Equivalent circuit analysis on organic solar cells: how can impedance spectroscopy results be interpreted?

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Equivalent circuit analysis on organic solar cells: how can

impedance spectroscopy results be interpreted?

Sofie ten Have

Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between

01-04–2017 and 04-07–2017

Studentnumber: 10507183

Supervisor: Elizabeth von Hauff

Second assessor: Rick Bethlem

Physics of Energy Faculty of Sciences Vrije Universiteit Amsterdam

The Netherlands July 4th, 2017

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Abstract

Organic photovoltaics (OPVs) are a promising alternative to conventional solar cells. To obtain their potential a better understanding of their performance is needed. Electrical prop-erties can be measured with impedance spectroscopy (IS), but these measurements are hard to interpret. To give the measurements physical meaning, an existing model from the literature is applied to various OPV systems and compared to a second model. It is found that the model can be applied to the IS measurements of most OPVs, however not to optimized cells or cells measured in the dark under low bias. Comparison with the second model did not explain this behaviour. The results demonstrate that current models do not adequately describe the photovoltaic response of OPVs and should be revised to get a better understanding of the performance of OPVs.

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Populair wetenschappelijke samenvatting

Het klinkt als een product met een keurmerk in de supermarkt, ditmaal geen biologische gehaktballen, maar organische zonnecellen. Wat is er dan beter aan een organische zonnecel in vergelijking met een conventionele zonnecel? Het verschil zit in het materiaal waarvan het gemaakt is. De meeste zonnepanelen zijn gemaakt van silicium [14], terwijl een organische zonnecel gemaakt is van, de naam zegt het al, organische materialen. Dat wil zeggen: de koolstofketens waar ook wij mensen, dieren en planten van gemaakt zijn. Het grote voordeel van dit materiaal is dat de grondstoffen oneindig verkrijgbaar zijn en ze ook nog eens makkelijker te verwerken zijn tot zonnepanelen en dus in potentie een stuk goedkoper [14, 6]. Ze zijn daarnaast flexibel en, ook heel belangrijk: ze kunnen zonlicht goed omzetten in elektriciteit [10]. Berekeningen laten zien dat ze in potentie wel 23 procent van het vermogen dat het zonlicht aanlevert, kunnen omzetten in elektrische energie [12]. Helaas lukt het wetenschappers nog niet om een organische zonnecel te maken die deze potentiële efficiëntie ook daadwerkelijk haalt. Op dit moment zijn de efficiëntste cellen zo rond de tien procent [14]. De grote vraag is natuurlijk: hoe maken we zonnecellen die wèl de 23 procent halen? Om die vraag te kunnen beantwoorden moet je weten hoe een organische zonnecel werkt. Met een methode waarbij een kleine wisselstroom op de zonnecel gezet wordt, impedance spectroscopy, kan men de reactie van de zonnecel op deze wisselstroom meten en informatie verkrijgen over de snelheid van de verschillende processen in een zonnecel. De geladen deeltjes in de zonnecel reageren namelijk iets vertraagd op de wisselstroom, afhankelijk van hoeveel weerstand zij ondervinden in het materiaal. In organische zonnecellen zijn er twee processen die de productie van stroom tegengaan: 1) traag transport van de elektronen en 2) het verdwijnen van elektronen doordat zij in een zogenaamd gat vallen in het materiaal, waarbij ze samen gaan met een positief deeltje en neutraal worden. Volgens een model door Basham et al [2] kun je deze twee processen terugvinden in de resultaten van impedance spectroscopy. Dit onderzoek laat echter zien dat deze interpretatie wellicht wat voorbarig is. Bij het analyseren van de metingen van vier typen organische zonnecellen met uiteenlopende efficiënties, bleek namelijk dat zij niet allemaal voldeden aan het voorgestelde model. Juist bij de meest efficiënte zonnecel wordt een onbekend effect gevonden dat niet te wijten valt aan transport of het wegvallen van elektronen. Alhoewel de andere zonnecellen wel grotendeels passen in het model, zijn er ook daar haken en ogen te vinden die de fysische interpretatie van het model in twijfel trekken. Kortom, dit onderzoek laat zien dat de exacte processen in organische zonnecellen niet zo gemakkelijk te achterhalen zijn. Wel is aan de hand van de metingen beter bepaald wat er wel en niet werkt in het model. Hoe de organische zonnecel dus precies werkt blijft nog even in het donker, maar als er in de toekomst met ander licht op wordt geschenen, is er goede hoop dat de cel ons in de toekomst veel energie zal gaan geven.

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1 Introduction

While the conventional photovoltaic panels can be seen on more and more rooftops [16], bulk-heterojunction (BHJ) organic photovoltaics (OPVs) are emerging as a promising alternative [14, 6]. They are very flexible and promising for potential low-cost production because of low-temperature and solution-based fabrication [14, 6, 19]. Besides, they have potential for a high light-to-energy conversion efficiency [10]. Efficiency of organic solar cells have already exceeded ten percent [14] and the Shockley-Quiesser limit has been calculated to be 23 percent [12]. Thus, there is a lot of room for improvement of these cells. The question raises: which physical processes are limiting organic solar cells [17, 24]? Which is a very relevant question as the energy demand in the world is rising [1] and the negative effects of fossil fuels ask for renewable alternatives. To understand the physical processes of OPVs several transient and small perturbation techniques can be used [5]. One of the existing small perturbation techniques, impedance spectroscopy (IS), is useful to study resistive processes taking place in an operating OPV device [10]. Gundlach and co-workers use IS as a method to predict the current-voltage curve in OPV devices [2]. They find an equivalent electrical circuit to model an OPV and from this model they calculate the rate of generation and recombination of carriers, the total amount of carriers and finally the current-voltage curve of an OPV device. They apply their model on an optimized, very efficient OPV, measured under illumination. In this thesis it is investigated whether Basham et al’s model can be applied to both optimized and non-optimized OPVs measured in the dark and under illumination to get deeper insight in the physical interpretation of the results of impedance spectroscopy. The influence of morphology and weight ratios of the material are also analyzed, by using crystalline and amorphous polymers in different weight ratios. Finally, comparison with Van Egmond’s [23] capacitance-frequency analysis on the data should give a more complete view of the physical meaning of Basham’s model.

After giving background on OPVs, impedance and Basham et al’s model, in this thesis the model is applied to IS measurements performed by Sichert [20]. The measurements are examined qualitatively, the model is fitted and the calculations from Basham et al’s method are performed. Based on the results the physical meaning of the equivalent circuit model proposed by Basham et al will be nuanced. A short comparison with Van Egmonds analysis tries to connect both methods. With the nuances on the model and a connection between several methods it is hoped to get a better understanding of the processes occurring in OPVs in order to optimize them further in the future.

2 Theory

2.1 Organic solar cells

As the name says, organic solar cells are mainly made of organic materials, i.e. hydrocarbon based compounds. The electronic structure of these organic semiconductors allow them to convert optical power of absorbed light into electrical power: a photocurrent.

2.1.1 Processes in an organic solar cell

Several important processes contributing to the production of a photocurrent in OPVs can be distin-guished: I) the absorption of photons, II) exciton generation, III) exciton diffusion and dissociation into free charges, IV) transport and V) charge extraction and collection (figure 1) [14].

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Figure 1: Schematic representation of the main processes in an OPV. The main processes are I) the absorption of photons, II) exciton generation, III) exciton diffusion and dissociation into free charges, IV) transport and V) charge extraction and collection [14]. 1) is geminate and 2) non-geminate recombination of electrons and holes.

A heterojunction organic solar cell consists of an electron donor material/ hole transport medium and an electron acceptor/electron transport medium, a polymer and fullerene respectively. Excitons are generated in the donor polymer (II), when an electron is excited from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) by an incoming photon (I) [13] [8]. An exciton then diffuses to a polymer-fullerene interface where the polymer donates the electron to the acceptor (III). This donation is motivated by the energy difference between the acceptor’s and donor’s LUMO; the acceptor’s LUMO has a slightly lower energy level [13]. After the transfer the electrons and holes, now free carriers, are transported towards the cathode, anode respectively (IV). Finally the carriers are extracted to produce a photocurrent (V). However, due to defects in the polymer, there is a chance that during transfer or transport of the electron, the electron recombines with a hole, called geminate (1) and non-geminate recombination (2) respectively [13]. These defect states are a result of energetic disorder, structural inhomogeneities and chemical impurities in the semiconducting material[11]. The defect states, also called traps, limit the loss of free carriers and they negatively affect charge separation and performance of the solar cell, as they lead to exciton annihilation and a carrier transport [25] [17].

Unlike the schematic representation of the processes shown in picture 1, actually the HOMO and LUMO are not at one energetic level, but have a distributed density of states (DOS) [17] [13] (figure 2).

Carriers are in localized electronic states. Due to disruptions and defects in organic semicon-ductors, carriers are transported via localized sites with slightly varying energy levels and distances [13] [20]. According to the Gaussian Disorder Model, transport of the carriers can be seen as hop-ping between the localized sites, where charge moves from one site to another by thermally assisted hopping and tunnelling [13] [20]. In figure 3 transport of charges is visualized.

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Figure 2: Schematic illustration of the density of states (DOS) distribution of the HOMO and LUMO level in an OPV with a Fermi energy indicated.

Figure 3: Visualization of transport of charges (figure from [20] after [7]).

charge carriers occupy a site in the center of the DOS (as shown in figure 2). In the center more sites of similar energetic level and spatial vicinity are available. On top of that, traps are being filled when the carrier density increases. Therefore a higher carrier density probably increases the mobility in an OPV. The mobility of electrons and holes can differ, also because their DOS distribution can differ [13]. Besides recombination, low mobility can be a plague to the performance of OPVs. It causes inefficient charge extraction, especially in systems with effective charge generation and/or large active layer thickness [24]. The efficiency of an OPV thus depends on the mobility of the charge carrier (µ) and its lifetime (τ ) [9]. The free carrier lifetime mobility product, the µτ -product, should be optimized to attain good cell performance.

2.1.2 Organic photovoltaic device

The efficiency of a cell also depends on the architecture of the cell. A schematic drawing of a bulk heterojunction organic solar cell can be found in figure 4. The polymer and fullerene are blended

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in a bulk heterojunction to form the active layer of the solar cell. This blend results in a spatially distributed interface of the two materials and enables the majority of the photogenerated excitons to reach a donor-acceptor interface within their lifetime [13]. With this architecture, a larger active layer can be used which increases the absorption efficiency and thereby the created current.

Figure 4: Schematic drawing of an organic BHJ solar cell.

2.1.3 Current-voltage characteristics

When optimizing an organic solar cell, the current density - voltage (j-V) characteristics of a cell play an important role. In figure 5 the j-V curve of a typical optimized solar cell is shown, both in the dark as under illumination.

Figure 5: The j-V curve of a typical optimized OPV device (rr-P3HT:PCBM 1:1). Jsc, short circuit

current, is the current at 0V bias, Voc, open circuit voltage, the voltage at which the photocurrent

and the diode current cancel each other and MPP is the maximum power point (after reference [20] and [13]).

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In the dark the solar cell acts like a diode [20]. Under illumination additional charge carriers are generated in the solar cell and a photocurrent is induced in opposite direction of the diode current. Important parameters are the short circuit current, Jsc, the current at 0V bias, and open circuit

voltage, Voc, the voltage at which the photocurrent and the diode current cancel each other and

the net current is zero [13]. The generated power of the solar cell, Pel = V · I, is maximum at

the biggest square under the J-V curve (dark grey), which gives a maximum power point (MPP) at the right bottom corner of the curve. The fill factor (FF) is defined as the difference of the maximum extractable power (dark grey square) and the product of Jscand Voc (light grey square)

and indicates the deviation from an ideal rectangular j-V curve,

F F = VM P P · JM P P V oc · Jsc

[13]. (1)

The power conversion efficiency (PCE), a measure for the performance of the cell, is given by the ratio of the maximum extractable power density and the power density of the incident light (PL) and given by:

P CE = Pel PL =VM P P · JM P P PL =Voc· Jsc· F F PL [13]. (2) 2.1.4 Materials

In order to study the effects of morphology and weight ratio, materials with the same chemical properties, but with different morphology and polymer:fullerene ratio are used to compose the OPVs examined in this study. The studied IS measurements are conducted both in the dark and under standard illumination conditions (see [20]), to be able to distinguish the effects of light on OPVs.

The polymer used for the experiments is poly(3-hexylthiophene-2,5-diyl) (P3HT) and the fullerene phenyl-C71-butyricacidmethylester (PCBM) [20], which are the basic materials of the most studied and best understood organic solar device [10, 13]. P3HT can be structured either regioregular (rr-P3HT) or regiorandom (rra-(rr-P3HT) (figure 6), where the sidechains of rra-P3HT are more disordered than those of rr-P3HT [17, 20] (figure 6.

The different morphologies of rr-P3HT and rra-P3HT impact their physical properties. For example, mobility of carriers in rr-P3HT is one order of magnitude larger than in rra-P3HT [17]. Also, exciton dissociation is very poor, resulting in a much lower efficiency of the rra-P3HT:PCBM blend than the rr-P3HT:PCBM blend [21]. The ratio of the blends also determines the efficiency of the solar cell. Measurements of the j-V characteristics have shown that a 1:1 polymer:fullerene weight ratio is more efficient than 1:2 for regioregular polymer (3.4% versus 2%), but a 1:2 ratio is more efficient for the regiorandom polymer (0.2% versus 0.1%)[20] (figure 7). For solar cells with regioregular polymer a balanced ratio of polymer and fullerene results in a balanced distribution of the two, thus allowing well-balanced charge transport which results in a high efficiency. For regiorandom P3HT however, it is merely the fullerene that takes care of charge transport, which is why the efficiency doubles when increasing the amount of fullerene [20].

Considering the properties described above, rr-P3HT-PCBM 1:1, rr-P3HT-PCBM 1:2, rra-P3HT:PCBM 1:1 and rra-rra-P3HT:PCBM 1:2 are appropriate blends to gain a better understanding of the influences of morphology, the polymer:fullerene ratio and illumination on the performance

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Figure 6: Schematic picture of a) regioregular P3HT and b) regiorandom P3HT. (After [17])

Figure 7: Scheme of the used blends (after [23])

of the cell. By examining the impedance spectroscopy results of these blends with Basham et al’s model it is hoped to get deeper insight in the DOS of the blends and the effect on the several processes in OPVs.

2.2 Impedance spectroscopy

The several polymer:fullerene blends described above are examined using impedance spectroscopy. This electrical technique is useful to study resistive processes taking place in an operating solar cell [10].

2.2.1 Impedance

Impedance spectroscopy measures impedance, which is the complex resistance. In circuits where a direct current (DC) is applied the resistance is given by Ohm’s law:

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R = V

I, (3)

with R the resistance in Ohm (Ω), V the Voltage in Volt (V ) and I the current in Amp`ere (A). In circuits with alternating current (AC) however, due to the dependence of the current and the resistance on the voltage, a more complex expression of Ohm’s law needs to be adopted. The AC-equivalent to equation 3 is,

Z(ω) =V (ω, t) I(ω, t) =

V0eiωt

I0ei(ωt+φ)

= Z0e−iφ, (4)

where i = √−1, Z is the impedance in Ω, Z0 ≡ V_I₀0, ω the angular frequency (related to the

frequency with ω = 2πf ) and t the time in seconds (s)[18]. Instead of expressing impedance with a modulus and phase angle, it can also be expressed rectangular as,

Z = R + iX (5)

(visualized in figure 8). Here X is the reactance in Ω. The reactance is determined by the inductance and capacitance of the measured circuit. In semiconductors there are no changing magnetic fields and therefore no inductance is expected. The imaginary component of the impedance is thus determined by the capacitance.

Figure 8: Plot of the real part of the impedance versus the imaginary part. The impedance can be described either by the modulus of Z and the phase angle φ or by the values of the reactance X and resistance R. [15] (figure by [22])

The impedance of a resistor is R and that of a capacitor is _iωC1 (visualized in figure 9). Ideal resistance and capacitance as shown in figure 9 do not exist in reality. Some of the electrons will always be travelling slower than others, resulting in a distributed relaxation time [22] (for more on relaxation times, section 2.2.3). To account for dispersion, one can take a Constant Phase Element (CPE) to model a non-ideal capacitor. The impedance of a CPE is described by,

ZCP E=

1 T(iω)

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Figure 9: Visual representation of the impedance of a) an ideal resistor and b) a capacitor. (After [22])

n is a dimensionless number between 0 and 1 and T a constant with dimension F sn−1_{. If n = 1}

equation 6 recovers a perfect capacitor. Van der Knaap [22] provides more details about dispersion found in the IS measurements from Sichert.

2.2.2 Method of impedance spectroscopy

Impedance spectroscopy is a method to measure the impedance of a substance and deduce electrical properties from this measurement. When applied to OPVs, the blend of polymer and fullerene is placed in between two electrodes with a DC voltage (bias voltage) [13]. On top of the bias a very small AC voltage enough to perturb the system, but not cause any changes to it, is applied, e.g. 20mV [10] . The frequency of the AC-voltage typically ranges from 1M Hz − 10Hz and the DC voltage usually ranges from the voltage at Jsc(0V ) until V = Voc [10], which is the region in which

the device is operating. The capacitive and resistive properties of the device will delay the flow of a current through the device. Measuring the current and its phase delay compared to the applied AC give the impedance of the device (equation 4).

2.2.3 Equivalent circuits

One way of analyzing the results of impedance spectroscopy is using equivalent circuit analysis. It involves fitting the impedance of an electrical circuit to the measured impedance [15]. The above described circuit elements can be used to form a circuit that fits the data.

The impedance of a circuit with a resistance and capacitance in parallel, an RC-element (figure 10a), is 1 ZRC = 1 ZR + 1 ZC = 1 R + 1 1 iωC =1 + iωRC R (7) ZRC= R 1 + iωRC[22]. (8)

When plotted in a Nyquist plot, where the real versus the imaginary part of the impedance is plotted, an RC-element looks like a semi-circle (figure 10b). When taking the absolute value squared of ZRC,

|ZRC|2=

1

R2 + (ωC)2

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the similarities between the formula of a circle and the impedance of an RC-element become clear and explain the shape of the plot. A 3D visualization gives an overview of how the real and imaginary part depend on the AC frequency with which the element is probed (figure 10c).

Figure 10: The impedance of an RC-element (a) visualized in b) a Nyquist plot and c) a 3D representation. This simulation was made with R = 1000Ω and C = 1 · 10−6F over a frequency range of 10Hz − 1M Hz.

Defining RC ≡ τ , the time constant of an RC-element in seconds (s) [15], gives ZRC =

R

1 + iωτ. (10)

The time constant indicates a relaxation process. Relaxation occurs when the frequency is too high for the carriers to respond and they return into an equilibrium state. The maximum of the semicircle has a characteristic angular frequency related to the relaxation time, ωmax = _τ1 = _RC1

[15].

Two circuits formed by placing in series an R and the above described RC-element are used to fit the IS measurements: the R-RC and R-RC-RC circuit, visualized in figure 11.

Figure 11: Circuits used to fit the impedance data: a) R-RC and b) R-RC-RC.

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circuit depends on the values of the R’s and C’s. When the R’s and C’s in the RC-elements are exactly the same, the shape is a perfect semicircle and cannot be distinguished from an R-RC circuit (figure 12 a). The more the values of the R’s and C’s in the RC-elements diverge however, the clearer two arcs can be seen in a Nyquist plot (figure 12 b and c).

Figure 12: The influence of changing the values of R and C in an R-RC-RC model. a) When both R’s and C’s in the RC-elements are equal, the plot cannot be distinguished from an R-RC model. b) When there are slight changes in R or C the shape changes slightly as well. c) When R and C diverge further, two arcs can be seen.

Say one of the mentioned circuits fits an impedance measurement, how can these be physically interpreted? When the results of an impedance measurement fit the R-RC circuit, this means that at least one resistive and capacitive process is taking place (assuming that no two different processes have exactly the same resistance and capacitance, as in figure 12 a)). However, because mathematically two RC-elements in parallel can be reduced to one RC-element, other processes could take place in parallel. In parallel physically means the processes are separated spatially. If the results of an impedance measurement fit the R-RC-RC circuit, this means that two processes are taking place in series, thus sequential in time. Note though, that again more processes could happen parallel to it (figure 13).

Besides being able to say whether and how many processes are taking place in series, also the time-scale in which a process takes place can be determined from the circuit. The time-scale is given by the time constants related to each semi-circle (τ = _RC1 ) and connected to the frequency by τ =_ω1

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Figure 13: Schematic representation of processes parallel and in series

to a process with a longer time scale and thus slower. An arc in the high frequency region (lower τ ) corresponds to a faster process (figure 14).

Figure 14: Faster processes can be found in the high frequency region (low τ ) and slower processes in the low frequency region (high τ ).

2.3 Impedance spectroscopy on organic solar cells

From equivalent circuit analysis sequential processes and their timescale can thus be determined. According to Clarke et al charge carrier lifetimes, insight in density of states, recombination behavior and diffusion lengths can also be deduced from impedance spectroscopy measurements[5]. What physical meaning is attributed to impedance spectroscopy measurements on organic solar cells?

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2.3.1 Physical meaning of equivalent circuits

In the literature specific processes of organic solar cells have been connected to the equivalent circuit model that fits IS measurements on OPV’s. Several studies find two semi-circles [2, 10, 17] or even three with a very wide frequency range [3] in the Nyquist plots of performed IS measurements. Those who found two semi-circles adopted the R-RC-RC model (figure 15). The first R of this model is a series resistance (Rs) attributed to the substrate and contacts of the device [2]. The

semicircle in the low frequency region is usually connected to recombination, the relaxation time τ being the mobile carrier lifetime [2]. The C from this RC-element can be associated with the chemical capacitance Cµ, which is the increase in charge and carrier density with a change in the

quasi Fermi-level[2]. Cµ is known to increase exponentially with increasing bias [8]. The R can be

seen as the recombination resistance Rrec[2, 10], i.e. the resistance of the system to electrical losses

due to recombination [10]. The semi-circle in the higher frequency region is supposed to represent transport effects, where τtr is the transit time, i.e. the time it takes to inject charge into the active

layer and transport it across the device [2].

Figure 15: The equivalent circuit diagram R-RC-RC, after [2]. Rsis the series resistance, Rtand

Ct represent transport effects, Rrec is the resistance of the system to recombination and Cµ the

chemical capacitance.

2.3.2 Modelling an OPV according to Basham et al

Using the R-RC-RC equivalent circuit diagram and the physical interpretation discussed above, Basham et al propose a formula to calculate the increase in the concentration of the faster carrier type (assumed to be the electron) [2]:

n = 1 qL

Z

CµαdV + n0, (11)

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constant that relates the voltage to the shift of the Fermi-level for one carrier type and n0 the

amount of carriers at the lower bound of the integration. n0 can be approximated by:

n0=

1

qLJscRtCt. (12)

The values for Cµ, Rt and Ct are all obtained from the fit of the R-RC-RC model over the

measured impedance data from the OPV. Filling in equation 6 from Basham et al gives a model for the j-V curve,

J (V ) = Jsc+

R CµαdV

RrecCµ

. (13)

To determine α the outcome of equation 13 can be fitted against the measured J (V ) curve, treating α as a fitting parameter. α relates the applied external voltage to the shift of the Fermi level. The shift is non-linear, because it is related to DOS. When the DOS is really broad the voltage needed to shift the Fermi level is higher and α will therefore be lower. A one to one shift of the Fermi-level with voltage results in α = 1, which is the maximum value of α.

Figure 16: Schematic presentation of how α is influenced by the DOS. When 0.1V is added and there’s no distribution α = 0.5 (blue) for each material. If there is a distribution, α = will be lower because energy is needed to fill the distribution (red).

Basham et al expects the parameter α to be a 0.5 if there is an equal shift in each material, as twice a half adds up to one. However, because the effective DOS is known to be higher in PCBM than in P3HT they expect less shift of the Fermi level in PCBM and a value less than a 0.5. For rr-P3HT:PCBM 1:1 they find α = 0.3.

2.3.3 Capacitance-frequency analysis

Besides equivalent circuit analysis based on the model of Basham et al, also a capacitance-frequency analysis (C-f analysis) has been performed on Sicherts data by Van Egmond [23] following the

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methods of Xu et al [25]. This analysis calculates the DOS defect distribution of the material by taking the derivative of the capacitance with respect to the frequency. The origin of the peaks that Van Egmond finds in his distribution could either be related to a defect state (traps) or a free carrier freeze out (when carriers have no time to shift in and out of depletion edge in response to the applied signal [17]) [23]. For more details on this analysis and Van Egmonds results see [23].

3 Methods

In this study the methods of Basham et al [2] are followed in order to examine whether the R-RC-RC model fits Sicherts data and what this physically means. For details about Sicherts measurements, see [20]. First, Sicherts data are analyzed qualitatively, both the shape and size of the Nyquist plots are discussed. Second, the R-RC-RC model is fitted on the data, using ZView2’s equivalent circuit tool. The quality of the fits are compared with fits of the R-RC model to see which of the models fits best. To compare the quality of the fits, the weighted sum of squares is used. The sum of squares is proportional to the average percentage error between the original data points and the calculated values from the model [26]. Since most of the measurements of both blends fit the R-RC-RC model, the values for the resistances and capacitors produced by these fits are used for further calculation. The fits with a sum of squares above five are neglected, as these are considered unreliable. Using the model’s values, the resistances, capacitors and the relaxation times (τ ≡ RC, see section 2.2.1) of both RC elements of the R-RC-RC model are calculated and plotted against voltage. To prevent confusion but without attributing any physical meaning to the model, the R’s and C’s in the model are named Rs− R1C1− R2C2, where Rsstands for series resistance, R1C1is

the RC-element in the low frequency region (corresponding to Rrec and Cµ according to Basham

et al [2]) and R2C2 is the RC-element in the high frequency region (corresponding to Rt and Ct

according to Basham et al [2]). Continuing to force the R-RC-RC model on the data, the carrier density of the several blends, including the values for n0 and α, are calculated (for details on the

calculation see section 2.3.2). For the error bars the values of the estimated error from the model are used. α is determined by plotting the outcome of equation 13 with α’s ranging from 0.1 to 1, next to the measured j-V curve and choosing the α that seems to fit best. Lastly, the results of this study are compared with the results of the C-f analysis done on the same data by Van Egmond [23].

4 Results

4.1 Qualitative results

The IS data delivered by Sichert have been examined qualitatively, comparing the form of the Nyquist plots. In appendix A all Nyquist plots of the measurements not included in this section can be found.

The size of the arcs in the Nyquist plots decreases rapidly over increasing bias, consistent with a dropping resistance over voltage. In figure 17 a typical Nyquist plot of a blend with regiorandom polymer in the dark is portrayed. Vertical zigzag graphs can be seen at the lower voltages (below 0.4V for 1:1 and below 0.6V for 1:2). As these zigzags are in the vertical direction and thus mainly in the imaginary domain, they can be attributed to capacitive behaviour. At higher voltages a

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Figure 17: Nyquist plots of rra-P3HT:PCBM 1:1 dark, typical shape of rra-P3HT:PCBM in the dark. At lower voltages the blend acts like a capacitor, at higher voltages a semi-circle can be seen, which hints towards an RC-element.

semicircle can be seen, which hints at an RC-element (figure 17). Under illumination the blends with regiorandom P3HT show a semi-circle for every measured voltage.

Figure 18: Nyquist plots of rr-P3HT:PCBM 1:1 dark, typical shape of rr-P3HT:PCBM in the dark. At lower voltages the blend acts like a capacitor, going towards one semicircle and at higher voltages two arcs can be seen.

In figure 18 a typical plot with regioregular polymer in the dark is depicted. The blends with regioregular P3HT in the dark, just like the regiorandom blends in the dark, show capacitive behaviour for low voltages (0.3V and lower), going towards a semicircle. At higher voltages however, not one but two arcs can be distinguished, hinting at two RC elements in series. Under illumination the 1:2 blend shows one semicircle at voltages lower than 0.4V and from 0.4V onward also two arcs

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appear. In the Nyquist plot of the 1:1 blend under illumination however, just one semicircle and a peculiar tail upwards in the high frequency range can be found for all measured voltages (figure 19). Another feature of this blend is that the plot from 0.2V is bigger than the one from 0.0V , while in all the other blends the size of the plot is decreasing over increasing bias.

Figure 19: Nyquist plots of rr-P3HT:PCBM 1:1 illuminated. The plots show a semicircle with an upward tail in the high frequency range.

4.2 Fitting the R-RC model and R-RC-RC model

Following from the qualitative analysis on the IS measurements, both an R-RC model (because its Nyquist plot is a semicircle, see section 2.2.1) and an R-RC-RC model (because its Nyquist plot has two arcs, see section 2.2.1) were tried on the data. To illustrate how well the models fit on the regiorandom and regioregular blends, in figure 20 and 21 Nyquists plots are shown including the fits of the R-RC and R-RC-RC model. In table 1 an impression of the sums of squares of the regiorandom blend is given. In table 2 the results from the regiorandom blend can be found. A complete set of results can be found in appendix B.

For the blends with regiorandom polymer measured in the dark, the R-RC model had sums of squares above five for almost all of the voltages, only 0.89V for the 1:2 blend and 0.9V for the 1:1 blend has sums of squares just below five. Under illumination, the R-RC model fits a bit better, with the sum of squares being in between one and five for most of the measured voltages. The 1:2 blend fit a bit worse than the 1:1, 0.0V , 0.2V and 0.4V had sums of squares above five. Again the highest voltages fit a bit better: for the 1:2 blend the fit was below one for 0.8V and 0.9V and for the 1:1 blend for 0.9V . The R-RC-RC model fits the regiorandom blends much better both in the dark and under illumination. In the dark, the sums of squares are mostly below five, only 0.0V and 0.2V for the 1:1 blend are above five. Under illumination, the model has sums of squares below one for all of the voltages in both blends and thus fits really well.

For the regioregular polymer, the R-RC model has sums of squares above five for all of the measurements, both in the dark and under illumination. The R-RC-RC model fits a lot better. In the dark, only at low voltages it doesn’t fit so well (1:2 blend sum of squares above five for voltages below 0.4V and in between one and five for 0.4V , 1:1 blend only the 0.2V between one and five), all

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Figure 20: rra-P3HT:PCBM 1:1 dark fitted with a) the R-RC model and b) the R-RC-RC model.

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Table 1: Impression of the sums of squares from the regiorandom blends. Overall the R-RC-RC model fits best, only at low voltages in the dark it does not fit very well. All calculated sums of squares can be found in appendix B.

regiorandom 1:1 1:2 Voltage (V) R-RC R-RC-RC Voltage(V) R-RC R-RC-RC 0 96.8 96.8 0 603 1.66 dark 0.6 124 1.84 0.6 642 1.09 0.9 2.67 0.187 0.89 3.58 0.134 Voltage (V) R-RC R-RC-RC Voltage (V) R-RC R-RC-RC 0 2.271 0.152 0 19.8 0.0837 illuminated 0.6 2.363 0.155 0.6 4.18 0.0481 0.9 0.795 0.0957 0.9 0.615 0.0368

Table 2: Impression of the sums of squares from the regioregular blends. Like the regiorandom blend, the R-RC-RC model fits the best and only under low bias voltage in the dark it doesn’t fit. An exception is the rr 1:1 illuminated blend, this blend cannot be fitted. All sums of squares can be found in appendix B. regioregular 1:1 1:2 Voltage (V) R-RC R-RC-RC Voltage(V) R-RC R-RC-RC 0 18.1 0.698 0 30.6 92.4 dark 0.4 28.1 0.634 0.4 114 3.75 0.6 19.6 0.115 0.6 70.4 0.254 Voltage (V) R-RC R-RC-RC Voltage (V) R-RC R-RC-RC 0 140 18.5 0 13 1.11 illuminated 0.4 115 6.46 0.4 36.8 0.817 0.6 119 8.93 0.6 42.9 0.0721

of the other measurements had sums of squares below one. Under illumination, the 1:2 blend has sums of squares in between one and five for the lower voltages and from 0.4V onward it has sums of squares below one. For the 1:1 blend under illumination, the blend with the upward capacitive tail in the high frequency range, all fits had sums of squares above five. When the frequency range for the fit was extended until 1015Hz, the R-RC-RC model did not fit as well. Also a capacitor was replaced by a CPE, but the R-RC-R(CPE) model does not yield better results: it gives sums of squares above five for almost all of the measurements (only 0.4V has a sum of squares in between one and five). Finally a fit with the R-RC-RC model was done with a frequency range (10Hz − 105Hz) that excludes the upward tail from the data. The sums of squares from these fits are all below one, except from the 0.0V , which has a sum of squares just above one.

All in all, for regiorandom and regioregular blends the R-RC-RC model has better fits than the R-RC model, the rest of the results are thus based on R-RC-RC. This means that probably two processes are taking place in series, which indicates two sequential processes. Only for the low voltages in the dark, the model does not fit so well. Probably because there is just capacitive behaviour when there is neither photocurrent nor high bias (see section 4.1). The 1:1 blend under

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illumination only fits the R-RC-RC model when neglecting the part of the high frequency range corresponding to the upward tail seen in its Nyquist plot.

4.3 Analyzing the results of the R-RC-RC model

Using the results from the R-RC-RC model, the values of the R’s and C’s and R1C1 = τ1 and

R2C2= τ2can be examined.

The relaxation times of the RC-elements are designated τ1 and τ2, where τ1 < τ2 so they

correspond to R1C1 and R2C2, respectively. The relaxation times are plotted in figure 22. Both

relaxation times are generally a bit lower in the regioregular blends compared to the regiorandom blends. τ1lies between 4.19 · 10−6− 0.0011s for regioregular, while for regiorandom τ1lies between

9.21 · 10−6− 0.052s. τ2 lies between 4.29 · 10−7− 3.35 · 10−6s for regioregular and between 9.2 ·

10−6− 1.63 · 10−5_{s for the regiorandom blends. Therefore the processes are faster in regioregular}

blends compared to regiorandom blends. Furthermore it can be noted that relaxation time 1 is higher in the dark than under illumination, and decreases more rapidly. This is probably due to a high resistance of the blend in the dark under low voltage. Relaxation time 2 seems to be constant in the regiorandom blends until voltages around Voc are reached. In the regioregular blends the 1:2

ratio shows a decrease in time over voltage, while the 1:1 blend does not show a clear decrease. Analyzing the values of the R’s and C’s a couple of things stand out (missing graphs can be found in appendix C). First, an analysis of the values for Rs(figure 23) shows that the values are

more or less constant in the dark, around 12 Ω. For the samples under illumination however, Rs

increases slightly until about 0.3V and then decreases over voltage. This voltage dependence of Rs

is interesting, as in literature Rsis related to resistance due to substrate and contacts, which is not

expected to be voltage dependent [2].

Second, the value of C1 increases exponentially over voltage in regioregular blends, but is

con-stant in regiorandom blends (figure 24). The value of C2stays low for all voltages and blends. The

diverging values of C1 and C2 due to the exponential increase of C1 in regioregular blends cause

the appearance of two arcs in the Nyquist plots of these blends at higher voltages. Furthermore, looking at the development of R1over voltage, R1is very big for blends measured in the dark under

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Figure 22: Relaxation time 1 τ1 plotted against voltage for a) the regiorandom blends and b) the

regioregular blends. Relaxation time 2 τ2plotted against voltage for c) the regiorandom blends and

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Figure 23: The values of the series resistance Rs of the Rs− R1C1− R2C2 model plotted against

voltage. The Rsof the illuminated samples increases slightly until about 0.3V and then decreases

over voltage. The Rsof the dark samples is more or less constant.

Figure 24: Value of C1 plotted against voltage. C1 increases exponentially in the regioregular

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4.4 Carrier density calculation

The carrier densities of all the blends in the dark and under illumination are calculated and plotted in figure 25. The carrier density of rra-P3HT:PCBM 1:1 under illumination could not be calculated because no α below one could be found to fit the measured j-V curve. Due to the exponential increase of C1 in regioregular blends, the carrier density from the regioregular blends increases

exponentially as well. Also note that in the dark, the carrier density at 0.0V is 0 because n0

only takes into account the created carriers by photocurrent. n0 for regiorandom 1:1, 1:2 and

regioregular 1:1 and 1:2 are 2.45 ± 0.47 · 1015_cm−3_{, 1.208 ± 0.061 · 10}15_cm−3_{, 2.1 ± 0.65 · 10}15_cm−3

and 3.33 ± 0.34 · 1015_cm−3 _{respectively. Although at zero bias the carrier density does not seem}

to be a measure for efficiency, at higher biases the more efficient solar cells have higher carrier densities.

Figure 25: The carrier density of the four different OPVs a) in the dark and b) under illumination. Besides the carrier density, the value of α for each blend is also interesting, as it should reveal something about the correlation of shifts in the Fermi energy Ef with applied voltage. The DOS of

the materials could effect this correlation (figure 16). According to Basham’s model α should be in between 0 and 0.5 [2]. They find a value of 0.3 for their regioregular 1:1 blend under illumination. An impression of the values for α found for the regiorandom and regioregular blends can be found in table 3 and 4 respectively. The complete list of estimated values of α can be found in appendix D. The regioregular 1:1 blend under illumination has an α that varies from 0.3 until 0.5 for voltages from 0.2V until 0.6V , quite close to Basham et al’s findings. For the other regioregular blends, α is almost one or one for all the voltages. The regiorandom blends have ranging α’s. The 1:1 illuminated blend has α’s above one, which would indicate something physically impossible. The 1:2 blend under illumination has α’s ranging from just above 1 until 0.7 with rising voltage, the 1:1 blend in the dark has α = 0.5 for almost all voltages and the 1:2 blend in the dark is 1 for voltages under 0.8V and is 0.3 above this. Based on these findings, it can be presumed that α is voltage dependent.

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Table 3: Impression of the values for α for regiorandom blends. There is no α for rra 1:1 0.2V in the dark, because the fit of the R-RC-RC model was not good enough.

regiorandom 1:1 1:2 Voltage (V) α Voltage(V) α 0.2 - 0.2 1 dark 0.6 >1 0.6 1 0.9 0.5 0.89 0.3 Voltage (V) α Voltage(V) α 0.2 >1 0.2 >1 illuminated 0.6 >1 0.6 1 0.9 >1 0.9 0.7

Table 4: Impression of the values for α for regioregular blends. There is no α for rr 1:2 0.2V and 0.4V in the dark, because the fit of the R-RC-RC model was not good enough.

regioregular 1:1 1:2 Voltage (V) α Voltage(V) α 0.2 >1 0.2 -dark 0.4 1 0.4 -0.6 1 0.6 1 Voltage (V) α Voltage(V) α 0.2 0.3 0.2 >1 illuminated 0.4 0.3 0.4 >1 0.6 0.5 0.6 1

4.5 Equivalent circuit analysis combined with capacitance-frequency

analysis

The analysis of the above results combined with the results presented by Van Egmond [23] yielded several interesting connections. First, whenever in the Nyquist plot two arcs can be seen, the area under the DOS peak in the high frequency region is above 10−9eV F . The only exception is the regioregular 1:1 blend under illumination, where the area is above 10−9eV F but there are not two arcs. Accumulation of many carriers in the high frequency region thus seems to be connected to the increase of C1. The frequency at which the DOS peak is positioned (ωDOS)and the angular

frequency on the maximum of the arc (ωmax) from R1C1 also seems to be connected: for most of

the blends the frequency at which they can be found differ at most one order of magnitude (rra 1:2 dark at most 2 orders and rr 1:1 dark goes from 4 orders difference to the same order of magnitude over voltage), figure 26 and 27.

Furthermore, Van Egmond fits the peaks seen in the DOS distribution with a Gaussian distri-bution. When analyzing the standard deviation of these Gaussian fits, a measure for the width of the peak, the standard deviation does not correlate with the α that this study finds. The expected relation between the width of the DOS and how fast the Fermi-energy rises with applied bias, therefore cannot be found.

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Figure 26: Comparing the position of the DOS peak (ωDOS) and the angular frequency at the

maximum of the R1C1 arc (ωmax) in the regiorandom blends.

Figure 27: Comparing the position of the DOS peak (ωDOS) and the angular frequency at the

maximum of the R1C1 arc (ωmax) in the regioregular blends.

5 Discussion

Based on the results a couple of conclusions can be drawn. First, the R-RC-RC equivalent circuit seems a good model for the measured OPVs. Even in the Nyquist plots that show one arc instead

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of two, the R-RC-RC model fits much better than the R-RC model. This means that two processes are in series, but the values of the resistors and capacitors are so close together that there is no clear distinction in the plot. The two sequential processes might be in parallel with other processes, but they cannot be distinguished with equivalent circuit analysis. Why the Nyquist plots of the regioregular blends under higher bias show two arcs, and other plots do not, can be explained by comparing the values of the R’s and C’s. The exponential increase of C1 over voltage in the

regioregular blends causes diverging values of the C’s when voltage rises. In the carrier density the results hereof can be seen as well: for regiorandom blends the carrier density is independent of bias, and therefore just depends on the photocurrent, while for regioregular blends the carrier density increases exponentially over voltage. That a rapid increase in capacitance has to do with the second arc showing up is also confirmed by the C-f analysis done by Van Egmond [23]. The area under the DOS peak that he finds, calculated with capacitance measurements, correlates to whether or not two arcs can be found in the Nyquist plots.

Exceptions to the R-RC-RC model are blends in the dark under low bias. The Nyquist plots of these blends have no arcs and the R-RC-RC model does not fit. Therefore in these solar cells the processes mentioned before are different or not taking place. The values of R1are very high in

the dark, which might suggest too much resistance to let any bias through. Also, the regioregular 1:1 blend under illumination is an exception to the model: only at frequencies below 105_{Hz the}

blend fits the R-RC-RC model. The very high C1and carrier density might cause other effects not

incorporated in the model.

Basham et al’s model is therefore confirmed by the results, but suggests a more careful interpre-tation. Not all OPV’s with all biases can be modeled with it and also Basham et al’s interpretation of α should be revised, as values higher than a 0.5 and even higher than 1 appear. The interpre-tation of Rs also needs revision, as under illumination Rs is voltage dependent. Another point of

discussion is that there is no way to find out if other processes are happening in parallel and which of these processes is actually measured. One should therefore be careful with attributing a physical process to an RC element in a certain frequency region. It should also be taken into account that the timescales of the processes vary slightly over voltage, so an RC-element cannot be pinned to a certain frequency. All in all, IS measurements analyzed with equivalent circuits are quite tricky to interpret. To continue giving these data physical interpretation it is suggested to analyze temper-ature dependent data. The tempertemper-ature dependence of several processes could reveal more about their nature. Further comparison of the equivalent circuit analysis with other analyses on OPVs could also yield a more firm physical interpretation of the equivalent circuit. Hopefully further research will eventually lead to a better understanding of the light to energy conversion of OPVs, so that one day our rooftops will be covered with cheap and efficient organic solar panels.

6 Conclusion

In summary, it has been shown that Basham et al’s model can be applied to non-optimized OPVs as well. The model does have its drawbacks though. Under low bias in the dark the model does not fit due to capacitive behaviour of the OPV. Also, the measurements of the illuminated optimized solar cell do not fit the model because of an unknown capacitive effect in the high frequency region. Besides these anomalies, the physical interpretation of the model can also be called into question because one of the parameters (α) reaches values that are physically impossible. Basham et al’s model thus seems correct, but needs revision on the physical interpretation. The influence of morphology is mainly found in the chemical capacitance. The chemical capacitance of the crystalline

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polymer increases exponentially over voltage, while that of the amorphous polymer does not. The influence of weight ratio can be seen in the carrier density, where the weight ratios which are known to produce a more efficient OPV have a higher carrier density. Finally, both the area under the DOS peaks found by C-f analysis and the location of these peaks seem connected to the RC-element in the low frequency region.

References

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A

Nyquist plots

Figure 28: Nyquist plot of rra-P3HT:PCBM 1:1 under illumination

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Figure 30: Nyquist plot of rra-P3HT:PCBM 1:2 under illumination

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B

Sums of squares of the fits

Unless otherwise specified, these fits are made over 10Hz − 1M Hz. Table 5: rra-P3HT:PCBM 1:1 dark

Voltage (V) R-RC R-RC-RC 0 96.8 96.8 0.2 543.1 373 0.4 95.6 3.83 0.6 124 1.84 0.8 34.1 0.739 0.9 2.67 0.187

Table 6: rra-P3HT:PCBM 1:1 illuminated Voltage (V) R-RC R-RC-RC 0 2.271 0.152 0.2 2.486 0.171 0.4 1.77·10−11 1.54·10−11 0.6 2.363 0.155 0.8 1.103 0.106 0.9 0.795 0.0957

Table 7: rra-P3HT:PCBM 1:2 dark Voltage (V) R-RC R-RC-RC 0 603 1.66 0.2 652 2.35 0.4 677 1.07 0.6 642 1.09 0.8 35.1 0.286 0.89 3.58 0.134

Table 8: rra-P3HT:PCBM 1:2 illuminated Voltage (V) R-RC R-RC-RC 0 19.8 0.0837 0.2 19 0.0842 0.4 13.7 0.0745 0.6 4.18 0.0481 0.7 1.69 0.0384 0.8 0.772 0.0341 0.9 0.615 0.0368

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Table 9: rr-P3HT:PCBM 1:1 dark Voltage (V) R-RC R-RC-RC 0 18.1 0.698 0.2 37.7 2.23 0.4 28.1 0.634 0.5 20.4 0.261 0.6 19.6 0.115 Table 10: rr-P3HT:PCBM 1:1 illuminated Voltage (V) R-RC R-RC-RC 10Hz-1MHz R-RC-RC 10Hz-·1015_Hz R-RC-RC 10Hz-·105_Hz 0 140 18.5 18.4 1.61 0.2 45.3 21.4 21.4 0.0459 0.4 115 6.46 6.45 0.0255 0.5 147 8.11 8.11 0.0213 0.6 119 8.93 8.75 0.0378 Table 11: rr-P3HT:PCBM 1:2 dark Voltage (V) R-RC R-RC-RC 0 30.6 92.4 0.2 57.6 12 0.3 89.3 9.71 0.4 114 3.75 0.5 86.6 0.836 0.6 70.4 0.254 Table 12: rr-P3HT:PCBM 1:2 illuminated Voltage (V) R-RC R-RC-RC 0 13 1.11 0.2 31.7 1.74 0.3 42.5 1.76 0.4 36.8 0.817 0.5 34.5 0.161 0.6 42.9 0.0721

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C

Values of resistances and capacitors

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D

Value of α

Table 13: Found values of α for rra-P3HT:PCBM 1:1 dark Voltage (V) α 0.6 above 1 0.8 0.5 0.89 0.5 0.9 0.5 0.91 0.5

Table 14: Found values of α for rra-P3HT:PCBM 1:1 illuminated Voltage (V) α 0 above 1 0.2 above 1 0.4 above 1 0.6 above 1 0.8 above 1 0.89 above 1 0.9 above 1 0.91 above 1

Table 15: Found values of α for rra-P3HT:PCBM 1:2 dark Voltage (V) α 0.2 1 0.4 1 0.6 1 0.8 0.3 0.88 0.3 0.89 0.3

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Table 16: Found values of α for rra-P3HT:PCBM 1:2 illuminated Voltage (V) α 0.2 above 1 0.4 above 1 0.6 1 0.7 0.7 0.8 0.8 0.85 0.5 0.88 0.7 0.89 0.7 0.9 0.7

Table 17: Found values of α for rr-P3HT:PCBM 1:1 dark Voltage (V) α 0.2 above 1 0.4 1 0.5 1 0.58 1 0.581 1 0.582 1 0.59 1 0.6 1

Table 18: Found values of α for rr-P3HT:PCBM 1:1 illuminated Voltage (V) α 0.2 0.3 0.4 0.3 0.5 0.3 0.58 0.5 0.6 0.5

Table 19: Found values of α for rr-P3HT:PCBM 1:2 dark Voltage (V) α 0.5 1 0.55 1 0.59 1 0.6 1 0.65 1

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Table 20: Found values of α for rr-P3HT:PCBM 1:2 illuminated Voltage (V) α 0.2 above 1 0.3 above 1 0.4 above 1 0.45 above 1 0.5 1 0.55 1 0.57 1 0.59 1 0.6 1 0.65 0.7

Equivalent circuit analysis on organic solar cells: how can impedance spectroscopy results be interpreted?