Efficiency Limits of Organic Bulk Heterojunction Solar Cells - The

17958

J. Phys. Chem. C 2009, 113, 17958–17966

Efficiency Limits of Organic Bulk Heterojunction Solar Cells Thomas Kirchartz,*,† Kurt Taretto,‡ and Uwe Rau† IEF5-PhotoVoltaik, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany, and Departamento de Electrotecnia, UniVersidad Nacional del Comahue-CONICET, Buenos Aires 1400, 8300 Neuquen, Argentina ReceiVed: July 3, 2009; ReVised Manuscript ReceiVed: August 7, 2009

We calculate the radiative efficiency limits of organic bulk heterojunction solar cells according to the theory of Shockley and Queisser and compare the results with experimental device performance. The difference between limiting theory (23% power conversion efficiency) and experimental data (4%) is explained and quantified by five reasons, namely the energy level misalignment at the donor/acceptor heterointerface of the bulk heterojunction, insufficient light trapping, low exciton diffusion lengths, nonradiative recombination, and low charge carrier mobilities. Comparison of the impact of the different loss mechanisms by numerical simulation reveals that efficiencies above 10% using PF10TBT/PCBM blends will require mostly a strong reduction of nonradiative recombination. The energy misalignment and the low carrier mobilities appear as a second-order restriction in this type of blend. I. Introduction The photovoltaic power conversion efficiency of nearly every single junction solar cell is fundamentally limited by the Shockley-Queisser (SQ) limit.1 The applicability of the SQ theory bases on the fact that the assumptionssnamely a stepfunction like quantum efficiency, perfect extraction of carriers, and radiative recombination as the only loss processsare all appropriate upper limits for many inorganic semiconductors used for photovoltaics. For instance, crystalline silicon as the dominant solar cell material has a short circuit current that comes very close to the one predicted by the SQ theory. For crystalline silicon solar cells, the main difference in efficiency, η, between reality (η ≈ 25%) and the radiative limit (η ≈ 33%) originates from the decrease in open circuit voltage due to nonradiative Auger recombination,2,3 being an intrinsic loss process independent of material quality. For organic bulk heterojunction (bhj) cells, the SQ limit is still an upper efficiency limit; however, some of the original assumptions of the SQ theory are no longer fair to assume in organic solar cells. In a first step, this paper applies the SQ theory to organic absorber layers using measured absorption coefficients calculating the radiative efficiency limits. However, since the difference between actual efficiencies of organic solar cells and the radiative limits are quite large, we investigate the reasons for this discrepancy. Therefore, we start to consider several of the features specific to organic bulk heterojunction solar cells and expand our model accordingly. Unlike in most inorganic semiconductors, photon absorption in organic semiconductors leads to the creation of excitons, which have an exciton binding energy, EB, that is much larger than the thermal energy, kT, at room temperature. The exciton will therefore not spontaneously split in the volume of the organic semiconductor and has to diffuse to the contacts to be collected. Typical exciton diffusion lengths, Lχ, however, are only on the order of Lχ ≈ 10 nm and therefore much smaller than the absorber thickness, d, required for efficient absorption of the incoming light (d ≈ 100 * To whom correspondence should be addressed. E-mail: t.kirchartz@ fz-juelich.de. † Forschungszentrum Ju¨lich. ‡ Universidad de Nacional del Comahue-CONICET.

nm). Due to these low exciton diffusion lengths,4 the absorber layers of efficient organic solar cells are not made up of one single material, but instead they consist of blends5-8 of two materials with different lowest unoccupied molecular orbitals (LUMOs). The energetic difference between the LUMO levels allows splitting of the exciton into free carriers at the interfaces between both materials; however, the energy step also decreases the energy of the free carriers. Thus, the high exciton binding energies, which create the need to split the excitons with the help of a band offset, create a fundamental loss process compared to single phase absorber materials. In addition, the use of large internal heterointerfaces creates regions with increased nonradiative recombination. This increased nonradiative recombination is visible as the photoluminescence quenching observed after blending two organic materials and constitutes another loss mechanism, typical for bulk heterojunction solar cells.9 Not only the exciton diffusion length is insufficient for the use of single phase materials, but also the free carrier mobilities of organic semiconductors are usually low10-12 compared to that of inorganic semiconductors. Thus, the assumption of complete carrier collection needs to be abandoned and replaced by mobility dependent efficiency limits.13,14 In the following, we start by calculating the SQ limit for idealized photovoltaic absorber layers and then proceed by introducing stepby-step features of organic bulk heterojunction solar cells. II. SQ Theory The radiative efficiency limit developed by Shockley and Queisser considers the solar cell as a body, emitting blackbody radiation, φbb, and absorbing solar radiation, φsun. Both absorbed and emitted photon flux only depend on the absorptance, a(E), which determines the short circuit current density

Jsc,SQ ) q

∫0∞ a(E)φsun(E) dE

(1a)

where E is the photon energy, q is the elementary charge. A step-function-like absorptance as used in the original SQ-theory, i.e., a(E) ) 1 (for E > Eg) and a(E) ) 0 (for E < Eg) yields

10.1021/jp906292h CCC: $40.75  2009 American Chemical Society Published on Web 09/17/2009

Efficiency Limits of Organic Bulk Heterojunction Solar Cells

Jsc,SQ ) q

∫E∞ φsun(E) dE

J. Phys. Chem. C, Vol. 113, No. 41, 2009 17959

(1b)

g

For an arbitrary absorptance, a(E), the SQ limit follows from eq 1a. In analogy to eq 1a,b for the short circuit current density, it follows for the diode saturation current density

J0,SQ ) q

∫0∞ a(E)φbb(E) dE

(2a)

likewise for the case of a step-function

J0,SQ ) q

∫E∞ φbb(E) dE

(2b)

g

The spectral dependence of the blackbody radiation in eq 2a,b is given by15

φbb(E) )

1 -E 2πE2 2πE2 exp ≈ 3 2 [exp(E/kT) - 1] 3 2 kT hc hc

( ) (3)

Here, h is Planck’s constant and c is the vacuum speed of light. From the J/V curve

[ ( qVkT ) - 1] - J

J ) J0,SQ exp

sc,SQ

(4)

the efficiency follows as the electrical power density P ) JmpVmp at maximum power point (Jmp/Vmp). The theory outlined above constitutes the efficiency limit for solar cells as long as a series of assumptions is applicable: The solar cell must be a single-junction solar cell and not a multijunction device with several band gaps. In addition, photons should not be allowed to create more than one electron/hole pair. However, in both cases the SQ theory is easily extended to cover such situations, too. A more critical restriction is the assumption that all carriers are thermalized and occupy their bands according to Fermi-Dirac statistics. Only in this case Wu¨rfel’s generalized Planck law16 is valid and eq 2a is applicable to calculate the saturation current. Therefore, hot carrier solar cells, as well as solar cells, where emission takes place via tail states that are not fully thermalized,17 cannot be described within the SQ theory. III. Radiative Efficiency Limits for Organic Solar Cells Figure 1 shows the device configurations, we study in the radiative limit. Figure 1a represents an ideal solar cell according to the SQ theory with well-defined band gaps and step-functionlike absorptances. Using such step-function-like absorptances, eqs 1-4) allow the calculation of the efficiency of single- and multijunction solar cells as a function of band gap.18,19 For solar cells made of organic absorber materials as sketched in Figure 1b, as well as for blends of organic absorber materials as sketched in Figure 1c, the SQ theory must be adapted. In general, any real photovoltaic absorber, be it inorganic or organic, has a finite absorption coefficient, a finite thickness and subsequently an absorptance with a smooth transition between the lowly absorbing region below the band gap and the strongly absorbing region above the band gap. To account for more realistic absorptances, a mathematical description of the absorption coefficient entering in eqs 1a and 2a is needed

Figure 1. Scheme of the solar cell configurations used for calculation of the radiative efficiency limits: (a) the idealized solar cell of the SQ theory with step-function-like absorptances and electronic bands, (b) a solar cell consisting of one organic material with a smeared out absorption edge and narrow bands, and (c) a bulk heterojunction or bilayer solar cell, where photogenerated carriers are mainly created in the electron donor and where emission of photons (the only loss process for a radiative efficiency limit) happens predominantly at the interface between the donor and acceptor molecules. The line thickness of the represented photons suggests the intensity of each process, i.e., absorption is strong in the donor and recombination and emission is strong at the interface.

that reproduces the typical shape of the absorption coefficients of the investigated materials. In principle, a Gaussian shape of the absorption coefficients seems reasonable since it can be understood by the typical Gaussian-like density of states of organic semiconductors.20,21 A convolution of Gaussian density of states directly leads to Gaussian-shaped absorption coefficients. For the absorption of charge transfer states the use of Gaussian functions is also backed by experimental results on Fourier transform photocurrent spectroscopy (FTPS) on various polymer-fullerene blends also showing Gaussian-shaped spectra.22 Figure 2 presents experimental data of photoluminescence (PL) and absorption coefficients, R, of two polymers (a and b) and one polymer-fullerene blend (c) that have been obtained from literature. Figure 2a shows data on regioregular poly(3hexylthiophene) (RR-P3HT) taken from ref 23. To obtain an absorption coefficient, R, that reproduces both the short circuit current density, Jsc, and the saturation current density, J0, we combine the experimental absorption coefficient (open squares) in the high-energy range with the absorption coefficient that follows from the PL emission (dashed line) under the assumption that Kirchhoff’s law holds.24 The resulting absorption coefficient, R, is presented as a solid line.

17960

J. Phys. Chem. C, Vol. 113, No. 41, 2009

Kirchartz et al.

Figure 2. Comparison of experimental data on PL and R (open squares) for two polymers and one polymer/fullerene blend. The absorption coefficients used for calculation of the radiative efficiency limits have been extrapolated using the PL data as an indication. For RR-P3HT (a) the absorption coefficient below the band gap was directly taken from the experimental PL data23 by assuming that Kirchhoff’s law holds. For the PF10TBT polymer (b) and PF10TBT /PCBM blend (c) R was taken from ref 25, and the PL data were taken from ref 9. Since PL and R belong therefore to different cells, we fitted the low-energy tail of R with a Gaussian function ensuring that the peak position of the resulting PL (dashed lines) fits to the experiment (open circles).

fit are a band gap Eg ) 2.2 eV, a bandwidth E0 ) 0.19 eV, and a peak absorption coefficient R0 ) 6.5 × 104 cm-1. Thus, compared to the polymer alone, R decreases in height by a factor of 3 which should be a result of blending, while the bandwidth E0 increases, reflecting the fact of the additional absorption by the charge transfer complex at the donor-acceptor interface as shown in refs 26-33. The main result of Figure 2 is that the low-energy part of the absorption coefficient, i.e., the energetic range below the first local maximum of the absorption coefficient, is best approximated by a Gaussian function. This result is the same for single-phase organic semiconductors or for blends with a large interfacial area that contributes to the absorption of photons and dominates the emission of photons as sketched in Figure 1c. In contrast to inorganic absorber materials, the absorption coefficient of organic absorbers does not increase monotonically like in most inorganic semiconductors but reaches a local maximum for finite energies and usually decreases for higher energies. Thus, it has been suggested to take a finite absorption window into account when determining the ultimate efficiency limits.34 However, typical absorption coefficients of polymers feature more than one peak, meaning that using one Gaussian peak for the absorption coefficient would be too pessimistic for calculations of efficiency limits. This is backed by quantum efficiencies of low-band-gap polymer solar cells, which show hardly any drop in quantum efficiency for higher energies.35,36 Thus, we model absorption coefficients R according to the form

R)

{

R0

Figure 2b shows R and PL emission of a second polymer, this time poly[9,9-didecanefluorene-alt-(bis-thienylene) benzothiadiazole] (PF10TBT). The data for the absorption coefficient (open squares) are taken from ref 25, while the experimental PL data (open circles) is taken from ref 9 and was measured on the same type of material. In this case, we use the PL data only as a rough hint for the peak position of the PL emission that corresponds to the absorption coefficient measured by Slooff et al.25 We extrapolate R toward lower energies with a Gaussian-shaped absorption coefficient (thin dashed line) following the relation

[(

R ) R0 exp -

E - Eg E0

)] 2

(5)

The fit parameters used are a band gap Eg ) 2.175 eV, a bandwidth E0 ) 0.14 eV, and a peak absorption coefficient R0 ) 2.1 × 105 cm-1. Finally, we obtain the resulting absorption coefficient (solid line) combining the Gaussian fit for low energies (E < 2 eV) and the experimental data of ref 25 for higher energies E > 2 eV. With a similar procedure, we obtain the absorption coefficient also for the case of a PF10TBT/PCBM([6,6]-phenyl C60 butyric acid methyl ester) blend. The experimental absorption coefficient (open squares) is again taken from ref 25 and the PL data (open circles) from ref 9. An important feature of the blend material is its visible emission from the charge transfer state which displays a maximum at a photon energy E ) 1.53 eV. This maximum is shifted to lower energies by almost 300 meV when compared to the maximum of the emission of the pure PF10TBT at E ) 1.81 eV (Figure 2b). The fit parameters for the Gaussian

[(

E - Eg E0 for E g Eg

R0 exp -

)] 2

for E < Eg

(6)

which has a Gaussian shape on the low-energy side but remains constant for higher energies. The definition of the absorption coefficient in eq 6 uses the same three characteristic parameters as eq 5: (i) Eg, being here defined as the energy at the peak of the Gaussian part of the absorption coefficient, (ii) E0, and (iii) the peak absorption coefficient, R0. From the absorption coefficient we calculate the most optimistic absorptance under the assumption of Lambertian light trapping37 with zero reflection at the front and unity at the back. To calculate the absorptance, we follow the scheme introduced by Green38 and use the equations outlined for instance in refs 39-42. We use a constant refractive index n ) 2 and a typical absorber layer thickness of d ) 100 nm for these calculations. Later, we will also discuss the efficiency loss caused by more realistic absorptances, where parasitic absorption in transparent conductive oxides and flat, reflecting surfaces are taken into account. Figure 3 shows the resulting (a) efficiencies, η, (b) short circuit current densities, Jsc, and (c) open circuit voltages, Voc, in the SQ limit for step-function-like absorptances (open squares) and in the radiative limit for R according to eq 6. Figure 3 varies E0 (E0 ) 0.05 eV, solid line; 0.1 eV, dashed line; 0.15 eV, dotted line; 0.2 eV, dash-dotted line; 0.25 eV, dash(2)-dotted line) and compares the result with the radiative limit of the three absorption coefficients presented in Figure 2. The peak absorption coefficient used for Figure 3 was chosen R0 ) 10/d, where d is the unspecified thickness of the absorber layer. This assumption implies that after one path through the absorber, only exp(-10) ) 4.5 × 10-5 of the light remains unabsorbed. Thus, the absorptance above the band gap is a(E) ≈ 1. Since the absorptance below the band gap is still higher than zero, Jsc increases with increasing E0, as shown in Figure 3b. In contrast,



Figure 4. Calculated maximum efficiency and optimum band gap as a function of E0. The values of ηmax and Eg,opt correspond to the abscissa and ordinate of the peaks of the curves in Figure 3. For low band widths, ηmax (solid line) approaches the SQ limit, while it decreases monotonically for higher band widths. In contrast, the corresponding optimum band gap (dashed line) increases monotonically. For realistic band widths of blends of E0 ) 0.19 eV, as derived from Figure 2c, follows an efficiency of ηmax ) 29% at a band gap of Eg,opt ) 1.8 eV.

Figure 3. Efficiency as a function of band gap for different values of E0 (lines) compared to the SQ limit (open squares) and the radiative limit for three state of the art organic semiconductors (single data points) according to the absorption coefficients presented in Figure 2. In all three materials, photovoltaic efficiencies above η ) 20% are reached at the radiative limit. Increasing E0 from 50 to 250 meV (in steps of 50 meV) leads to an increase of sub-band-gap absorption and thus to an increase of Jsc while at the same time leading to an emission peak at lower photon energies and thus to smaller Voc values. This results in an efficiency vs band gap characteristic shifted to higher band gaps and lower efficiencies if compared to the SQ limit.

Voc (cf. Figure 3c) decreases since the emission shifts to lower energies. The combined effect of Jsc and Voc leads to an efficiency (cf. Figure 3a), whose peak shifts to higher band gaps and lower efficiencies, when increasing E0. It is interesting to note that the general trend seen in Figure 3 is similar to what is predicted by analyzing the dependence of the radiative efficiency limit of solar cells with band gap fluctuations43 where the standard deviation, σg, of the fundamental band gap plays a role that is somewhat analogous to E0. Also shown in Figure 3 are data points corresponding to the three absorption/emission spectra shown in Figure 2a-c. The slightly lower efficiency of the blended material originates in a lower Voc, as a consequence of the broadened absorption peak that results from blending (see discussion above). Figure 4 displays the dependence of peak efficiency, ηmax, and optimum band gap, Eg,opt, on E0. For low band widths, the shape of the absorptance becomes similar to a step function, and ηmax approaches the SQ limit at 33%. For higher band widths, it decreases monotonically, while the corresponding Eg,opt increases monotonically. For realistic band widths of blends of E0 ) 0.19 eV, as derived from Figure 2c, follows an efficiency of ηmax ) 29% at a band gap of Eg,opt ) 1.8 eV. IV. Reasons for the Efficiency Gap between Radiative Limit and Experiment The radiative efficiency limits discussed above have the advantage that hardly any internal properties of the photovoltaic energy conversion process have to be known. Any charge carrier is assumed to be collected directly after photogeneration and every injected carrier recombines radiatively leading to an emission defined by the absorptance of the device and the Planck spectrum. The radiative efficiencies η > 20% that follow from

the state of the art organic semiconductors are much higher than current experimental single-junction solar cell efficiencies at η ≈ 5%. This observation directly leads to the question, where this factor of 4 in efficiency drop originates. There are in principle four reasons for this efficiency gap: (1) Optical losses, meaning that light may be parasitically absorbed in solar cell layers that are not active in charge carrier collection or that the current organic solar cells, have no effective light trapping scheme as most inorganic solar cells.44-46 (2) Exciton losses due to insufficient transport of excitons to the next donor-acceptor interface or due to inefficient exciton dissociation at the interface. (3) Recombination losses, meaning that only a small part of the carriers recombines radiatively, but most of the carriers recombine nonradiatively at interfaces or defects in the absorber and at the contacts. (4) Collection losses, meaning that due to insufficient mobilities only part of the free carriers reach the contacts. In order to quantify the importance of these loss mechanisms, we need to have a simple device model capable of reproducing experimental current/voltage curves. In a real organic bulk heterojunction solar cell, charge collection is a multistep process,47,48 starting with the exciton diffusion and dissociation at the donor-acceptor interface. The exciton dissociation is followed in most models by the creation of a charge-transfer state at the interface, which then either recombines or dissociates to form free electrons and holes. The electrons and holes are then transported by drift and diffusion to their respective contacts, where they are collected, or recombine along their transport paths. Typical models used for electronic simulations of organic bulk heterojunction solar cells implement various details of the charge collection process, such as temperature- and field-dependent dissociation of the charge transfer state, or lateral transport of electrons from a donorrich phase to the acceptor phase.49,50 Such detailed studies may be necessary for meaningful simulations of experimental data. However, for our purposesthe discussion of efficiency limitssit is most appropriate to use a simple model and show the general trends. To describe the dynamics of dissociation and recombination of the charge transfer state, we need to define two rate constants and two equilibrium concentrations as shown schematically in Figure 5a. The equilibrium concentration for the excitons in the charge transfer state are denoted by ξ0, the combined

17962


Kirchartz et al.

Figure 5. Scheme of organic bulk heterojunction solar cells used for calculation of efficiency limits showing (a) the rate constants for the interaction of the charge transfer state with the free carriers and (b) how Figure 1c extends to the third dimension, when we have to include transport of charge carriers into the model.

equilibrium concentration of the free carriers (the hole in the donor and the electron in the acceptor phase) is denoted by ni. The rate constant for dissociation of the charge transfer exciton and for recombination of the free carriers forming again an exciton is denoted by krec, and the recombination constant of the charge transfer exciton is denoted by kf. Then, the dissociation probability pdiss of the charge transfer exciton is given by51

pdiss )

krecni2 krecni2 + kfξ0

) 1+

1 kfξ0

(7)

krecni2

The result of eq 7 is equivalent to the result for the dissociation probability in refs 52-57; however, we explicitly express pdiss in terms of equilibrium concentrations and rate constants as required by the principle of detailed balance.58-61 From the dissociation probability defined in eq 7 follow51 both the recombination rate R of free charge carriers

R ) krecnp(1 - pdiss) )

krecnpkfξ0 krecni2 + kfξ0

(8)

as a function of the product np of electron and hole concentration, and the generation rate, G, of free charge carriers

G ) Goptpdissηχd ) Gopt

krecni2 krecni2 + kfξ0

ηχd

(9)

as a function of the optical generation rate, Gopt, and the efficiency, ηχd, for the diffusion and dissociation of excitons.51

To achieve efficient dissociation of the charge transfer state according to eq 7, either the coupling of the charge transfer state to the free electrons and holes (quantified by krec) must be more efficient than the nonradiative recombination (quantified by kf), or the energetic position of the charge transfer state must be high compared to that of the free electrons and holes, such that ξ0 , ni. In the case of a large difference in equilibrium concentrations, the energy offset between charge transfer state and free carriers allows efficient extraction even for very short nonradiative decay times kf-1. After creation of free carriers has happened, the free carriers must be separated in the built-in electric field as sketched in Figure 5b. The transport of charge carriers in an organic semiconductor is usually described by drift-diffusion transport like in inorganic materials. The relevant quantity, characterizing the material quality with respect to charge carrier separation is the electron and hole mobility, µ. To discuss the difference between current experimental device characteristics of organic bulk heterojunction solar cells, we want to simulate measured current/voltage curves to estimate the losses due to the absence of special light trapping schemes, nonradiative recombination, and insufficient collection of carriers. A very simple model of the whole solar cell needs only a small set of parameters, namely the mobility, µ, of electrons and holes (which we assume to be identical in the following), the rate constants kf and krec, the equilibrium concentrations ni and ξ0, and the boundary conditions at the contacts. These boundary conditions are defined by the built in voltage, Vbi, fixing the potential difference between the contacts, and the surface recombination velocities. These should be as high as possible for majorities and as low as possible for minorities. For simplicity, we restrict recombination to the bulk of the material only, setting the surface recombination for the minorities at the contacts to zero. Together with the absorption coefficient shown in Figure 2c, ref 25 also discusses the performance and the current/voltage curves of solar cells made from PF10TBT/PCBM. We will use these electrical measurements to determine meaningful parameters for our simple model that needs to be able to reproduce the experimental current/voltage curve from ref 25. To determine the electrical parameters of this material system, we set Vbi ) Eg/q ) 1.3 V,62 and kf ) 2.5 × 108 s-1 (following from timeresolved PL measurements of the charge transfer state9). From the optical data (refractive index and absorption coefficient) and thicknesses of the blend layer (d ) 186 nm) and the ITO and PEDOT:PSS window layers follows the absorptance as shown in refs 25 and 51. From the internal quantum efficiency (as shown in refs 25 and 51), information about the exciton dissociation is obtained. The internal quantum efficiency for short wavelengths λ < 420 nm, where absorption takes place in the fullerene, is ∼86%. If one assumes that due to the higher exciton diffusion length in fullerenes compared to polymers all excitons created in the short wavelength range are absorbed, the dissociation probability pdiss ≈ 0.86 is obtained. For higher wavelengths, the internal quantum efficiency depicted in refs 25 and 51 has a constant value around 75%, implying that in the polymer about 87% () 0.75/0.86) of the photogenerated excitons are converted into excitons in the charge transfer state. Under open circuit conditions, no net current flows and thus the quasi-Fermi levels are flat throughout the device. Thus, the spatially integrated generation and recombination rates have to be equal, leading to


kfξ0d

krecnp krecni2 + kfξ0

)

∫0 Gopt(x) dx k d


krecni2

2 recni

+ kfξ0

(10) using eqs 8 and 9. In eq 10, krec cancels out, np/n2i ) exp(qVoc/ kT), and all other parameters except ξ0 are known. Thus, from the value of the open circuit voltage Voc ) 999 mV, the equilibrium concentration

ξ0 )

∫0d Gopt(x) dx × exp(-qVoc/kT)/(kfd) ) 3 × 10-4 cm-3

(11)

of the charge transfer state is obtained. The values of mobility µ ) 8 × 10-5 cm2/(V s) and intrinsic charge carrier concentration ni ) 1.4 × 108 cm-3 are chosen such that they lead to a good fit to the experimental current/ voltage curve. The value of the intrinsic charge carrier concentration corresponds to an effective density of states NC and NV for the conduction and valence band of NC ) NV ) 1019 cm-3 at a band gap Eg ) 1.3 eV, using the relation ni2 ) NCNV exp(-Eg/kT). Finally, using eq 7, krec results from the fixed values of pdiss, ξ0, ni, and kf as krec ) 2.4 × 10-11 cm3/s. Figure 6 compares the (a) power density/voltage and (b) current density/voltage curves at the radiative limits of the pure PF10TBT polymer (A, solid line, using the optical data presented in Figure 2b) and the PF10TBT/PCBM blend (B, dashed line, using the optical data presented in Figure 2c) with the fit (F, open triangles) to the experimental current/voltage curve taken from ref 25. From Figure 6b, we immediately see that the dashed line displays the lower open-circuit voltage of the blend compared to the pure polymer (solid line), as the main fundamental disadvantage caused by the broadened absorption peak. In addition, Figure 6 shows the radiative limit (C, dotted line) for the blend assuming the same optical properties as used for the fit to the experimental data. The only difference between the two radiative limits for the blend (dashed line vs dotted line) is the way the absorptance is calculated. The calculation leading to the dashed line assumes that no light is absorbed outside the absorber layer and that the interfaces scatter the light isotropically. This is difficult to achieve in practice due to the low refractive index mismatch between the absorber layer and the window layer. In contrast, the simulation using the dotted line assumes flat interfaces, a coherent superposition of the waves and parasitic absorption in the window layers. These optical losses amount to an absolute efficiency loss of 6.4%. Apart from the radiative limits for polymer and blend, as well as the experimental curve, there are two other lines shown in Figure 6 that are labeled with D and E and which are simulated using the parameters given in Table 1. The dash-dotted line corresponds to the situation, where finite exciton diffusion lengths and thus a finite efficiency of the creation of charge transfer excitons have been taken into account. The finite exciton diffusion efficiency ηχd ) 0.87 leads to an absolute efficiency loss of 2% compared to the dotted line (C) only due to the reduced short circuit current. Note here, that a recent paper of Park et al.63 showed that at least in some polymer/fullerene solar cells, the internal quantum efficiency and thus also the exciton diffusion efficiency approach 100%. If we allow for nonradiative recombination in addition to radiative recombination but still keep the mobilities sufficiently high that all carriers that are optically created are also collected at the contacts under short-circuit conditions, the line with open

Figure 6. (a) Power density/voltage and (b) current density/voltage curves of organic solar cells in six stages of idealization. The solid line represents the radiative efficiency limit for PF10TBT (using the optical data presented in Figure 2b), while the dashed line shows the radiative efficiency limit for the PF10TBT/PCBM blend (using the optical data presented in Figure 2c). Including realistic estimates for parasitic absorption in the inactive solar cell layers and considering only flat surfaces without light scattering leads to the performance characterized by the dotted line. The dash-dotted line corresponds to the situation, when a finite efficiency, ηχd, of exciton diffusion is taken into account, which is in our example ηχd ) 87% (cf. Table 1). Taking into account also nonradiative recombination and finite mobilities reduces the efficiency such that the resulting P/V and J/V curves are given by the line with open squares and the open triangles. The latter characteristic is a fit to experimentally determined data taken from ref 25.

TABLE 1: List of the Configurations (A-F) Discussed in the Text and the Parameters Associated with These Configurationsa D

E

F

efficiency of exciton diffusion ηχd 0.87 ∞ ∞ 8 × 10-5 carrier mobility µn,p [cm2(Vs)-1] decay constant kf [s-1] kf ) kf,rad ) 2 2.5 × 108 2.5 × 108 recombination constant krec [cm3s-1] 2.4 × 10-11 intrinsic carrier concentration 1.4 × 10-8 ni [cm-3] equilibrium concentration of the 3 × 10-4 charge transfer state ξ0 [cm-3] thickness d of active layer [nm] 186 a The current density/voltage and power density/voltage curves of these configurations are depicted in Figure 6. A defines the radiative limit for the polymer and B the radiative limit of the blend. In both cases, light trapping is assumed to be perfect, i.e., Lambertian. C defines the radiative limit for planar surfaces and includes parasitic absorption in the optical layers. D is the same as C but including losses due to low exciton diffusion lengths. In addition, E includes nonradiative recombination via the charge transfer state and F finite mobilities of electrons and holes. The efficiency for configurations D-F is calculated by solving the drift-diffusion equation in one dimension, using the following electrical parameters.

squares follows (curve E). All parameters are chosen identical to the ones used to fit the experimental current/voltage curve (open triangles) except for the mobilities of electrons and holes (cf. Table 1). The mobility was assumed to be µ ) 104 cm2/ (V s), being a good approximation to infinity. The efficiency loss that is only due to the introduction of nonradiative recombination is roughly 5.6%. Finally, the difference between the dash-dotted line and the open triangles represents the loss due to the inclusion of finite mobilities. This loss amounts to

17964


Kirchartz et al. we need to calculate the value of kf in the radiative limit. Assuming flat quasi-Fermi levels in the radiative limit, we obtain the radiative decay constant kf,rad from the radiative saturation current density J0,SQ for the blend (dashed line in Figure 6) that follows from the emission spectrum via eq 2a. According to ideal diode theory, the saturation current density J0 for flat quasiFermi levels is given by

J0 ) qRd/exp(qV/kT)

(12)

Using eq 8 for the recombination rate R, we obtain Figure 7. Efficiency as a function of the recombination constant kf for high mobilities (line+open circles) and for realistic mobilities that have been used to fit the experimental J/V curves (line+open squares). For low values of kf, the efficiency is not sensitive to mobility since extraction of carriers is efficient due to the long lifetimes. The efficiency decreases monotonically due to the increase in recombination and the subsequent decrease in open circuit voltage. High values of kf limit the extraction of carriers and thus Jsc. The efficiency drops now more drastically since Jsc, Voc, and FF suffer now. A higher mobility shifts the position of this stronger drop toward higher values of kf, since higher mobilities aid in collecting the carriers. The three situations, where the J/V curves are shown in Figure 6, are indicated by the corresponding capital letters D, E, and F as used in the legend of Figure 6.

Figure 8. Efficiency as a function of electron and hole mobility for kf ) 2.5 × 108 s-1, i.e., for the value of kf used to fit the experimental current voltage curve. The drop in efficiency is due to insufficient collection of charge carriers at lower mobilities. The absorber layer thickness is assumed to be d ) 186 nm as for the experimental data in ref 25, which was used to extract all parameters. For infinite mobilities, the efficiency saturates and corresponds to the J/V curve E as shown in Figure 6, while at µn,p ) 8 × 10-5 cm2/(V s), the efficiency corresponds to situation F, i.e., the experimental curve in Figure 6.

3.0% absolute, leading to the experimental efficiency of 4.2% for the PF10TBT/PCBM solar cell. Figures 7 and 8 discuss the efficiency losses due to nonradiative recombination and decreased charge carrier mobilities in more detail. Figure 7 shows the efficiency as a function of the recombination constant kf for a constant value of krec ) 2.4 × 10-11 cm3/s and for a very high mobility µ ) 104 cm2/(V s) (line and open circles) as well as a lower mobility µ ) 8 × 10-5 cm2/(V s) (line and open squares; this mobility is the same that was used to fit the experimental J/V curve). In both cases, we assume that only 87% of the excitons in the polymer are able to diffuse to the donor/acceptor interface and form charge transfer excitons as we did for the simulation of the experimental data. In order to estimate the highest physically possible value for the decay constant kf and the difference between the experimental value kf ) 2.5 × 108 s-1 and the highest value,

J0,SQ ) qkf,radξ0d

krecni2 krecni2 + kf,radξ0

(13)

giving kf,rad ≈ 2 s-1 for krec ) 2.4 × 10-11 cm3/s, ξ0 ) 3 × 10-4 cm-3, ni ) 1.4 × 108 cm-3, and d ) 186 nm. This radiative limit for the decay constant corresponds to the simulation parameters assumed for curve D in Figure 6. From this radiative limit as indicated by the left vertical dashed line, the efficiency first decreases monotonically with a constant slope independent from the choice of mobility. This is due to a decrease of opencircuit voltage caused by the increase of the recombination constant. For recombination constants kf > 106 s-1, the efficiency starts to drop more rapidly in case of the lower mobility. Now, the collection of carriers becomes inefficient and the short circuit current density Jsc, as well as the fill factor FF also start to drop. In case of the higher mobility, the charge carrier collection suffers only when the recombination constant increases even more (kf > 108 s-1). The ratio between the radiative decay constant and the experimental one is approximately kf,rad/(kf + kf,rad) ≈ 10-8. This quantity matches the quantum efficiency QLED of a light emitting diode. The difference in open circuit voltage between the experimental Voc and the radiative limit Voc,rad is given by60,61

kT ln[QLED] ) q kf,rad kT - ln ≈ 470mV (14) q kf + kf,rad

∆Voc ) Voc,rad - Voc ) -

[

]

It is important to note that this difference between the opencircuit voltage in the radiative limit and the experimentally determined open-circuit voltage is considerably higher than that of most inorganic solar cells like crystalline silicon,64,65 Cu(In, Ga)Se2,66 or III/V67 compounds. Thus, from our estimation of the ratio of decay constants, we can conclude that although pure polymers are efficient light emitting diodes, the blended device is expected to be a worse light emitting diode than most inorganic solar cells. Let us now have a closer look at the dotted vertical line at the right in Figure 7, which indicates the realistic value of the recombination constant kf. For very high mobilities (µ ) 104 cm2/(V s)), the efficiency is η ≈ 7.2%, while it is only η ≈ 4.2% for the parameters that fit the experimental current/voltage curve. The question arises, which mobility is actually necessary to achieve efficient carrier collection in our simple onedimensional model? Figure 8 displays the efficiency, η, as a function of mobility, µ, for the same values of the recombination constants as used to fit the experimental current/voltage curve (kf ) 2.5 × 108


J. Phys. Chem. C, Vol. 113, No. 41, 2009 17965 and from radiative to nonradiative recombination. The leftmost column (a) corresponds to the sequence of loss mechanisms chosen in Figure 6. If one compares the efficiency gain, resulting from an idealization of one parameter starting from the experimental efficiency, it becomes obvious that especially the elimination of nonradiative recombination would guarantee the largest increase in efficiency (from 4.2% to 12.8%). The impact of mobility is also rather high when starting from the experimental limit (leftmost column), although with radiative recombination only, the collection of charge carriers is sufficiently high and carrier mobility stops limiting cell efficiency (cf. Figure 7 or Figure 9 columns b and c). Additionally, we notice that the impact on efficiency of carrier mobility is considerably larger than the impact of exciton diffusion efficiency. V. Conclusions

Figure 9. Comparison of efficiencies and efficiency losses calculated by four different sequences of introduction of loss mechanisms, separated by dashed lines and labeled a-d. All calculations start from the SQ limit (top line) to the experimental efficiency of η ) 4.2% (bottom line), calculated with the parameters of case F in Table 1. The leftmost column (a) corresponds to the sequence of loss mechanisms chosen in Figure 6. Starting from the experimental efficiency, especially the elimination of nonradiative recombination would guarantee the largest increase in efficiency (from 4.2% to 12.8%). Note that with radiative recombination only (columns b and c), the collection of charge carriers is sufficiently high and carrier mobility stops limiting cell efficiency (cf. Figure 7).

s-1 and krec ) 2.4 × 10-11 cm3/s). It becomes obvious that mobilities as high as µ ) 104 cm2/(V s) are not necessary for complete extraction of charge carriers from the device. Already for mobilities µ > 10-2 cm2/(V s), the efficiency saturates, meaning that a mobility improvement of roughly 2 orders of magnitude would be very valuable for device performance. These 2 orders of magnitude difference depend critically on the amount of nonradiative recombination and the thickness of the device. A reduction of kf by 2 orders of magnitude would compensate for lower mobilities as shown by the vanishing gap between the high and low mobility curve in Figure 7. However, a larger mobility always enables thicker absorber layers, increasing absorption in the active layer. Finally, we intend to estimate the relative importance of different loss mechanisms. To do so, the approach used in Figure 6 is not sufficient. In Figure 6 we chose one sequence of loss processes and went step by step from the experimental current/ voltage curve to a highly idealized radiative limit. However, the loss associated for instance with the transition from high mobilities to realistic mobilities will be largely different, depending on the other parameters. When you increase mobility starting from the experimental situation, the benefit is large, while it will be negligible, when you start from a situation where nonradiative recombination is switched off. Thus, in order to make a fair comparison of loss mechanisms and in order to show the direct potential for improvement, Figure 9 introduces the loss mechanisms in four sequences, where each sequence starts with a different loss mechanism. From top to bottom, each step replaces an ideal by a realistic physical process, e.g., from pure polymer to blend (emission at lower energy), from fully light scattering to flat interfaces (optics), from high to low carrier mobility, from efficient to realistic exciton diffusion lengths,

We have adapted the theory of Shockley and Queisser to calculate radiative efficiency limits of organic bulk heterojunction solar cells. The radiative efficiencies that follow from the optical properties of typical polymers are on the order η ≈ 23%, that of polymer/fullerene blends (in this case PF10TBT/PCBM) are on the order of η ≈ 21%, while current laboratory solar cells from the same material achieve η ≈ 4%. In order to explain this large gap between experiment and theory, we identify and quantify five principal loss mechanisms. The (i) energetic difference between the energy of the exciton in the polymer and the charge transfer state at the donor/acceptor heterointerface used by bulk heterojunction solar cells only leads to a small efficiency loss of 2%, while most losses are due to (ii) optical losses due to insufficient light trapping and parasitic absorption in layers that do not collect photocarriers, (iii) losses due to inefficient collection of photogenerated excitons, (iv) recombination losses due to a large amount of nonradiative recombination at the distributed heterointerfaces, and (v) losses due to insufficiently high charge carrier mobilities. Quantitative comparison of these loss mechanisms reveals that the by far most dominant loss mechanism in organic solar cells is the nonradiative recombination at the donor-acceptor interface, which would allow an increase in efficiency from 4.2% to 12.8%. Nonradiative recombination accounts for the loss in open circuit voltage, as well as for part of the loss due to carrier collection. Interestingly, the great advantage of organic semiconductors in light emitting diodes, namely its favorable high ratio between radiative and nonradiative recombination rates, are sacrificed in bulk heterojunction solar cells in favor of carrier collection. The necessary introduction of the donor-acceptor interface makes the ratio between radiative and nonradiative recombination the biggest obstacle for high photovoltaic efficiencies in bulk heterojunction solar cells. For the given high nonradiative recombination rates found in experimental devices, large efficiency improvements are also feasible by increasing the charge carrier mobility, thereby also improving carrier collection and fill factor. An increase in exciton mobilities will further enhance the photocurrent by increasing the amount of excitons that form free carriers and lead to smaller efficiency improvements on the order of 0.5% absolute. Due to the low refractive index of organic absorber materials, only smaller direct improvements may be obtained by introducing Lambertian light trapping schemes. Acknowledgment. We thank B. E. Pieters (Forschungszentrum Ju¨lich) for many fruitful discussions on solar cell modeling.

17966


References and Notes (1) Shockley, W.; Queisser, H. J. J. Appl. Phys. 1961, 32, 510. (2) Tiedje, T.; Yablonovitch, E.; Cody, G. C.; Brooks, B. G. IEEE Trans. Elec. DeV. 1984, ED-31, 711. (3) Green, M. A. IEEE Trans. Elec. DeV. 1984, ED-31, 671. (4) Forrest, S. R. MRS Bull. 2005, 30, 28. (5) Sariciftci, N. S.; Smilowitz, L.; Heeger, A. J.; Wudl., F. Science 1992, 258, 1474. (6) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J. Science 1995, 270, 1789. (7) Hoppe, H.; Sariciftci, N. S. J. Mater. Res. 2004, 19, 1924. (8) Blom, P. W. M.; Mihailetchi, V. D.; Koster, L. J. A.; Markov, D. E. AdV. Mater. 2007, 19, 1551. ¨ .; Meskers, S. C. J.; Sweelssen, J.; Koetse, (9) Veldman, D.; Ipek, O M. M.; Veenstra, S. C.; Kroon, J. M.; van Bavel, S. S.; Loos, J.; Janssen, R. A. J. J. Am. Chem. Soc. 2008, 130, 7721. (10) Pivrikas, A.; Juska, G.; Mozer, A. J.; Scharber, M.; Arlauskas, K.; ¨ sterbacka, R. Phys. ReV. Lett. 2005, 94, 176806. Sariciftci, N. S.; Stubb, H.; O ¨ sterba¨cka, R.; (11) Dennler, G.; Mozer, A. J.; Juska, G.; Pivrikas, A.; O Fuchsbauer, A.; Sariciftci, N. S. Org. Elec. 2006, 7, 229. ¨ sterbacka, R. Prog. (12) Pivrikas, A.; Sariciftci, N. S.; Juska, G.; O PhotoVolt: Res. Appl. 2007, 15, 677. (13) Mattheis, J.; Werner, J. H.; Rau, U. Phys. ReV. B 2008, 77, 085203. (14) Kirchartz, T.; Rau, U. Thin Solid Films 2008, 516, 7144. (15) Planck, M. Vorlesungen u¨ber die Theorie der Wa¨rmestrahlung; Barth: Leipzig, 1906. (16) Wu¨rfel, P. J. Phys. C: Solid State Phys. 1982, 15, 3967. (17) Street, R. A. Appl. Phys. Lett. 2008, 93, 133308. (18) Henry, C. H. J. Appl. Phys. 1980, 51, 4494. (19) Martı´, A.; Araujo, G. L. Sol. Energy Mat. Sol. Cells 1996, 43, 203. (20) Tessler, N.; Roichman, Y. Org. Elec. 2005, 6, 200. (21) Hulea, I. N.; Brom, H. B.; Houtepen, A. J.; Vanmaekelbergh, D.; Kelly, J. J.; Meulenkamp, E. A. Phys. ReV. Lett. 2004, 93, 166601. (22) Vandewal, K.; Gadisa, A.; Oosterbaan, W. D.; Bertho, S.; Banishoeib, F.; van Severen, I.; Lutsen, L.; Cleij, T. J.; Vanderzande, D.; Manca, J. V. AdV. Funct. Mater. 2008, 18, 2064. (23) Kim, Y.; Cook, S.; Tuladhar, S. M.; Choulis, S. A.; Nelson, J.; Durrant, J. R.; Bradley, D. D. C.; Giles, M.; McCulloch, I.; Ha, C.-S.; Ree, M. Nat. Mater. 2006, 5, 197. (24) Daub, E.; Wu¨rfel, P. Phys. ReV. Lett. 1995, 74, 1020. (25) Slooff, L. H.; Veenstra, S. C.; Kroon, J. M.; Moet, D. J. D.; Sweelsen, J.; Koetse, M. M. Appl. Phys. Lett. 2007, 90, 143506. (26) Hasharoni, K.; Keshavarz-K., M.; Sastre, A.; Gonza´lez, R.; BellaviaLund, C.; Greenwald, Y.; Swager, T.; Wudl, F.; Heeger, A. J. J. Chem. Phys. 1997, 107, 2308. (27) Kim, H.; Kim, J. Y.; Park, S. H.; Lee, K.; Jin, Y.; Kim, J.; Suh, H. Appl. Phys. Lett. 2005, 86, 183502. (28) Benson-Smith, J. J.; Goris, L.; Vandewal, K.; Haenen, K.; Manca, J. V.; Vanderzande, D.; Bradley, D. D. C.; Nelson, J. AdV. Funct. Mater. 2007, 17, 451. (29) Loi, M. A.; Toffanin, S.; Muccini, M.; Forster, M.; Scherf, U.; Scharber, M. AdV. Funct. Mater. 2007, 17, 2111. (30) Morteani, A. C.; Sreearunothai, P.; Herz, L. M.; Friend, R. H.; Silva, C. Phys. ReV. Lett. 2004, 92, 247402. (31) Offermans, T.; van Hal, P. A.; Meskers, S. C. J.; Koetse, M. M.; Janssen, R. A. J. Phys. ReV. B 2005, 72, 045213. (32) Benson-Smith, J. J.; Wilson, J.; Dyer-Smith, C.; Mouri, K.; Yamaguchi, S.; Murata, H.; Nelson, J. J. Phys. Chem. B 2009, 113, 7794. (33) Hallermann, M.; Haneder, S.; Da Como, E. Appl. Phys. Lett. 2008, 93, 053307.

Kirchartz et al. (34) Minnaert, B.; Burgelman, M. Prog. PhotoVolt.: Res. Appl. 2007, 15, 741. (35) Peet, J.; Kim, J. Y.; Coates, N. E.; Ma, W. L.; Moses, D.; Heeger, A. J.; Bazan, G. C. Nat. Mater. 2007, 6, 497. (36) Kim, J. Y.; Lee, K.; Coates, N. E.; Moses, D.; Nguyen, T.-Q.; Dante, M.; Heeger, A. J. Science 2007, 317, 222. (37) Yablonovitch, E. J. Opt. Soc. Am. 1982, 72, 899. (38) Green, M. A. Prog. PhotoVolt: Res. Appl. 2002, 10, 235. (39) Mattheis, J. Mobility and homogeneity effects on the power conVersion efficiency of solar cells, PhD Thesis: University of Stuttgart, 2008, p 140. (40) Kirchartz, T.; Helbig, A.; Rau, U. Sol. Energy Mat. Sol. Cells 2008, 92, 1621. ¨ pping, J.; Peters, M.; Kirchartz, T.; (41) Ulbrich, C.; Fahr, S.; U Rockstuhl, C.; Wehrspohn, R. B.; Gombert, A.; Lederer, F.; Rau, U. Phys. Stat. Solidi A 2008, 205, 2831. (42) Kirchartz, T.; Seino, K.; Wagner, J.-M.; Rau, U.; Bechstedt, F. J. Appl. Phys. 2009, 105, 104511. (43) Rau, U.; Werner, J. H. Appl. Phys. Lett. 2004, 84, 3735. (44) Campbell, P.; Green, M. A. J. Appl. Phys. 1987, 62, 243. (45) Fahr, S.; Rockstuhl, C.; Lederer, F. Appl. Phys. Lett. 2008, 92, 171114. (46) Rockstuhl, C.; Lederer, F.; Bittkau, K.; Carius, R. Appl. Phys. Lett. 2007, 91, 171104. (47) Rau, U.; Kron, G.; Werner, J. H. J. Phys. Chem. B 2003, 107, 13547. (48) Kirchartz, T.; Mattheis, J.; Rau, U. Phys. ReV. B 2008, 78, 235320. (49) Maturova´, K.; van Bavel, S. S.; Wienk, M. M.; Janssen, R. A. J.; Kemerink, M. Nano Lett. 2009, 9, 3032. (50) Maturova´, K.; Kemerink, M.; Wienk, M. M.; Charrier, D. S. H.; Janssen, R. A. J. AdV. Funct. Mater. 2009, 19, 1379. (51) Kirchartz, T.; Pieters, B. E.; Taretto, K.; Rau, U. J. Appl. Phys. 2008, 104, 094513. (52) Braun, C. L. J. Chem. Phys. 1983, 80, 4157. (53) Mihailetchi, V. D.; Koster, L. J. A.; Hummelen, J. C.; Blom, P. W. M. Phys. ReV. Lett. 2004, 93, 216601. (54) Koster, L. J. A.; Smits, E. C. P.; Mihailetchi, V. D.; Blom, P. W. M. Phys. ReV. B 2005, 72, 085205. (55) Sievers, D. W.; Shrotriya, V.; Yang, Y. J. Appl. Phys. 2006, 100, 114509. (56) Gommans, H. H. P.; Kemerink, M.; Kramer, J. M.; Janssen, R. A. J. Appl. Phys. Lett. 2005, 87, 122104. (57) Kotlarski, J. D.; Blom, P. W. M.; Koster, L. J. A.; Lenes, M.; Slooff, L. H. J. Appl. Phys. 2008, 103, 084502. (58) Bridgman, P. W. Phys. ReV. 1928, 31, 101. (59) Nelson, J.; Kirkpatrick, J.; Ravirajan, P. Phys. ReV. B 2004, 69, 035337. (60) Rau, U. Phys. ReV. B 2007, 76, 085303. (61) Kirchartz, T.; Rau, U. Phys. Status Solidi A 2008, 205, 2737. (62) Moet, D. J. D.; Slooff, L. H.; Kroon, J. M.; Chevtchenko, S. S.; Loos, J.; Koetse, M. M.; Sweelssen, J.; Veenstra, S. C. Mater. Res. Soc. Symp. Proc. 2007, 974, CC03–09. (63) Park, S. H.; Roy, A.; Beaupre´, S.; Cho, S.; Coates, N.; Moon, J. S.; Moses, D.; Leclerc, M.; Lee, K.; Heeger, A. J. Nat. Photon. 2009, 3, 297. (64) Kirchartz, T.; Rau, U.; Kurth, M.; Mattheis, J.; Werner, J. H. Thin Solid Films 2007, 515, 6238. (65) Kirchartz, T.; Helbig, A., Reetz, W., Reuter, M., Werner, J. H., Rau, U. Prog. PhotoVolt. Res. Appl. 2009, 17, 394. (66) Kirchartz, T.; Rau, U. J. Appl. Phys. 2007, 102, 104510. (67) Kirchartz, T.; Rau, U.; Hermle, M.; Bett, A. W.; Helbig, A.; Werner, J. H. Appl. Phys. Lett. 2008, 92, 123502.

JP906292H

Efficiency Limits of Organic Bulk Heterojunction Solar Cells - The

Recommend Documents