2/3 + · · · ), (2.9)

24


Figure 2.2: Schematic representation of the charge carrier separation at the interface betweendonor (D) and acceptor (A). Upon excitation of the donor, an exciton is created which diffusesthrough the donor until it reaches the interface. At the interface, the electron is transfered to theacceptor, thus forming a bound electron-hole (e/h) pair. This pair can either dissociate into freecarriers or decay to the ground state.

where b = q3F/(8πεk2BT2) and J1 is the Bessel function of order one. Fuoss and Ac-

cascina have estimated that the equilibrium constant at zero field K(0) for dissociationof a weak electrolyte is given by [24,26]

K(0) =3

4πa3e−Eb/kBT =

kdiss(y, T, 0)

kr, (2.10)

where Eb is the electron-hole pair binding energy. Combining Eqs. (2.9) and (2.10) gives

kdiss(y, T, F) =3kr

4πa3e−Eb/kBT J1(2

√−2b)/

√−2b. (2.11)

The decay rate of the bound electron-hole pair to the ground state k f is used as a fitparameter.

Suppose that Ge−h is the generation rate of bound electron-hole pairs. The number ofbound electron-hole pairs created per unit volume and time R from free charge carrierswill be equal to

R = kr(np − n2int), (2.12)

where nint = Ncv exp(

− Eeffgap

2qVt

)

is the intrinsic carrier density of electrons and holes. Then,

the number X of bound electron-hole pairs per unit volume is changed in time by∗

dX

dt= Ge−h − k f X − kdissX + R. (2.13)

∗Note that this equation implies that k−1f is not equal to the lifetime of the bound electron-hole pair and

that the decay of electron-hole pairs is not exponential in time.

25


The same equation was used by Barker et al., [11] when they considered dissociation ofbound charge carrier pairs in a bilayer photovoltaic device. In steady-state this worksout to be

Ge−h − k f X = kdissX − R, (2.14)

which is the net number density of generated free carriers. Therefore, the continuityequation for electrons will read

dn

dt=

1

q

∂

∂xJn + kdissX − R. (2.15)

Using Eq. (2.8), kdiss = [p/(1 − p)]k f , therefore, Eq. (2.14) becomes

kdissX = pGe−h + pR (2.16)

and the continuity equation may be written as

1

q

∂

∂xJn = p(y, T, F)Ge−h − [1 − p(y, T, F)]R. (2.17)

As polymer systems are subject to disorder, it is reasonable to assume that theelectron-hole pair distance is not constant throughout the system. [27] As a result, thedissociation probability p(y, T, F) in Eq. (2.17) should be integrated over a distributionof electron-hole pair distances, i.e.,

P(a, T, F) =∫

∞

0p(y, T, F) f (a, y)dy, (2.18)

where f (a, y) is a normalized distribution function, see Fig. 2.3, given by [1]

f (a, y) =4√πa3

y2e−y2/a2. (2.19)

In the remainder of this article the a, T, F dependencies of P are dropped for notationalconvenience. With this new expression for the dissociation probability P, the continuityequation is given by

1

q

∂

∂xJn = PGe−h − (1 − P)R. (2.20)

This implies that after bimolecular recombination the carriers are not necessarily lost, butfirst form a bound electron-hole pair, which can either again dissociate into free carriersor decay to the ground state, in which case the carriers are lost. The net generation rateU becomes

U = PGe−h − (1 − P)kr(np − n2int). (2.21)

26


0 1 2 30.0

0.5

1.0

f(a,y

)

y/a

Figure 2.3: The distribution function f (a, y) given by Eq. (2.19).

2.2.3 Bimolecular recombination in bulk heterojunctions

Up to now, the bimolecular recombination constant kr has not been specified. Bimolec-ular recombination in organic semiconductors is known to follow the Langevin expres-sion, i.e., the rate of recombination depends on the sum of the mobilities of both carriers.This expression has to be adapted when applied to BHJ solar cells, as a consequence ofthe confinement of both types of carriers to two different phases. For pristine materialsthe recombination strength kr is given by [15]

kr =q

ε(µn + µp). (2.22)

The sum of the mobilities of both carriers appears in Eq. (2.22) since both carriers are freeto move toward each other. Thus, if one of the carriers is much faster than the other, e.g.,µn ≫ µp, it is the fastest carrier that determines the recombination rate, as schematicallyindicated in Fig. 2.4(a). The validity of the Langevin theory towards organic semiconduc-tors was confirmed by investigations of the electron-hole recombination in anthraceneby Karl and Sommer in 1971. [28] Furthermore, it was demonstrated that the bimolecularrecombination processes in conjugated polymers is correctly described by the Langevinexpression. [29]

However, the major difference between a pristine semiconductor and a blend is thatin the latter the electrons and holes are confined to different phases, and the recombina-tion of free electrons and holes now mainly occurs across the interface of the materials, asschematically indicated in Fig. 2.4(b). As a first approach, it was proposed by Braun [24]

that the recombination constant kr has to be adapted to

kr =q

〈ε〉〈µn + µp〉, (2.23)

27


Figure 2.4: Schematic semiconductor energy diagram showing bimolecular recombination in pris-tine semiconductors (a) and in bulk heterojunction solar cells (b). Due to the energy barrier at thedonor/acceptor interface, the charge carriers are confined to two different materials.

where 〈. . .〉 denotes the spatial average. The rationale for using a spatial average valueis to compensate for eventual mobility differences between electrons and holes in thecomponents of the blend. However, consider the situation as depicted in Fig. 2.4(b),where the electron is at distance re from the interface, while the hole is at a distance rh.Depending on the initial values of re and rh, the carriers have to travel a certain distanceand, subsequently, are stuck at the interface due to the energy offsets between the energylevels of the donor and acceptor. When µn ≫ µp the electron reaches the interface muchfaster than the hole and the total time needed for both carriers to reach the interface isdominated by the hole. Thus in contrast to a pristine material [Fig. 2.4(a)] the bimolecularrecombination in a blend is now governed by the slowest charge carrier. Therefore, it isexpected that for bulk heterojunction solar cells the recombination constant should notdepend on µn + µp, as in Eq. (2.22), but will be close to

kr =q

〈ε〉 min(µn, µp), (2.24)

i.e., the recombination constant is dominated by the slowest carrier, in contrast to theoriginal Langevin result Eqs. (2.22) and (2.23). In subsection 2.4.1, the applicability ofEqs. (2.23) and (2.24) will be discussed by comparing the numerical results to the exper-imental data on a suitable system.

2.3 Numerical method

How does one obtain a current-voltage characteristic from the equations described inthe previous section? This section briefly summarizes how these equations are solvedon a computer. The basic equations are discretized on a finite number of points, thusgiving a finite number of equations that can be solved by using well-known numericaltechniques. The set of these points forms the grid, which is specified by hi = (xi+1 − xi),where i is the index of the grid point. To discretize the basic equations, the method offinite differences is used. [14]

28

2.3. Numerical method

Finite difference approximations of derivatives are obtained by using truncated Tay-lor series. Consider, for example, the series expansion of ψi±1 = ψ(xi±1),

ψi+1 = ψi + hi∂ψ

∂x+

h2i

2

∂2ψ

∂x2+ O(h3

i ) (2.25a)

ψi−1 = ψi − hi−1∂ψ

∂x+

h2i−1

2

∂2ψ

∂x2+ O(h3

i−1). (2.25b)

The first order derivative of ψ can be approximated by subtracting Eq. (2.25a) fromEq. (2.25b), which yields

∂ψ

∂x≈ ψi+1 − ψi−1

hi + hi−1. (2.26)

Notice that the terms corresponding to the second order derivatives in the Taylor expan-sions [Eqs. (2.25)] do not cancel exactly, due to the non-uniformity of the grid intervals.Equation (2.26) is called the centered difference approximation.

To obtain an approximation for the second order derivative, it is convenient to in-volve the mid-points of the grid, defined as xi+1/2 = (xi + xi+1)/2. The first orderderivative [Eq. (2.26)] can now be expressed as

∂ψ

∂x≈ ψi+1/2 − ψi−1/2

hi+hi−12

. (2.27)

By taking the derivative of this equation, one has

∂2ψ

∂x2≈

∂ψ∂x |i+1/2 − ∂ψ

∂x |i−1/2

hi+hi−12

. (2.28)

The derivatives ∂ψ/∂x|i±1/2 can be approximated with centered differences by usingEq. (2.26). By doing so, the Poisson equation [Eq. (2.1)] can be rewritten as

ψi+1−ψihi

− ψi−ψi−1hi−1

hi + hi+1=

q

2ε(ni − pi). (2.29)

Discretization of the continuity equations requires a little care since it is necessary toapproximate the carrier densities between two grid points in order to get a finite differ-ence equation. This is a consequence of the derivative in Eq. (2.2). Simply approximatingthe carrier densities by averaging between grid points can lead to serious instabilitiessince the carrier concentrations can change very rapidly between grid points. Schar-fetter and Gummel have proposed a solution to this problem. [30] In their approach thefield is assumed constant between grid points and an exponential variation of the carrierdensities results. For example, one has for the electron density

n(x ∈ [xi, xi+1]) = [1 − gi(x, ψ)]ni + gi(x, ψ)ni+1, (2.30)

29


where the growth function gi(x, ψ) is defined as

gi(x, ψ) =1 − exp

(

ψi+1−ψiVt

x−xihi

)

1 − exp(

ψi+1−ψiVt

) . (2.31)

A similar expression holds for the hole density.The resulting discretized Poisson and continuity equations are solved in an iterative

manner based on the work of Gummel: [31] First, a guess is made for the potential andthe carrier densities. With this guess, a correction δψ to the guessed potential is calcu-lated from the Poisson equation. This new potential is then used to update the carrierdensities by solving the continuity equations. This process is repeated until convergenceis reached, i.e., until the corrections get very small. A flow diagram of this method isshown in Fig. 2.5.

2.4 Simulation results and discussion

In this section, the numerical model is put to the test: First, the bimolecular recombina-tion strength is established by comparing the numerical results to current-voltage char-acteristics of P3HT/PCBM solar cells, completing the description of the model. Subse-quently, the model is applied to MDMO-PPV/PCBM devices, which have been studiedextensively, making them ideal for modeling purposes. After fitting the current-voltagecharacteristics, the field and charge carrier density profiles are discussed extensively.

2.4.1 Bimolecular recombination: Comparison with experimental re-sults

Let us return to the matter of the bimolecular recombination strength discussed in sub-section 2.2.3. A major experimental difficulty to investigate bimolecular recombinationin donor/acceptor blends is that the (photo)luminescence, which can serve as a finger-print of free electron and hole recombination, is completely quenched due to the ultra-fast electron transfer. Recently, a new technique has been developed, called photoin-duced charge carrier extraction in a linearly increasing voltage (Photo-CELIV). [32,33] Inthis technique, charge carriers are generated by a short laser flash and, after an adjustabledelay time, are extracted under a voltage ramp. Photo-CELIV enables one to study themobility of charge carriers and their lifetime simultaneously. Fortunately, the temporalvariation of the charge carrier density can also be extracted, enabling one to determinethe bimolecular recombination constant.

However, the fill factor is also strongly dependent on the bimolecular recombina-tion strength. In order to discriminate between the spatially averaged Langevin result[Eq. (2.23)] and the recombination rate dominated by the slowest carrier [Eq. (2.24)], alarge difference between the mobilities of electrons and holes is required. Only in that

30

2.4. Simulation results and discussion

Figure 2.5: Flow diagram of the simulation program. To solve the basic equations, Gummel itera-tion is used. First a guess is made for the potential and carrier densities. Subsequently, a correctionδψ to the guess for the potential is calculated from the Poisson equation. This correction is added tothe guessed potential and this is repeated until convergence is reached. Next, the carrier densitiesare calculated from the new potential by solving the continuity equations. This entire procedureis repeated until the change in the carrier densities is smaller than a pre-set tolerance η, i.e., untilconvergence is reached.

31


-0.2 0.0 0.2 0.4 0.6 0.8

-1.0

-0.5

0.0

J L/J s

c

Va [V]

Figure 2.6: Simulated current-voltage characteristics of a bulk heterojunction photovoltaic de-vice. The solid line corresponds to the recombination constant of Eq. (2.23), while the dashedline denotes the result for Eq. (2.24), i.e., taking only the slowest charge carrier into account. Themaximum obtainable FF would be 84%, corresponding to no recombination (dotted line).

case are the predicted recombination constants clearly different. The hole mobility in theP3HT phase of a P3HT/PCBM device may be varied over several orders of magnitudeby thermally annealing the film, at the same time only slightly changing the electronmobility in the PCBM phase, [34] see Fig. 1.3.

The fill factor is a measure of the shape of the current-voltage characteristic of a pho-tovoltaic device and has a marked dependence on recombination strength. Figure 2.6shows simulated current-voltage characteristics for various recombination strengths anda two orders of magnitude difference between electron and hole mobility (µn = 3 × 10−7

m2/Vs and µp = 3 × 10−9 m2/Vs). Without bimolecular recombination the fill factor is84% (Fig. 2.6, dotted line), but FF drops from 57% to 44% when increasing kr from takingonly the slowest carrier [Eq. (2.24)] to the spatial average result [Eq. (2.23)]. This sensi-tivity of FF on the recombination strength enables one to determine whether the slowestcarrier determines the Langevin rate instead of the spatial average of the mobilities.

What else determines the fill factor? Apart from bimolecular recombination there aretwo other known processes that influence the fill factor of this type of solar cell: Build-upof space charge and a large series resistance due to the electrodes or substrate. The seriesresistance of our devices typically amounts to 30–40 Ω, and due to the relatively lowcurrents of polymer/fullerene devices this effect does not play a role. The occurrenceof space-charge-limited photocurrents (see subsection 4.1.2) is known to reduce the fillfactor to about 42%. [35] Space charge in bulk heterojunction solar cells is caused by ei-ther a large difference in mobilities combined with a high illumination intensity, or asubstantial amount of traps. Since the transport characteristics of electrons and holes inP3HT/PCBM blends bear all the characteristics of trap-free conduction, [34] space chargedue to charge trapping does not occur. To avoid space-charge-limited photocurrents

32


0.0 0.1 0.2 0.3 0.4 0.5 0.6

-1.0

-0.5

0.0

J L/J s

c

Va [V]0.0 0.1 0.2 0.3 0.4 0.5 0.6

-1.0

-0.5

0.0

Va [V]

J L/J s

c

(b)(a)

Figure 2.7: Current-voltage characteristics normalized to the short-circuit current (symbols) oftwo P3HT/PCBM solar cells annealed at 52°C (a) and 70°C (b). The solid lines denote simulationsusing Eq. (2.24), while the dashed lines correspond to simulations using Eq. (2.23).

due to the large difference in mobilities, [35] the measurements were performed at lowintensity (approximately 8 W/m2).

Figure 2.7 shows current-voltage measurements performed on P3HT/PCBM devices,for two different annealing temperatures and, hence, mobility ratios. The mobility dif-ference ranges from two (70°C) to more than three orders of magnitude for the device an-nealed at 52°C. The currents in Fig. 2.7 have been normalized to the short-circuit currentin order to make the comparison between the two different recombination mechanismseasier: When the recombination strength is increased, the current will drop slightly be-cause of higher recombination losses. The lines in Fig. 2.7 denote simulation results forboth recombination mechanisms. The values of the parameters used in our simulationsand resulting FFs are listed in Table 2.1.∗ The parameters a and k f are determined sepa-

rately from the field dependence of the dissociation probability P.†

From Fig. 2.7 and the calculated FF it is obvious that the recombination strength tak-ing only the slowest carrier into account reproduces the experimental data very well, incontrast to the (spatially averaged) Langevin rate. Furthermore, the model calculationsconfirm that at the low light intensities used here the photocurrent is not limited by spacecharge. It should be noted that due to the absence of free parameters it is impossible toobtain a good fit with the spatially averaged Langevin recombination rate. The spatiallyaveraged recombination leads in all cases to too low values for the FF. This observationclearly supports the slowest-carrier-dominated recombination rate. As the difference inelectron and hole mobility becomes smaller, the spatially averaged Langevin rate ap-proaches the slowest-carrier-dominated model, eventually leading to identical results

∗The uncertainty in the values is approximately 50%, except for the value of a, which is sensitive to changesof 10%.

†The way a and k f are determined is discussed in greater detail in subsection 2.4.2

33


Table 2.1: Overview of parameters used in the fits to the data of Fig. 2.7, and corresponding fillfactors.

parameter unit annealed at 52°C annealed at 70°C

µn m2/Vs 2.0 × 10−8 1.0 × 10−7

γn (m/V)0.5 3.4 × 10−4 2.0 × 10−4

µp m2/Vs 8.0 × 10−12 2.0 × 10−10

γp (m/V)0.5 5.0 × 10−4 2.5 × 10−4

a nm 1.8 1.8k f (slow) s−1 1.0 × 104 1.2 × 104

k f (spat. av.) s−1 1.1 × 107 2.5 × 106

FF (exp.) % 40 61FF (slow) % 39 60FF (spat. av.) % 32 48

when the electron- and hole mobilities are equal. Figure 2.8 shows the experimental FF(symbols) as a function of µp/µn, as well as the result for both recombination models.As already demonstrated, the difference in the FF is large when µp ≪ µn, but when thedifference in mobility is in the order of a factor of 10 or less, it is hard to distinguishbetween both models.

Further direct proof of the slowest-carrier-dominated bimolecular recombinationcomes from recent Photo-CELIV experiments in which the recombination constant hasbeen measured directly in a polymer/fullerene blend. In blends of MDMO-PPV withPCBM Mozer et al. [33] found 6 × 10−17 m3 s−1 for the recombination constant in this ma-terial system. Using µp = 2 × 10−8 m2/Vs [6] and P = 0.46,∗ one has in case of slowest-

carrier-dominated recombination (1 − P)kr = 5.8 × 10−17 m3 s−1, in excellent agreementwith the experimental result by Mozer et al. The recombination constant based on thespatial average amounts to (1 − P)kr = 4.2 × 10−16 m3 s−1, which exceeds the measuredvalue by almost an order of magnitude. In the remainder of this thesis the slowest-carrier-dominated recombination mechanism, i.e., Eq. (2.24), will be used.

2.4.2 Modeling MDMO-PPV/PCBM bulk heterojunction solar cells

Now that the bimolecular recombination mechanism has been established, what can besaid about the carrier densities and field distributions in a device? This subsection dealswith the modeling of the well known MDMO-PPV/PCBM system, which will be inves-tigated in detail.

Figure 2.9 shows the current-voltage characteristics of a 120-nm-thick MDMO-PPV/PCBM (1:4 by weight) BHJ solar cell. In this graph the effective photocurrent den-sity Jph, obtained by subtracting the dark current from the current under illumination, is

∗This value of P is taken at the maximum power point (Va = 0.685), as discussed on page 38.

34


10-3 10-2 10-130

40

50

60

70

FF [%

]

p/ n

Figure 2.8: The experimental FF (symbols) corresponding to different annealing temperaturesas a function of the ratio of hole to electron mobility. The solid line denotes simulations usingEq. (2.24), while the dashed line corresponds to simulations using Eq. (2.23).

plotted as a function of effective applied voltage V0 − Va, where V0 is the compensationvoltage defined by Jph(Va = V0) = 0. [1] In this way, V0 − Va reflects the internal electricfield in the device. It should be noted that Va = 0.884 V is very close to the open-circuitvoltage (0.848 V).

For low effective voltages V0 − Va the photocurrent increases linearly with effectivevoltage, and subsequently tends to saturate. Mihailetchi et al. [1] demonstrated that thislow voltage part can be described with an analytical model developed by Sokel andHughes [36] for zero recombination (see subsection 4.1.1), as indicated by the dashed linein Fig. 2.9. The linear behavior at low effective voltage is the result of a direct compe-tition between diffusion and drift currents. In the model by Sokel and Hughes, all freecharge carriers are extracted at higher effective voltage and the photocurrent saturatesto qGL, where G = PGe−h is the generation rate of free charge carriers. The fact thatthe experimental photocurrent does not saturate, but gradually increases for large ef-fective voltages has been attributed to the field dependence of the generation rate G.The two parameters governing the field- and temperature-dependent generation rate,the electron-hole pair distance a and the decay rate k f , can be determined by equatingthe high field photocurrents to qGL. The value of a determines the field at which thedissociation efficiency saturates, and hence a can be determined independently of k f . Itis evident from Fig. 2.9 that the calculated photocurrent fits the experimental data overthe entire voltage range. For comparison, in Fig. 2.10 the experimental and calculatedJL are also shown in a conventional linear plot focusing on the fourth quadrant. The ex-cellent agreement between experimental and calculated data now enables one to furtheranalyze the device behavior under different bias conditions in more detail.

35


0.01 0.1 1 101

10

J ph

[A/m

2 ]

V0-Va [V]

Figure 2.9: Photocurrent density Jph as a function of effective applied voltage (V0 − Va). Thesymbols represent experimental data of MDMO-PPV/PCBM devices at room temperature. Thesolid line denotes the simulation, while the dashed line represents the result of Sokel and Hughes.

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8-40

-20

0

20

40

J L [A

/m2 ]

Va [V]

Figure 2.10: The current density under illumination of an MDMO-PPV/PCBM device (symbols)and the numerical result (line).

36


Table 2.2: Overview of the parameters used in the fit to the data shown in Figs. 2.9 and 2.10.

Parameter Symbol Numerical value

effective band gap Eeffgap 1.29 eV

electron mobility µn 2.0 × 10−7 m2/Vshole mobility µp 2.0 × 10−8 m2/Vseff. density of states Ncv 2.5 × 1025 m−3

generation rate of bound pairs Ge−h 2.7 × 1027 m−3 s−1

dielectric constant 〈ε〉 3.0 × 10−11 F/me/h pair distance a 1.3 nmdecay rate k f 2.5 × 105 s−1

0 20 40 60 80 100 120

-0.6

-0.3

0.0

0.3

0.6

-30

-20

-10

0

x [nm]

[V]

J n,p

[A/m

2 ]

0 20 40 60 80 100 1201017

1019

1021

1023

1025

0

1

2

x [nm]

n, p

[m-3

]

U [x

1027

m-3

s-1]

(a) (b)

Figure 2.11: The device at short-circuit, (a) shows the potential (solid line) and the electron andhole current densities (dashed and dotted lines, respectively), (b) shows the electron and holedensities (dashed and solid lines, respectively) and the net generation rate (dotted line).

The device at short-circuit

The calculated potential, current densities, carrier densities, and net generation rate atshort-circuit are depicted in Fig. 2.11. The potential under illumination is virtually equalto the potential in dark (not shown). The number of photogenerated charges is not suf-ficient to change the potential significantly. Apart from band bending near the contacts,a consequence of the large amount of charge, the field is constant throughout the deviceand, concomitantly, so is the dissociation rate of bound electron-hole pairs. As a result,the current densities exhibit a linear dependence on the position in the device. The onlyexception to this is found near the contacts where both electron and hole densities arehigh, especially near the top contact (x = 0), and recombination becomes important.

In the bulk of the device, the hole density is roughly one order of magnitude higherthan the electron density. This is a result of the difference in mobility between electronsand holes. Since the holes are slower, they pile up in the device. If the difference inmobility becomes large enough and the hole mobility is indeed quite low, then the hole

37


Table 2.3: An overview of voltage, current density, average dissociation probability, and relativenumber of free carriers lost due to recombination at short-circuit, maximum power, and open-circuit conditions.

Va JL 〈P〉 rec. loss[V] [A/m2] [%] [%]

short-circuit 0 28.3 57 2maximum power 0.685 19.6 46 14

open-circuit 0.845 0 42 91

density will become so large that the potential under illumination differs from the po-tential in dark, due to space charge (see subsection 4.1.2). However, with a mobilitydifference of only a factor of ten, the overall carrier densities are rather low as comparedto other devices such as LEDs (at operating voltages) or FETs. This is because the fieldin the device is quite large at short-circuit and carriers are readily extracted. For thisreason space charge effects only play a minor role, leading to a nearly constant field inthe device.

The average dissociation rate 〈P〉 of bound electron-hole pairs is equal to 57%. Thisimplies that a significant improvement to the device performance could be made by fa-cilitating this dissociation. The number of charge carriers lost due to bimolecular recom-bination can be computed in the following way. The average net generation rate 〈U〉 canbe obtained by integrating Eq. (2.2a), which yields

〈U〉 =1

L

∫ L

0U(x)dx =

Jn(0)− Jn(L)

qL. (2.32)

The total number of generated charges is calculated from the average dissociation rateand hence the recombination rate is known. It turns out that at short-circuit conditionsonly 2% of the free charge carriers are lost due to bimolecular recombination and sub-sequent decay. The low loss of charge carriers (with the mobilities of this system) is aconsequence of the high field strength, which ensures good charge extraction and lowcarrier densities at short-circuit. Since the carrier densities are low, bimolecular recom-bination is weak and hence the recombination lifetime of the charge carriers is relativelylong. On the other hand, due to the high field strength, the time needed for the chargecarriers to exit the device is quite small, and therefore only few charge carriers are lost.An overview of dissociation rate and loss of carriers at various applied voltages is givenin Table 2.3.

The device at maximum power output

The maximum power output occurs at Va = 0.685 V for the simulated device. At this bias,the field in the device is smaller than at short-circuit, resulting in a smaller dissociation

38


0 20 40 60 80 100 120

-0.3

0.0

0.3

0.6

-4

-3

-2

-1

0

1

2

x [nm]

[V]

J n,p

[A/m

2 ]

0 20 40 60 80 100 1201017

1019

1021

1023

1025

0

1

2

x [nm]

n, p

[m-3

]

U [x

1027

m-3

s-1]

(a) (b)

Figure 2.12: The device at open-circuit, (a) shows the potential (solid line) and the electron andhole current densities (dashed and dotted lines, respectively), (b) shows the electron and holedensities (dashed and solid lines, respectively) and the net generation rate (dotted line). Note thatthe scales of part (a) are changed as compared to Fig. 2.11(a).

efficiency being 46%, see Table 2.3. Another consequence of the reduction of the fieldstrength is the increase of lost free carriers; as much as 14% of all free carriers recombineand subsequently decay. This increase of losses is for the largest part a consequence of aless favorable extraction of charges. First, the time it takes for a charge carrier to exit thedevice is longer as compared to short-circuit, due to a lower electric field strength in thedevice. This leads to larger charge carrier densities, thereby increasing the probability ofbimolecular recombination of charge carriers. Furthermore, the dissociation probabilityof a bound electron-hole pair slightly decreases, leading to larger loss of carriers oncebimolecular recombination has taken place.

The device at open-circuit

At open-circuit conditions, it is evident that the field in the device is smaller than atshort-circuit, see Fig. 2.12. Since there is no net current, there exists a balance betweendrift and diffusion of charge carriers. Therefore, the field in the device cannot be zeroand, consequently, Voc is smaller than Eeff

gap. [37]

The current densities are almost zero throughout the device. The reason for the cur-rent densities not being zero everywhere lies in the field dependence of the generationrate. In case of a constant generation rate the current densities would be zero every-where. From Fig. 2.12 it can be seen that the densities are almost symmetrical. In fact, ifthe generation rate would be constant, then from Eq. (2.3),

1

Vt

∂

∂xψ =

1

n

∂

∂xn = − 1

p

∂

∂xp. (2.33)

39


Integration yields

lnn(xb)

n(xa)= ln

p(xa)

p(xb)∀ xa, xb ∈ [0, L] (2.34)

and hencen(xb)

n(xa)=

p(xa)

p(xb). (2.35)

Using Eqs. (2.5a) and (2.6b), we have

n(L − x) = p(x), (2.36)

showing that the carrier densities are indeed symmetrical.Another striking feature of Fig. 2.12 is the fact that the carrier densities are much

higher at open-circuit than at short-circuit. This is because of the much lower field whichmakes extraction of charge carriers more difficult. This also results in a much smaller netgeneration rate, in fact, 91% of all free charge carriers are lost due to recombination. ∗

Only in the immediate vicinity of the contacts there is a significant net generation ofcharge carriers. This is not due to an increase in generation due to a higher electric field,but due to the fact that at the contact there is an enormous difference in charge carriersdensities and the free carriers have no counter part to recombine with.

2.5 Summary and conclusions

In this chapter, the MIM model was introduced, which consistently describes the current-voltage characteristics of polymer/fullerene BHJ solar cells. Central to this model is thedescription of the active layer as one effective medium, described by basic semiconduc-tor equations. Drift and diffusion of charge carriers, the effect of charge density on theelectric field, bimolecular recombination and a temperature- and field-dependent gener-ation mechanism of free charges are incorporated.

With regard to the bimolecular recombination in BHJs, it was shown that the recom-bination constant is not dominated by the fastest charge carrier but by the slowest one,as a consequence of the confinement of the respective carriers to different materials. Thesensitivity of the fill factor and the infeasibility to obtain a good fit with the originalLangevin rate made it possible to discriminate between the original (spatially averaged)Langevin rate and the rate dominated only by the slowest charge carrier. Moreover,direct measurements of the recombination rate in MDMO-PPV/PCBM blends quantita-tively confirm this reduction of the Langevin recombination constant.

Subsequently, this model has been used to simulate experimental data of an MDMO-PPV/PCBM bulk heterojunction photovoltaic device. At short-circuit only 57% of all

∗Because of the boundary condition Eq. (2.6b), there is a huge hole concentration gradient near the cathode,as is clear from Fig. 2.12(b), causing the holes near the cathode to move toward it. A similar reasoning appliesto electrons near the anode. These electrons and holes counteract the normal current flow (they go the wrongway) and, consequently, not all charge carriers have to recombine in order to have JL = 0.

40

2.5. Summary and conclusions

created bound electron-hole pairs dissociate into free carriers. Once separated though,only 2% of all charge carriers are lost due to bimolecular recombination and subsequentdecay. At maximum power output the situation is different, less electron-hole pairs areseparated (46%) and, more importantly, the loss of free carriers is much higher (14%).Therefore, bimolecular recombination, although not important at short-circuit, is a sig-nificant loss mechanism in photovoltaic devices. At open-circuit, the charge carrier den-sities become larger due to the slower extraction of charge carriers. Since the diffusionof charges has to be opposed by drift of charges, the field in the device is nonzero.

41


References

[1] V. D. Mihailetchi, L. J. A. Koster, J. C. Hummelen, and P. W. M. Blom, Phys. Rev. Lett. 93,216601 (2004).

[2] I. Riedel, J. Parisi, V. Dyakonov, L. Lutsen, D. Vanderzande, and J. C. Hummelen,Adv. Funct. Mater. 14, 38 (2004).

[3] P. Schilinsky, C. Waldauf, and C. J. Brabec, Appl. Phys. Lett. 81, 3885 (2002).

[4] A. M. Goodman and A. Rose, J. Appl. Phys. 42, 2823 (1971).


[6] C. Melzer, E. Koop, V. D. Mihailetchi, and P. W. M. Blom, Adv. Funct. Mater. 14, 865 (2004).

[7] E. A. Katz, D. Faiman, S. M. Tuladhar, J. M. Kroon, M. M. Wienk, T. Fromherz, F. Padinger,C. J. Brabec, and N. S. Sariciftci, J. Appl. Phys. 90, 5343 (2001).

[8] V. Dyakonov, Physica E 14, 53 (2002).

[9] C. J. Brabec, S. E. Shaheen, C. Winder, N. S. Sariciftci, and P. Denk, Appl. Phys. Lett. 80, 1288(2002).

[10] P. Schilinsky, C. Waldauf, J. Hauch, and C. J. Brabec, J. Appl. Phys. 95, 2816 (2004).

[11] J. A. Barker, C. M. Ramsdale, and N. C. Greenham, Phys. Rev. B 67, 075205 (2003).

[12] C. J. Brabec, N. S. Sariciftci, and J. C. Hummelen, Adv. Funct. Mater. 11, 15 (2001).


[14] S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer-Verlag, Wien, 1984).

[15] P. Langevin, Ann. Chim. Phys. 28, 433 (1903).

[16] H. Hoppe, N. Arnold, D. Meissner, and N. S. Sariciftci, Thin Solid Films 451-452, 589 (2004).

[17] M. A. Green, Solar Cells Operating Principles, Technology and System Applications (Prentice Hall,Englewood Cliffs, NJ, 1982).


[19] J. C. Scott, S. Ramos, and G. G. Malliaras, J. Imaging Sci. Technol. 43, 234 (1999).

[20] V. D. Mihailetchi, L. J. A. Koster, P. W. M. Blom, C. Melzer, B. de Boer, J. K. J. van Duren, andR. A. J. Janssen, Adv. Funct. Mater. 15, 795 (2005).

[21] Y. Roichmann and N. Tessler, Appl. Phys. Lett. 80, 1948 (2002).


42


[23] L. Onsager, Phys. Rev. 54, 554 (1938).

[24] C. L. Braun, J. Chem. Phys. 80, 4157 (1984).

[25] L. Onsager, J. Phys. Chem. 2, 599 (1934).

[26] R. M. Fuoss and F. Accascina, Electrolytic conductance (Interscience, New York, 1959), p. 213.

[27] T. E. Goliber and J. H. Perlstein, J. Chem. Phys. 80, 4162 (1984).

[28] N. Karl and G. Sommer, Phys. Status Solidi A 6, 231 (1971).

[29] A. Pivrikas, G. Juska, R. Osterbacka, M. Westerling, M. Viliunas, K. Arlauska, and H. Stubb,Phys. Rev. B 71, 125205 (2005).

[30] D. L. Scharfetter and H. K. Gummel, IEEE Trans. Electron Devices 16, 64 (1969).

[31] H. K. Gummel, IEEE Trans. Electron Devices 11, 455 (1964).

[32] G. Juska, N. Nekrasas, K. Genevieius, J. Stuchlik, and J. Koeka, Thin Solid Films 451-452, 290(2004).

[33] A. J. Mozer, N. S. Sariciftci, D. Vanderzande, R. Osterbacka, M. Westerling, and G. Juska,Appl. Phys. Lett. 86, 112104 (2005).


[35] V. D. Mihailetchi, J. Wildeman, and P. W. M. Blom, Phys. Rev. Lett. 94, 126602 (2005).

[36] R. Sokel and R. C. Hughes, J. Appl. Phys. 53, 7414 (1982).

[37] V. D. Mihailetchi, P. W. M. Blom, J. C. Hummelen, and M. T. Rispens, J. Appl. Phys. 94, 6849(2003).

43

CHAPTER

THREE

Open-circuit voltage of bulkheterojunction solar cells

Summary

In this chapter, two models for the open-circuit voltage are introduced: First, amodel formulated for p-n junctions is examined. By studying the dependency of theopen-circuit voltage on light intensity, it is concluded that this model does not correctlydescribe the open-circuit voltage of BHJ solar cells. Whereas the experimental datashow a slope S of Voc as a function of ln(I) is equal to Vt, the p-n junction model predictsa slope of nVt, where n ranges from 1.35 to 1.98. This phenomenon is observed fortwo different PPV derivatives as donor material. The main cause of this discrepancylies in the fact that the strong voltage dependence of the photocurrent is not taken intoaccount.

Within the framework of the MIM model an alternative explanation for the open-circuit voltage is presented. Based on the notion that the quasi-Fermi potentials areconstant throughout the device, a formula for Voc is derived that consistently describesthe open-circuit voltage. Not only is the light intensity dependence predicted by theexpression in accordance with the experimental data, but the numerical value of Voc isalso correct. Next, the predictions of the MIM model and its relation to other types ofsolar cells are discussed.

45

Chapter 3. Open-circuit voltage of bulk heterojunction solar cells

3.1 Open-circuit voltage in p-n junction based models

One of the key parameters of photovoltaic devices is the open-circuit voltage, which isthe voltage for which the current in the external circuit equals zero. In polymer/fullerenecells limitations of the open-circuit voltage have been attributed to Fermi level pinning [1]

and to band bending at the contact due to the injection of charges. [2] For further opti-mization of solar cell performance fundamental understanding of the mechanisms gov-erning the photovoltaic performance is indispensable.

For a conventional inorganic p-n junction solar cell the dark current is given by [3]

JD = Js(

eVanVt − 1

)

, (3.1)

where Js is the (reverse bias) saturation current density and n is the ideality factor. Thecurrent density under illumination (JL) is subsequently described by

JL = Js(

eVanVt − 1

)

− Jph, (3.2)

where Jph is the photogenerated current density. For an ideal solar cell it is assumed thatthe photogenerated current density Jph is voltage-independent, meaning that Jph = Jsc

at any applied voltage. Under this assumption, Eq. (3.2) directly provides an expressionfor Voc, given by

Voc = nVt ln(Jsc/Js + 1). (3.3)

This expression for Voc, derived for conventional inorganic solar cells, has also been usedto analyze the temperature dependence of Voc of polymer/fullerene BHJ solar cells. [4,5]

Both Jsc and Js depend on temperature, while Js is not directly measurable. This makesthe verification of Eq. (3.3) — and the p-n junction model — complicated. A more directway of testing Eq. (3.3), and consequently Eq. (3.2), is to investigate the dependenceof Voc on light intensity since then only Jsc changes. Since it has been demonstratedthat Jsc is nearly linearly dependent on light intensity, [6,7] it follows from Eq. (3.3) thatVoc should exhibit a slope of nVt, when plotted as a function of the logarithm of lightintensity. Figure 3.1 confirms that Jsc is indeed linear in light intensity for the devicesdiscussed here.

Figure 3.2 shows the current density in dark JD as a function of applied voltage Va

for MDMO-PPV/PCBM and BEH-PPV/PCBM based solar cells at different tempera-tures. By fitting the exponential part of the current-voltage characteristics to Eq. (3.1)the ideality factors are determined. The results are summarized in Table 3.1. At roomtemperature the ideality factor n typically amounts to 1.4 and then increases furtherwith decreasing temperature to 2.0 for MDMO-PPV/PCBM devices and 1.5 for BEH-PPV/PCBM devices at 210 K.

The current voltage characteristic (JL − Va) of an illuminated (800 W/m2) MDMO-PPV/PCBM device at room temperature is shown in Fig. 3.3(a), together with the cur-rent predicted by Eq. (3.2). It is clear that there is a large discrepancy between the predic-tions of the model and the experimental data: around zero bias the predicted current is

46

3.1. Open-circuit voltage in p-n junction models

10 100 10000.1

1

10

295 K = 1.00 250 K = 1.00 210 K = 1.00

J sc [A

/m2 ]

I [W/m2]

MDMO-PPV/PCBM (1:4 wt.)

10 100 10000.1

1

10

J sc [A

/m2 ]

I [W/m2]

295 K = 0.98 250 K = 0.97 210 K = 0.94

BEH-PPV/PCBM (1:1 wt.)

(b)(a)

Figure 3.1: The short-circuit current density (symbols) as a function of light intensity for varioustemperatures, the lines denote fits to Jsc ∝ Iα. Part (a) shows data on an MDMO-PPV/PCBM (1:4wt.) device, while part (b) presents data on a BEH-PPV/PCBM (1:1 wt.) solar cell.

0.4 0.6 0.8 1.0

0.01

0.1

1

10

100

295 K 250 K 210 K

MDMO-PPV/PCBM(1:4 wt.)

J D [A

/m2 ]

Va [V]0.4 0.6 0.8 1.0

0.01

0.1

1

10

100

295 K 250 K 210 K

BEH-PPV/PCBM (1:1 wt.)

J D [A

/m2 ]

Va [V]

(b)(a)

Figure 3.2: Experimental dark current of MDMO-PPV/PCBM (symbols) and fit to the exponen-tial part (lines) at various temperatures. Part (a) shows data on an MDMO-PPV/PCBM (1:4 wt.)device, while part (b) presents data on a BEH-PPV/PCBM (1:1 wt.) solar cell.

Table 3.1: Overview of ideality factors n obtained from Fig. 3.2 and slopes S obtained from Fig. 3.4,for two different photovoltaic devices.

295 K 250 K 210 KMDMO-PPV n 1.34 1.62 1.98

S [Vt ] 1.03 1.01 0.90BEH-PPV n 1.35 1.47 1.65

S [Vt ] 1.04 1.01 1.00

47


0.01 0.1 1 101

10

J ph

[A/m

2 ]

V0-Va [V]

Jph=Jsc

0.0 0.2 0.4 0.6 0.8

-30

0

30

60

90

Va [V]

J L [A

/m2 ]

(b)(a)

Figure 3.3: (a) Experimental current density under illumination of an MDMO-PPV/PCBM deviceat 295 K (symbols) and the current density predicted by Eq. (3.2) (line) with n = 1.34 (see Table3.1). (b) The photocurrent density Jph of an MDMO-PPV/PCBM device (symbols) as a functionV0 − Va. The line denotes the short circuit current density corresponding to the assumption of Jph

being constant.

basically constant, in contrast to the experimental current, while near Voc the predictedcurrent is much too high. These observations already strongly indicate that the p-n junc-tion model is not applicable to polymer/fullerene bulk heterojunction devices. Fig. 3.4shows Voc as a function of the logarithm of light intensity at various temperatures forboth devices. The experimental data are fitted with a linear function with slope S whichis given in Table 3.1 in units of Vt. Surprisingly, the experimental slopes are within exper-imental error equal to Vt instead of nVt [Eq. (3.3)] for both devices and all temperatures.Thus, next to the photocurrent (Fig. 3.3) also the light intensity dependence of Voc is notin agreement with the p-n junction model.

The main reason for this disagreement is that Eq. (3.3) is based on the assumptionof a voltage independent photocurrent density Jph. Recently, it has been shown by Mi-

hailetchi et al. [8] that the photogenerated current shows a very different behavior (seesubsection 2.4.2): In Fig. 3.3(b) the photocurrent of an MDMO-PPV/PCBM device is plot-ted as a function of effective applied voltage, V0 − Va. Near the compensation voltage,a linear dependence of the photogenerated current upon applied voltage is observed,followed by saturation at high fields. This behavior is caused by the opposite effect ofdrift and diffusion of charge carriers. Consequently, the assumption of a constant pho-tocurrent is not valid. When the photocurrent near the open-circuit voltage is equated toJsc (solid line) it is clear from Fig. 3.3(b) that the photocurrent is strongly overestimated,hence Eqs. (3.2) and (3.3) cannot be expected to reproduce the experimental data. The fitof Eq. (3.2) to experimental photocurrent data is often improved by including series andshunt resistances. [9] However, the physical meaning of these quantities is not clear.

48

3.2. Open-circuit voltage in the MIM model

10 100 10000.6

0.7

0.8

0.9

1.0BEH-PPV/PCBM (1:1 wt.)

295 K 250 K 210 K

V oc [

V]

I [W/m2]10 100 1000

0.6

0.7

0.8

0.9

1.0MDMO-PPV/PCBM (1:4 wt.)

295 K 250 K 210 K

V oc [

V]

I [W/m2]

(b)(a)

Figure 3.4: Voc as a function of light intensity. The lines denote the linear fits to the experimentaldata. Part (a) shows data on an MDMO-PPV/PCBM (1:4 wt.) device, while part (b) presents dataon a BEH-PPV/PCBM (1:1 wt.) solar cell.

3.2 Open-circuit voltage in the metal-insulator-metal

model

3.2.1 General considerations

Which factors govern the open-circuit voltage in the MIM model? Consider the situationas depicted in Fig. 3.5: Due to the fast charge transfer process after exciton formation andsubsequent energetic relaxation, the maximum potential that a BHJ solar cell can sustainis limited to the difference between the LUMO level of the donor and the HOMO levelof the acceptor, viz.,

Voc ≤ Eeffgap/q. (3.4)

Clearly, this even holds for electrodes with a difference in work function Φm1,2 larger

than Eeffgap. In practice, however, Voc is significantly smaller than this upper limit. The

highest value for Voc is found when Ohmic contacts are used, i.e., Φm1 ≤ LUMO(A) andΦm2 ≥ HOMO(D). As these contacts cause high carrier densities in the semiconductor,at least in the vicinity of the electrodes, Voc is typically 0.4 V less than Eeff

gap/q. [10] In thenext subsection the Voc in the case of Ohmic contacts will be extensively studied.

3.2.2 Formula for the open-circuit voltage

Would it be possible to derive a formula for Voc based on the MIM model, as an alterna-tive for Eq. (3.3)? In this subsection, we shall see that this is indeed possible and that theresulting expression does explain the intensity dependence of Voc .

49


Figure 3.5: Band diagram of a BHJ device sandwiched between two electrodes with work func-tions Φm1,2. After photogeneration (1) and electron transfer to the acceptor phase (2), the electronrapidly thermalizes.

As a first step, the quasi-Fermi potentials φn,p are introduced as [11]

n(p) = nint exp[

(−)ψ − φn(p)

Vt

]

. (3.5)

The quasi-Fermi potentials are a measure of the deviation from equilibrium of the sys-tem. In equilibrium np = n2

int, however,

np = n2int exp

(φp − φn

Vt

)

, (3.6)

when the system is not in equilibrium.∗ The current densities, given in Eq. (2.3), can berewritten in terms of the quasi-Fermi potentials as

Jn(p) ∝ n(p)∂

∂xφn(p). (3.7)

As we have seen in chapter 2 (see Fig. 2.12), at open-circuit the current densities are(virtually) zero, consequently, the quasi-Fermi potentials are constant (see Fig. 3.6). Sinceit is assumed that at the contacts the metal electrodes are in thermal equilibrium with thesemiconductor blend, the quasi-Fermi potentials have to be equal to the potential at thecontacts. This implies that the difference φp − φn is constant throughout the device andequal to the applied voltage at open-circuit, therefore

np = n2int exp

(

Voc/Vt

)

. (3.8)

∗Although the definition of φn,p in Eq. (3.5) is inspired by the Boltzmann equation, this definition is notlimited to circumstances that justify the use Boltzmann’s equation, however, Eq.(3.7) is.

50

3.2. Open-circuit voltage in the MIM model

0 20 40 60 80 100 120

-0.4

-0.2

0.0

0.2

0.4

[V]

x [nm]

Voc

Figure 3.6: The quasi-Fermi potentials φn,p for electrons (dashed line) and holes (solid lines) underopen-circuit conditions. These values correspond to the fit of an MDMO-PPV/PCBM device asdiscussed in subsection 2.4.2.

The continuity equation for electrons is given by Eq. (2.20), i.e.,

1

q

∂

∂xJn(x) = PGe−h − (1 − P)R. (3.9)

To a very good approximation, the recombination rate R, given by Eq. (2.12), can be writ-ten as R = krnp. Since the current densities are zero, so are their derivatives and hencerecombination and generation cancel everywhere in the device. Hence from Eq. (2.20) itfollows that

Ge−h = krnp1 − P

P. (3.10)

Therefore, using Eq. (3.8) and solving for Voc, one has∗

Voc =Eeff

gap

q− Vt ln

[

(1 − P)kr N2cv

PGe−h

]

. (3.11)

A similar formula was derived for amorphous silicon p-i-n junction solar cells. [12]

What does this equation tell us? Since the dissociation probability P depends on volt-age, this is not a strictly explicit relation. However, P only shows a relatively small varia-tion in the voltage range of interest here. Therefore, this equation gives insight into howparameters such as the generation rate (and hence light intensity) affect the open-circuitvoltage. Moreover, this formula predicts the right slope S of Voc versus light intensity,

∗One might wonder what happens when n and p have their maximum value Ncv. In this case, Eq. (3.10)reduces to PGe−h = (1 − P)krN2

cv and hence the argument of the logarithm in Eq. (3.11) is equal to unity,

thereby ensuring that Voc ≤ Eeffgap/q, which also follows from Eq. (3.8).

51


viz., Vt. Furthermore, Eq. (3.11) is consistent with the notion of a field-dependent pho-tocurrent, in contrast to Eq. (3.3), since both drift and diffusion of charge carriers havebeen taken into account through the use of Eq. (3.7). Equation (3.11) shows that for asmall generation rate (corresponding to low light intensity), qVoc can be much smallerthan the effective band gap Eeff

gap. When the incident light intensity is increased, Voc in-

creases logarithmically, but it cannot exceed Eeffgap/q, as required by the conservation of

energy.At room temperature, using the values for P, G, and Eeff

gap given in subsection 2.4.2,we have Voc = 0.85 V, in excellent agreement with the experimental data on MDMO-PPV/PCBM as presented in subsection 2.4.2. This implies that Voc is 0.44 V lower thanEeff

gap/q, which is due to a voltage loss at the contacts because of band bending. [10] Ofcourse, the magnitude of this loss depends on light intensity and material parameters,as is clear from Eq. (3.11).

Returning to the temperature dependence of Voc , it should be mentioned that the factthat there is no sharply defined band gap strongly complicates the use of Eq. (3.11) to ex-plain temperature dependent measurement of Voc . Due to the presence of energetic dis-order in both materials, their HOMO and LUMO levels exhibit a Gaussian broadeningof typically 0.1 eV. [13,14] Since the exact distribution of energy levels in the PPV/PCBMblend is not known, the uncertainty in Eeff

gap is of the same order of magnitude as thevariation of Voc with temperature, thereby prohibiting an exact quantitative analysis.

Influence of other types of recombination

The inclusion of another type of non-geminate recombination changes Eq. (3.11) andthe light intensity dependence of Voc. One example would be the recombination of freeholes with trapped electrons in polymer/polymer solar cells, [15] which is described byShockley-Read-Hall (SRH) recombination.∗ Figure 3.7 shows the calculated intensitydependence of Voc when SRH recombination is included, using mobilities and trap den-sities typical of polymer/polymer solar cells. It follows that the intensity dependenceof Voc is much stronger when SRH recombination is included: the slope S increases to1.66 Vt. The change of slope S when SRH recombination plays a role was also notedfor amorphous silicon p-i-n junction solar cells. [12] This result suggests that the intensitydependence of Voc may be used as an experimental tool for studying the transport andpossible trapping in BHJ solar cells. In this way, Mandoc el al. were able to discriminatebetween trap-free and trap-limited electron transport in polymer/polymer solar cells. [15]

In their experiments, S = 1.55 Vt was found, in accordance with SRH recombination. Itis important to note that the ideality factor of these devices is even higher (> 2), so thep-n junction based model (see section 3.1) cannot explain the experimentally observedintensity dependence.

∗One might think of introducing another recombination term in Eq. (3.10) and proceeding with the deriva-tion of an expression for Voc. However, due to the strongly varying density of trapped electrons, the rateconstant for SRH recombination will strongly vary throughout the device, and, consequently, the assumptionof generation and recombination canceling everywhere breaks down.

52

3.3. Comparison with other solar cells

10 100 1000

1.2

1.3

1.4

V oc [

V]

I [W/m2]

without SRH with SRH

Figure 3.7: Calculated intensity dependence of Voc with and without the inclusion of SRH recom-bination. When SRH recombination is included, the intensity dependence is much stronger (S =1.66 Vt).

3.3 Comparison with other solar cells

3.3.1 Influence of non-homogeneity

Up to now, we have only considered BHJs which are homogeneous in their composition.The MIM model predicts that the open-circuit voltage of such BHJs cannot be largerthan the difference between the work functions of the electrodes. However, this is notgenerally true for all types of solar cells. By inducing a concentration gradient of donorand acceptor materials on the molecular scale, a so-called graded BHJ can be realized. [16]

Mihailetchi et al. have shown that the mobility of electrons and holes in MDMO-PPV/PCBM BHJs depends on the volume ratio of both materials, finding that the mobil-ity through a phase is enhanced when the relative volume of that phase is increased. [17]

Additionally, they showed that the highest generation rate of electron-hole pairs Gmax

occurs in a 1:1 (by volume) mixture of MDMO-PPV and PCBM. These results imply thatin a graded BHJ both the charge carrier mobilities and the generation rate are highlynon-uniform, making it possible to tailor the properties of the active layer in such a waythat the zone with the highest charge generation efficiency (i.e., the region with highconcentrations of both components) coincides with the maximum of the optical field,while providing efficient carrier transport to the electrodes. The behavior of Voc will bedifferent for such a non-uniform system.∗

∗In the MDMO-PPV/PCBM system, the dissociation efficiency of bound electron-hole pairs is also stronglydependent on the volume ratio of both components, as discussed in Ref. [17]. As the MDMO-PPV/PCBMsystem only serves as an illustration, this effect is ignored in the present analysis.

53


Figure 3.8: A homogeneous BHJ solar cell with electrodes made of the same metal. When no biasvoltage is applied, there exists no preferential direction for the charge carriers to go to.

Consider a homogeneous BHJ solar cell with contacts made of the same metal, andassume that the work function is equal to the mean value of the HOMO and LUMO en-ergies, see Fig. 3.8.∗ In the case of constant (but not necessarily equal) electron and holemobilities, the short-circuit current would be zero since there is no preferential directionfor the charge carriers, consequently, Voc = 0. When the profile of the carrier generationrate is strongly asymmetrical, see Fig.. 3.9(a), the open-circuit voltage is still very small.

In the case of electron and hole mobilities which are not constant, the predictions bythe MIM model are different: Suppose that near the left electrode (x = 0), the electronmobility is much higher than near the right electrode (x = L), and that the opposite ap-plies to the hole mobility. Now there does exist a preferential direction for the chargecarriers, making the carriers flow in opposite directions and hence generating a net (fi-nite) current. It follows that Voc 6= 0. Therefore, it is not generally true that the MIMmodel predicts that the open-circuit voltage is always lower than (or equal to) the differ-ence in work functions of the electrodes. To illustrate this, consider

µn(0)

µn(L)=

µp(L)

µp(0)= 1000, (3.12)

with µn exponentially decreasing and µp exponentially increasing with x. The calcu-lated current-voltage characteristics for such a device are shown in Fig. 3.9(b). Indeed,the MIM model predicts a finite open-circuit voltage (0.25 V), even in the absence of adifference in work function of the electrodes. Moreover, the results depicted in Fig. 3.9demonstrate that a nonzero Voc is not so much induced by a non-uniform generationrate, but rather by non-uniform transport properties of the active layer. In principle,this effect may be used advantageously to improve the open-circuit voltage of poly-mer/fullerene BHJ solar cells with Ohmic contacts.

The most extreme example of a non-homogeneous device structure is a bilayer solarcell. [18] Ramsdale et al. demonstrated that the Voc of such a device can indeed be muchlarger (> 1 V) than the work function difference between the electrodes. [19] This effect

∗The exact work function of is of no consequence, the most important requirement, however, is that thework functions of both metals be equal.

54

3.3. Comparison with other solar cells

0.0 0.1 0.2 0.3 0.4

-20

0

20

40

60

J L [A

/m2 ]

Va [V]0.0 0.1 0.2 0.3 0.4

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

G [x

1027

m-3

s-1]

x/L

Va [V]

(a) (b)

Figure 3.9: (a) Simulated current-voltage characteristics of an illuminated device with equal con-tacts and constant mobilities. The dashed line corresponds to a constant generation rate G offree charge carriers, while the solid line is calculated by taking the generation profile depicted inpart (b) into account. (b) Current-voltage characteristics calculated with mobilities according toEq. (3.12). The dashed line corresponds to a constant G, while the solid line has been obtainedusing a generation rate profile as shown in the inset (solid line). The inset also shows the constantgeneration rate (dashed line) corresponding to average generation rate of the profile.

was attributed to accumulation of charge at the heterojunction giving rise to a diffusioncurrent that must be counterbalanced by a drift current at open-circuit.

3.3.2 Comparison with (in)organic low mobility solar cells

Having discussed the device characteristics of polymer/fullerene BHJ cells it is inter-esting to compare their operating mechanism with other types of solar cells that alsoemploy low mobility semiconductors as amorphous silicon based p-i-n devices. [12,20,21]

These devices consist of a thin layer of intrinsic material sandwiched between heav-ily doped p and n layers, which function as electrodes. First, the photogeneration ofcharges in these p-i-n devices is fundamentally different: light absorption directly cre-ates free charge carriers since geminate recombination of photogenerated charge car-riers is of no importance. [22] In a BHJ device an exciton is created upon light absorp-tion, which subsequently dissociates across the donor/acceptor interface, creating abound electron-hole pair. This bound pair can either dissociate into free charges con-tributing to the photocurrent or decay to the ground state, resulting in a strongly field-and temperature-dependent geminate recombination process. [8] This difference is at theheart of our model. Furthermore, bimolecular recombination, and not trapping and sub-sequent recombination, is the prime loss mechanism of free carriers (i.e., carriers thatalready escaped the bound electron-hole pair).

55


The operational principle of dye-sensitized solar cells (DSSCs) has received muchattention as well. Such a DSSC consists of a nanoporous titanium dioxide electrode, cov-ered with a monolayer of dye molecules, immersed in a liquid electrolyte containing,e.g., the I−/I−3 redox couple. [23] Light is absorbed by the dye layer and consequently, theelectrons are transported through the titanium dioxide phase, while the dye molecule isreduced by the I−/I−3 redox couple. There has been much debate about whether the driv-

ing mechanism of charge transport of these devices is the same as for p-n junctions. [24–26]

Whereas p-n junctions need a built-in field to transport the charges, DSSCs do not seemto need an internal field. Rather, the hole transport through the electrolyte is driven bydiffusion, since the electrolyte cannot sustain an electric field. The transport of electronsis less well understood and two models have been proposed: [27] The so-called junctionmodel assumes that electrons are field driven, thereby limiting the open-circuit voltageto the difference in work function of the substrate electrode and the solution redox po-tential. The kinetic model on the other hand, assumes that the electrons are also drivenby diffusion, as are the holes, and that the electric field in the titanium dioxide networkis zero due to screening by the electrolyte. Therefore, the charge carriers diffuse awayfrom the interface where they were created, and hence an internal electric field is notneeded for photovoltaic action. As a result even a device with equal redox potential andsubstrate electrode work function can exhibit a nonzero open-circuit voltage. This hasindeed been observed for DSSCs, [27] as well as for bilayer devices consisting of conju-gated polymers [19] with top and bottom electrodes with the same work function.

It has been asserted that the same should hold for BHJ devices, but that the electricfield must also be taken into account since there is no mobile electrolyte to screen thefield. [28,29] As a consequence, the open-circuit should not be limited to the differencein electrode work function. However, by varying the work function of the PEDOT:PSSanode, Frohne et al. have shown that when the work function of the anode coincides withthe LUMO of PCBM, thereby yielding a symmetric device, the open-circuit voltage iszero. [30] Furthermore, Mihailetchi et al. also demonstrated, by varying the top electrode(cathode), that the open-circuit voltage is determined by the difference in electrode workfunction. [10] These results are in full agreement with the metal-insulator-metal pictureused here. Since we treat the BHJ as one effective semiconductor, there is no interfacefor the charge carriers to diffuse away from and the transport is modeled as taking placein only one material, thereby resembling the p-i-n junction model. The success of themetal-insulator-metal picture suggests that such diffusion is not important in BHJ solarcells. Not only is this model successful in describing the effect of various electrodes onthe open-circuit voltage, [10,30] it also describes, e.g., its light intensity dependence. [31]

3.4 Conclusions

In this chapter, a model for the open-circuit voltage was introduced. By studying thedependency of Voc on incident light intensity, it has been demonstrated that the Voc ofBHJ solar cells is inconsistent with p-n junction models: Whereas the experimental data

56

3.4. Conclusions

showed that the slope S of Voc as a function of ln(I) is equal to Vt, the p-n junction modelpredicts a slope of nVt, where n ranges from 1.35 to 1.98. This phenomenon was observedfor two different PPV derivatives as donor material. The main cause of this discrepancylies in the fact that the strong voltage dependence of the photogenerated current is nottaken into account.

An alternative model for the open-circuit voltage has been presented, based on thenotion that the quasi-Fermi potentials are constant throughout the device. Subsequently,a formula for Voc was derived, consistently explaining the light intensity dependence ofthe open-circuit voltage of polymer/fullerene bulk heterojunction devices with Ohmiccontacts. Next, the predictions of the MIM model and its relation to other types of solarcells have been discussed.

57


References

[1] C. J. Brabec, A. Cravino, D. Meissner, N. S. Sariciftci, T. Fromherz, M. T. Rispens, L. Sanchez,and J. C. Hummelen, Adv. Funct. Mater. 11, 374 (2001).


[3] S. M. Sze, Physics of semiconductor devices (Wiley, New York, 1981).

[4] E. A. Katz, D. Faiman, S. M. Tuladhar, J. M. Kroon, M. M. Wienk, T. Fromherz, F. Padinger,C. J. Brabec, and N. S. Sariciftci, J. Appl. Phys. 90, 5343 (2001).

[5] D. Chirvase, Z. Chiguvare, M. Knipper, J. Parisi, V. Dyakonov, and J. C. Hummelen,J. Appl. Phys. 93, 3376 (2003).

[6] P. Schilinsky, C. Waldauf, and C. J. Brabec, Appl. Phys. Lett. 81, 3885 (2002).

[7] I. Riedel, J. Parisi, V. Dyakonov, L. Lutsen, D. Vanderzande, and J. C. Hummelen,Adv. Funct. Mater. 14(1), 38 (2004).

[8] V. D. Mihailetchi, L. J. A. Koster, J. C. Hummelen, and P. W. M. Blom, Phys. Rev. Lett. 93,216601 (2004).

[9] P. Schilinsky, C. Waldauf, J. Hauch, and C. J. Brabec, J. Appl. Phys. 95, 2816 (2004).


[11] S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer-Verlag, Wien, 1984).

[12] E. A. Schiff, Sol. Energy Mater. Sol. Cells 78, 567 (2003).



[15] M. M. Mandoc, W. Veurman, L. J. A. Koster, M. M. Koeste, J. Sweelssen, B. de Boer, andP. W. M. Blom (unpublished).

[16] B. Pradhan and A. J. Pal, Synth. Met. 155, 555 (2005).


[18] C. W. Tang, Appl. Phys. Lett. 48, 183 (1986).

[19] C. M. Ramsdale, J. A. Barker, A. C. Arias, J. D. MacKenzie, R. H. Friend, and N. C. Greenham,J. Appl. Phys. 92, 4266 (2002).

58


[20] P. Chatterjee, J. Appl. Phys. 76, 1301 (1994).

[21] F. A. Rubinelli, R. Jimenez, J. K. Rath, and R. E. I. Schropp, J. Appl. Phys. 91, 2409 (2002).

[22] F. Carasco and W. E. Spear, Philos. Mag. B 47, 495 (1983).

[23] M. Gratzel, Prog. Photovolt. Res. Appl. 8, 171 (2000).

[24] K. Schwarzburg and F. Willig, J. Phys. Chem. B 107, 3552 (2003).

[25] G. Kron, T. Egerter, J. H. Werner, and U. Rau, J. Phys. Chem. B 107, 3556 (2003).

[26] B. A. Gregg, J. Phys. Chem. B 107, 13540 (2003); J. Bisquert, ibid. 107, 13541 (2003); J. Au-gustynski, ibid. 107, 13544 (2003); K. Schwarzburg and F. Willig, ibid. 107, 13546 (2003); U. Rau,G. Kron, and J. H. Werner, ibid. 107, 13547 (2003).

[27] F. Pichot and B. A. Gregg, J. Phys. Chem. B 104, 6 (2000).

[28] B. A. Gregg and M. C. Hanna, J. Appl. Phys. 93, 3605 (2003).

[29] B. A. Gregg, J. Phys. Chem. B 107, 4688 (2003).

[30] H. Frohne, S. E. Shaheen, C. J. Brabec, D. C. Muller, N. S. Sariciftci, and K. Meerholz,ChemPhysChem 9, 795 (2002).

[31] L. J. A. Koster, V. D. Mihailetchi, R. Ramaker, and P. W. M. Blom, Appl. Phys. Lett. 86, 123509(2005).

59

CHAPTER

FOUR

Short-circuit current of bulkheterojunction solar cells

Summary

A typical feature of polymer/fullerene based solar cells is that the short-circuitcurrent density does not scale exactly linearly with light intensity. Instead, a powerlaw relationship is found given by Jsc ∝ Iα, where α ranges from 0.85 to 1. In anumber of reports this deviation from unity is speculated to arise from the occurrenceof bimolecular recombination. In this chapter, simple analytical models are discussed,showing that the short-circuit current should show a power law behavior on intensity,Jsc ∝ Iα, with 0.75 < α < 1, depending on the mobility of both carriers. By applyingthe numerical model as outlined in chapter 2, it is demonstrated that the experimentallyobserved intensity dependence is indeed caused by space charge effects and does notoriginate from bimolecular recombination losses. This explanation is verified for anexperimental model system with a mobility difference that can be tuned from one tothree orders of magnitude by changing the post-production annealing treatment.

61

Chapter 4. Short-circuit current of bulk heterojunction solar cells

4.1 Introduction

One of the key parameters of any solar cell is the short-circuit current density Jsc, and itsoptimization is of great importance for the further improvement of organic photovoltaicdevices. What determines the short-circuit current? An important issue, in this respect,is the dependence of Jsc on incident light intensity I. Several authors have reported apower law dependence of Jsc, i.e. Jsc ∝ Iα, where α ranges typically from 0.85 to 1 forpolymer/fullerene based solar cells. [1–7] Thus far, the deviation from α = 1 has beenconjectured to arise from a small loss of carriers via bimolecular recombination. [2,4,7] Asa first step, this section deals with three simple analytical expressions for the photocur-rent generated by a BHJ solar cell. The concept of space-charge-limited photocurrent,introduced in this section, will prove to be of use in explaining the intensity dependenceof Jsc.

4.1.1 Uniform field approximation

Consider the simple case of a photoconductor with non-injecting contacts and a uniformelectric field distribution. Goodman and Rose derived that, under the assumption ofnegligible recombination of charge carriers, [8]

Jph = qGL, (4.1)

meaning that in this case all photogenerated charge carriers are simply extracted andthe current density depends only on the generation rate G. In their derivation, Goodmanand Rose took only drift of charge carriers into account and neglected the contribution ofdiffusion. Sokel and Hughes carried this analysis one step further by including diffusionof carriers, finding [9]

Jph = qGL[ exp(V/Vt) + 1

exp(V/Vt) − 1− 2

Vt

V

]

, (4.2)

where V is the voltage drop across the active layer. For a BHJ device this voltage dropis given by V0 − Va. The result by Sokel and Hughes shows two regimes: A lineardependence of Jph on voltage for small biases, while reducing to Eq. (4.1) at moderatelyhigh bias (including short-circuit conditions), see Fig. 4.1.

On the basis of Eqs. (4.1) and (4.2) one expects that Jsc = qGL and, hence, that Jsc

is proportional to the incident light intensity. From experiments it is evident that thisis not always true; for some systems Jsc is clearly sublinear in light intensity. The factthat Eqs. (4.1) and (4.2) do not depend on charge carrier mobility is a consequence ofthe assumption of no recombination, thereby ensuring that all carriers exit the deviceand that the electric field is uniform. In this chapter, we shall see that it is exactly thisassumption of a uniform electric field that fails when Jsc shows sublinear behavior.

62

4.1. Introduction

0.01 0.1 11

10

J ph

[A/m

2 ]

V0-Va [V]

Jsc

I II

Figure 4.1: The photocurrent data (symbols) of an MDMO-PPV/PCBM device, as discussed insubsection 2.4.2, together with the Sokel and Hughes result Eq. (4.2) (solid line) and the predictionby Goodman and Rose [dashed line, Eq. (4.1)]. At low effective applied voltages Jph ∝ V0 − Va

(regime I), while at V0 − Va ' 0.3 V, the photocurrent saturates to qGL (regime II).

4.1.2 Space-charge-limited photocurrents

Mihailetchi et al. have demonstrated that a large difference in electron- and hole mobility,accompanied by a low mobility of the slowest carrier, may lead to space-charge-limitedphotocurrents. [10] The extraction of photogenerated carriers is governed by the meancarrier drift length w, which is the mean distance a carrier travels before recombinationoccurs. When both the electron (wn) and hole (wp) drift lengths are larger than the activelayer thickness, then the charges will readily flow out without distorting the field inthe device, see Fig. 4.2(a). However, in case wn ≫ wp and wp < L there will be anet positive space charge near the anode, see Fig. 4.2(b). There exist three regimes inthe device: Near the cathode the electron density is much larger than the hole density;this is a small region (I). Next to this region, there exists a balance between electronand hole density, yielding a neutral region (II). Near the anode, the holes dominate thedevice (III), resulting in a large net space charge and concomitant large voltage drop, asindicated in Fig. 4.2(b). The large field strength in region III facilitates the extraction ofholes, ensuring that the extraction current of holes and electrons is equal.

When the photocurrent is space-charge-limited, the following relation holds: [8,10]

Jph ∝ G0.75√

V0 − Va. (4.3)

Thus, fully space-charge-limited photocurrents are characterized by a square-root de-pendence on voltage and are proportional to I0.75, irrespective of the amount of bimolec-ular recombination. On the other hand, non space-charge-limited devices have a lineardependence of Jph on I. Therefore, it is expected that the three-quarter power intensitydependence gradually increases to a linear dependence if the difference between the mo-

63


Figure 4.2: (a) Band diagram of a BHJ solar cell with balanced electron and hole mobilities; bothtypes of charge carriers can readily flow out of the device and the field in the device is uniform.(b) Band diagram in the case of hole accumulation and concomitant space-charge-limited behavior.Near to the cathode, a small region dominated by electrons (I) exists next to a large neutral region(II), where electron and hole densities are comparable. Most of the potential drops across the holeaccumulation layer (III) in order to facilitate the extraction of holes.

bility of electrons and holes is reduced. Since the occurrence of space charge is sensitiveto the mobility difference, it is easy to understand that various material combinationswill give different exponents α describing the intensity dependence of Jsc.

The experimental observation of space-charge-limited photocurrents in BHJ solarcells makes a very strong point in favor of the MIM model and cannot be explainedby a p-n junction model. The fact that Eq. (4.3) describes the experimental data so well,even though Goodman and Rose originally derived it for one material instead of a blend,shows that the assumption of an effective medium holds.

4.2 Intensity dependence of the short-circuit current

As mentioned in the introduction, several authors have reported a power law depen-dence of Jsc upon light intensity I, i.e. Jsc ∝ Iα, where α ranges typically from 0.85 to1. [1–7] Thus far, the deviation from α = 1 has been conjectured to arise from a small lossof carriers via bimolecular recombination. [2,4,7] It has been argued that pure bimolecularrecombination would lead to a square-root dependence of Jsc on intensity, i.e., α = 0.5.The argument is based on the assumption that generation and recombination of carrierscancel and that n = p. Under these assumptions, it follows from Eq. (3.10) that

n = p ∝√

Ge−h. (4.4)

When only drift of charge carriers is taken into account, one has

JL ∝√

Ge−hF, (4.5)

which implies that α = 0.5. However, as we have seen in section 3.2, Eq. (3.10) only holdsat open-circuit, i.e., when JL = 0, clearly this does not apply to short-circuit conditions.

64

4.2. Intensity dependence of the short-circuit current

70 80 90 100 110 120

0.92

0.96

1.00

Annealing temperature [oC]

Figure 4.3: The experimentally determined exponent α as a function of annealing temperature(open symbols) and the prediction by the numerical model (filled symbols), the solid lines aredrawn as a guide to the eye, while the dashed line corresponds to α = 1.

Therefore, the influence of bimolecular recombination on the intensity dependence ofJsc has not yet been established. In this section, it is shown that bimolecular recombi-nation does not account for the observed values of α, but that the true cause lies in thebuild-up of net space charge due to imbalanced transport of charge carriers.

How can one put this hypothesis to the test? P3TH/PCBM devices are highly suitedfor this purpose, since the ratio of electron to hole mobility can be changed over severalorders of magnitude by a simple post-production annealing treatment, see Fig. 1.3. Thisenables one to verify whether the occurrence of space-charge-limited photocurrents isaccompanied by α < 1. Figure 4.3 shows the experimentally obtained exponents α,ranging from 0.94 (device annealed at 70°C) to 1.00 (device annealed at 120°C). Clearly,α is close or equal to unity for a small difference between the electron and hole mobil-ity (corresponding to high annealing temperatures), while α deviates significantly fromunity for large differences in mobility (low annealing temperatures).

In order to check whether the photocurrent is space-charge-limited, let us take acloser look at the voltage dependence of Jph. Figure 4.4(a) shows Jph versus effective

applied voltage V0 − Va of a device annealed at 70°C under 1.15 kW/m2 illumination.Such a plot typically shows three regimes: a linear dependence of Jph on V0 −Va for smallfields, a square-root part (the space-charge-limited regime due to the large mobility dif-ference), and a gradual transition to saturation of Jph at high fields corresponding to highreverse-bias (where Jph = qGL). Figure 4.4(b) shows the intensity dependences of thephotocurrent at three different V0 −Va corresponding to the various regimes; clearly, theintensity dependence changes when going from low to high V0 − Va. The extent of thespace-charge-limited regime depends on the thickness of the active layer and on light in-tensity: when the light intensity is increased, the space-charge-limited regime grows andextends to higher V0 − Va. In the present case the photocurrent is space-charge-limited

65


100 1000

10

100

J p

h [A

/m2 ]

I [W/m2]

0.1 V short-circuit 1.5 V

0.01 0.1 11

10

100

01.ph IJ

940.ph IJ

80.ph IJ

J ph

[A/m

2 ]

V0-Va [V]

(b)(a)

Figure 4.4: (a) Experimental photocurrent density Jph as a function of effective applied voltageV0 − Va for a device annealed at 70°C under illuminated at various intensities (symbols). Thedashed line corresponds to a square-root dependence of Jph on V0 − Va, while the arrows indicatethe intensity dependence at 0.1 V, short-circuit, and 1.5 V, respectively. The corresponding currentdensities as a function of light intensity are shown in (b) (symbols), together with linear fits to thedata (lines).

for the effective applied voltage range 0.1–0.35 V (the square-root regime with Jph ∝ I0.8),as shown in Fig. 4.4.

Furthermore, Fig. 4.4 shows that Jph indeed saturates at high enough (reverse bias)voltages as indicated by the linear dependence on intensity at V0 − Va = 1.5 V. It shouldbe noted that as V0 is typically 0.03 V larger than the open-circuit voltage, short-circuitconditions correspond to V0 − Va = 0.64 V. Therefore, the short-circuit current, at V0 −Va = 0.64 V, corresponds to a regime where the transition from the space-charge-limitedregime to the saturation regime occurs and, as a consequence, 0.8 < α < 1.

4.2.1 Numerical results

The numerical model introduced in chapter 2 enables one to investigate theoretically theintensity dependence of Jsc. Figure 4.5 shows fits to the current-voltage characteristicsof devices annealed at different temperatures illuminated at 1.15 kW/m2 (no filter). Thevalues of the parameters used in obtaining these fits are listed in Table 4.1. The modelingof P3HT/PCBM devices is discussed in greater detail in Ref. [11]. By decreasing thegeneration rate G proportionally to the intensity, the value of α predicted by the modelcan be determined. The filled symbols in Fig. 4.3 denote the simulation results; clearlythese results are in good agreement with the experimental data, showing that the MIMmodel describes the intensity dependence of Jsc correctly.

What about the influence of bimolecular recombination on α? The numerical devicemodel enables one to address this influence by increasing the recombination strength in

66

4.2. Intensity dependence of the short-circuit current

1 10 100

0.75

0.80

0.85

0.90

0.95

1.00

kr/kr slowest

0.0 0.2 0.4 0.6

-60

-30

0

J L [A

/m2 ]

Va [V]

70oC 100oC

90oC 120oC

(b)(a)

Figure 4.5: (a) Current density under illumination JL as a function of applied bias Va of devicesannealed at various temperatures (symbols). The lines denotes the fits made with the numericalmodel. (b) The exponent α as a function of recombination strength kr normalized to the value usedin the fit to the experimental data [Eq. (2.24)], showing that α is only weakly dependent on kr.

Table 4.1: Overview of parameters used in the fits to the data of Fig. 4.5(a).

parameter unit 70°C 90°C 100°C 120°C

µn m2/Vs 1.1 × 10−7 1.5 × 10−7 2.0 × 10−7 2.0 × 10−7

γn (m/V)0.5 2.0 × 10−4 1.0 × 10−4 5.0 × 10−5 5.0 × 10−5

µp m2/Vs 1.2 × 10−10 1.5 × 10−9 3.0 × 10−9 1.0 × 10−8

γn (m/V)0.5 2.5 × 10−4 5.0 × 10−5 5.0 × 10−5 5.0 × 10−5

Ge−h m−3 s−1 5.9 × 1027 5.6 × 1027 5.7 × 1027 5.7 × 1027

a nm 1.8 1.8 1.8 1.8k f s−1 1 × 104 2 × 104 2 × 104 2 × 104

67


0.0 0.2 0.4 0.6 0.8 1.0

1020

1022

1024

n, p

[m-3

]

x/L

70oC

120oC

0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.2

0.0

0.2

0.4

[V]

x/L

70oC

120oC

(b)(a)

Figure 4.6: Numerical results for devices annealed at two different temperatures at a bias of Va =0.3 V. Part (a) shows the electron (gray lines) and hole (black lines) densities, while part (b) showsthe potential.

the numerical calculations for the device annealed at 70°C. Figure 4.5 shows the result-ing α when the recombination strength kr is increased up to two orders of magnitude. Itappears that α is only weakly dependent on kr ; even increasing the bimolecular recom-bination strength by a factor of 100 does not change α. This observation confirms thatbimolecular recombination does not account for the observed sublinear dependence ofJsc on intensity. Note, however, that this does not imply that bimolecular recombinationis not an important loss mechanism with respect to the performance. [12]

To illustrate the effects of space charge build-up, Fig. 4.6 shows the simulated carrierdensities and potential of devices annealed at 70°C and 120°C at a bias Va = 0.3 V. Atthis bias the former device is space-charge-limited (see Fig. 4.4(a) with V0 − Va = 0.34 V),while the latter is not. As discussed in subsection 4.1.2, there exist three regimes inthe space-charge-limited device, see Fig. 4.6(a): Near the cathode the electron density ismuch larger than the hole density. Next to this region, there exists a balance betweenelectron and hole density, yielding a neutral region. Near the anode, the holes dominatethe device, resulting in a large net space charge and concomitant large voltage drop,as indicated in Fig. 4.6(b). The field distribution in this device bears the characteristicsof space charge effects as shown in Fig. 4.2(b).∗ In the case of more balanced chargetransport, as for the device annealed at 120°C, the extraction of holes is not the limitingfactor, at least at normal (1 Sun) intensities. Therefore, the field in the device is muchmore homogeneous and the absolute difference between electron and hole density is notas pronounced.

∗Note that Fig. 4.2(b) shows the electron energy, while Fig. 4.6(b) shows the potential.

68

4.3. Conclusions

4.3 Conclusions

In this chapter, two simple analytical models have been introduced. It follows that theshort-circuit current should show a power law behavior on intensity, Jsc ∝ Iα, with0.75 < α < 1, depending on the mobility of both charge carriers: When both typesof charge carriers have a sufficiently high mobility, α will be close to unity. On the otherhand, when (only) one of the charge carriers has a very low mobility, the solar cell willsuffer from a build-up of net space charge, resulting in α < 1.

The application of the numerical model, as outlined in chapter 2, confirms that theexperimentally observed intensity dependence is indeed caused by space charge effects.Moreover, increasing the bimolecular recombination strength does not change α, hence,recombination losses per se do not account for the intensity dependence.

4.4 Experimental

The solar cells addressed in this chapter are bulk heterojunctions consisting of a 97 nmthick blend of P3HT as electron donor and PCBM as electron acceptor in a 1:1 weightratio. This blend is sandwiched between a hole-conducting layer of PEDOT:PSS, and anevaporated lithium fluoride (LiF) (1 nm)/aluminum (100 nm) top electrode. Dependingon post-production thermal annealing treatment, the ratio of electron- and hole mobil-ity ranges from three orders of magnitude to a factor of 20, [11] see Fig. 1.3, making thisan ideal model system for our purpose of tuning the mobility difference. After fabri-cation the current-voltage characteristics of these devices were measured in a nitrogenatmosphere both in dark and under illumination. A white light halogen lamp set at ap-

proximately 1.15 kW/m2 was used to illuminate the devices. To obtain light intensitydependent measurements, a set of neutral density filters was used, yielding an intensityvariation of slightly more than one order of magnitude. It is important to note that theshape of the transmission spectrum is very nearly constant for the filters used, since thisensures that the intensity is proportional to the number of absorbed photons. Therefore,the generation rate of electrons and holes is expected to be proportional to the intensity.

69


References

[1] J. Gao, F. Hilde, and H. Wang, Synth. Met. 84, 979 (1997).

[2] P. Schillinsky, C. Waldauf, and C. J. Brabec, Appl. Phys. Lett. 81, 3885 (2002).

[3] J. K. J. van Duren, X. N. Yang, C. W. T. Bulle-Lieuwma, A. B. Sieval, J. C. Hummelen, andR. A. J. Janssen, Adv. Funct. Mater. 14, 425 (2004).

[4] I. Riedel, N. Martin, F. Giacalone, J. L. Segura, D. Chirvase, J. Parisi, and V. Dyakonov, ThinSolid Films 451-452, 43 (2004).

[5] V. Dyakonov, Thin Solid Films 451-452, 493 (2004).

[6] I. Riedel, J. Parisi, V. Dyakonov, L. Lutsen, D. Vanderzande, and C. J. Brabec,Adv. Funct. Mater. 14, 38 (2004).

[7] D. Gebeyehu, M. Pfeiffer, B. Maenning, J. Drechsel, A. Werner, and K. Leo, Thin Solid Films451-452, 29 (2004).

[8] A. M. Goodman and A. Rose, J. Appl. Phys. 42, 2823 (1971).

[9] R. Sokel and R. C. Hughes, J. Appl. Phys. 53, 7414 (1982).



[12] L. J. A. Koster, E. C. P. Smits, V. D. Mihailetchi, and P. W. M. Blom, Phys. Rev. B 72, 085205(2005).

70

CHAPTER

FIVE

Hybrid organic/inorganic solar cells

Summary

In this chapter, solar cells with active layers consisting of MDMO-PPV and inor-ganic semiconductor materials are discussed. One method to obtain such blends is tospin cast a co-solution of the polymer and an organic precursor for either TiO2 or ZnO.The performance of MDMO-PPV/prec-TiO2 devices was quite poor, probably due to alack of crystallinity of the TiO2. Although the efficiency of the MDMO-PPV/prec-ZnOdevices was quite reasonable, the hole transport through the polymer phase clearlysuffered from the addition or formation of the ZnO.

Another approach is to make nanocrystalline ZnO ex situ and add this to the MDMO-PPV solution. It is demonstrated that the hole transport through the thus-formedblends is not affected by the presence of nc-ZnO. The electron mobility in these MDMO-PPV/nc-ZnO blends also is quite decent (one order of magnitude larger than the holemobility), consequently, the hole transport through the polymer phase is identified asthe limiting factor in these devices.

71

Chapter 5. Hybrid organic/inorganic solar cells

5.1 Introduction to hybrid organic/inorganic solar cells

So far, several electron acceptors have been shown to yield efficient devices: conjugatedpolymers, [1] fullerenes, [2] and inorganic nanocrystals. [3] In the class of inorganic accep-tors, metal oxides are among the most studied materials. Titanium dioxide has beenstudied in several forms: nanoparticles, [4,5] porous networks, [6] and in situ formation oftitanium dioxide from a precursor. [7] Recently, zinc oxide nanoparticles (nc-ZnO) havealso been used as electron accepting material, in combination with MDMO-PPV, with anAM1.5 efficiency of 1.6%. [8,9] Zinc oxide has several merits: Zinc oxide is a cheap andenvironmentally friendly material that can be made in crystalline form at low temper-ature. Furthermore, it displays good transport properties, even in films consisting ofnanoparticles. [10]

5.2 Hybrid solar cells with acceptors from a precursor

One of the challenges in making BHJ solar cells is obtaining a suitable morphology; thisis especially so for hybrid devices since a common solvent for both components is noteasily found.

5.2.1 Using a precursor for titanium dioxide

Van Hall et al. chose an attractive approach to circumvent this problem by mixing a so-lution of MDMO-PPV with an organic precursor for TiO2. [7] By spin casting the filmin ambient conditions, the precursor (titanium(IV) isopropoxide) reacts with moisturefrom the air, thereby forming TiO2 (at least 65% was converted) in an MDMO-PPV ma-trix. They found that efficient charge transfer from the polymer phase to the TiO2 phaseoccurs, although the photoluminescence was not fully quenched; [7] the residual emis-sion was attributed to photoexcitations that do not reach the interface with TiO2. This issupported by scanning electron microscopy measurements on these blends performedby Slooff et al. [11] They demonstrated that the TiO2 phase in blend with MDMO-PPVshows typical dimensions of 10–20 nm, which is somewhat larger than the exciton diffu-sion length in PPV. [12]

Figure 5.1(a) shows the current-voltage characteristics of a solar cell based onMDMO-PPV and TiO2 formed by the precursor route (prec-TiO2). The efficiency ofthese devices is rather limited, typically 0.2%. A strong correlation was found betweenthe performance of these devices and the relative humidity during processing, whichis not surprising given the hydrolysis reaction necessary to form TiO2. Slooff et al. havedemonstrated that the best devices are obtained at a relative humidity of around 50%. [13]

Unfortunately, MDMO-PPV/prec-TiO2 films made at this level of humidity were inho-mogeneous and very rough, with root-mean-square roughnesses approximately equalto half the active layer thickness. Moreover, the best performing solar cells displayed

72

5.2. Hybrid solar cells with acceptors from a precursor

100 1000

0.5

0.6

0.7

1

10

V oc [

V]

I [W/m2]

J sc [

A/m

2 ]

0.0 0.2 0.4 0.6 0.8

-8

-4

0

4

8

J L [A

/m2 ]

Va [V]

(b)(a)

Figure 5.1: (a) Current-voltage characteristics of an MDMO-PPV/prec-TiO2 (4:1 by volume, 120nm thick active layer) solar cell under illumination. The relative humidity during sample fab-rication was 63 %. (b) The symbols indicate the intensity dependence of Voc (line has a slopeS = 2.0 Vt) and Jsc (fitted to Jsc ∝ Iα, where α = 1.00 ± 0.03).

strong hysteresis in their current-voltage characteristics in dark, therefore, charge trans-port studies could not be executed. The intensity dependence of Jsc gives a hint though:Figure 5.1(b) shows that Jsc is linearly dependent on intensity. As demonstrated in chap-ter 4 this indicates that the mobilities of electrons and holes cannot differ much. Figure5.1(b) also shows the intensity dependence of Voc. Surprisingly, the intensity dependenceof Voc is much stronger than what is to be expected on basis of Eq. (3.11) (S = 2.0Vt),which may indicate that recombination is not only bimolecular. It should be mentionedthat the p-n junction model, Eq. (3.3), also cannot explain the observed behavior sincethe ideality factor is equal to 2.4.

Why is the efficiency of these devices so modest? One possible explanation is thatthe conversion of titanium(IV) isopropoxide does not yield crystalline TiO2, since thisreaction has to be performed at low temperature due to the presence of the polymer. Ascrystalline TiO2 is only obtained at temperatures of more than 350°C, [14] it is to be ex-pected that only amorphous TiO2 is formed, thereby limiting the transport of electrons.Moreover, Van Hall et al. observed a blue-shift in the absorption spectrum of MDMO-PPV upon addition of TiO2, [7] suggesting that the polymer is also affected.

5.2.2 Using a precursor for zinc oxide

In contrast to TiO2, zinc oxide (ZnO) is known to crystallize at much lower tempera-tures. [15] Beek et al. have shown that BHJ solar cells based on MDMO-PPV and ZnOformed from a precursor (prec-ZnO) can formed by using diethylzinc. [16] As the hydrol-

73


-0.3 0.0 0.3 0.6 0.9 1.2-30

-15

0

15

30

J L [A

/m2 ]

Va [V]

Figure 5.2: Current-voltage characteristics of an illuminated (I = 877 W/m2) MDMO-PPV/prec-ZnO device with an efficiency of 1.0%, Jsc = 21.5 A/m2, Voc = 1.00 V, and FF = 42%. The thicknessof the active layer is 100 nm, with a root-mean-square roughness of 8 nm.

ysis and condensation of diethylzinc take place very rapidly when diethylzinc is exposedto air, it is necessary to moderate these reactions by adding tetrahydrofuran, which canstabilize diethylzinc by coordination of the zinc atom. By spin casting a co-solutionof diethylzinc and MDMO-PPV and subsequent thermal annealing at a moderate tem-perature (110°C), crystalline ZnO is formed in the MDMO-PPV matrix. Although thephotoluminescence of the resulting films is not completely quenched, long-lived photo-generated charges are indeed formed. [16]

Figure 5.2 shows the current-voltage characteristics of an illuminated MDMO-PPV/prec-ZnO (15 vol.-% ZnO, assuming full conversion) with an efficiency of 1.0%.

It is remarkable that the optimal devices are obtained with only 15 vol.-% prec-ZnO.When the concentration of prec-ZnO is increased, the films become very rough and in-homogeneous. Fortunately, devices with 15 vol.-% prec-ZnO are relatively smooth (thedevice of Fig. 5.2 has a roughness of 8 nm) and do not show hysteresis in the JD-Va char-acteristics. Therefore, charge transport studies were undertaken. Altough it was possibleto study the transport of holes, unfortunately, it turned out to be very difficult to makereproducable electron-only devices.

In order to study the transport of holes through the MDMO-PPV/prec-ZnO layer,the standard LiF/Al cathode was replaced by palladium. Figure 5.3 shows the current-voltage characteristics of such a device, together with the characteristics of a pristineMDMO-PPV hole-only device with a comparable thickness of the active layer. Thestrong bias dependence of the current through the MDMO-PPV/prec-ZnO device isstriking. When fitted to Eq. (1.8), a zero-field mobility of 1.4 × 10−12 m2/V s and afield activation parameter γ = 1.35 10−3 (m/V)0.5 are found (fit not shown). Such highvalues of γ are difficult to rationalize within the framework of trap-free space-charge-

74

5.2. Hybrid solar cells with acceptors from a precursor

0 1 2 3 410-4

10-2

100

102

104

J D [A

/m2 ]

Vint [V]

Figure 5.3: Hole-only diodes of MDMO-PPV/prec-ZnO (L = 95 nm, squares) and pristine MDMO-PPV (L = 91 nm, circles)

limited currents. Furthermore, as is evident from Fig. 5.3, the currents through the pris-tine MDMO-PPV film and the MDMO-PPV/prec-ZnO device seem to converge at highbias. That this difference in current is not caused by the annealing treatment of theMDMO-PPV/prec-ZnO blend, was confirmed by annealing and processing a pristineMDMO-PPV device in the same way as the blend devices. No difference between anannealed and a not annealed MDMO-PPV device was observed, therefore, the strongbias dependence of MDMO-PPV/prec-ZnO devices must be linked to the presence orformation of ZnO.

What can be the cause of the strong dependence on bias? One possible explanationmight be the presence of neutral traps for charge carriers, since this is known to cause astrong dependence on field strength. For example, one has for traps with an exponentialdistribution in energy of width Etr,

[17]

J ∝ µVr+1

L2r+1, (5.1)

where r = Etr/kBT. However, it was found that in order to obtain a good fit to the data,it was still necessary to incorporate a high field dependence of the mobility. In addition,the predicted temperature and active layer thickness dependence was not in accordancewith the measurements.∗ The possibility of a barrier to hole injection was dismissed forthe same reasons.

Another possible candidate is field-assisted detrapping of charge carriers, the so-called Poole-Frenkel (PF) mechanism. [18] The PF mechanism describes the enhancementof the escape rate of a carrier from an oppositely charged trap by the presence of an elec-tric field, see Fig. 5.4(a). This detrapping results in a larger free carrier density, thereby

∗Varying the active layer thickness is somewhat cumbersome, since this may affect the chemical reactionof the precursor.

75


0 1 2 3 4 5

10-3

10-1

101

103

295 K 275 K 255 K 235 K 215 K

J D [A

/m2 ]

Vint [V]0

0

U(r)

r

(b)(a)

Figure 5.4: (a) Potential energy in the case of Coulomb interaction (solid line). When an externalfield is applied (dotted line), the escape probability of the charge carrier from the trap is increasedin the direction of the applied field (dashed line). (b) Current-voltage characteristics of an MDMO-PPV/prec-ZnO device in the hole-only configuration (symbols) at various temperatures. The linesdenote fits to the PF mechanism, Eq. (5.2).

increasing the conductivity of the film. The resulting current density JPF is given by

JPF ∝ F exp

(

η√

F

kBT

)

, (5.2)

where η is the detrapping parameter. In the PF theory, η is equal to [18]

ηPF =

√

q3

πε. (5.3)

Figure 5.4(b) shows current-voltage data of an MDMO-PPV/prec-ZnO hole-only diodeat various temperatures fitted to Eq. (5.2). Table 5.1 lists the thus obtained values forη. Clearly, the values obtained for η are close, albeit somewhat smaller, than those pre-dicted by Eq. (5.3). That there is a difference between the PF model and the data is notsurprising: The PF model, as presented here, considers only a Coulomb potential in onedimension, while the actual potential may be different and the escape of the trappedcarrier will be a three dimensional process.

What can be learnt from the seeming success of the PF model? In any case, theblue-shift of the absorption spectrum of MDMO-PPV/prec-ZnO films as compared topristine MDMO-PPV observed by Beek et al. is notable. [16] As this blue-shift was alsoobserved for MDMO-PPV redissolved from an MDMO-PPV/prec-ZnO film, it is con-nected to degradation of the polymer and indicates that the conjugation of the backboneis partly destroyed. Possibly, the double bond of the vinylene group in the PPV backbone(see Fig. 1.2) reacts with ZnO – anions, thereby breaking the conjugation. As a side-effect,

76

5.3. Polymer solar cells with zinc oxide nanoparticles

Table 5.1: The values of η, relative to ηPF as given by Eq. (5.3), used in the fits shown in Fig. 5.4(b)

T [K] η/ηPF

295 0.92275 0.88255 0.87235 0.87215 0.84

negative charges may be present in the polymer phase, which can act as charged trapsfor holes. Alternatively, there may be negative groups on the surface of the ZnO phasein the blend with MDMO-PPV. P3HT is expected to be more stable against a chemicalreaction with ZnO – anions and may lead to better performing solar cells. These obser-vations indicate that considerable care is required when designing polymer solar cellswith an acceptor formed in situ from a precursor.

5.3 Polymer solar cells with zinc oxide nanoparticles

By preparing the inorganic acceptor ex situ, it is possible to have greater control over thereaction conditions and purity of the obtained materials. This comes at the cost of havingto provide a means for mixing the acceptor with the polymer. Beek et al. have developeda method for making polymer solar cells with nc-ZnO as acceptor material. [8] Zinc oxidenanoparticles approximately 5 nm in diameter (see Fig. 5.5) were synthesized by hydrol-ysis and condensation of zinc acetate dihydrate by potassium hydroxide in methanol,using the method of Pacholski et al. [8,19] The zinc oxide particles were dispersed in amixture of methanol and chlorobenzene without the aid of additional surfactants or lig-ands.

Figure 5.6(a) shows the current-voltage characteristics of an MDMO-PPV/nc-ZnOsolar cell under illumination. Typically, the open-circuit voltage ranges from 0.7 Vto 0.8 V. As the conduction band of nc-ZnO (4.2 eV) lies deeper than the LUMO ofPCBM (approximately 3.8 eV), it is easy to understand that the open-circuit voltageof MDMO-PPV/nc-ZnO solar cells is slightly lower than the open-circuit voltage ofMDMO-PPV/PCBM devices (Voc = 0.80–0.85 V). [20,21] Since the effective masses of elec-trons and holes in ZnO is relatively low, quantum confinement effects already start toplay a role at a diameter of approximately 8 nm. [22] Moreover, considerable influence ofsurface conditions is to be expected, rendering the exact positions of the electronic lev-els of nc-ZnO quite sensitive to the circumstances during synthesis. Figure 5.6(b) showsthe short-circuit current density as a function of light intensity of an MDMO-PPV/nc-ZnO solar cell. When fitted to Jsc ∝ Iα, α = 1.03 ± 0.02 is obtained, showing that theshort-circuit current density is linear in the intensity.

77


Figure 5.5: Transmission electron micro-graph of nc-ZnO. The length of the scale bar correspondsto 20 nm.

(b)

100 1000

1

10

J sc [

A/m

2 ]

I [W/m2]0.0 0.2 0.4 0.6 0.8

-20

-10

0

10

20

J L [A

/m2 ]

Va [V]

(a)

Figure 5.6: (a) Current-voltage characteristics of an illuminated MDMO-PPV/nc-ZnO solar cellwith an active layer thickness of 130 nm (symbols); the line denotes the numerical modeling re-sult. (b) The incident light intensity dependence of the short-circuit current density of an MDMO-PPV/nc-ZnO photovoltaic device (symbols) and a fit to the relation Jsc ∝ Iα, where α = 1.03 ± 0.02(line).

78


(b)

0 1 2 3 40.1

1

10

100

1000

J D [A

/m2 ]

Vint [V]0 1 2 3 4

0.01

0.1

1

10

100

1000

J D [A

/m2 ]

Vint [V]

pristine MDMO-PPV MDMO-PPV/nc-ZnO

(a)

Figure 5.7: (a) Current-voltage characteristics of hole-only diodes of pristine MDMO-PPV (activelayer thickness 90 nm) and MDMO-PPV/nc-ZnO (active layer thickness 130 nm). The lines denotefits to Eq. (1.8). (b) Current-voltage characteristics of electron-only diodes of MDMO-PPV/nc-ZnO(active layer thickness 115 nm). The line denotes a fit to Eq. (1.8).

5.3.1 Charge transport in MDMO-PPV/nc-ZnO blends

In order to assess to the transport of holes in MDMO-PPV/nc-ZnO solar cells, the cath-ode is replaced by a high work function electrode, thereby blocking the injection of elec-trons from the contact. This results in space-charge-limited current flow by holes. Figure5.7(a) shows current-voltage characteristics of a pristine MDMO-PPV hole-only diodewith an active layer of 90 nm thickness. By fitting the experimental data to Eq. (1.8) andusing Vbi = 0.4 V, a value of 4.0 × 10−10 m2/V s is obtained for the zero-field mobility,together with γ = 3.5 × 10−4 (m/V)0.5. Note that this batch of MDMO-PPV, synthesizedvia the sulfinyl route, has a ten times higher hole mobility than previously reported forMDMO-PPV. [23] Figure 5.7(a) also shows current-voltage measurements of an MDMO-PPV/nc-ZnO hole-only diode, with an active layer thickness of 130 nm. Although theblend layer is somewhat thicker than the layer of MDMO-PPV discussed previously, thecurrent densities are very similar. In fact, using Vbi = 0.3 V a zero-field mobility of 5.5× 10−10 m2/V s is obtained and γ = 3.5 × 10−4 (m/V)0.5, showing that, within experi-mental error, the hole mobility in the polymer phase of the blend is not affected by thepresence of nc-ZnO.

To study the transport of electrons, the bottom- and top electrode were made ofsamarium, a metal with a low work function. Spin casting on samarium requires somecare, since it is a reactive metal, but extensive testing showed no significant degradationof the bottom electrode. However, in this configuration the samarium bottom electrodeis not used to inject electrons into the active layer, but only to suppress the injection of

79


holes, while the top electrode supplies the electrons.∗

The current-voltage characteristics of an electron-only device with a 115 nm thickactive layer consisting of the MDMO-PPV/nc-ZnO blend is depicted in Fig. 5.7(b). Nobuilt-in voltage is subtracted, since the bottom and top electrode are made of the samemetal. Using Eq. (2), we find µ0 = 2.8 × 10−9 m2/Vs and γ = 0.5 × 10−4 (m/V)0.5, so theelectron mobility is an order of magnitude larger than the hole mobility of the polymerphase. Transport of electrons in nc-ZnO films has also been studied using a electrochem-ically gated transistor [10,15] showing that the electron mobility in these films shows astrong dependence on the number of electrons per particle. In these measurements, themobility ranged from 10−7 m2/Vs to 10−5 m2/Vs. These values are in good agreementwith photocurrent measurements performed on electrochemical cells. [24] However, it isdifficult to compare these values to the values reported here, since the volume fraction ofZnO present in the film is much lower in our case (25 vol.%). Furthermore, the electronconcentration in a bulk heterojunction solar cell under operating conditions (see chap-ter 2) is several orders of magnitude lower than those reported in Refs. [10] and [15].On the basis of these observations, it is reasonable to expect that the mobilities found inRefs. [10], [15], and [24] represent an upper limit to electron mobility through the nc-ZnOphase in MDMO-PPV/nc-ZnO devices.

In chapter 4 it was shown that the intensity dependence of the short-circuit currentis determined by the ratio of electron- to hole mobility, leading to different values of theexponent α (Jsc ∝ Iα). When the electron mobility is much larger (typically more thantwo orders of magnitude) than the hole mobility, buildup of net space charge results in0.75 < α < 1. [25] On the other hand, if the mobilities of electrons and holes are com-parable, the transport is balanced and a is equal to unity. The linear dependence of theshort-circuit current density on light intensity (see Fig. 5.6(b), α = 1.03± 0.02), supportsour findings of the electron- and hole mobilities. It should be noted that Beek et al. havereported a lower value for a, i.e., 0.93. [8] However, in their investigation, the MDMO-PPV was synthesized via a different route, probably leading to a lower hole mobility,thereby leading to a lower value of α.

5.3.2 Improving the efficiency of MDMO-PPV/nc-ZnO solar cells

In order to identify the factors limiting the performance of MDMO-PPV/nc-ZnO solarcells, the numerical model presented in chapter 2 was applied to the data of Fig. 5.6.Note, that a field-dependent generation rate of free electrons and holes was not con-sidered, since it is not expected that this results in a significant field-dependence in thelimited voltage range considered here, due to the high dielectric constant of ZnO. Usingµp0 = 5.5 × 10−10 m2/Vs and γ = 3.5 × 10−4 (m/V)0.5 for the mobility of holes, µn0 = 3.7

× 10−9 m2/V s and γ = 0.5 × 10−4 (m/V)0.5 for the electrons, and a generation rate of

∗Solar cells with an MDMO-PPV/nc-ZnO or P3HT/PCBM blend with samarium as a top contact (insteadof LiF (1 nm)/Al) show a good performance and, most importantly, an open-circuit voltage equal to deviceswith LiF/Al as top electrode.

80


free carriers G = 1.26 × 1027 m−3 s−1, a good agreement between experimental data andnumerical modeling is obtained (see Fig. 5.6 (a)), allowing for a detailed investigation ofthe factors governing the performance of these solar cells.

Comparing MDMO-PPV/PCBM with MDMO-PPV/nc-ZnO solar cells

A striking feature of the MDMO-PPV/PCBM system is that the best performing solarcells contain 80 wt.-% PCBM (corresponding to 70 vol.-% PCBM, using the densitiesof MDMO-PPV and PCBM of Ref. [26]), although PCBM hardly contributes to the ab-sorption of light. Two main reasons for the need for such high PCBM loadings can begiven: [27] Surprisingly, it has been demonstrated that the hole mobility of the MDMO-PPV/PCBM blend actually increases upon addition of PCBM. At 80 wt.-% PCBM, thehole mobility amounts to 2.0 × 10−8 m2/V s, which is an increase of more than twoorders of magnitude as compared to pristine MDMO-PPV. [28,29] Additionally, the per-formance of MDMO-PPV/PCBM solar cells benefits from a higher dielectric constantassociated with the addition of PCBM, since this facilitates the dissociation of boundelectron-hole pairs across the polymer-PCBM interface. [27]

Interestingly, the performance of MDMO-PPV/PCBM solar cells with only 25 vol.-%PCBM, corresponding to the composition of the best performing MDMO-PPV/nc-ZnOcells, is markedly worse with an efficiency of only 0.2%. [27] Moreover, at that composi-tion, the electron mobility in the PCBM phase is equal to approximately 3 × 10−10 m2/Vs and the hole mobility equals the pristine MDMO-PPV value. Therefore, the electronmobility of the MDMO-PPV/nc-ZnO system is higher at this composition, as is the ef-ficiency (1.6%). The generation of free charge carriers under operating conditions in theMDMO-PPV/nc-ZnO system is more efficient (G = 1.26 × 1027 m−3 s−1, for the deviceof Fig. 5.6) than in the MDMO-PPV/PCBM (3:1 by volume) devices, where G = 5 × 1026

m−3 s−1. [27] Since the volume ratio of MDMO-PPV in both systems is the same (75 vol.-%), this lowering of the carrier generation is attributed to the less efficient electron-holepair dissociation due to the lower dielectric constant of PCBM. Model calculations showthat this changes the dissociation efficiency by more than a factor of 2.

Improving the performance of MDMO-PPV/nc-ZnO solar cells

As already mentioned, the open-circuit voltage of MDMO-PPV/nc-ZnO is lower thanthe open-circuit voltage of MDMO-PPV/PCBM devices due less favorable energetic po-sition of the conduction band of nc-ZnO. However, as will be demonstrated below, themain cause for a lower efficiency, as compared to optimized MDMO-PPV/PCBM de-vices, lies in the lower charge carrier mobilities.

The concentration of nc-ZnO in these blends is limited by the film forming prop-erties: when more than 33 vol.-% of nc-ZnO is added, the film quality becomes verypoor. [8] The fact that one is limited to rather low nc-ZnO content, complicates a goodcomparison between both systems. For example, it is at this moment unclear whetherthe spectacular enhancement of the hole mobility upon addition of PCBM will also be

81


0.0 0.2 0.4 0.6

-20

-10

0

10

J L [A

/m2 ]

Va [V]

Figure 5.8: Simulated current-voltage characteristics showing the influence of the charge carriermobilities. The solid line is the fit to the experimental data shown in Fig. 5.6(a). The dashed linedenotes the numerical result for the case when the hole mobility is increased to 2.0 × 10−8 m2/Vs,the MDMO-PPV/PCBM (1:4 by weight) value, while the dotted line corresponds to what wouldhappen when the electron mobility is also increased to the standard MDMO-PPV/PCBM value of2.0 × 10−7 m2/Vs.

induced by nc-ZnO addition, if it were possible to maintain a good morphology. Ad-ditionally, in view of the high mobilities reported for nc-ZnO electrodes, [10,15,24] it isreasonable to assume that the electron mobility through the nc-ZnO phase would alsobenefit from a larger volume percentage of nc-ZnO. Additionally, Beek et al. have shownthat the photoluminescence of an MDMO-PPV/nc-ZnO containing 25 vol.% nc-ZnO isnot completely quenched, probably due to large polymer domains in the film morphol-ogy. [8] The need for a better control over the morphology of the blend is obvious, andone option would be the use of additional ligands that improve the dispersability ofthe nanocrystals. However, Greenham et al. have demonstrated that the use of a lig-and can seriously hamper the charge transfer from conjugated polymers to inorganicnanocrystals. [3] Huynh et al. were able to control the morphology of films consisting ofcadmium selenide nanocrystals blended with poly(3-hexylthiophene) through the useof the weakly binding ligand pyridine. [30] After deposition of the blend film, the ligandcould be removed by heating the sample under vacuum. Another approach is to usean electroactive ligand, which mediates the electron transfer between cadmium selenidenanoparticles and conjugated polymers. [31,32] These results show the potential of the useof ligands for controlling the properties of polymer/inorganic nanoparticles blends.

To show that higher efficiencies can indeed be obtained once the hole mobility isimproved, the effect is calculated of enlarging the hole mobility up to the MDMO-PPV/PCBM (1:4 by weight) value, 2.0 × 10−8 m2/V s, on the current-voltage charac-teristics of an MDMO-PPV/nc-ZnO solar cell, see Fig. 5.8. As expected, the efficiency ofMDMO-PPV/nc-ZnO solar cells benefits from this improvement of the charge transport,

82


and the efficiency would be enhanced by 35%. The fact that the hole mobility is equal tothe pristine MDMO-PPV value represents a limit to the efficiency that may be relievedby replacing MDMO-PPV with another, more suitable, polymer.

Although bulk ZnO is a very good electron conductor, the electron mobility in thenc-ZnO phase is lower than the electron mobility of PCBM. Since electron mobilitiesthat are at least comparable to or higher than the electron mobility of PCBM have beenreported, [10,25] it is to be expected that by fine-tuning the processing conditions, the elec-tron mobility in the nc-ZnO phase can be improved. However, since the hole mobilityis lower than the electron mobility, it is to be expected that not much is to be gainedby improving the mobility of electrons. Therefore it comes as no surprise, that also in-creasing the electron mobility to the PCBM value (2.0 × 10−7 m2/V s) yields an onlyslightly higher efficiency, which is 44% higher than the efficiency of the actual devices(see Fig. 5.8).

The main increase in the efficiency for the system with enhanced mobilities lies in anincrease in fill factor due to better transport of charges. As the open-circuit voltage de-pends on the bimolecular recombination strength [see Eq. (3.11)], Voc decreases slightlywhen the charge carrier mobilities are increased, see Fig. 5.8. Because of this increaseof the mobilities, the carrier densities in the bulk of the device are lowered, since thecarriers flow out of the device more easily. The field and carrier densities in the device,therefore, come closer to their values in dark (in other words, the quasi-Fermi potentialsplitting becomes less) and hence the open-circuit voltage decreases.∗ This implies thatthere is an optimum for the charge carrier mobilities, depending on light intensity andactive layer thickness. At intensities around 1 Sun, the optimal values of the mobilitiesis in the order of 10−8–10−6 m2/Vs, according to the numerical model.

In a recent study, P3HT has been used to replace MDMO-PPV as the electron donormaterial. [33] It is well known that, depending on processing conditions, the hole mobilityin the P3HT phase of P3HT/PCBM solar cells can be very high, resulting in very efficientdevices. [34,35] In the case of P3HT/nc-ZnO solar cells, it was found that the efficiencyincreased up to 0.9% upon thermally annealing of the devices, which is not an improve-ment as compared to MDMO-PPV/nc-ZnO devices, despite the supposedly higher holemobility. It is, however, unclear whether the hole mobility in the P3HT phase of the hy-brid device is as high as in the P3HT/PCBM devices, since the presence of nc-ZnO mayinfluence the crystallization of P3HT. The sublinear (α = 0.9) intensity dependence ofthe short-circuit current [33] suggests that there exists at least a large difference betweenelectron and hole mobility. Additionally, it was found that not all of the P3HT was inclose proximity to ZnO, because of an unfavorable morphology, which limits the excitonquenching process and thereby the charge generation process. This observation clearlydemonstrates the need for greater control over the film morphology.

∗As can be seen from Eqs. (2.2) and (2.3), when µn,p : G = constant the current-voltage characteristics havethe exact same shape and only the magnitude of the current changes. This already implies that Voc decreaseswhen the mobilities become larger, since this is equivalent to a solar cell with lower mobilities but illuminatedwith a lower intensity. It is, therefore, not desirable to have near-infinite charge carrier mobilities.

83


Modifying the ZnO nanoparticles

The exact positions of the energy levels of nc-ZnO are sensitive to processing circum-stances through their size and surface conditions. It was tried to advantageously use thissensitivity to enhance the performance of MDMO-PPV/nc-ZnO solar cells. Attempts toobtain smaller particles, giving rise to higher open-circuit voltages, were largely un-successful. Although obtaining smaller particles is in itself not difficult, the growth cansimply be quenched, for example, by cooling the reaction mixture, it proved very hard tokeep a small particle size while washing the ZnO particles. This washing step is crucial,since the material purity plays an important role in photovoltaic devices.

Beek et al. observed that the use of a small ligand, in this case n-propylamine, im-proves the film forming properties of the MDMO-PPV/nc-ZnO mixture. [9] Since this issuch a small molecule, the use of n-propylamine may not be detrimental to the chargetransfer processes. Unfortunately, they found that this ligand resulted in a decrease ofJsc and Voc. The decrease in Jsc could be slightly restored by increasing the amount ofnc-ZnO in the blend, but this reduced Voc even further. In order to try to circumvent thisloss in Voc, the effect of ligands with an opposite dipole (with respect to n-propylamine)was studied (see Experimental). Although the film quality indeed improved, this didnot lead to an enhancement of the efficiency.

5.4 Conclusions

In this chapter, solar cells with active layers consisting of a polymer (MDMO-PPV) andinorganic semiconductor materials were investigated. One method to obtain such blendsis to spin cast a co-solution of the polymer and an organic precursor for either TiO2 orZnO. Although the efficiency of the MDMO-PPV/prec-ZnO devices was quite reason-able, the hole transport through the polymer phase clearly suffered from the addition orformation of the ZnO.

Another approach is to make nanocrystalline ZnO ex situ and add this to the MDMO-PPV solution. It is demonstrated that the hole transport through the thus-formedblends is not affected by the presence of nc-ZnO. The electron mobility in these MDMO-PPV/nc-ZnO blends also is quite decent, although not as high as reported for electro-chemical cells. By replacing the MDMO-PPV by a polymer with a higher hole mobility,while maintaining a favorable morphology, and by further optimizing the processing ofnc-ZnO, it should be possible to reach significantly higher efficiencies.

5.5 Experimental

In order to make MDMO-PPV/prec-TiO2 blends, a 3 mg/mL solution of MDMO-PPV intoluene was prepared in nitrogen. To this solution, titanium(IV) isopropoxide (Aldrich)was added, such that the final ratio MDMO-PPV:prec-TiO2 corresponded to 4:1 by vol-ume (assuming full conversion). After spin casting the active layer in air, the samples

84

5.5. Experimental

were kept in dark for one hour in order to let the of the precursor take place. This wasfollowed by one hour in vacuum at 40°C to remove any residual solvents or reactionproducts.

Blends of MDMO-PPV/prec-ZnO were made as follows: A stock solution of 0.4 Mdiethylzinc in toluene and tetrahydrofuran was prepared by adding 1.8 mL of diethylz-inc (1.1 M in toluene, Aldrich) to 3.2 mL of dry tetrahydrofuran under a nitrogen atmo-sphere. Assuming full conversion, 1 mL of this stock solution gives 32 mg ZnO. The cor-rect amount of diethylzinc stock solution was added to a 3mg/mL MDMO-PPV solutionin chlorobenzene. Spin casting, and subsequent aging and annealing, of the active layerwas performed in a nitrogen atmosphere with a relative humidity of approximately 40%.The samples were aged for 15 minutes, followed by annealing at 110°C for 30 minutes.

The nc-ZnO sols were used within a week after synthesis. Typically, 2.95 g of zincacetate dihydrate was dissolved in 125 mL methanol, kept at 60°C. A solution of 1.48g potassium hydroxide (87%) in methonal was prepared at room temperature. The hy-droxide containing solution was added to the zinc ions containing solution in 8 min-utes, under constant stirring. All glassware was dried in an oven. After addition ofthe potassium hydroxide solution, a clear solution was obtained. The solution was al-lowed to react for 135 minutes, and was subsequently allowed to cool down for 2 h toroom temperature without stirring, causing a white powder to precipitate. Precipateand mother-liquid were separated and the precipitate was washed twice with 50 mL ofmethanol. In order to disperse the particles, 10 mL of chlorobenze was added to thewashed precipitate. After filtration (1 µm), the concentration of nc-ZnO was determinedby weighing the solid residue after solvent evaporation. Typically, the concentration ofnc-ZnO was 60 mg/mL. A 6 mg/mL MDMO-PPV solution in chlorobenze was prepared.A mixture of methanol and chlorobenzene (1:9 by volume) and the appropriate amountof nc-ZnO sol were added to bring the MDMO-PPV concentration to 3 mg/mL. Thisblend was spin cast in a nitrogen-filled glove box. The following substances were triedas ligands: 2,3,4,5,6-pentafluoraniline, 2,2,3,3,3-pentafluoropropylamine, and 1H,1H-pentadecafluorooctylamine.

85


References

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[2] G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, Science 270, 1789 (1995).

[3] N. C. Greenham, X. Peng, and A. P. Alivisatos, Phys. Rev. B 54, 17628 (1996).

[4] A. C. Arango, S. A. Carter, and P. J. Brock, Appl. Phys. Lett. 74, 1698 (1999).

[5] C. Y. Kwong, A. B. Djurisic, P. C. Chui, K. W. Cheng, and W. K. Chan, Chem. Phys. Lett. 384,372 (2004).

[6] K. M. Coakley, Y. Liu, M. D. McGehee, K. L. Frindell, and G. D. Stucky, Adv. Funct. Mater. 13,301 (2003).

[7] P. A. van Hal, M. M. Wienk, J. M. Kroon, W. J. H. Verhees, L. H. Slooff, W. J. H. van Gennip,P. Jonkheijm, and R. A. J. Janssen, Adv. Mater. 15, 118 (2003).

[8] W. J. E. Beek, M. M. Wienk, and R. A. J. Janssen, Adv. Mater. 16, 1009 (2004).

[9] W. J. E. Beek, M. M. Wienk, M. Kemerink, X. Yang, and R. A. J. Janssen, J. Phys. Chem. B 109,9505 (2005).

[10] A. L. Roest, J. J. Kelly, D. Vanmaekelbergh, and E. A. Meulenkamp, Phys. Rev. Lett. 89, 036801(2002).

[11] L. H. Slooff, M. M. Wienk, J. M. Kroon, Thin Solid Films 451–452, 634 (2004).

[12] D. E. Markov, J. C. Hummelen, P. W. M. Blom, A. B. Sieval, Phys. Rev. B 72, 045216 (2005).

[13] L. H. Slooff, J. M. Kroon, J. Loos, M. M. Koetse, and J. Sweelssen, Adv. Funct. Mater. 15, 689(2005).

[14] M. Okuya, K. Nakade, and S. Kaweke, Sol. Energy Mater. Sol. Cells 70, 425 (2002).

[15] E. A. Meulenkamp, J. Phys. Chem. B 102, 5566 (1998).

[16] W. J. E. Beek, L. H. Slooff, M. M. Wienk, J. M. Kroon, and R. A. J. Janssen,Adv. Funct. Mater. 15, 1703 (2005).

[17] K. C. Kao and W. Hwang, Electrical transport in solids (Pergamon Press, Oxford, 1981).

[18] J. Frenkel, Phys. Rev. 54, 647 (1938).

[19] C. Pacholski, A. Kornowski, and H. Weller, Angew. Chem. Int. Ed. 41, 1188 (2002).

[20] S. E. Shaheen, C. J. Brabec, N. S. Sariciftci, F. Padinger, T. Fromherz, and J. C. Hummelen,Appl. Phys. Lett. 78, 841 (2001).

[21] V. Dyakonov, Physica E 14, 53 (2002).

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[22] Z. Hu, G. Oskam, and P. C. Searson, J. Colloid Interface Sci. 263, 454 (2003).


[24] V. Noack, H. Weller, and A. Eychmuller, J. Phys. Chem. B 106, 8514 (2002).


[26] C. W. T. Bulle-Lieuwma, W. J. H. van Gennip, J. K. J. van Duren, P. Jonkheijm, R. A. J. Janssen,and J. W. Niemantsverdriet, Appl. Surf. Sci. 203–204, 547 (2003).


[28] C. Melzer, E. Koop, V. D. Mihailetchi, and P. W. M. Blom, Adv. Funct. Mater. 14, 865 (2004).

[29] S. M. Tuladhar, D. Poplavskyy, S. A. Choulis, J. R. Durrant, D. D. C. Bradley, and J. Nelson,Adv. Funct. Mater. 15, 1171 (2005).

[30] W. U. Huynh, J. J. Dittmer, W. C. Libby, G. L. Whiting, and A. P. Alivisatos,Adv. Funct. Mater. 13, 73 (2003).

[31] D. J. Milliron, A. P. Alivisatos, C. Pitois, C. Edder, and J. M. J. Frechet, Adv. Mater. 15, 58(2003).

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87

CHAPTER

SIX

Improving the efficiency of bulkheterojunction solar cells

Summary

In this chapter, various ways to improve the efficiency of bulk heterojunction so-lar cells are identified by using the MIM model as outlined in chapter 2. A muchpursued way to enhance the performance is to increase the amount of photons absorbedby the film by decreasing the band gap of the polymer. Calculations based on theMIM model confirm that this would indeed improve the performance. However, it isdemonstrated that the effect of minimizing the energy loss in the electron transfer fromthe polymer to the fullerene derivative is even more beneficial. By combining these twoeffects, it turns out that the optimal band gap of the polymer would be 1.9 eV. Withbalanced charge transport, polymer/fullerene solar cells can reach power conversionefficiencies of at least 10.8%.

89

Chapter 6. Improving the efficiency of bulk heterojunction solar cells

6.1 Introduction

How efficient can bulk heterojunction solar cells be? Which material requirements mustbe fulfilled? These are the most important questions in this thesis. For p-n junction basedsolar cells, the former question was addressed in the 1950’s. Shockley and Queisserstudied the detailed balance limit to the efficiency of p-n junction solar cells by treatingthe sun and the solar cell as two black bodies at temperatures Tsun = 6000 K and Tcell =300 K, respectively. [1] Araujo and Martı generalized these arguments and found that theoptimal band gap E

opgap is equal to 1.3 eV, with a maximal efficiency of 31%. [2] Loferski

pointed out that atmospheric conditions change the value of Eopgap because the spectrum

of the incident light is affected. Therefore, the optimal band gap for p-n junction solarcells under AM1.5 illumination is equal to 1.4 eV. [3] The voltage V

opMPP corresponding to

the maximum power point for an optimized band gap (under full concentration) is givenby [2]

VopMPP =

Eopgap

q

(

1 − Tcell

Tsun

)

, (6.1)

showing that VopMPP is very close to E

opgap/q. Under normal intensity (1 Sun), it can easily

be shown that

VopMPP ≈ 0.95

Eopgap

q− 0.27. (6.2)

This limit cannot be directly applied to BHJ solar cells: Due to the offset betweenthe LUMOs of the donor and the acceptor, necessary for charge transfer, the Voc of BHJsolar cells is limited to Eeff

gap/q (see chapter 3), even for an idealized situation. As a

consequence, VopMPP for a BHJ device will be smaller than the value predicted by Eq. (6.2).

Therefore, the detailed balance limit for BHJ solar cells is significantly lower than thevalue predicted for p-n junctions and it follows that the optimal value of the band gap ofthe absorbing polymer will be significantly larger than 1.4 eV.∗

In this chapter, a calculation of the detailed balance limit of BHJ solar cells will notbe attempted. Instead, ways to improve existing devices will be identified. As a firstapproximation Coakley and McGehee predicted that an efficiency of 10% may be withinreach. [4] In their calculation it is assumed, among other things, that the fill factor is equalto unity and recombination, either geminate or bimolecular, is neglected. By using thenumerical model outlined in chapter 2, a more detailed calculation can be performed.The starting point of this investigation will be the P3HT/PCBM system, with an effi-ciency of 3.5%. By combining charge carrier mobility measurements [5] with current-voltage measurements performed on illuminated solar cells, the experimental current-voltage characteristics of these solar cells are quantitatively modeled. The thus obtained

∗As the Shockley equation for p-n junctions directly follows from this detailed balance analysis, it is easy tosee that, since the detailed balance limit for BHJs is different, the current-voltage characteristics of BHJs cannotbe described by the Shockley equation.

90

6.2. Improving polymer/fullerene solar cells

theoretical description of P3HT/PCBM solar cells enables one to investigate the enhance-ment of the efficiency when a number of solar cell parameters is varied.

Scharber et al. have also predicted that an efficiency of 10% is achieveable, [6] withan optimal polymer band gap of 1.4 eV. In their calculation it is assumed that the fillfactor is equal to 65%, regardless of the HOMO and LUMO positions of the materials.However, if the effective band gap Eeff

gap is increased, the fill factor also increases, therebyfavoring larger values of the band gap of the polymer. Therefore, the optimal value ofthe polymer’s band gap is underestimated in their calculation.

6.2 Improving polymer/fullerene solar cells

Let us take a closer look at two important parameters: the energy loss in electron transferand the band gap of the absorber. The effect of minimizing the energy loss in the electrontransfer from donor to acceptor material is found to be of paramount importance; anefficiency of 8.4% is predicted by minimizing this loss. This comes as no surprise whenone considers that only photons with an energy larger than 2 eV are absorbed, whileVoc = 0.6 V. Subsequently, the effect of decreasing the polymeric band gap is studied.Several research groups have put a lot of effort in the synthesis and application of thesepolymers. [7–11] At a first glance a small band gap polymer seems beneficial. Due toan improved overlap with the solar spectrum the absorption is enhanced, leading toefficiencies larger than 6%. Surprisingly, it is found that once the energy loss in electrontransfer is minimized, the best performing solar cell comprises a polymer with a bandgap of around 2 eV, clearly not a small band gap. In these cells a reduction of the bandgap is accompanied by a decrease of the open-circuit voltage, canceling the benfit ofan absorption increase. With energy levels, band gaps and mobilities simultaneouslyoptimized polymer/fullerene solar cells can reach nearly 11% efficiency.∗

The devices used in this chapter are BHJs of P3HT and PCBM annealed at 110°C withan active layer thickness of 97 nm. Figure 6.1 shows the current density under illumi-nation (JL) as a function of applied bias (Va) of a P3HT/PCBM solar cell. To describethe current-voltage characteristics of polymer/fullerene solar cells the MIM model, asoutlined in chapter 2, is used, see Fig. 6.1.

The inset of Fig. 6.1 shows the positions of the LUMO and HOMO of P3HT andPCBM. Due to the large offset between the LUMO of the donor, LUMO(D), and theLUMO of the acceptor, LUMO(A), electron transfer from the donor onto the acceptortakes place, thereby breaking up the exciton. However, the excess energy of the electronand the hole is quickly dissipated. This energy loss is reflected in the open-circuit volt-age, which is limited by the difference between the HOMO of the donor and the LUMOof the acceptor, see chapter 3. [12,13] Concomitantly, the need for a LUMO(A)-LUMO(D)offset reduces the output power (and hence efficiency) of the solar cell.

∗Although this is not a strictly limiting value.

91


-0.4 -0.2 0.0 0.2 0.4 0.6

-90

-60

-30

0

30

3.8

2.7

J L [A

/m2 ]

Va [V]

PCBM

P3HT4.8

6.1

Figure 6.1: The current-voltage characteristics of a P3HT/PCBM bulk heterojunction solar cell(symbols) illuminated at 1 kW/m2 and the fit to the data (line). The inset shows the energy levelsof P3HT and PCBM (energies given in eV with respect to vacuum).

Experimental and theoretical investigations of polymer/polymer BHJs show thatelectron transfer occurs provided that the difference in LUMO levels is larger than thebinding energy of the intrachain exciton, [14] which is known to be approximately 0.4eV. [15] Since the difference in LUMO levels is much larger than the exciton bindingenergy, it should be possible to decrease the LUMO(A)-LUMO(D) offset without de-creasing the electron transfer efficiency and thereby increasing the energy differencebetween the HOMO of the donor and the LUMO of the acceptor. Figure 6.2 showsthe influence of the LUMO(A)-LUMO(D) offset on the device efficiency when all otherparameters are kept the same as for the P3HT/PCBM device. The performance of thephotovoltaic devices is greatly enhanced by lowering the LUMO(A)-LUMO(D) offset,primarily caused by an increase in open-circuit voltage. For the P3HT/PCBM system,the LUMO(A)-LUMO(D) offset amounts to 1.1 eV, leading to 3.5 % efficiency. To beon the safe side, the LUMO(A)-LUMO(D) offset is not lowered below 0.5 eV, althoughBrabec et al. have shown that efficient charge transfer takes place in a small band gappolymer/fullerence device with a LUMO(A)-LUMO(D) offset of only 0.3 eV. [9] The pos-sibilty of triplet formation from the charge transfer state, which can become more prob-able when the LUMO(A)-LUMO(D) offset is decreased, is ignored. [16] By lowering thisoffset to 0.5 eV the device effiency would increase to more than 8 %, showing the greatimportance of matching the electronic levels of donor and acceptor.

Now we turn to the influence of the polymer’s band gap. Since P3HT has a rela-tively large band gap (2.1 eV), improvement of the overlap of the absorption spectrumof the materials used with the solar spectrum may also increase device performance. Theeffect of lowering the polymer band gap is studied by shifting the P3HT part in the ab-sorption spectrum of a P3HT/PCBM blend film down in energy. In this way, a realistic

92


0.5 0.6 0.7 0.8 0.9 1.0 1.13

4

5

6

7

8

9

Effic

ienc

y [%

]

LUMO(A) - LUMO(D) [eV]

2.7

Acceptor

Donor4.8

6.1

Figure 6.2: The influence of the offset between the LUMO of the donor and the acceptor (symbols),the line is drawn as a guide to the eye.

absorption spectrum for the polymer is taken, both in shape and in magnitude and theassumption that all above band gap photons are absorbed and contribute to the pho-tocurrent is not made.∗ The HOMO level of the polymer phase is taken constant, so theopen-circuit voltage is not affected by the decrease in band gap, and the energy levelsof PCBM remain unchanged. Subsequently, the resulting increase in absorption is cal-culated and the exciton generation rate is modified accordingly. By using this as inputfor the numerical model, together with the parameters obtained in fitting the current-voltage data of the real P3HT/PCBM device (see Fig. 6.1), the resulting device efficiencyis calculated, see Fig. 6.3. Clearly, the device performance benefits from lowering theband gap, reaching 6.6 % for a 1.5 eV band gap. The band gap is not lowered beyond1.5 eV, which corresponds to a LUMO(A)-LUMO(D) offset of 0.5 eV, to ensure efficientelectron transfer from the polymer to PCBM. The increase in performance is accountedfor by enhancement of the short-circuit current. This calculation shows that the effectof only tuning the LUMO(A)-LUMO(D) offset is more beneficial than only lowering thepolymeric band gap.

As a next step the combined effect of lowering the band gap of the polymer whilstkeeping the LUMO(A)-LUMO(D) offset to 0.5 eV is studied, see Fig. 6.4. For a bandgap of 1.5 eV the efficiency amounts to 6.6%, corresponding to the maximum of Fig. 6.3.However, when the band gap is increased the now fixed LUMO(A)-LUMO(D) offsetleads to an increase of the open-circuit voltage, therebye enhancing the efficiency in spiteof reducing the absorption. As shown before in Fig. 6.2, the efficiency corresponding toa 2.1 eV band gap is more than 8%. However, the efficiency shows a broad maximum

∗As the band gap of the polymer is decreased, the generation of charges will be due to longer wavelengths,which in turn need a thicker active layer to be absorbed. This optical effect is ignored in the present analysis,overestimating the efficiency of small band gap devices.

93


1.5 1.6 1.7 1.8 1.9 2.0 2.13

4

5

6

7

Effic

ienc

y [%

]

Polymer bandgap [eV]

Acceptor

Donor4.8

6.1

3.8

Figure 6.3: The influence of the band gap of the polymer on device efficiency (symbols). The lineis drawn as a guide to the eye.

1.5 1.6 1.7 1.8 1.9 2.0 2.16

7

8

9

Effic

ienc

y [%

]

Polymer bandgap [eV]

Acceptor

Donor4.8

6.1

Figure 6.4: The combined effect of tuning the LUMO(A)-LUMO(D) offset to 0.5 eV and changingthe polymer band gap (symbols). The line is drawn as a guide to the eye.

94


100 150 200 250 300

6

8

10

12

Effic

ienc

y [%

]

Active layer thickness [nm]

Figure 6.5: The influence of the active layer thickness on the efficiency taking the hole mobility asis (squares) or increasing it to 2.0 × 10−7 m2/Vs (circles). The lines are drawn as guides to the eye.

with the optimal band gap in between 1.9 eV and 2.0 eV, reaching an efficiency of 8.6%.Surprisingly, the optimal band gap when the LUMO(A)-LUMO(D) offset is kept at 0.5eV is very close to the present P3HT value of 2.1 eV, demonstrating that the usage ofsmall band gap polymers is not the most efficient way of increasing the performance.

Up to this point we have not considered the influence of charge carrier mobility. Thethickness of current polymer/fullerene BHJs is limited by the rather low hole mobilityof the polymer phase as compared to the electron mobility of the fullerene. Typically,increasing the thickness of the active layer beyond 150 nm leads to a decrease in fill factor.Lenes et al. have shown that the decrease in fill factor is due to a combination of chargerecombination and space charge effects. [17] On the other hand, device performance isexpected to be enhanced by a thicker active layer since more light is absorbed. Therefore,the effect of increasing the hole mobility to the value for the electron mobility, i.e., 2.0× 10−7 m2/V s is studied, in combination with a polymeric band gap of 1.9 eV anda LUMO(A)-LUMO(D) offset of 0.5 eV, corresponding to an optimal situation. A 97 nmthick device with these specifications would yield an efficiency of 9.2%, see Fig. 6.5. Sucha high value of the hole mobility in polymer systems is not unrealistic: By optimizingthe processing conditions, an even slightly higher value has been obtained. [18]

In order to vary the active layer thickness, it is necessary to recalculate the volumegeneration rate of electron-hole pairs. The absorption at each wavelength is calculatedfrom the absorption coefficient, taking into account the reflection of the aluminum elec-trode. By integrating this over the AM1.5 spectrum, one gets the relative value for thegeneration rate. [5] By performing this calculation for various layer thicknesses, the re-sulting efficiency can be estimated. It should be noted that this is a simplified procedureand it would be more accurate to incorporate optical interference effects in the device, [19]

however, the inclusion of an absorption profile as found by Hoppe et al. influences theoutcome by less than 0.2%.

95


Figure 6.5 shows the variation of the efficiency with active layer thickness for bothvalues of the hole mobility. As expected, the optimal thickness for the situation with thecurrent hole mobility is around 100–150 nm, as observed experimentally. Increasing thehole mobility causes the optimum to shift toward 200 nm. The efficiency at this thicknessis 10.8%, showing the great potential of polymer/fullerene based solar cells.

It is worthy of note, that in the present analysis—by taking P3HT/PCBM as a startingpoint and only changing the parameters under investigation—several (implicit) assump-tions are made. First of all, in this calculation, the absorption of the fullerene is neglected.Depending on the chemical structure of the fullerene, this may or may not be a seriousomission. Furthermore, it is assumed that the dissociation of electron-hole pairs is notaffected by changing the energy levels of the materials: Neither the possibility of tripletformation, [16] nor the possible influence of the LUMO(A)-LUMO(D) offset on the sepa-ration distance a has been included. All in all, the P3HT/PCBM system functions only asan example of a generic strategy and, therefore, is not as general as the detailed balancelimit for p-n junctions.

6.3 Conclusions

In this chapter, it was shown that the device efficiency of P3HT/PCBM bulk heterojunc-tion solar cells would greatly benefit from tuning of the LUMO level of PCBM in sucha way that the LUMO(A)-LUMO(D) offset would be 0.5 eV. In that case the efficiencycould be as high as 8%. Another, much pursued, way to improve the performance is toincrease the amount of photons absorbed by the film by decreasing the band gap of thepolymer. Calculations based on the MIM model confirm that this would indeed enhancethe performance. However, the best efficiency is reached when both effects are com-bined, i.e., favourable LUMO’s of both donor and acceptor and tuning of the polymericband gap. The optimal band gap lies rather close to the present value, however. This in-dicates that, although lowering the polymeric band gap enhances the efficiency, it wouldbe more benefical to either lower the LUMO of PCBM or find another acceptor with amore favourable LUMO level combined with good charge transporting properties. Withbalanced charge transport, polymer/fullerene solar cells can reach power conversionefficiencies of at least 10.8%.

96


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[3] J. J. Loferski, J. Appl. Phys. 27, 777 (1956).

[4] K. M. Coakley and M. D. McGehee, Chem. Mater. 16, 4533 (2004).


[6] M. C. Scharber, D. Muhlbacher, M. Koppe, P. Denk, C. Waldauf, A. J. Heeger, and C. J. Brabec,Adv. Mater. 18, 789 (2006).

[7] J. Roncali, Chem. Rev. 97, 173 (1997).

[8] C. Winder and N. S. Sariciftci, J. Mater. Chem. 14, 1077 (2004).

[9] C. J. Brabec, C. Winder, N. S. Sariciftci, J. C. Hummelen, A. Dhanabalan, P. A. van Hal, andR. A. J. Janssen, Adv. Funct. Mater. 12, 709 (2002).

[10] S. E. Shaheen, Synth. Met. 121, 1583 (2001).

[11] J. K. J. van Duren, A. Dhanabalan, P. A. van Hal, and R. A. J. Janssen, Synth. Met. 121, 1587(2001).

[12] L. J. A. Koster, V. D. Mihailetchi, R. Ramaker, and P. W. M. Blom, Appl. Phys. Lett. 86, 123509(2005).

[13] C. J. Brabec, A. Cravino, D. Meissner, N. S. Sariciftci, T. Fromherz, M. T. Rispens, L. Sanchez,and J. C. Hummelen, Adv. Funct. Mater. 11, 374 (2001).

[14] J. J. M. Halls, J. Cornill, D. A. dos Santos, R. Silbey, D. -H. Hwang, A. B. Holmes, J. L. Bredas,and R. H. Friend, Phys. Rev. B 60, 5721 (1999).

[15] S. Barth and H. Bassler, Phys. Rev. Lett. 79, 4445 (1997).

[16] T. A. Ford, I. Avilov, D. Beljonne, and N. C. Greenham, Phys. Rev. B 71, 125212 (2005).

[17] M. Lenes, L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Appl. Phys. Lett. 88, 243502(2006).

[18] V. D. Mihailetchi, H. Xie, B. de Boer, L. M. Popescu, J. C. Hummelen, P. W. M. Blom, andL. J. A. Koster, Appl. Phys. Lett. 89, 012107 (2006).

[19] H. Hoppe, N. Arnold, N. S. Sariciftci, and D. Meissner, Solar Energy Mater. & Solar Cells 80,105 (2003).

97

Publications

1. L. J. A. Koster, W. J. van Strien, W. J. E. Beek, and P. W. M. Blom, Device operation ofconjugated polymer/zinc oxide bulk heterojunction solar cells, Adv. Funct. Mater. (to bepublished).

2. P. W. M. Blom, V. D. Mihailetchi, L. J. A. Koster, and D. E. Markov, Device physics ofpolymer:fullerene bulk heterojunction solar cells, Adv. Mater. (to be published).

3. V. D. Mihailetchi, H. Xie, B. de Boer, L. J. A. Koster, and P. W. M. Blom, Trans-port and Photocurrent Generation in Poly(3-hexylthiophene):Methanofullerene Bulk-Heterojunction Solar Cells, Adv. Funct. Mater. 16, 699 (2006).

4. L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Ultimate efficiency of poly-mer/fullerene bulk heterojunction solar cells, Appl. Phys. Lett. 88, 093511 (2006).

5. M. M. Mandoc, L. J. A. Koster, and P. W. M. Blom, Optimum charge carrier mobilityin organic solar cells (submitted).

6. L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Improving the efficiency of plasticsolar cells, SPIE Newsroom, DOI: 10.1117/2.1200606.0289 (2006).

7. L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, The optimal band gap for plasticsolar cells, SPIE Newsroom (to be published).

8. M. Lenes, L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Thickness dependenceof the efficiency of polymer:fullerene bulk heterojunction solar cells, Appl. Phys. Lett. 88,243502 (2006).

9. L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Bimolecular recombination inpolymer/fullerene bulk heterojunction solar cells, Appl. Phys. Lett. 88, 052104 (2006).

10. V. D. Mihailetchi, H. Xie, B. de Boer, L. J. A. Koster, L. M. Popescu,J. C. Hummelen, and P. W. M. Blom, Origin of the enhanced performance in poly(3-exylthiophene):methanofullerene solar cells using slow drying, Appl. Phys. Lett. 89,012107 (2006).

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11. L. J. A. Koster, E. C. P. Smits, V. D. Mihailetchi, and P. W. M. Blom, Device model forthe operation of polymer/fullerene bulk heterojunction solar cells, Phys. Rev. B 72, 085205(2005).

12. L. J. A. Koster, V. D. Mihailetchi, R. Ramaker, and P. W. M. Blom, Light intensitydependence of open-circuit voltage of polymer:fullerene solar cells, Appl. Phys. Lett. 86,123509 (2005).

13. L. J. A. Koster, V. D. Mihailetchi, H. Xie, and P. W. M. Blom, Origin of thelight intensity dependence of the short-circuit current of polymer/fullerene solar cells,Appl. Phys. Lett. 87, 203502 (2005).

14. V. D. Mihailetchi, L. J. A. Koster, and P. W. M. Blom, Effect of metal electrodes on theperformance of polymer:fullerene bulk heterojunction solar cells, Appl. Phys. Lett. 85, 970(2005).

15. V. D. Mihailetchi, L. J. A. Koster, P. W. M. Blom, C. Melzer, B. de Boer,J. K. J. van Duren, and R. A. J. Janssen, Compositional dependence of the per-formance of poly(p-phenylene vinylene):methanofullerene bulk-heterojunction solar cells,Adv. Funct. Mater. 15, 795 (2005).

16. V. D. Mihailetchi, L. J. A. Koster, J. C. Hummelen, and P. W. M. Blom, Photocurrentgeneration in polymer-fullerene bulk heterojunctions, Phys. Rev. Lett. 93, 216601 (2004).

17. L. J. A. Koster, V. D. Mihailetchi, J. C. Hummelen and P. W. M. Blom, Performanceenhancement of poly(3-hexylthiophene):methanofullerene bulk-heterojunction solar cells,Proceedings of SPIE 6334 (to be published).

18. L. J. A. Koster, V. D. Mihailetchi, R. Ramaker, H. Xie, and P. W. M. Blom, Lightintensity dependence of open-circuit voltage and short-circuit current of polymer/fullerenesolar cells, Proceedings of SPIE 6192, 122 (2006).

19. M. Lenes, V. D. Mihailetchi, L. J. A. Koster, and P. W. M. Blom, Space-charge forma-tion in thick MDMO-PPV:PCBM solar cells, Proceedings of SPIE 6192, 120 (2006).

20. L. J. A. Koster, V. D. Mihailetchi, B. de Boer, and P. W. M. Blom, Modeling of poly(3-hexylthiophene):methanofullerene bulk-heterojunction solar cells, Proceedings of SPIE6192, 10 (2006).

21. V. D. Mihailetchi, B. de Boer, C. Melzer, L. J. A. Koster, and P. W. M. Blom, Electronand hole transport in poly(para-phenylenevinylene):methanofullerene bulk heterojunctionsolar cells, Proceedings of SPIE 5520, 20 (2004).

22. L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Extraction of photo-generatedcharge carriers from polymer-fullerene bulk heterojunction solar cells, Proceedings ofSPIE 5464, 239 (2004).

23. L. J. A. Koster, V. D. Mihailetchi, and P. W. M. Blom, Modeling the photocurrentof poly-phenylene vinylene/fullerene-based solar cells, Proceedings of SPIE 5520, 200(2004).

Summary

Harvesting energy directly from the Sun is a very attractive, but not an easy way ofproviding mankind with energy. Efficient, cheap, lightweight, flexible, and environmen-tally friendly solar panels are very desirable. Conjugated polymers bear the potentialof fulfilling these requisites. Due to their unique chemical makeup, these polymers canbe used as optoelectronically active materials, e.g., they can be optically excited and cantransport charge carriers.

As compared to inorganic materials, polymers have (at least) one serious drawback:upon light absorption excitons are formed, rather than free charge carriers. A secondmaterial is needed to break up these excitons. A much used way of achieving this is tomix the polymer with a material that readily accepts the electrons, leaving the holes inthe polymer phase. As excitons in the polymer phase only move around for a couple ofnanometers before they decay to the ground state, it is vital to induce a morphology thatis characterized by intimate mixing of both materials (a so-called bulk heterojunction orBHJ).

A typical BHJ solar cell, see Fig. 1(a), consists of a glass substrate coated with a trans-parent electrode, the active layer, and a metallic top electrode. The active layer is formedby spin casting a co-solution of the polymer and the electron accepting material. Figure1(b) shows a typical current-voltage curve of a BHJ solar cell. The voltage for which thecurrent in the external circuit is zero is called the open-circuit voltage Voc. The currentdensity that flows out of the solar cell at zero bias is named the short-circuit currentdensity Jsc. These two important quantities are described in the following.

Although significant progress has been made, the efficiency of current BHJ solar cellsstill does not warrant commercialization. Targeted improvement is hindered by lim-ited understanding of the factors that determine the performance. The main theme ofthis thesis is to introduce a simple model for the electrical characteristics of BHJ solarcells relating their performance to basic physics and material properties such as chargecarrier mobilities. The metal-insulator-metal (MIM) model, as introduced in this work,describes the generation and transport processes in the BHJ as if occurring in one virtual

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0.0 0.3 0.6 0.9

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Voc

Jsc

J L [A

/m2 ]

Va [V]

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Figure 1: (a) Schematic layout of a BHJ solar cell. (b) Typical current (JL) - voltage (Va) curve of aBHJ solar cell. The open-circuit voltage and the short-circuit current density are also indicated.

semiconductor. Drift and diffusion of charge carriers, the effect of charge density on theelectric field, bimolecular recombination, and a temperature- and field-dependent gener-ation mechanism of free charges are incorporated. By using (values close to) measuredcharge carrier mobilities, the experimental current-voltage characteristics are regainedby the MIM model, showing the soundness of this approach.

Although bimolecular recombination in organic semiconductors can be adequatelydescribed by Langevin’s equation, meaning that the recombination strength depends onthe sum of the charge carrier mobilities, BHJs behave differently. As is known from directmeasurements, the bimolecular recombination strength in BHJs is significantly smallerthan predicted by the Langevin equation. From the modeling of current-voltage charac-teristics, it is found that the bimolecular recombination strength is indeed significantlyreduced, and is governed by the mobility of the slowest charge carrier and not by thesum of the mobilities.

The MIM model sheds new light on two key parameters of BHJ solar cells: the open-circuit voltage and the short-circuit current. By studying the dependence of Voc on in-cident light intensity, it is established that BHJs behave differently than inorganic p-njunctions. Within the framework of the MIM model, an alternative explanation for theopen-circuit voltage is presented. Based on the notion that the quasi-Fermi potentialsare constant throughout the device, a formula for Voc is derived that consistently de-scribes the open-circuit voltage. In short, if suitable electrodes are applied to the activelayer, Voc is determined by the energy levels of both materials. The energy needed todissociate excitons represents an important loss in Voc.

Simple analytical expressions for the current that can be drawn from a photoconduc-tor indicate that the short-circuit current density should be equal to qGL, where q is the

elementary charge, G the generation rate of free electrons and holes, and L is the thick-ness of the active layer. In this case, Jsc is proportional (through G) to the intensity Iof light incident on the device. This linear dependence has been observed in many sys-tems. A small deviation from linearity, in which case Jsc ∝ Iα with 0.85 ≤ α ≤ 1, wasalso reported for various systems. This sublinear behavior was ascribed to bimolecularrecombination.

In the 1970’s Goodman and Rose pointed out that the photocurrent can become lim-ited by space charge, provided that the active layer be thick enough and there exists adifference between electron and hole mobilities. Under these premises, the photocurrentis expected to be proportional to I0.75. This suggests that the exponent α is a function ofthe charge carrier mobilities and that the sublinear behavior is caused by space-charge-buildup. Numerical modeling and measurements on a suitable BHJ system confirm thatthe intensity dependence of Jsc is indeed governed by space charge rather than by bi-molecular recombination per se.

Hybrid organic/inorganic solar cells, are an auspicious alternative to poly-mer/fullerene devices. In this case, an inorganic semiconductor, either titanium dioxideor zinc oxide, is used as the electron acceptor. One way of making these cells is the pre-cursor route: A precursor for the inorganic semiconductor is mixed with the solution ofthe polymer. Upon spin casting of the active layer in ambient conditions, the precursorreacts with moisture from the air and the inorganic semiconductor is formed. Althoughpromising, this method seems to harm the transport of holes through the polymer phasein the active layer.

Alternatively, the inorganic semiconductor, in this case zinc oxide, can be formedex situ in the form of nanoparticles. This enables one to control the reaction conditionsand purity of the material better. It is demonstrated that the hole transport through thethus-formed blends is not affected by the presence of the zinc oxide nanoparticles. Theelectron mobility in blends with the often used conjugated polymer MDMO-PPV is quitedecent and, consequently, the hole transport through the polymer phase is identified asthe limiting factor in these devices.

A much pursued way to increase the performance is to increase the amount of pho-tons absorbed by the film by decreasing the band gap of the polymer. Calculations basedon the model presented in this work confirm that this would indeed enhance the perfor-mance. However, it is demonstrated that the effect of minimizing the energy loss in theelectron transfer from the polymer to the acceptor phase is even more beneficial. Bycombining these two effects—under the premise of just sufficient driving force for exci-ton dissociation—it turns out, that the optimal band gap of the polymer is 1.9 eV. Thisis significantly higher than what is predicted for p-n junction solar cells (1.4 eV). Withbalanced charge transport, polymer/fullerene solar cells can reach power conversionefficiencies of at least 10.8%.

Samenvatting

Het direct uit zonlicht oogsten van energie is een aantrekkelijke, maar zeker niet een-voudige, manier om de mensheid van energie te voorzien. Efficiente, goedkope, lichte,flexibele en mileu-vriendelijke zonnepanelen zijn zeer begerenswaardig. Geconjugeerdepolymeren hebben het in zich om aan deze voorwaarden te voldoen. Dankzij hun uniekechemische structuur kunnen deze polymeren gebruikt worden als optoelektronisch ac-tieve materialen; ze kunnen bijvoorbeeld optische geexciteerd worden en ze kunnen lad-ingsdragers transporteren.

In vergelijking met anorganische materialen hebben polymeren (tenminste) een be-langrijk nadeel: Lichtabsorptie leidt tot de creatie van excitonen in plaats van vrije lad-ingsdragers. Een tweede materiaal is nodig om deze excitonen te splitsen. Een veelgebruikte methode om dit te bereiken is het mengen van het polymeer met een materi-aal dat makkelijk elektronen accepteert, terwijl de gaten op het polymeer achterblijven.Aangezien excitonen slechts een paar nanometer heen en weer bewegen voordat ze ver-vallen naar de grondtoestand, is het cruciaal om een morfologie te bewerkstelligen diegekenmerkt wordt door een zeer fijne mening van beide materialen (een zogenaamdebulk heterojunctie of BHJ).

Een typische BHJ zonnecel, zie Fig. 1(a), bestaat uit een glazen substraat dat be-dekt is met een transparante elektrode, de actieve laag en een metalen bovenelektrode.De actieve laag wordt gevormd door het substraat te bedekken met een oplossing vanhet polymeer en een elektronen acceptor, waarna het oplosmiddel verdampt. Figuur1(b) laat een typische stroom-spanning curve zien van een BHJ zonnecel. De span-ning waarbij de stroom in het externe circuit nul is, heet de openklemspanning Voc. Destroomdichtheid die uit de zonnecel stroomt bij afwezigheid van een aangelegde span-ning wordt de kortsluit stroomdichtheid Jsc genoemd. Deze twee belangrijke groothedenzullen in het vervolg worden beschreven.

Niettegenstaande de belangrijke vooruitgang die is geboekt, is de effientie van dehuidige BHJ zonnecellen te laag om commerciele productie te rechtvaardigen. Gerichteverbetering wordt bemoeilijkt door het beperkte inzicht in de factoren die de efficientie

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0.0 0.3 0.6 0.9

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Figuur 1: (a) Schema van een BHJ zonnecel. (b) Typische stroom (JL) - spanning (Va) curve vaneen BHJ zonnecel. De openklemspanning en de kortsluit stroomdichtheid zijn ook aangegeven.

bepalen. Het belangrijkste doel van dit proefschrift is de introductie van een invoudigmodel voor de elektrische eigenschappen van BHJ zonnecellen, dat hun prestaties re-lateert aan basale fysica en materiaaleigenschappen zoals de mobiliteiten van ladings-dragers. Het metaal-isolator-metaal (MIM) model, zoals geıntroduceerd in dit werk,beschrijft de generatie- en transportprocessen in de BHJ alsof die zich voordoen in eenvirtuele halfgeleider. Drift en diffusie van ladingsdragers, de invloed van ladings-dichtheid op het elektrisch veld, bimoleculaire recombinatie en een temperatuurs- enveldafhankelijk generatiemechanisme zijn in het model omvat. Door (waarden dichtbij)gemeten ladingsdragersmobiliteiten te gebruiken, kunnen de experimentele stroom-spanning curven worden gereproduceerd, hetgeen de redelijkheid van deze aanpak laatzien.

Ondanks dat bimoleculaire recombinatie in organische halfgeleiders goed beschre-ven wordt door de vergelijking van Langevin, wat inhoudt dat de recombinatiesterkteafhangt van de som van de ladingsdragersmobiliteiten, gedragen BHJs zich anders.Zoals blijkt uit directe metingen, is de bimoleculaire recombinatiesterkte in BHJs sig-nificant lager dan voorspeld door de Langevin uitdrukking. Uit het modelleren vanstroom-spanning curven kan geconcludeerd worden dat de bimoleculaire recombinati-esterkte inderdaad significant zwakker is, en dat die gedomineerd wordt door de mo-biliteit van de langzaamste ladingsdrager en niet door de som van de mobiliteiten.

Het MIM model werpt nieuw licht op twee zeer belangrijke parameters: de open-klemspanning en de kortsluit stroomdichtheid. Door de afhankelijkheid van Voc op delichtintensiteit te bestuderen, is duidelijk dat BHJs zich anders gedragen dan anorgan-ische p-n juncties. Binnen het raamwerk van het MIM-model wordt een alternatieveverklaring voor de Voc gepresenteerd. Gebaseerd op de notie dat de quasi-Fermi po-

tentialen constant zijn binnen de zonnecel, wordt een formule voor Voc afgeleid die deopenklemspanning consistent beschrijft. Kort samengevat, als geschikte elektrodes wor-den gebruikt, dan wordt Voc bepaald door de energie niveaus van beide materialen. Deenergie die nodig is om de excitonen te dissocieren beperkt Voc in belangrijke mate.

Eenvoudige analytische uitdrukkingen voor de stroom die uit een fotogeleidergetrokken kan worden geven aan, dat de kortsluit stroomdichtheid gegeven zou moetenworden door qGL, waarbij q de elementaire lading is, G de generatiesnelheid van vrijeelektronen en gaten en L de dikte van de actieve laag. In dat geval is Jsc evenredig (doorG) met de intensiteit I van het licht dat op de zonnecel valt. Deze lineaire afhankelijkheidis voor vele systemen waargenomen. Een kleine afwijking van dit lineaire gedrag,waarbij Jsc ∝ Iα met 0.85 ≤ α ≤ 1, is echter ook gerapporteerd voor vele systemen.Dit sublineaire gedrag werd toegeschreven aan bimoleculaire recombinatie.

In de jaren ’70 van de vorige eeuw hebben Goodman en Rose er op gewezen datde fotostroom kan worden gelimiteerd door ruimtelading, vooropgesteld dat de actievelaag dik genoeg is en dat er een verschil bestaat tussen de mobiliteit van gaten en elektro-nen. Onder deze voorwaarden, wordt verwacht dat de fotostroom evenredig is met I0.75.Dit suggereert dat de exponent α een functie is van de ladingsdragersmobiliteiten en dathet sublineaire gedrag veroorzaakt wordt door opbouw van ruimtelading. Numeriekeberekeningen en metingen aan een geschikt BHJ systeem bevestigen dat de intensiteit-safhankelijkheid van Jsc inderdaad bepaald wordt door ruimtelading, in plaats van doorbimoleculaire recombinatie per se.

Hybride organische/anorganische zonnecellen zijn een veelbelovend alternatief voorpolymeer/fullereen zonnecellen. In dit geval wordt een anorganische halfgeleider, tita-niumdioxide of zinkoxide, gebruikt als elektron acceptor. Een manier om deze cellente maken is door het een voorloper van het anorganische materiaal te mengen met deoplossing van het polymeer. Door dit mengsel in lucht te verwerken, wordt de voor-loper omgezet in een anorganische halfgeleider. Hoewel veelbelovend, lijkt het erop dathet deze methode het transport van gaten door de polymeerfase van de actieve laagverslechtert.

De anorganische halfgeleider, in dit geval zinkoxide, kan ook ex situ gesynthetiseerdworden, bijvoorbeeld in de vorm van nanodeeltjes. Hierdoor is het mogelijk om meercontrole uit te oefenen over de de reactie omstandigheden en de zuiverheid van hetmateriaal. Het transport van gaten door de polymeerfase blijkt niet te lijden onderde aanwezigheid van de zinkoxide nanodeeltjes. De elektronenmobiliteit in mengselsmet het veel gebruikte geconjugeerde polymeer MDMO-PPV is heel behoorlijk en, dien-tengevolge, blijkt het gatentransport de beperkende factor te zijn in deze zonnecellen.

Een veel onderzochte manier om de prestaties te verbeteren, is het verhogen van hetaantal fotonen dat door de film wordt geabsorbeerd door middel van het verkleinenvan de energiekloof van het polymeer. Berekeningen gebaseerd op het model zoalshier gepresenteerd laten zien dat dit inderdaad leidt tot een verhoging van de ef-ficientie. Het blijkt echter, dat het minimaliseren van het energieverlies in de elek-tronenoverdracht van het polymeer naar de acceptor nog voordeliger is. Door dezetwee effecten te combineren—onder de premisse van juist voldoende drijvende kracht

voor excitondissociatie—kan worden aangetoond dat de optimale energiekloof vanhet polymeer 1,9 eV is. Dit is significant hoger dan wat voorspeld wordt voor p-njunctie zonnecellen (1.4 eV). Indien het ladingstransport evenwichtig is, kunnen poly-meer/fullereen zonnecellen efficienties halen van tenminste 10,8%.

Dankwoord

Mijn dank gaat allereerst uit naar mijn promotor en begeleider Paul Blom, omdathij mij de kans heeft gegeven mijn jongensdroom te vervullen. Daarnaast heefthij mij altijd kundig begeleid en geleerd hoe een experimenteel fysicus te zijn. Bert deBoer wil ik ook hartelijk danken voor zijn vele adviezen, zowel op chemisch als op al-gemeen wetenschappelijk gebied. I am very grateful for the valuable comments madeby the members of the reading committee, Laurens Siebbeles, Kees Hummelen, and NeilGreenham.

Lenneke Slooff en Waldo Beek hebben ook een belangrijke rol in mijn onderzoek naarhybride zonnecellen gespeeld, waarvoor ik hun veel dank ben verschuldigd. During myresearch, I have had a very pleasant and fruitful collaboration with Valentin Mihailetchi.I am very grateful for his valuable input, without which this thesis would have been alot shorter. I would also like to thank all other (former) colleagues at the Physics of Or-ganic Semiconductors group: Teunis, Cristina, Denis, Ronald, Magda, Maria, Francesco,Irina, Kamal, Margo, Andre, Martijn L., Martijn K., Herman, Yuan, Auke, Jan K., andWelmoed. Hylke Akkerman en Afshin Hadipour wil ik bedanken voor onze goedegesprekken en mooie avondturen in binnen- en buitenland. De afstudeerstudenten dieik heb mogen begeleiden nemen een aparte plaats in; Edsger Smits, Wouter van Strienen Date Moet wil ik hartelijk danken voor hun inzet. Hun bijdragen aan mijn onder-zoek zijn bijzonder nuttig geweest. Zonder de ondersteuning van Minte Mulder, JurjenWildeman, Jan Harkema, Frans van der Horst, Renate Slagter en Linda Schneider zouhet een stuk moeilijker zijn geweest om tot resultaten te komen.

Bij het doen van wetenschappelijk onderzoek is ontspanning zeer belangrijk, in hetleven is vriendschap onontbeerlijk. Ik ben mijn vrienden, en met name Gerdi, Petra,Daan Hein, Mei-Nga en Paul, dan ook zeer erkentelijk voor hun rol hierbij. In ditgezelschap nemen Hans en Coen een bijzondere plek in, daar zij ook hebben bijgedra-gen aan mijn wetenschappelijke vorming. Mijn paranimfen, Jan D. en Jaap Jan wil ikhartelijk danken voor hun vriendschap door de jaren heen en hun hulp bij het volbren-gen van mijn promotie.

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Mijn ouders ben ik onnoemelijk veel dank verschuldigd voor alle steun, liefde engeduld die zij mij gedurende mijn leven hebben gegeven. Ook mijn broer en zus dank ikvoor de bijzondere band die wij hebben.

Dat er buiten de natuurkunde nog een hele wereld bestaat wist ik al, maar alleendankzij Anke weet ik hoe mooi die wereld is; een ontdekking die alle wetenschap teboven gaat.

Jan Anton

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