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Efficiency of organic solar cells:

Optimizing a model for the fill factor using machine learning

Author

Elisa C. O

OSTWAL

Supervisors Prof. Dr. Michael B

IEHL

Prof. Dr. L. Jan Anton K

OSTER

July, 2018

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Contents

1 Introduction 4

I Organic solar cells 6

2 Theory: Organic solar cells 7

2.1 Organic semiconductors . . . 7

2.2 Fill Factor . . . 9

2.3 Competition between recombination and extraction . . . 9

2.4 Transient simulations (ZimT) . . . 11

2.4.1 Extraction rate . . . 11

2.4.2 Recombination rate . . . 12

2.5 Steady-state simulations (SIMsalabim) . . . 14

2.5.1 Recombination rate . . . 14

2.5.2 Extraction rate . . . 14

3 Research & Results: Organic solar cells 16 3.1 Steady-state simulations . . . 16

3.2 Transient simulations . . . 19

II Machine learning 20

4 Theory: Machine learning 21 4.1 Initial model . . . 21

4.2 Exponents . . . 21

4.3 Offset . . . 22

4.4 Using other parameters and introducing extra parameters . . . 23

4.5 Range of the parameters . . . 23

5 Research & Results: Machine learning 24 5.1 Changing the exponents . . . 24

5.2 Offset . . . 26

5.3 Using other parameters . . . 27

5.4 Using extra parameters . . . 28

5.5 Range of parameters . . . 29

6 Conclusion 31

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1 | Introduction

By far the largest part of the world’s electricity is generated using fossil fuels. In the past few decades these methods of generating electricity have become a controversial topic as there is strong evidence that the large quantities of carbon-dioxide that are emitted in the process are affecting the climate. Although there are strong indications that confirm this idea, the subject is still being debated. One thing is however inevitable: fossil resources will run out, and, with the rate they are being consumed in present day, this may happen in the near future.

The need for renewable ways of generating energy is therefore urgent. Photovoltaic energy con- version (more commonly known as solar energy) appears to be a promising technique, which however still needs improvement.

The solar panels that are currently commercially available are made of a silicon-based or inor- ganic material. These have an efficiency of around 20%. To produce well-functioning inorganic solar cells a high purity is required which makes the fabrication process costly and requiring a lot of energy [1], making the material not very sustainable. An alternative would be to use carbon- based or organic materials which have an easy fabrication process with low cost. On top of that organic solar cells have potentially (commercially) interesting features such as their mechanical flexibility (inorganic materials on the other hand are generally rather rigid). [2]

Organic photovoltaics (OPVs) are however not yet on the same level as inorganic photovoltaics in terms of their efficiency. To compare, state-of-the-art inorganic solar cells have an efficiency of almost 45% [3], while the most recent developments in OPV enable solar cells with an efficiency of 15% [4]. To enhance the efficiency, we need to get a better understanding of what determines the efficiency of OPVs. In this study we will try to do this by means of studying the so-called fill factor, which is directly related to the efficiency of solar cells. We will use an analytic approach to devise a model which describes the fill factor, building on the theory available. We would like to optimize this model so that we can predict the fill factor of a device based on its characteristics.

In recent years machine learning has proven itself to be well suited for this purpose. Although the foundations of machine learning were already laid in the 1950s, its techniques have been em- ployed in several fields of science only during the past two decades. In the field of astronomy machine learning turns out to be an excellent toolkit for tasks such as classification of galaxies.

[5] Another application of machine learning that came forth from astronomy is identifying which characteristics play an important role in classification. [6] We could then apply similar techniques to select which material- and device parameters determine the performance of our solar cell. This shows that machine learning appears to be a promising approach in optimizing our model.

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Part I

Organic solar cells

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2 | Theory: Organic solar cells

2.1 Organic semiconductors

Most organic materials are insulators due to the formation of σ bonds between the constituent carbon atoms. These bonds cause a large bandgap Egap, which makes it difficult for electrons to get promoted from the valence band into the conducting band. However, a particular type of organic materials, namely conjugated polymers and conjugated molecules, do not have these properties. In conjugated polymers the bonds between the carbon atoms are alternating single or double so each carbon atom binds to only three adjacent atoms, leaving one electron per carbon atom unbound in a pzorbital. This allows for the formation of π bonds, consisting of a bonding π-band and an anti-bonding π-band. The bonding π band is energetically favourable, so at a temperature of K = 0, when no extra energy gets added to the system, the electrons fully fill up the π band, leaving the anti-bonding πband unoccupied. The highest molecular orbital of the π band is referred to as the highest occupied molecular orbital (HOMO), whereas the low- est molecular orbital of the π band is referred to as the lowest unoccupied molecular orbital (LUMO). The difference (gap) in energy between these two orbitals is closely related to Egap. In conjugated polymers the HOMO/LUMO gap is small enough for an electron to get promoted from HOMO into LUMO upon optical excitation. This makes the material an intrinsic semicon- ductor. This also explains why these materials can produce electricity when exposed to light. [7]

Organic materials have a low dielectric constant. Due to this low dielectric constant, the proba- bility of forming free charge carriers upon light absorption is very low. Instead, strongly bound excitons are formed. [2] When a negatively charged electron gets excited by a photon, it moves from the HOMO level to the LUMO level, leaving behind a positively charged hole in the HOMO level. The electron is attracted to this hole by the electrostatic Coulomb force, and experiences repulsive Coulomb forces from the neighbouring electrons. This makes the electron-hole pair strongly bound, having slightly less energy than the unbound electron-hole pair. We refer to this bound electron-hole pair as an exciton. (Figure 2.1)

The exciton can get dissociated, generating free charges. If these free charges get extracted, an electric current is created. In order for the charges to become dissociated, two different materials are required: a donor material, which will "donate" its electron, and an acceptor material, which will accept the donated electron. In order for this to happen the energy of the LUMO level of the acceptor needs to be slightly less than the energy of the LUMO level of the donor material. Fur- thermore, the HOMO-LUMO gap of the donor material will be such that exciton formation may occur, the HOMO-LUMO gap of the acceptor material will be slightly larger such that excitons will not form in this layer. (Figure 2.2)

There are two processes which limit the number of free charges that can be extracted. Firstly, the exciton may decay: the electron goes back to the HOMO level, filling up its associated hole.

(Figure 2.3) This prevents generation of free charges. Secondly, it is possible that as the free or unbound electron traverses the material it encounters an unbound hole. The electron may then go from the conduction band into the valence band, filling up the hole. This process is called re- combination. (Figure 2.4) In organic semiconductors bimolecular recombination dominates over other types of recombination. [7]

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Figure 2.1: Exciton formation: a photon excites an electron from the HUMO level into the LUMO level, leaving behind a positively charged hole in the HOMO level. The electron and hole are strongly bound, forming an exciton.

Figure 2.2: Extraction: the electron moves from the LUMO level of the donor material to the LUMO level of the acceptor material. When the exciton dissociates, the unbound charges can be extracted.

Figure 2.3: Exciton decay: the electron decays back to the HOMO level, eliminating the exciton.

Figure 2.4: Recombination: the free electron decays from the LUMO level of the acceptor the HOMO level of the donor material.

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2.2 Fill Factor

When we apply a voltage V to a solar cell it will produce an electric current J. At 0 V the solar cell produces the maximum current, referred to as the short circuit current JSC. The current decreases with applied voltage until at some point it vanishes. We refer to the voltage at which this happens as the open circuit voltage VOC. A typical course of the decrease in current as a function of voltage is shown in Figure 2.5. Such a plot is referred to as a JV curve.

The applied voltage and output current together generate an electric power P = J·V. The maximum power obtained would occur when both the voltage and current are maximum. These points however do not overlap: the current is maximal if we apply no voltage at all (JSC), while there is no output current when the voltage is at its maximum (VOC). The maximum power would therefore be the product of the two: JSC·VOC. This would occur if there was no decrease in current as a function of the applied voltage, or if there would be a sharp cutoff of the output current at VOC. Graphically, one can represent this as the area marked by the red border in the JV curve. It is important to note that this does not agree with how a solar cell works: there will always be a decrease in current as a function of the applied voltage, however slight this may be. The area marked by the red border will therefore never be completely filled. Instead there will be an intermediate point (combination of voltage and current) at which the power is maximum, referred to as the maximum power point (MPP). The power this produces can again be represented by an area under the JV curve, indicated by the blue border in Figure 2.5. The ratio between the power at MPP and the theoretical maximum power is called the Fill Factor (FF) and is a measure of the efficiency of a solar cell.

FF= JMPP·VMPP

JSC·VOC (2.1)

Figure 2.5: A typical JV curve. FF is defined as the ratio between the power at MPP and the theoretical maximum power. Image taken from [8].

2.3 Competition between recombination and extraction

There are three main processes in solar cells which influence the number of free charge carriers in the material and hereby the produced current: generation, extraction, and recombination. These processes have been described in section 2.1. Free charge carriers are generated at a rate G. These will flow towards either end of the material (electrons towards the cathode and holes towards the anode) where they can be extracted. We define the extraction rate kexto be the number of charges that can be extracted per volume per charge. We distinguish two extraction rates: one for the electrons (knex) and one for the holes (kpex).

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While moving towards either end of the layer, part of the free charge carriers may recombine, and can therefore no longer be extracted. We define the recombination rate krecto be the number of recombination events per volume per charge. Unlike for the extraction rates, we do not have separate recombination rates for the electrons and holes since they recombine as a pair. The recombination rate is dependent on the bimolecular recombination strength γ which is given by:

γ = kLγpre, where γpreis a dimensionless reduction prefactor and kL is the classical Langevin recombination coefficient, given by:

kL= q e0eR

(µp+µn), (2.2)

in which q is the elementary charge, e0is the vacuum dielectric constant, eRis the relative dielec- tric constant of the material, and µpand µnare the hole and electron mobilities.

Recombination thus has a negative influence on the efficiency, or in other words there is com- petition between recombination and extraction. If recombination is strong, few electrons can be extracted, lowering the output current, which results in a bad performance. This idea has been worked out and studied in the paper by Bartesaghi et al. [8] They pose that FF is determined by the ratio between krecand kex. They approximate krecand kexby:

krec= γGL

2

pVint , kex= nVint

L2 (2.3)

where γ the strength of bimolecular recombination, G the volume rate at which holes are gener- ated, L the thickness of the active layer, µnthe electron mobility, µpthe hole mobility, and Vintthe internal voltage. The ratio between the two is therefore:

krec

kex

=krec· 1 kex

= γGL

2

pVint

· L

2

nVint

= γGL

4

nµpVint2

They introduce a factor θ which is defined as this ratio, omitting the constant factor18:

θγGL

4

µnµpVint2 (2.4)

From their plots (both simulated data and experimental data) there does appear to be a corre- spondence between θ and FF. Note that θ is plotted on a log-scale, meaning that FF∼log(θ).

(a) (b)

Figure 2.6: θ vs FF. (a) Simulations. (b) Experimental data. Image taken from [8].

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The formulae for the extraction rate kexand the recombination rate krecby Bartesaghi are (reason- able) approximations. However, the fact that they are approximations and not exact may be the cause of the spread in the plot. Moreover, we note that Barthesaghi’s model relies on approxima- tions made at SC, while FF is determined at MPP. We would therefore like to calculate the rates exactly and at MPP such that we can test the initial assumption, namely that the fill factor FF is determined by the ratio between the two rates. If this assumption is true we can then examine whether the spread is caused by approximations made in defining kex or krec. We expect that, if this initial assumption is correct, the spread will be due to inaccuracies in the formula for the recombination rate since this is a rather complex process which is still being studied. The model could in that case be improved by correcting the formula for calculating the recombination rate.

We will use two approaches to calculate the extraction rate and recombination rate exactly. The first will be to use an already existing procedure which requires running transient simulations.

[10] We would however prefer to use steady-state simulations since these have a shorter execu- tion time. For this we will have to set up a model of our own to calculate the rates.

2.4 Transient simulations (ZimT)

There is an already existing procedure which can be used to determine the rates. [10] This method requires the results from transient drift-diffusion simulations (ZimT) and uses another program (OriginPro) to derive the rates.

2.4.1 Extraction rate

In the extraction experiment light is directed at a solar cell with an initial intensity. When the material absorbs photons this creates a number of free charge carriers per volume, denoted by the generation rate G. We apply a bias voltage to the solar cell. The device is in steady state at this point. Then, at t= 0, the light intensity is slightly reduced, which causes a reduction in G:

G+∆G. The bias voltage is kept constant during this process. To compensate for the change in G and reach a steady state with this lower G, charges will be extracted: G+∆G ≈ E+∆E1, where∆E the extraction of the excess charges. The extraction of charges will result in a decay in current. (Figure 2.7) The current does not drop at once, but decays exponentially over a short time, proportionally to the extraction of charges [10]:

Jdecay∝ exp(−kex·t) (2.5)

By fitting an exponential curve to the decay in current we can determine the extraction rate kex.

Figure 2.7: Decay in current as a result of a slight reduction in light intensity.

1G+∆G=E+∆E+R+∆R, where ∆R the recombination of excess charges. However ∆R is small compared to ∆E.

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It is important to note that the extraction rate depends on the voltage that is applied to the solar cell. Under SC conditions there is no applied voltage and so no external electric field to coun- teract the internal electric field. The free electric charges are driven out of the material partially by diffusion, but mostly by the internal electric field. The latter phenomenon is referred to as drift. At a certain applied voltage the bias voltage will cancel the internal voltage Vint, effectively nullifying the internal electric field. As a result there will be no drift. Instead, the charges only get extracted by diffusion. At any other voltage the extraction rate will stem from a distribution from both drift and diffusion [10]:

kex=kdri f t+kdi f f usion (2.6)

In the model the assumption is made that the contribution of drift is due to a uniform field F generated by the effective voltage drop over the device Ve f f[10]:

kdri f t=2µF

L , where F= Ve f f

L (2.7)

Because of this, we cannot use all possible devices for the simulations: using a device with a large thickness L (>150 nm), highly unbalanced mobilities (µn µpor µn µp), or large γpre

violates this assumption. ([11], figure S1)

2.4.2 Recombination rate

Similar experiments exists for finding the recombination rate krec. Here we simulate a solar cell under open circuit (OC) conditions: we set up the solar cell in such a way such that there is no output current J. In real life experiments this is done by putting a large resistance in between the solar cell and the measurement apparatus (oscilloscope).

In the simulations we can measure the recombination rate at two different applied voltages: VOC and VMPP. The latter would be the most interesting since FF is defined at MPP. In both cases we need to ensure that there is no output current. When we apply a voltage of VOCthis is auto- matically assured. Simulations under these conditions are referred to as transient photovoltage or TPV experiment.

Like in the experiment for finding the extraction rate, we begin our measurement by directing light at the material with an initial intensity. This induces free charge carriers, generated at a rate G. The solar cell is in steady state with charges recombining at a rate R. We reduce the light intensity by a small amount, causing a drop in G: G+∆G. Now, to balance out the change in G,∆G, the excess charges will recombine: G+∆G = R+∆R. [12] We cannot have a reduction in current as a result: there is no output current after all. Instead, we will have an exponential decay in voltage, proportional to the recombination rate:

Vdecay∝ exp(−krec·t) (2.8)

To determine the recombination rate krecwe then fit an exponential curve to the decay in voltage.

In the case that we apply any other bias voltage such as VMPPwe need to ensure that no charges are injected in order to establish no output current. This is guaranteed by putting insulators at the end points, effectively "removing" the electrodes. These conditions are simulated in the transport and recombination via the displacement current or TRDC experiment.

Again, light is aimed on the surface of our solar cell, stimulating the generation of free charges with a rate G. The device is in a steady state and the charges recombine at a rate R. When we slightly decrease the light intensity we will have a reduction G: G+∆G. To get back into a state of equilibrium, the change in G,∆G, needs to be nullified which is done by recombination:

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G+∆G = R+∆R. [12]. Consequently, we will not have a decline in output current nor a drop in voltage, but a decay in displacement current Jd, proportional to the recombination rate:

Jd,decay∝ exp(−krec·t) (2.9)

To determine the recombination rate krec we then fit an exponential curve to the decay in dis- placement current.

In both simulation models there is a trade-off to make. In the TPV experiment we can only apply a voltage of VOC. Note however that for good FF VOC should be close to VMPP. In the TRDC experiment we can apply any voltage including VMPP, but since we remove the electrodes this can change the behaviour of the remaining device like a chain reaction, and we are not studying a real solar cell anymore since a real solar cell has no insulators at the ends. Neither of the methods thus use the same model assumptions as Bartesaghi, which will create a discrepancy in results.

In the formula for kexand krecderived by Bartesaghi the electric field is assumed to be uniform.

[8] In practice this is often not the case. In the model of ZimT the electric field is not assumed to be constant, and is taken into consideration in the simulations. This should give more accurate results than the formulae, but note that there will thus be a discrepancy between the models.

Figure 2.8: TPV experiment: decay in voltage V as a result of a slight reduction in light intensity.

Figure 2.9: TRDC experiment: decay in displacement current Jdas a result of a slight reduction in light intensity.

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2.5 Steady-state simulations (SIMsalabim)

We would prefer to extend the steady-state simulation program SIMsalabim to calculate the ex- traction rate and recombination rate directly. These steady-state simulations require significantly less time to execute, and are slightly more user-friendly since only one program is used. For this purpose we will set up an analytical, discrete 1D model of our solar cell.

We have a solar cell of length (thickness) L and split up this length in NP small sections (grid points) so as to create a grid. (Figure 2.10) We can evaluate properties of the solar cell at each grid point and use this to evaluate what happens over the total length.

2.5.1 Recombination rate

At any grid point i we can calculate the number of recombination events per volume and time, let this be Ri. The recombination rate at a grid point i is defined as the number of recombination events per volume and time and charges:

krec,i= Ri

ni (2.10)

The total number of recombination events per volume and time and charges over the length L is then the sum over all grid points:

krec=

NP

i=1

krec,i =

NP

i=1

Ri

ni (2.11)

We assume the internal electric field to be uniform, and only consider the effects of bimolecular recombination. Riis then defined as:

Ri =γi∗ (ni pi−qi2), (2.12) where γprethe aforementioned Langevin prefactor, n the number of free electrons, p the number of free holes, and qi the total number of intrinsic charge carriers. [12]

Figure 2.10: We split up a solar cell of length L into NP sections so as to create a grid.

A charge q moves from gridpoint i to the next one i+1 until it reaches the end i =NP.

2.5.2 Extraction rate

The extraction rate is defined as the number of charges that can be extracted per volume per charge. This is dependent on the time texneeded for an electron to exit the material: kex = t1

ex. The time an electron takes to leave the material depends on where it starts (gets generated), but an electron can be generated at any grid point in the material. We therefore should average over all grid points:

tex= 1 NP

NP

i=1

tex,i→NP, (2.13)

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where tex,i→NPthe time between the electron getting created in grid point i and leaving the mate- rial at the last grid point NP. Note that we assume electrons to go from i to NP and not the other way around, which need not be the case.

The time tex,i→NP required for an electron to move from a grid point i to the final grid point NP is then the sum of times it takes to move from one grid point to the next:

tex,i→NP =

NP

i=1

tex,i→i+1 (2.14)

When an electron traverses from a grid point i to the next, the time it needs is defined as the distance in between the two points divided by the velocity in between the two grid points:

tex,i→i+1= xi+

1 2

vi+1 2

= hi+

1

2 L

vi+1 2

(2.15)

where x the width, h the size of the grid, and v the velocity of the electron. The velocity v is dependent on the current J:

vi+1

2 =

Ji+1 2

q ni+1 2

, (2.16)

where q the elementary charge (constant).

Finally, the number of charges at the intermediate grid point i+12 needs to be determined. Tak- ing it to be the average of the charge densities over the two grid points is a bad approximation.

Instead we will use the growth function G to express the charge density at i+12, as described by Selberherr [13]:

ni+1

2 =ni· (G−1) +ni+1·G (2.17)

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3 | Research & Results: Organic solar cells

We want to test the initial hypothesis, namely that the fill factor FF is proportional to the ratio be- tween the recombination rate krecand the extraction rate kex. To test this assumption we want to use the values of these two rates given by the analytical model we have set up. We should how- ever first verify whether our model is correct: does it give reasonable results. We shall compare the results to the three reference values for this: the values given by the formulae by Bartesaghi (Eq. 2.3), and the two results obtained from the transient simulations (TPV/TRDC) (Section 2.4).

The values should be of a similar order. We will also plot θ against FF and compare the spread the four different methods give.

For the transient simulations only 10 selected devices were used because both the simulations and extraction of data is rather time-consuming. This is a very small number of points compared to the 1,000 data points that were generated for the other methods.

A positive correlation between θ and FF was preferred. Therefore θ has been defined as kex

krec

rather than the inverse. This definition of θ is used for the four different models.

3.1 Steady-state simulations

From Table 3.1 we can infer a number of interesting features. First, note that the range of the parameters of row ’Zimt’ in Table 3.1 is small compared to the other methods. This is because we have only used a small number of devices as well as from a selected range.

In the analytical model kexappears to be on the low side compared to the values found using Bartesaghi’s formula (Eq. 2.3). Especially at OC and MPP the analytical method yields remark- ably low minimum values. On the other hand, krecis relatively high in the analytical model. This together results in small values of θ in the analytical model. We note that in the analytical model the values for both kexand krecat SC are the closest to the rates found by Bartesaghi’s model. This makes sense as Bartesaghi’s model is constructed at SC.

kex(s1) krec(s1) θ

Bartesaghi 1.6 · 103− 2.9 · 106 1.9 · 101− 3.8 · 107 1.7 · 103− 3.0 · 105 Analytical SC 5.6 · 101− 1.7 · 106 3.6 · 102− 9.5 · 107 3.2 · 105− 1.9 · 102 Analytical OC 3.2 · 100− 3.5 · 104 1.0 · 103− 1.2 · 108 2.7 · 107− 6.4 · 101 Analytical MPP 3.0 · 100− 2.7 · 105 7.1 · 102− 1.1 · 108 8.5 · 106− 9.7 · 100 ZimT 1.4 · 105− 9.0 · 106 8.2 · 103− 3.7 · 105(TPV) 2.8 · 102− 4.5 · 101

1.0 · 105− 1.6 · 106(TRDC) 1.8 · 101− 1.1 · 100 Table 3.1: Comparing the different models

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In Figure 3.1 we have plotted θ against FF. Clearly the analytical model shows a much bigger spread than Bartesaghi’s model, at every point (SC,OC,MPP). From the R2and RMSE values in Table 3.2 it also becomes evident that Bartesaghi’s model is the best out of the four. Interestingly, out of the three possible measuring points the analytical model seems to perform best at OC.

(a) (b)

(c) (d)

Figure 3.1: The analytical model at three points: (a) short circuit, (b) open circuit,

(c) maximum power point. We want to compare the spread of the models to (d) the initial model.

R2 RMSE

θ 0.932 0.0430

θA,SC 0.726 0.0852 θA,OC 0.766 0.0787

θA,MPP 0.663 0.0945

Table 3.2: Goodness of fit for the different models

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The question arises which part of the analytical model – kexor krec– creates the large spread. To find out which component contributes to the spread the most, we have to eliminate the spread of the other (as much as possible). We do this by constructing two "hybrid" θ, which use kex from one of the models (Bartesaghi or analytical), and krecfrom the other. The spread in Bartesaghi’s model is relatively small, so this should eliminate a large portion of the spread from either kexor krec. We then define

θH= kex,B

krec,A and θ0H= kex,A

krec,B, (3.1)

where A denotes the analytical model and B denotes Bartesaghi’s model. Because the spread in the analytical model for the three measuring points (SC, OC, MPP) is of similar order, we will only consider one of the three for constructing and plotting (3.1). We pick MPP as this is the point we are ultimately interested in.

From Figure 3.2 we immediately observe that the spread in the plot of θ0H(Figure 3.2b) is consid- erably smaller than in the plot of θH(Figure 3.2a). More importantly, in the plot of θ0Hthe spread is comparable to the spread in the plot of θ (Figure 3.1d), and certainly smaller than in the plot of θA(Figure 3.1a-c). Since in θ0Hthe recombination rate krecis computed using Bartesaghi’s model, we infer that the spread in θAis due to errors in the model constructed for krec. This leads us to conclude that the analytical model we have set up for the recombination rate in section 2.5.1 is incomplete or incorrect.

(a) (b)

Figure 3.2: Cause of the spread in the analytical model. (a) θH uses the extraction rate as com- puted by Bartesaghi’s formula, and the recombination rate as computed by the analytical model.

(b) θ0Huses the extraction rate as computed by the analytical model, and the recombination rate as computed by Bartesaghi’s formula.

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3.2 Transient simulations

For the transient simulations we used 5 pairs of devices. (Appendix, Table 6.1) Each pair consists of two devices with a FF which is approximately equal. If the model is of good predictive quality, we would expect the two devices belonging to one pair to have the same value for θ.

In Figure 3.3a we have plotted the four different models in one graph to illustrate how the spread of each model compares to the other models. Each model has a different range of values for θ (Appendix, Figure 6.1). We have therefore plotted the normalized values of θ, so we can compare the spread of each model to the other models.

From the limited number of points it is difficult to identify one model as superior to the oth- ers. The θ stemming from the TPV experiments appears to give the largest spread however, marking it as the worst model out of the four. The remaining three models (Figure 3.3b) show an equally large spread, approximately. This is somewhat remarkable, as from our previous results we would expect the analytical model to behave worse. What is also noteworthy is that for these three models the spread seems to become larger with increasing FF. In contrast, the spread of the TPV model appears to be "random": it does not become larger with increasing FF.

We note that for the TRDC experiment the second device of pair 1 (Appendix, Table 6.1) gave a remarkably high θ compared to the first device. We therefore mark it as an outlier and left it out of the graph to get a better idea of its spread. (This device is however included in Figure 6.1d in the Appendix.) The most remarkable characteristic of this device is a large γpre(∼ 0.7), indicating strong bimolecular recombination. As described by Le Corre et al. this could cause a violation of the assumption that the contribution of drift is due to a uniform field F generated by the effective voltage drop over the device Ve f f, giving inaccurate results. ([11], Figure S1) We conclude that from the two transient models, the TRDC model performs better or in other words is more accurate than the TPV model. This implies that the approximations made in the TPV experiment introduce a larger error than the ones made in the TRDC experiment. We want to stress however that we cannot make any strong conclusions since we have used only very few data points.

(a) (b)

Figure 3.3: (a) Comparing the spread of the four different models. (b) TPV has been left out to show the increase of the spread with increasing FF for the other three models.

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Part II

Machine learning

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4 | Theory: Machine learning

4.1 Initial model

We want to predict the Fill Factor (FF) based on the device parameters. Rather than starting from scratch, we decide to take the model developed by Bartesaghi et al. as the initial model. We therefore restrict ourselves to the following parameters to predict the FF: the mobilities of the charge carriers involved (the electron mobility µn, the hole mobility µp), the internal voltage Vint, the strength of bimolecular recombination γ, the volume rate at which holes are generated G, and the thickness of the active layer L. We summarize the influence of the device parameters by a factor θ. The θ suggested by Bartesaghi et al. is:

θ= γGL

4

µnµpVint2 (2.4)

Note that θ can be generalized to:

θ=γaGbLcµdnµepVintf , (4.1) where a=1, b=1, c=4, d= −1, e= −1, f = −2 in the θ found by Bartesaghi.

We aim for θ to be dimensionless so as to define a dimensionless scaling law, which is preferred in the field of physics. Note that the form of the current θ – that is: a product of input variables raised to a power – can satisfy this condition; a sum of input variables raised to a power (polynomial) on the other hand could not. Moreover, the variables show to be not directly correlated to FF ([9], Figure 7). Both of these facts suggest that a polynomial would not be a good guess for our model. On the contrary, the form of the current θ does seem promising.

4.2 Exponents

We want to try improving θ by altering the exponents (a... f ). We will attempt finding the values of the exponents using regression.

From Figure 2.6 we can deduce that FF scales with the logarithm of θ. Furthermore, within a certain range of FF (approximately 0.4 to 0.7) FF appears to depend linearly on log(θ). The idea then arises to perform linear regression on this part of the data: FF∼log(θ). Note that:

log(θ) = log(γaGbLcµdnµepVintf )

= a·log(γ) +b·log(G) +c·log(L) +d·log(µn) +e·log(µp) + f·log(Vint) Therefore, if we perform linear regression on the data using the model

FF=β0+

6 i=1

Xiβi with XT= log(γ), log(G), log(L), log(µn), log(µp), log(Vint) (4.2) then the slopes β1...β6correspond to the values of the exponents a... f .

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Note that we are not looking for an exact scaling law, that is FF = log(θ), or in other words have a one-to-one correspondence between the two. Rather, we are searching for a measure of approximating FF by log(θ)or FF ∼ log(θ): if we have a θ of 102we have a FF of 0.3, if we have a θ of 10−4we have a FF of 0.7, and so on. This notion allows us to multiply the exponents by a constant scalar, so to obtain nicer (possibly rational but preferably integer) values for the exponents. It is the ratio between the exponents that determines the fit after all.

4.3 Offset

The experimental data suggests that the different devices (combination of parameters (µnp,γ,Vint)) follow the same trend of θ against FF but each with a different horizontal offset. The idea arises to introduce a device-dependent parameter ’offset’ in our model: θ0 =κ·θ, where κ the device- dependent offset. Again, FF∼log(θ0), so:

log(θ0) = log(κ·θ)

= log(κγaGbLcµdnµepVintf )

= log(κ) +a·log(γ) +b·log(G) +c·log(L) +d·log(µn) +e·log(µp) +f ·log(Vint) Thus log(κ)corresponds to the intercept β0when performing linear regression using (4.2). When we perform linear regression using this model we will also obtain the values of the exponents a... f for every device. We would like to fix the exponents in θ0, but this is only acceptable to do if the exponents a... f are (approximately) the same for every device.

For every device we fix µn, µp, γ (the material properties), and Vint, and vary L and G. We thus fix four out of six input parameters. If we were to perform linear regression on this data set, we would have a problem with the rank of the system. Linear regression uses the least squares method to find the best fit. This method requires that the input data, an N× (p+1)matrix with N the number of measurements and p the number of input variables (in our case p= 6), has a full column rank; this assures that we have a unique solution. [16] If we fix four out of the six input parameters, we will not have a full rank of the input data. To compensate for this, we will add a small amount of noise to the four fixed parameters.

Figure 4.1: Horizontal offset κ between two devices.

(’device 2’ and ’device 4’ refer to the devices in Appendix, Table 6.2)

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4.4 Using other parameters and introducing extra parameters

Kirchartz et al. claim to have found a θ which explains FF better. [14] Rather than using µnand µpto predict FF, they make the distinction between the slowest of the two mobilities µsand the fastest of the two mobilities µf. They also put different weights on the mobilities:

θK= γGL

4

µ1.6s µ0.4f Vint2 (4.3)

We will verify whether θKindeed explains FF better than θ as suggested by Bartesaghi (Eq. 2.4).

It is possible that the model found by Bartesaghi is incomplete or in other words is missing parameters to fully explain the system, and that the spread is caused by this. There are two parameters which appear to be of predictive value, but which are not (directly) included in the model: the earlier mentioned Langevin prefactor γpre, and µrthe ratio between the slowest of the two mobilities µsand the fastest of the two mobilities µf:

µr = µs

µf (4.4)

There may be more parameters possible which could add to the model, but we will restrict our- selves to examine the effect of these two parameters.

4.5 Range of the parameters

Some parameters may cause a bigger spread in the data than others. We should therefore inspect which parameters cause (what part of) the spread. We need to take these results into account when performing linear regression on our data. If one or more parameters are responsible for a big spread in the data then it could be a good idea to limit their range to reduce the spread in the data points. We can then perform linear regression more accurately on the (limited) data set. After performing linear regression on this control data we should test the new model on the complete data set to verify it works in this range as well.

For future reference, we have summarized the range of the device parameters in Table 4.1.

Symbol Description Range

γpre Recombination pre-factor 10−3−100 G Generation rate of free charges 1025−1028m−3s−1

L Thickness 60−260 nm

µn Electron mobility 10−10−10−7m2V−1s−1

µp Hole mobility 10−10−10−7m2V−1s−1

Vint Internal voltage 0.6−0.9 V

Table 4.1: Range of the parameters used in the drift-diffusion simulations

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5 | Research & Results: Machine learning

5.1 Changing the exponents

Using the full range of our parameters, we perform linear regression on our simulated data set.

The model we will be fitting is:

FF=β0+

6 i=1

Xiβi with XT= log(γ), log(G), log(L), log(µn), log(µp), log(Vint) (4.2) The slopes β0...β6will then correspond to the values of the exponents, as explained in section 4.2.

In Table 5.1 the results are shown from performing linear regression using the model. We have at- tempted to make the values of the exponents close to integers by multiplying them by a constant scalar (35 in this case). According to this θ should then approximately be:

θLRγ−1G−1L−4µ1nµ1pVint3 = µnµpV

3 int

γGL4 (5.1)

This is very close to θ suggested by Bartesaghi, the only difference being the value of the exponent of Vint ≡ f . We know however that Vintonly has a minor influence because of its small magnitude and small range. To illustrate this we have plotted θ obtained by Bartesaghi ( f =2), θLRobtained by linear regression ( f = 3), and θexcl which excludes the influence of Vint( f = 0). (Figure 5.1) There are negligible differences between the three models. This is confirmed by the quality factors of the three different models, which are summarized in Table 5.2. From this table we identify θ as marginally better than the other two. Remarkably θexcl, which does not include the influence of Vint, has a slightly better quality than θLR which was found using linear regression. One of the reasons to include Vintin the model nonetheless is the objective of making θ a dimensionless scaling factor. Merely because of this we prefer Bartesaghi’s θ over the other two.

γ G L µn µp Vint

exponents -0.0275 -0.0353 -0.1155 0.0317 0.0310 0.0776 exponents*35 -0.9628 -1.2346 -4.0422 1.1100 1.0866 2.7163

Table 5.1: Exponents obtained by performing linear regression. The second row contains the exponents multiplied by a constant scalar (35) to obtain values close to integers.

R2 RMSE

θ 0.936 0.0411

θLR 0.929 0.0441 θexcl 0.932 0.0432

Table 5.2: Goodness of fit for the different models.

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(a)

(b)

(c)

Figure 5.1: Changing the exponent f in θ=γ−1G−1L−4µ1nµ1pVintf . (a) f =2, (b) f =3, (c) f =0.

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5.2 Offset

Next, we try to improve our model by introducing a new factor which accounts for the apparent offset in the horizontal direction between the different devices in the experimental data. As ex- plained in section 4.3, if we perform linear regression on our experimental data using the model

FF=β0+

6 i=1

Xiβi with XT= log(γ), log(G), log(L), log(µn), log(µp), log(Vint) (4.2) we will get a slope βjwhich corresponds to the exponent for every input parameter Xj, but also an intercept β0which corresponds to the horizontal offset κ. Note that we will obtain a different κfor every device: the offset is device-dependent. We will not use the device-specific exponents obtained from performing linear regression.

Because the experimental data contains measurement errors, we prefer to simulate the data in- stead. We use a number of the devices as provided by Bartesaghi. We could only use the devices from the graph showing the experimental data that were visible. (Appendix, Table 6.2) For every device we simulate 500 data points, varying L and G. For each device we then perform linear regression using the model, storing the device-dependent offset parameter so that we can use it when plotting θ0=κθ.

We have plotted the results in Figure 5.2. It immediately becomes evident that introducing the prefactor κ causes a greater difference between the devices, creating a bigger spread. When we inspect the values of the exponents of every device (Appendix, Table 6.3), we notice that they differ a lot and that some have very small values, indicating a small weight or little influence, while others have very large values, indicating a large weight or great influence. This may be due to that the curves of the different devices do not all have the same shape: some are S-shaped while others are J-shaped or show a linear trend, some have a steep curve while others are almost flat. The initial assumption that was needed for the device-dependent offset parameter to be of value is thus violated, namely that the exponents of all devices need to have roughly the same value. We therefore conclude that including κ does not improve the model.

(a) (b)

Figure 5.2: θ vs FF (a) excluding the device-dependent offset κ, and (b) including κ.

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5.3 Using other parameters

We wish to verify whether using µs and µf instead of µpand µn indeed improves the model, as proposed by Kirchartz [14]. To get a first impression, we plot θ as suggested by Kirchartz:

θK= µ

1.6s µ0.4f Vint2

γGL4 (4.3)

To get an idea of its predictive quality we have calculated its R2 and RMSE value (Table 5.3).

Both from these values and the graph (Figure 5.3a) the model does not seem to be better than the model by Bartesaghi.

When we perform linear regression using the model

FF=β0+

6 i=1

Xiβi with XT= log(γ), log(G), log(L), log(µs), log(µf), log(Vint) (5.2) we obtain

θµ=γ−1G−1.3L−4µs1µ1.3f Vint3 = µsµ

1.3 f Vint3

γG1.3L4 (5.3)

Both its quality factors are better than those of θKand θ. We were curious as to what the influence of b the exponent of G is: is the quality affected if we change b from−1.3 to−1. We find the R2 value to decrease from 0.941 to 0.936, and the RMSE factor to go up from 0.0395 to 0.0411. We note that these new quality values are exactly the same as those of θ. This suggests that µs and µf have an equally predictive power as µpand µn.

We note however that unlike in Kirchartz’ model, µf gets a bigger weight (larger exponent) than µs. Our result is not in accordance with several studies: these suggest a dominant influence of the charge-carriers with slower mobility. [15]

(a) (b)

Figure 5.3: θ uses µpand µn, (a) θKand (b) θµuse µsand µf.

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R2 RMSE

θ 0.936 0.0411

θK 0.915 0.0476 θµ 0.941 0.0395

Table 5.3: Goodness of fit for the different models.

5.4 Using extra parameters

We are interested in the predictive value of two other parameters which are not (directly) in- cluded in the model: Langevin prefactor γpre, and the ratio between the slowest of the two mobilities µs and the fastest of the two mobilities µf: µr = µs

µf. We therefore perform linear regression on the data using the following models:

FF=β0+

8 i=1

Xiβi, XT= log(L), log(G), log(Vint), log(γ), log(γpre), log(µn), log(µp), log(µr) (1)

FF=β0+

7 i=1

Xiβi, XT= log(L), log(G), log(Vint), log(γ), log(γpre), log(µn), log(µp) (2)

FF=β0+

6 i=1

Xiβi, XT= log(L), log(G), log(Vint), log(γ), log(γpre), log(µr) (3)

FF=β0+

5 i=1

Xiβi, XT= log(L), log(G), log(Vint), log(γpre), log(µr) (4)

We have summarized the results in table 5.4. We have rounded off the exponents to integers where possible or fractions otherwise.

Models (1), (2), and (3) perform slightly better than the model found by Bartesaghi, while model (4) clearly performs worse. From the latter we can deduce that we cannot replace both γ by γpre

and µn,pby µr. Model (1) and (2) perform equally well. While in model (1) we have introduced both γpreand µr, in model (2) we have only introduced γpre. If we compare the weight (value of the exponent) of γprebetween model (1) and (2), we notice that γpregets assigned a considerably bigger weight in model (2). This, together with the fact that the model gets slightly worse if we replace µn,pby µr (comparing model (2) and (3)) we deduce that µrdoes not add to the model.

This reasoning leads us to conclude that out of the 4 models (2) would be best.

model θ R2 RMSE

(1) L−4G−4/3Vint3 γ−6/5γpre1/5µ5/4n µ5/4p µ−1/4r 0.941 0.0395 (2) L−5G−3/2Vint7/2γ−2/3γ−1/2pre µnµp 0.941 0.0395 (3) L−4G−5/4Vint3 γ5/4γ−11/5pre µ1/2r 0.940 0.0398 (4) L−4G−5/4Vint3 γ−5/4pre µ1/5r 0.896 0.0525

Table 5.4: Goodness of fit for the different models.

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5.5 Range of parameters

Until thus far we have used the full range of the parameters when performing linear regression.

We now wish to restrict the range in an attempt to limit the variance. This will create a smaller search space, and thus hopefully increase our chances of finding a good model. The idea is to gradually extend this range to one which is as close as possible to the full range without the ac- curacy of the model suffering.

We start with the range used in our transient simulations. This range is rather restricted and we know that for this range the assumption made in section 2.4.1 holds. In the end we can ex- pand the range to one which is almost the same as the original range without the quality of the model getting (significantly) worse, the only exception being the lower limits of the mobilities µn, p: instead of 10−10we have 10−9as our lower bound. We have listed both ranges in table 5.5.

We have named the θ obtained by performing linear regression on this set of parameters θs:

θs =L−4G−4/3Vint2 γ−5/6µn µp= µnµpV

int2

γ5/6G4/3L4 (5.4)

We have summarized its quality factors and compared it with θ (Bartesaghi, eq. 2.4) and θK(Kir- chartz, eq. 4.3) in Table 5.6. We have tried to round off the exponents found for θsto integers. We could not do this for the exponents of G and γpreas this significantly reduced the quality. (Note that rounding off both exponents to integers would result in θ as found by Bartesaghi.) Instead we have rounded them off to fractions. We have included two versions of θs: one obtained di- rectly from performing linear regression which yields the highest quality, and a second one in which we have sacrificed a bit of its quality by setting the exponent of Vintto 2 to make θsalmost dimensionless. Comparing this new-found θs to the previous models, we conclude that this has the highest quality out of all attempts. This also becomes evident when comparing the graphs of FF as a function θ and FF as a function of θs(Figure 5.4). Figure 5.4b shows considerably less spread in the region of a large FF (FF>0.5) compared to Figure 5.4a, confirming that θsexplains FF better than θ.

Range start Range final Range original Units γpre 10−4−100 10−3−100 10−3−100 - G 1.5·1027−1.5·1028 1025−1028 1025−1028 m−3s−1

L 80−120 60−260 60−260 nm

µn 10−8.5−10−6 10−9−10−7 10−10−10−7 m2V−1s−1 µp 10−8.5−10−6 10−9−10−7 10−10−10−7 m2V−1s−1

Vint 0.6−0.8 0.6−0.9 0.6−0.9 V

Table 5.5: Range of the parameters used in linear regression.

θ R2 RMSE

θ L−4G−1Vint2 γ−1µnµp 0.932 0.0430 θK L−4G−1Vint2 γ−1µ1.6s µ0.4f 0.910 0.0496 θs L−4G−4/3Vint3 γ−5/6µnµp 0.947 0.0380 L−4G−4/3Vint2 γ−5/6µnµp 0.946 0.0385 Table 5.6: Goodness of fit for the different models.

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(a)

(b)

Figure 5.4: The initial model (a) θ vs FF, and the improved model (b) θsvs FF.

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6 | Conclusion

We have made an attempt to set up a model to directly calculate the extraction rate and recom- bination rate. While the calculations for the extraction rate seem solid, the results found from the equations used to calculate the recombination rate do not correspond well to simulations and cause a big spread in the data. The numerical model we have set up for the recombination rate in section 2.5.1 is therefore incorrect or incomplete.

In our pursuit to obtain a better θ we have performed transient simulations in the hope of more accurate numbers for the extraction rate and recombination rate. Unfortunately we were only able to obtain ten data points, which could not provide us with much information. We found the TRDC experiment to give better results than the TPV experiment, indicating that better approxi- mations are made in the TRDC experiment. We note however that since this finding is based on only ten data points this statement is not solid and is therefore a preliminary conclusion.

The transient simulations were rather time-consuming as it proved to be somewhat difficult to choose the right set of parameters and as the fitting had to be done manually. The latter process can be facilitated by sending the data obtained from the transient simulations to a fitting program which automatically fits the data to an exponential decay function (piping the transient simula- tions to a fitting program). What hinders this however is that the program parameters used in the transient simulations turn out to be largely device-dependent which makes it difficult to predict whether they are correct; it usually takes a couple of tries before we obtain the right parameters.

If we can find a suitable way to predict and alter these parameters however then piping the pro- cesses should not be a problem.

Using linear regression, an elemental technique of machine learning, we have finally obtained θs which performs better than the original θ by Bartesaghi. The key to attaining this improved model was to restrict the range of the parameters to a range in which the solar cell behaved well.

We have shown that changing the parameters or adding extra parameters did not (significantly) enhance the quality of the model.

The final model is summarized by:

θs =L−4G−4/3Vint2 γ−5/6µn µp= µnµpV

int2

γ5/6G4/3L4 (5.4)

If we compare the spread of the two graphs (Figure 5.4) we notice that, although there is in gen- eral a smaller spread in the plot of θsvs FF, θsappears to perform slightly worse than the original model θ in the low FF region. One may argue that from a practical point of view we are more interested in the high FF region since this entails devices with a higher efficiency. On the other hand we would prefer a model which performs better overall as this would enrich our under- standing of the underlying processes and could improve already existing theory.

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