University of Groningen
Device physics of donor/acceptor-blend solar cells Koster, Lambert Jan Anton
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solar cells
Lambert Jan Anton Koster
Lambert Jan Anton Koster PhD thesis
University of Groningen, The Netherlands MSC PhD thesis series 2007-04
ISSN 1570-1530
ISBN-13: 9789036729413
The research described in this thesis forms part of the research program of the Dutch Polymer Institute (DPI), project #323.
Device physics of donor/acceptor-blend solar cells
Proefschrift
ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen
aan de Rijksuniversiteit Groningen op gezag van de
Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op
vrijdag 23 februari om 16.15 uur
door
Lambert Jan Anton Koster
geboren op 24 mei 1975 te Hoogeveen
Beoordelingscommissie : Dr. N. C. Greenham Prof. dr. J. C. Hummelen Prof. dr. L. D. A. Siebbeles
1 Introduction to organic solar cells 1
1.1 Solar energy . . . 2
1.2 Conjugated polymers . . . 2
1.3 Transport of charges in conjugated polymers . . . 3
1.3.1 Hopping transport in disordered systems . . . 4
1.3.2 Transport in conjugated polymers . . . 5
1.3.3 Measuring the charge carrier mobility . . . 6
1.3.4 Conjugated polymers used in this thesis . . . 6
1.4 Organic photovoltaics in a nutshell . . . 7
1.5 Device fabrication and characterization . . . 10
1.6 Objective and outline of this thesis . . . 11
2 Metal-insulator-metal model for bulk heterojunction solar cells 19 2.1 Introduction . . . 20
2.2 Description of the model . . . 21
2.2.1 Basic equations . . . 22
2.2.2 Generation of free charge carriers . . . 24
2.2.3 Bimolecular recombination in bulk heterojunctions . . . 27
2.3 Numerical method . . . 28
2.4 Simulation results and discussion . . . 30
2.4.1 Bimolecular recombination: Experimental results . . . 30
2.4.2 Modeling MDMO-PPV/PCBM bulk heterojunction solar cells . . . 34
2.5 Summary and conclusions . . . 40
3 Open-circuit voltage of bulk heterojunction solar cells 45 3.1 Open-circuit voltage in p-n junction models . . . . 46
3.2 Open-circuit voltage in the MIM model . . . 49
3.2.1 General considerations . . . 49
3.3 Comparison with other solar cells . . . 53
3.3.1 Influence of non-homogeneity . . . 53
3.3.2 Comparison with (in)organic low mobility solar cells . . . 55
3.4 Conclusions . . . 56
4 Short-circuit current of bulk heterojunction solar cells 61 4.1 Introduction . . . 62
4.1.1 Uniform field approximation . . . 62
4.1.2 Space-charge-limited photocurrents . . . 63
4.2 Intensity dependence of the short-circuit current . . . 64
4.2.1 Numerical results . . . 66
4.3 Conclusions . . . 69
4.4 Experimental . . . 69
5 Hybrid organic/inorganic solar cells 71 5.1 Introduction to hybrid organic/inorganic solar cells . . . 72
5.2 Hybrid solar cells with acceptors from a precursor . . . 72
5.2.1 Using a precursor for titanium dioxide . . . 72
5.2.2 Using a precursor for zinc oxide . . . 73
5.3 Polymer solar cells with zinc oxide nanoparticles . . . 77
5.3.1 Charge transport in MDMO-PPV/nc-ZnO blends . . . 79
5.3.2 Improving the efficiency of MDMO-PPV/nc-ZnO solar cells . . . . 80
5.4 Conclusions . . . 84
5.5 Experimental . . . 84
6 Improving the efficiency of bulk heterojunction solar cells 89 6.1 Introduction . . . 90
6.2 Improving polymer/fullerene solar cells . . . 91
6.3 Conclusions . . . 96
Publications 99
Summary 101
Samenvatting 105
Dankwoord 109
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 chapter several aspects of polymer solar cells will be introduced. After discussing the transport of charge in conjugated polymers, the electro-optical processes in bulk heterojunction solar cells are discussed. Finally, an overview of this thesis is given.
1.1 Solar energy
What can be a more attractive way of producing energy than harvesting it directly from sunlight? The amount of energy that the Earth receives from the sun is enormous: 1.75
×1017W. As the world energy consumption in 2003 amounted to 4.4×1020J, 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 to 39%,[1]while commerically available solar panels, have a significantly lower efficiency of 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 solar cells allow. Although significant progress has been made, the efficiency of converting solar energy into electrical power obtained with plastic solar cells still does not warrant commercialization: the most efficient devices have an efficiency of 4-5%.[2]To improve the efficiency of plastic solar cells it is, therefore, crucial to understand what limits their performance.
1.2 Conjugated polymers
Since Shirakawa, MacDiarmid, and Heeger demonstrated in 1977 that the conductivity of conjugated polymers can be controlled by doping,[3]a new field has emerged. They were 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 the formation of σ bonds between the constituent carbon atoms. In conjugated polymers, e.g., polyacetylene, the situation is different: In these polymers, the bonds between the carbon 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 in a pzorbital. The mutual overlap between these pzorbitals results in the formation of π bonds along the conjugated backbone, thereby delocalizing the π electrons along the entire 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 highest occupied 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
Figure 1.1: In polyacetylene, the bonds between adjacent carbon atoms are alternatingly single or double.
together by the σ bonds, falling apart. Therefore, it is possible to promote an electron from the HOMO to the LUMO level upon, for example, light absorption.
As the band gap (energy difference between the HOMO and LUMO) of a conjugated system depends on its size,[9] any disturbance of the conjugation along the polymer’s backbone will change the local HOMO and LUMO positions. Real conjugated polymers are therefore subject to energetic disorder. The density of states of these systems is often approximated 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 have a three dimensional periodical lattice structure, charge transport in polymers cannot be described by standard semiconductor models. As these systems show energetic and spatial disorder, the concept of band conduction of free charge carriers does not apply. In this section, a summary is given of how charge carrier transport in conjugated polymers and 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 the 1970s the charge transport in molecularly doped polymers was studied by performing time-of-flight (TOF) measurements. In this type of experiment, a sample is sandwiched between two non-injecting electrodes.∗A short light pulse is used to illuminate one side of 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 traverse the 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 described by[11–15]
µ=µ0exp(γ√
F), (1.1)
where µ0 is the zero-field mobility, F is the field strength, and γ is the field activation parameter.
∗Note, that no direct physical contact between the electrodes and the sample is necessary.
1.3.1 Hopping transport in disordered systems
How can the results summarized in Eq. (1.1) be rationalized? As these materials are disordered, the concept of band conduction does not apply. Instead, localized states are formed and charge carriers proceed from one such a state to another (hopping), thereby absorbing 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 ǫito an unoccupied state j with energy ǫjis described by[18]
Wij=ν0exp(−2γRij)
(exp
−ǫkj−ǫBTi ǫi<ǫj
1 ǫi≥ǫj, (1.2)
where ν0is the attempt-to-jump frequency, Rij is the distance between the states i and j, γ is the inverse localization length, kBis Boltzmann’s constant, and T is temperature.
The wave function overlap of states i and j is described by the first exponential term in Eq. (1.2), while the second exponential term accounts for the temperature dependence of the phonon density.
In his pioneering work, B¨assler described the transport in disordered organic systems as a hopping process in a system with both positional and energetic disorder.[10]The hopping rates between sites were assumed to obey Eq. (1.2) and the site energies varied according to a Gaussian distribution with a standard deviation σ. Such a system cannot be solved analytically. By performing Monte Carlo simulations, the following expression for the charge carrier mobility µ was proposed[10]
µ=µ∞e− 3kBT2σ
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 lattice spacing, and Σ describes the positional disorder.
Although Eq. (1.3) predicts a functional dependence on field strength similar to Eq. (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 spatial correlations between site energies into account.[19]In this model, the mobility takes the form[20,21]
µ=µ∞exp
"
−
3σ 5kBT
2
+0.78
σ kBT
3/2
−Γ
!rqaF σ
#
, (1.4)
∗Since, Eq. (1.3) is an expression which describes the outcome of Monte Carlo simulations, this is a purely mathematical definition of µ∞and does not mean that it has the physical meaning of the mobility at infinite temperature. At best, it may be interpreted as the mobility if there would be no barriers to hopping at all.
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 of charges 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 also observed 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 organic systems.[25]
In the foregoing discussion, only the dependence of the mobility on temperature and field strength was taken into account. When the applied voltage is increased in a TOF experiment, 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 an important dependence on charge carrier density.[26–29]Moreover, it was shown that the increase of the mobility with increasing bias voltage (and concomitant increase in carrier density) 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 is interpreted as an effect of the field only. It should be noted, however, that the polymers used 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 described here is quite weak. Additionally, as we will see in chapter 2, the carrier density in solar cells is fairly modest.
Several alternative models exist for explaining charge transport; one of them is the so-called polaron model which was first applied to inorganic crystals[30] and later to conjugated polymers.[31]An excess charge carrier in a solid causes a displacement of the atoms in its vicinity thus lowering the total energy of the system. This displacement of atoms results in a potential well for the charge carrier, thereby localizing it. The charge carrier 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
√ErTexp
−(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=µ0exp
−4kEr
BT − (aF)2 4ErkBT
sinh(aF/2kBt)
aF/2kBT . (1.6)
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 due to disorder.
1.3.3 Measuring the charge carrier mobility
When an insulator is contacted by an electrode that can readily inject a sufficiently large number of charge carriers — a so-called Ohmic contact — and another electrode that can 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 to determine the mobility in an experimental configuration that is relevant for solar cells.
Considering only one charge carrier (either electrons or holes), the SCL current density JSCLflowing across a layer with thickness L is given by[34]
JSCL= 9 8εµVint2
L3 , (1.7)
where ε is the dielectric constant of the material and Vint is the internal voltage drop across the active layer. When the mobility is of the form as given in Eq. (1.1), one can approximate JSCLby[35]
JSCL= 9
8εµ0e0.891γ√Vint/LVint2
L3 . (1.8)
The internal voltage in an actual device is related to the applied voltage Vaby
Vint=Va−Vbi−VRs, (1.9)
where Vbi is the built-in voltage which arises from the difference in work function of the bottom and top electrode and VRs is the voltage drop across the series resistance of the 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 becomes quadratic, corresponding to the SCL regime.
By judiciously choosing the electrode materials, the injection of either carrier type can be suppressed or enhanced, thereby enabling one to selectively assess either the hole or electron mobility. The way to do this, is to make sure that the work function of one of the electrodes is close to the energy level of the transport band under investigation, while there 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 used as 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. The polymer 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
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 for modeling purposes. Recently, another polymer has emerged: poly(3-hexylthiophene) (P3HT),∗ which is used in the most efficient polymer solar cells to date.[2] The final polymer 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, the zero field mobility amounts to 5×10−11m2/V s.[36] Surprisingly, the hole mobility of MDMO-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 blend consists of PCBM, the hole mobility of the polymer phase is equal to 2×10−8m2/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 its succes as a donor in BHJ solar cells with PCBM.
P3HT is unique in its own right: Padinger et al. observed that solar cells made from P3HT 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 the mobility:[40]In its pristine form the hole mobility amounts to 10−8m2/V s, see Fig. 1.3.
For comparison, Fig. 1.3 also shows the electron mobility of the PCBM phase in these blends. When blended with PCBM, the hole mobility initially decreases, however, upon annealing the hole mobility in the P3HT phase of the blend with PCBM is restored to its pristine 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 the efficiency solar cells based on a single organic material (a so-called homojunction) were unsuccessful, mainly because of the low dielectric constant of organic materials (typ-
∗In this research, only regio-regular P3HT is used
20 40 60 80 100 120 140 160 10
-11 10
-9 10
-7
pristine P3HT
holes
P3HT:PCBM
electrons
holes
a s-ca st
[m
2/Vs]
Annealing Temperature [ o
C]
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, the probability 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 case of 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 too weak 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. By carefully matching these materials, electron transfer from the donor to the acceptor, or hole transfer from the acceptor to the donor, is energetically favored. In 1992 Sariciftci et 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 to be dissociated the excitons must be generated in close proximity to the donor/acceptor interface, since the diffusion length is typically 5–7 nm.[47–49] This need limits the part of the active layer that contributes to the photocurrent to a very thin region near the donor/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 be overcome? In 1995 Yu et al. devised a solution:[7]By intimately mixing both components the interfacial area is greatly increased and the distance excitons have to travel in order to reach 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 breakthrough in terms of power conversion efficiency was reached by Shaheen et al. who showed that the solvent used has a profound effect on the morphology and performance of BHJ solar cells.[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]
Figure 1.4: Organic photovoltaics in a nutshell: Part (a) shows the process of light absorption by the polymer yielding an exciton which has to diffuse to the donor/acceptor interface. If the exciton reaches this interface, electron transfer to the acceptor phase is energetically favored, as shown in part (b), yielding a Coulombically bound electron-hole pair. The dissociation of the electron-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 electrodes in order to be extracted (d). Exciton decay is one possible loss mechanism, see part (e), while geminate recombination of the bound electron-hole pair and bimolecular recombination of free charge carriers (f) are two other possibilities.
Figure 1.5: Schematic layout of a BHJ solar cell. A part of the active layer is enlarged to illustrate the processes of light absorption and charge transport.
The main steps in photovoltaic energy conversion by organic solar cells are depicted in Fig. 1.4. The foremost process is light absorption by the polymer, yielding an exciton which 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-hole pairs, 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 substrate is coated with indium-tin-oxide (ITO) which is a transparent conductive electrode with a high work function, suitable to act as an anode. To reduce the roughness of this ITO layer 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 in vacuum through a shadow mask.
In order to determine the performance and electrical characteristics of the photo- voltaic devices, current-voltage measurements are performed (positive Vacorresponds to 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 current density under illumination at zero applied voltage Vais called the short-circuit current density Jsc.∗ The maximum voltage that the cell can supply, i.e., the voltage where the
∗Jscis taken positive throughout this thesis, as is customary.
0.0 0.3 0.6 0.9 -60
-30 0 30
J sc
V oc V
oc
J sc JL
[A/m
2]
V a
[V]
FF =
|J L
V a
| 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 voltage Voc. The fill factor FF is defined as
FF= |JLVa|max
VocJsc , (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 three quantities by
χ= JscVocFF
I , (1.11)
where I is the incident light intensity. Because of the wavelength and light intensity dependence 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 light intensity (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 the atmosphere).[51]
1.6 Objective and outline of this thesis
Although significant progress has been made, the efficiency of current BHJ solar cells still 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 for the electrical characteristics of BHJ solar cells, relating their performance to basic physics and material properties such as charge carrier mobilities.
The basis of this research is laid down in chapter 2, which describes the MIM model used 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 diffusion of charge carriers, the effect of charge density on the electric field, bimolecular recom- bination and a temperature- and field-dependent generation mechanism of free charges are incorporated. From the modeling of current-voltage characteristics, it is found that the bimolecular recombination strength is significantly reduced, and is governed by the slowest charge carrier. Subsequently, the numerical model is successfully applied to ex- perimental data on MDMO-PPV/PCBM solar cells, showing field and carrier density profiles.
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 correctly describe the open-circuit voltage of BHJ solar cells. 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 are constant throughout the device, a formula for Vocis derived that consistently describes the open-circuit voltage. Next, the predictions of 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 solar cells 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 ranges from 0.85 to 1. In this chapter, it is shown that this behavior does not originate from bimolecular 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, either titanium dioxide or zinc oxide, is used as the electron acceptor. One way of making these cells is the precursor route: A precursor for the inorganic semiconductor is mixed with 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 is formed. Although promising, this method seems to harm the transport of charge carri- ers through the active layer. Alternatively, the inorganic semiconductor, in this case zinc oxide, can be formed ex situ. This enables one to better control the reaction conditions and purity of the material. The transport of charge carriers as well as limitations to the efficiency are investigated in detail.
In chapter 6, various ways to improve the efficiency of bulk heterojunction solar cells are identified by using the MIM model as outlined in chapter 2. A much pursued way to increase the performance is to increase the amount of photons absorbed by the film by decreasing the band gap of the polymer. Calculations based on the MIM model confirm that this would indeed enhance the performance. However, it is demonstrated that the effect of minimizing the energy loss in the electron transfer from the polymer to the
fullerene derivative is even more beneficial. By combining these two effects, it turns out that the optimal band gap of the polymer would be 1.9 eV. Ultimately, with balanced charge transport, polymer/fullerene solar cells can reach power conversion efficiencies of 10.8%.
Table 1.1:List of symbols and abbreviations used in this thesis.
Symbol description
A acceptor
a electron-hole pair distance α exponent in Jsc∝Iα
AM1.5 air mass 1.5
BEH-PPV poly(2,5-bis(2’-ethylhexyloxy)-p-phenylene vinylene) BHJ bulk heterojunction
D donor
Dn(p) electron (hole) diffusion coefficient DSSC dye-sensitized solar cell
Eeffgap effective band gap
ε dielectric constant
η Poole-Frenkel detrapping parameter
F field strength
FF fill factor
φn(p) electron (hole) quasi-Fermi potential G generation rate of free charge carriers Ge−h generation rate of bound electron-hole pairs γ field activation parameter of mobility
hi grid spacing
HOMO highest occupied molecular orbital I incident light intensity
ITO indium tin oxide
JD current density in dark
JL current density under illumination Jn(p) electron (hole) current density Jph photogenerated current density Jsc short-circuit current density
kB Boltzmann’s constant
kdiss electron-hole pair dissociation rate kf electron-hole pair decay rate kr bimolecular recombination rate
L active layer thickness
LUMO lowest unoccupied molecular orbital
MDMO-PPV poly(2-methoxy-5-(3’,7’-dimethyl octyloxy)-p-phenylene vinylene) MIM model metal-insulator-metal model
µn(p) electron (hole) mobility
n electron density
Ncv effective density of states of valance and conduction bands nc-ZnO nanocrystalline zinc oxide
nint intrinsic carrier density
p hole density
P electron-hole pair dissociation probability
ψ potential
Continued on next page
Symbol description
P3HT poly(3-hexylthiophene)
PCBM 6,6-phenyl C61-butyric acid methyl ester
PEDOT:PSS poly(3,4-ethylene dioxythiophene):poly(styrene sulfonate)
Photo-CELIV photoinduced charge carrier extraction in a linearly increasing voltage PPV poly(phenylene vinylene)
prec-ZnO zinc oxide by precursor route
q elementary charge
R recombination rate of charge carriers S slope of Vocvs. ln(I)
SCL space-charge-limited
SRH Shockley-Read-Hall
σ width of Gaussian distribution energy distribution
T absolute temperature
TOF time-of-flight
U net generation rate of free carriers
V0 compensation voltage
Va applied voltage
Vbi built-in voltage
Vint internal voltage across active layer Voc open-circuit voltage
Vt thermal voltage
wn(p) electron (hole) drift length
x position
X density of bound electron-hole pairs χ power conversion efficiency h. . .i spatial average
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TWO
Metal-insulator-metal model for bulk heterojunction solar cells
Summary
In this chapter, a numerical device model is introduced that consistently describes the current-voltage characteristics of polymer/fullerene BHJ solar cells. This numerical model hinges on the description of the active blend layer as one effective medium (the so-called metal-insulator-metal model), described by basic semiconductor equations.
Drift and diffusion of charge carriers, the effect of charge density on the electric field, bimolecular recombination and a temperature- and field-dependent generation mechanism of free charges are incorporated. From the modeling of current-voltage characteristics, it is found that the bimolecular recombination strength is significantly reduced, and is governed by the slowest charge carrier.
Subsequently, the numerical model is successfully applied to experimental data on MDMO-PPV/PCBM (1:4 by weight) solar cells. As a result, it is demonstrated that in these devices space charge effects only play a minor role, leading to a relatively constant electric field in the active layer. Furthermore, at short-circuit conditions only a small fraction of free carriers is lost due to bimolecular recombination.
2.1 Introduction
An accurate and reliable numerical description of BHJ solar cells is highly desirable when 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-PPV and PCBM as donor and acceptor, respectively. At high reverse bias saturation of the photocurrent is observed, indicating that all electron-hole pairs are separated. From this observation 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 been suggested[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 large difference 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 such a way that the transport of the slowest carrier is enhanced. This results in a nonuniform field, since the charges of photogenerated electrons and holes do not cancel. Conse- quently, the slowest charge carrier will dominate the device because the faster carrier can 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 in MDMO-PPV.[5,6]This raises the question of whether the photocurrent will be dominated by a nonuniform electric field and resulting space charge. So far, no detailed description of bulk heterojunction solar cells clarifying field distribution and carrier densities has been 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 presented a numerical model describing the current-voltage characteristics of bilayer conjugated polymer photovoltaic devices. However, since the electronic structures of bilayers and bulk heterojunctions are distinct, their operational principles are fundamentally differ- ent.
In this chapter a device model is presented that quantitatively addresses the role of contacts, 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 relevant equations. Because of the particular morphology of BHJ solar cells, a new expression for the bimolecular recombination strength is proposed. Subsequently, the generation mechanism of free charge carriers is described, which completes the description of the model. 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 the simulations are presented, showing field and carrier density distributions for this type of
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 hole are transported through the respective materials and collected by the electrodes. As the anode a layer of poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) deposited on top of 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 in the metal-insulator-metal representation with positive applied bias Va, under operating conditions (Vasmaller than open-circuit voltage).
devices. An important loss process in solar cells is the bimolecular recombination of free charge carriers. The numerical simulations show that the recombination losses in BHJ solar cells strongly depend on the bias conditions; at short-circuit only a small fraction of free charge carriers is lost due to bimolecular recombination. At the maximum power output, 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? The metal-insulator-metal (MIM) model used in this thesis, is based on an effective medium approach, treating the blend of both components as one intrinsic semiconducting mate- rial. The lowest unoccupied molecular orbital (LUMO) of the acceptor and the highest occupied molecular orbital (HOMO) of the donor act as valence and conduction band of this virtual semiconductor, respectively (see Fig. 2.1 for the relevant energy levels of both materials).[8,12] The energy difference between the LUMO of the acceptor and the HOMO of the donor functions as the effective band gap (Eeffgap) of the semiconductor.
Note, that in a disordered system, like an organic solar cell, the band gap will not be a rigorously defined quantity due to the Gaussian density of states of both the acceptor and the donor material.[13]
The model used in this thesis contains drift and diffusion of charge carriers, and the effect of space charge on the electric field in the device. It should be mentioned, that the influence of disorder on the transport of carriers is only taken into account through the magnitude and field/temperature dependence of the mobility, leaving the equations themselves unaltered. The resulting basic equations describing transport through semi- conductors are solved self-consistently.[14]Recombination is described as a bimolecular
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 is known not to be strictly correct, the incorporation of an exponential dependence of the generation rate on distance, resulting from absorption of light by the active layer, does not significantly influence our results. As the devices considered in this study are very thin (120 nm), the assumption of uniform generation of electron-hole pairs does not give rise to serious inconsistencies.[16]
The generation of free charge carriers is a two-step process: exciton dissociation across 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) exciton dissociation, driven by the difference in LUMO levels of MDMO-PPV and PCBM, has a quantum efficiency of almost unity[12] and is assumed to be field independent. The resulting electron-hole pair is metastable (up to milliseconds at 80 K) and its dissociation is 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 based hole-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 characteristic for a space-charge-limited (SCL) current. The occurrence of SCL current enables one to directly determine the hole mobility from the current-voltage characteristics. It should be noted that a material with shallow traps would also exhibit an identical voltage and thickness dependence, and the observed mobility would be an effective mobility in that case, including trapping effects. However, transient measurements demonstrated that the measured mobility does not show any evidence of trapping effects.[19] The same holds for the electron transport in bulk PCBM.[5] Additionally, the electron- and hole mobility in the blend of both materials have been addressed, both showing trap-free SCL 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 neglected in the model.
2.2.1 Basic equations
The equations[14] used to describe the transport through the virtual semiconductor are the 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 potential ψ(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)and p(x), it is implicitly assumed that space charge caused by electrons in one phase can be neutralized by holes in the other phase, which is not obvious considering the actual morphology of a BHJ. However, this assumption is supported by the consistent description of space charge effects given in chapter 4.
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 for notational convenience, unless stated otherwise. Only one spatial dimension is consid- ered, since the device has a planar structure with a very small thickness (typically 100 nm) compared to the lateral dimensions (typically several mm).
In order to solve the basic equations, a set of equations is needed relating the current densities to the carrier densities and the potential. Incorporating both drift and diffusion of charge carriers, one has∗
Jn= −qnµn∂x∂ ψ+qDn ∂
∂xn, (2.3a)
Jp= −qpµp ∂
∂xψ−qDp ∂
∂xp, (2.3b)
where Dn,pare the carrier diffusion coefficients, which are assumed to obey the Einstein relation[14]
Dn,p=µn,pVt, (2.4)
with Vtthe thermal voltage, i.e., Vt=kBT/q, where kBis Boltzmann’s constant and T is the absolute temperature. However, it should be noted that at high carrier densities the diffusion coefficient may be increased.[21]
To obtain a unique solution of the system of equations formed by Eqs. (2.1)–(2.4) it is necessary to specify the carrier densities and potential at both contacts. The contact at x=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 the conduction band of the semiconductor and is therefore Ohmic as far as electron injection is considered. Using Boltzmann statistics
n(0) =Ncv, (2.5a)
p(0) =Ncvexp
−E
effgap
qVt
, (2.5b)
where Ncvis the effective density of states of both the conduction and valence band. This assumption 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 is reversed, as is customary in this field.
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 (2.5×1025m−3)[22]
for both bands is used. Similarly, the bottom contact is assumed to be an Ohmic hole contact, thus
n(L) =Ncvexp
−E
gapeff
qVt
, (2.6a)
p(L) =Ncv. (2.6b)
The Ohmic nature of both contacts is supported by current-voltage measurements on both materials, which clearly show space-charge-limited behavior.[5,6] The boundary condition for the potential reads
ψ(L) −ψ(0) = 1
qEeffgap−Va, (2.7)
where Vais the applied voltage.
2.2.2 Generation of free charge carriers
As outlined in chapter 1, exciton dissociation at the donor/acceptor interface does not directly yield free charge carriers, but rather bound electron-hole pairs. The dissociation of 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 this theory by pointing out that the bound electron-hole pair, which acts as a precursor for free charge carriers, has a finite lifetime, see Fig. 2.2. The bound electron-hole pair may decay to the ground state with a decay rate kf or dissociate into free carriers. The sep- aration into free carriers is a competition between dissociation (rate constant kdiss) and recombination (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 these two models is in the boundary conditions: Whereas Onsager’s 1934 theory has a source at 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) +kf(T), (2.8) depending on both temperature T and field strength F. The ratio of kdiss(y, T, F) to kdiss(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)
Figure 2.2: Schematic representation of the charge carrier separation at the interface between donor (D) and acceptor (A). Upon excitation of the donor, an exciton is created which diffuses through the donor until it reaches the interface. At the interface, the electron is transfered to the acceptor, thus forming a bound electron-hole (e/h) pair. This pair can either dissociate into free carriers 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 dissociation of a weak electrolyte is given by[24,26]
K(0) = 3
4πa3e−Eb/kBT= kdiss(y, T, 0)
kr , (2.10)
where Ebis the electron-hole pair binding energy. Combining Eqs. (2.9) and (2.10) gives
kdiss(y, T, F) = 3kr
4πa3e−Eb/kBTJ1(2√
−2b)/√
−2b. (2.11)
The decay rate of the bound electron-hole pair to the ground state kf is used as a fit parameter.
Suppose that Ge−his the generation rate of bound electron-hole pairs. The number of bound electron-hole pairs created per unit volume and time R from free charge carriers will be equal to
R=kr(np−n2int), (2.12)
where nint=Ncvexp
−E2qVgapefft
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−kfX−kdissX+R. (2.13)
∗Note that this equation implies that k−f1is not equal to the lifetime of the bound electron-hole pair and that the decay of electron-hole pairs is not exponential in time.
