Bachelor thesis as of Dec 30 2020

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\textsc{University of Blabla}\\
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\textsc{Faculty of Mathematics, Informatics and Natural Sciences}\\
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\textsc{Department of Physics}\\

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    \Large\textsc{Bachelor Thesis}
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\LARGE\textbf{Analysis of spatially resolved current-voltage dependence in CIGS solar cells}
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\Large submitted by:\\
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Engel von Reims\\
Matriculation number: 123456
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First Supervisor: Dr. Smart\\
Second Supervisor: Prof. Dr. Alsosmart
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Submitted at: January 8, 2021
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%\title{Analysis of spatially resolved current-voltage dependence in CIGS solar cells\\ \small{Supervisor: Dr. Smart}}
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\begin{abstract}
The IV curve of a solar cell carries information about its performance and about limiting performance factors. Hitherto, this kind of analysis has only been done by measuring the total current of a solar cell under well-defined illumination conditions and applying a varying bias voltage to simulate different environment conditions. However, not all regions of a solar cell manifest the same performance due to inhomogeneities in the elemental distribution and thickness. Techniques such as X-ray fluorescence resolve the solar cell in its spacial dimension and help determining the relative elemental distribution.
The examined solar cell has a copper indium gallium diselenide (CIGS) thin-film material which over the years has become a competitive alternative to established materials such as crystalline silicon. Lock-in amplification filters out the voltage-induced current, leaving only the X-ray beam induced current (XBIC). This way, single regions can be examined under different bias voltages.

In this analysis, the performance of the solar cell is both resolved in its spatial and voltage dimension, yielding an IV curve for each scanned pixel with a resolution of 50 nm edge length. A twofold distinction into high- and low-performing pixels is made. It is found that high-performing pixels for the most part remain high-performing over different bias voltages and the same is found for the low-performing pixels. This leads to the thesis, that absorption can be made responsible for this behaviour. However, a voltage-dependent trend is observed for the forward bias, where a distinction into high- and low-performing regions can be made solely on the data basis of the current difference in the forward bias range.

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\section{Introduction}
\subsection{Motivation}
Within the last decades, global warming and anthropogenic CO\textsubscript{2} emissions caught the attention of scientists, policy-makers and the media.\cite{rich-2018} Since then, the Intergovernmental Panel on Climate Change (IPCC) has published its Fifth Assessment Report in which it asserts with high confidence the anthropogenic CO\textsubscript{2} gases as the main cause for global warming.\cite{pachauri-2014-IPCC} In 2016, the world's energy usage took up 73\% of the total greenhouse gas emissions \cite{climatewatchdata-2016}, most of which stem from coal and gas.\cite{iea-2020} Heat and electricity mainly determine the energy usage, but renewable energy sources still play a comparatively small role. They have been on the rise in the global electricity mix, where their contribution has increased by a rate more than twice as their contribution to the global energy mix \cite{our-world-in-data-electricity}\cite{our-world-in-data-energy}, but this can be rather attributed to renewable energies substituting nuclear energy; fossil fuels have barely been reduced over the last 20 years.
Still, even slight improvements of solar cell conversion efficiencies from sunlight into electric energy can have an important impact on both economics and the environment. The type of material examined is copper indium gallium diselenide (CIGS) which has several interesting properties. Being a thin-film material and not being dependent on a glass substrate, it can be rolled during manufacturing and application. Of all thin-film materials, CIGS has the highest efficiency\cite[p.195]{smets-2016}, with a current record efficiency of over 23\%.\cite{green-2020-pip-january}
Due to its thin material, it can be produced cheaply. The scarcity of indium however limits the advantages of CIGS.

\subsection{Outline}
In this thesis, the X-ray beam induced current of a CIGS solar cell is examined as spatially resolved into pixels of 50 nm edge length and under different bias voltages. This allows to differentiate between voltage-dependent and voltage-independent effects, determining factors for high- and low-performing regions.
In \textbf{Chapter 2}: \textit{Theory of solar cells}, the basic structure of a solar cell, as well as the specific structure of CIGS, and its behaviour under bias voltage and illumination are explained.
In \textbf{Chapter 3}: \textit{Experiment}, the experimental setup environment at DESY's beamline and the design of the experiment of 2018 are sketched. Also, lock-in amplification is shortly summarized, since it plays a big role in extracting the XBIC signal from the directly measured current.
In \textbf{Chapter 4}: \textit{Data preparation}, it is explained how image registration and Gaussian filtering is applied to the data and which part of the data had to be discarded for proper IV curve analysis.
In \textbf{Chapter 5}: \textit{Results and Discussion}, the current distribution for each bias voltage is shown. It appears that the distribution remains rather constant, leading to the question whether there are pixels that could be voltage-independent. For this, both a separation into high- and low-performing pixels and k-means clustering for different inputs is performed. It is here confirmed that the majority of pixels retains its allocation into the group of high or low performance. Also, it is found that the slope at positive bias determines the grouping into higher and lower performing pixels.
In \textbf{Chapter 6}: \textit{Conclusions}, conclusions are drawn with respect to the type of analysis employed here and with regards to the results.

\section{Theory of solar cells}
\subsection{The p-n junction in equilibrium}
The p-n junction is the heart of a solar cell. It is crucial for generating electric current under illumination. A p-n junction is composed of two different layers stacked together. Those layers differ in their \textit{doping} and can also differ in their materials being used.
Doping means that additional \textit{donor} or \textit{acceptor} atoms are being added to the layer. In n-type doping, donor atoms are added in a way that the layer's free electrons amount to \textit{n}, whereas in p-type doping, acceptor atoms are added in a way that the layer's free holes amount to \textit{p}. These additional atoms are also referred to as \textit{impurity atoms}. A semiconductor with insignificantly many impurity atoms still has   'more' than the intrinsic number \textit{n\textsubscript{i}} of free holes/electrons. Electrons are called the \textit{majority carriers} in the n-type layers, since they constitute the majority of free charge carriers there. In the p-type layer, they are the \textit{minority carriers}, and the same goes for holes vice versa.

In the language of band theory, electrons are situated in the conduction band (CB), whereas holes are situated in the valence band (VB). It follows from the law of mass action and band theory\footnote{The derivation is shortly summarized in \cite[p.56]{smets-2016}.} that
\begin{equation}
\label{eqn:np}
np = n_{i}^{2} = N_{C}N_{V}\cdot exp\left(\frac{E_{V}-E_{C}}{kT}\right),
\end{equation}
where N\textsubscript{C} and N\textsubscript{V} is the number of states in the CB and VB respectively. E\textsubscript{C}$-$E\textsubscript{V} is the band gap energy and kT the thermal energy. Interesting to note about this equation is that the product $np$ is independent of the Fermi energy. Since doping shifts the Fermi energy level, the equation will still be valid for semiconductors that are doped. For example, if we dope 1 cm\textsuperscript{3} of Si with 10\textsuperscript{17} P atoms, we can calculate from Equation \ref{eqn:np} that there will be about $2.22\cdot 10^3$ holes. This huge difference underlines the distinction made between majority and minority carriers.
Both, free holes and free electrons, can move inside the layer and also enter the adjacent layer. The movement, however, is restricted by the \textit{diffusion length} and \textit{lifetime}. The diffusion length is defined for minority carriers only.\footnote{The majority charge carriers are much less likely to recombine, so their diffusion length would exceed by far the dimension of the p-n junction. Therefore, minority charge carriers determine the current.} If the layer materials only differ by the elements injected during doping, then the junction is called a \textit{homojunction}. A typical example would be a crystalline silicon solar cell (c-Si).\footnote{Amorphous silicon solar cells (a-Si) have a p-i-n junction, where there is an intermediary intrinsic layer which is not doped, because the absence of an ordered crystalline lattice structure introduces dangling bonds that limit the diffusion length of charge carriers.\cite[p.181]{smets-2016}} If the layers differ by the materials being used, or more precisely their band gaps differ, then the junction is called a \textit{heterojunction}. As we will see later, the layers in CIGS solar cells are composed of different materials.
\\
\\
Now, let us consider an n-type and p-type layer forming a p-n junction. For clarity of reading, the processes are explained here and in the following subsection for the flow of electrons only, noting that the same process\footnote{By principle only, since the level of n-type doping is often different from the level of p-type doping.} is happening with the holes, only vice versa.

A free electron leaves a positively ionized donor atom. Both charges are equally opposite to each other and therefore the bulk of both the n- and p-type region remains electrically neutral during doping. However, the huge difference in concentration of electrons allows for a significant diffusion across the p-n junction, where the free electrons recombine with the holes that are abundant in the p-type region. Because of this recombination, this flow of electrons is called the \textit{recombination current} $I\textsubscript{rec}$. The electrons close to the depletion region will probably be the first to recombine right at the opposite side of the junction. The recombination will create a positively ionized \textit{space charge} in the p-type region. "Space charge" refers to the fact that, in contrast to the majority of ionized acceptor atoms in the p-type region, this charge will remain fixed at its position and remain electrically charged, for there aren't enough holes in its vicinity that the electron could again recombine with.
This process will repeat until the space charges have built up an electric field that is exactly opposite to the recombination current. Now, that there is a region depleted of free charge carriers, we refer to it as the \textit{depletion region} and to the electric field as the \textit{internal electric field}. Due to thermally generated free electrons in the n-type region\footnote{Also in the depletion region, but that is rather irrelevant, because the size of the depletion region is very small compared to the bulk regions.}, that will randomly move into the depletion region where they experience the force of the internal electric field, there must be a separate flow of electrons be accounted for. It is called the \textit{drift current} or \textit{thermal-generation current} $I\textsubscript{gen}$.\footnote{The electrons of the recombination current will experience the force of the internal electric field of course as well, but this decrease in current is already included in the $I\textsubscript{rec}$ variable.} The p-n junction is now in equilibrium, when there is no net current $I$ flowing through the depletion region:
\begin{equation}
\label{eqn:currents-in-equilibrium}
I = I_{rec} - I_{gen} = 0
\end{equation}

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{Figures/Fig8.2_p-n_junction_in_equilibrium.png}
\caption{Schematic of the p-n junction in equilibrium with an internal electric field. Two import remarks must be made about the illustration: First, the minus in the p-type region symbolizes local charges, but it is not fixed in space; unlike the minus at the space charge region that symbolizes a fixed charge. (Since electrons are fermions and therefore indistinguishable, it can be said without loss of generality that the charge is fixed, even if one likes to imagine that there are short changes in the local charge as if an electron would 'move'.) Second, the space charge region is not exactly sharp, but has an exponential fall-off which can be approximated to a sharp edge, neglecting however certain effects mentioned in subsection \ref{ssec:important-parameters}. Image source: \cite[p.85]{smets-2016}}
\label{fig:pn-junction-equilibrium}
\end{figure}
\subsection{The p-n junction under bias voltage and illumination}
A solar cell with a p-n junction as displayed in Figure \ref{fig:pn-junction-equilibrium} is, of course, not used for producing power, but the solar cell will be part of an array of solar cells in both series and parallel circuits and experience external potentials from the adjacent solar cells. More on that at the end of the subsection.

If an external \textit{bias voltage} is applied to the front and back contact of the solar cell, the internal electric field will become smaller or larger, depending on whether the voltage is positive (\textit{forward bias}) or negative (\textit{reverse bias}) on the p-doped side with respect to the n-doped side.  The generation current will not be affected by the voltage, as long as it is in a moderate range, whereas the recombination current increases with the Boltzmann factor:
\begin{subequations}
\begin{align}
\label{eqn:dark-IV-I_gen}
I_{gen}(V) &= I_{gen}(V=0) = I_{0}\\
\label{eqn:dark-IV-I_rec}
I_{rec}(V) &= I_{rec}(V=0)\cdot exp\left(\frac{eV}{kT}\right)
\end{align}
\end{subequations}
$I\textsubscript{0}$ is called the \textit{saturation current} and is supposed to be as low as possible.

Under forward bias condition, a negative electrode will be attached to the contact of the n-type region. This means that electrons flow through the cathode into the n-type layer. Since there are now more electrons than before, the diffusion across the depletion region will become larger and the recombination current $I\textsubscript{rec}$ increases. The external electric field adds to the internal electric field, such that the electric field across the depletion region is decreased. Due to the increased diffusion current, many more electrons will be present in the p-type region; the amount of ionized dopants will become larger than the number of majority charge carriers. Electrons are now the majority charge carriers in the p-type region and the negative space charges are excited to a level that breaks the bond. Therefore, the depletion region will shrink. Since the generation current only depends on charge carriers reaching the depletion region within the minority charge carrier diffusion length and shrinkage of the depletion region is minor compared to the dimension of the p-n junction, $I\textsubscript{gen}$ remains constant. A current flows despite the circumstance that there are so many ionized dopants which could act as a countering force, because the anode will constantly supply holes for the electrons to recombine with.
The forward bias will move the Fermi energy for the n-type region $E_{FC}$ closer to the CB edge.

Under reverse bias condition, electrons are injected into the p-type region, where holes are the majority carriers, and recombine with the holes. This eliminates free holes from the negatively charged ionized dopants. Electrons are pushed from the cathode and from the negatively charged edge of the depletion region which poses as a potential barrier for electrons. This way, the electrons will be more likely to recombine at the edge of the depletion region and thereby widen it. The larger the reverse bias voltage, the less diffusion will take place. Since diffusion leads to restoration of equilibrium by recombination, it is detrimental for the overall flow of current. A reverse bias voltage will lead to less recombination by diffusion and the recombination current $I\textsubscript{rec}$ will be reduced. A larger electric field will not affect the generation current, since the only purpose of it is to transport the charge carriers over to the other side; the speed of that is irrelevant. For that reason, for high reverse bias voltages, there is a small remaining saturation current.
The reverse bias will move the Fermi energy for the p-type region band $E_{FV}$ closer to the VB edge.

Under illumination, additional energy is given to the thermal excitation energy and more electron-hole pairs are created. This significantly increases the number of minority charge carriers and adds to the generation current $I\textsubscript{gen}$. Putting all of these contributions together finally yields the \textit{diode} or \textit{Shockley equation}:
\begin{equation}
\label{eqn:shockley}
I = I_{ph} + I_{gen} - I_{rec} = I_{ph} + I_{0}\cdot \left[1 - exp\left(\frac{eV}{kT}\right)\right]
\end{equation}
The equation is visualized in Figure \ref{fig:I-V-characteristics}. There it can be seen that photo-generated current shifts the dark I-V curve upwards.
It depends on the energy of the photon, how much an electron will be excited. It can be excited beyond the CB edge, but then it will dissipate its energy through \textit{thermalization}. A currently researched future technology are hot-carrier solar cells (HCSC) which seek to slow down thermalization and make use of that additional thermalization energy. The maximum energy of such an electron could then exceeds $e \cdot V_{OC}$, but in normal (non-hot carrier) solar cells this is a negligible rare process.
\\
How does all of this pertain to the real-world application of a solar cell? As briefly mentioned above, solar cells are merged into a series and parallel circuit array. Where the solar cells are connected in series, the voltages add up. Variation in the load will change the voltage of the individual cell, between 0 (for zero resistance)\footnote{Diffusion would no longer happen across the depletion region, but through the conducting wire and recombination would not take place in the other doped region, but in the wire. If this was not the case, the recombination in the other doped region would leave even more ionized dopants and thereby increase the voltage drastically.} and the voltage that is created by the illumination. But this is only a partial contribution to the overall voltage that an individual solar cell will experience inside the array: The adjacent solar cells will also produce an additional voltage under illumination which is contrary to the internal electric field voltage, because the photo-generated electron-hole pairs will elicit a counteracting recombination current that, just like during forward-biasing, will lower the potential difference across the depletion region.
The series circuit will connect the p-doped region with the n-doped region. This comes akin to reverse-biasing the solar cell. So, under ideal working circumstances, the array's solar cells will be reverse-biased. However, under non-ideal circumstances, there can be two detrimental effects of reverse-bias induced by adjacent solar cells. One is \textit{partial shading}: If one of the cell does not receive any light, it will not create any power, but because its p-side is attached to an n-side and its n-side to an p-side, it is still forced to have a voltage. This will inevitably lead to heat dissipation.\cite[p.256]{smets-2016} Similarly, reverse currents occur due to ohmic shunt resistances which is the most frequent source of detrimental reverse currents.\cite{breitenstein-2011}

\subsection{Important parameters for characterization of solar cells}
\label{ssec:important-parameters}
When the p- and n-type region are short-circuited, the voltage will be zero and the current will be determined by the photo-generated electron-hole pairs. As $I$ approaches $I\textsubscript{SC}$, it is the maximum current of a solar cell that still produces power.
In contrast, when the p- and n-type region are not connected or the resistance is infinitely high, then the voltage will be the \textit{open-circuit voltage} $V\textsubscript{OC}$. The $V\textsubscript{OC}$ could also be defined as the voltage established by the internal electric field or the bias voltage that will be necessary to make $I$ zero. As $V$ approaches $V\textsubscript{OC}$, it is the maximum voltage of a solar cell that still produces power. It can be calculated from rearranging Equation \ref{eqn:shockley}:
\begin{subequations}
\begin{align}
\label{eqn:ISC}
I_{SC} &= I_{ph}\\
\label{eqn:VOC}
V_{OC} &= \frac{kT}{e}\cdot ln\left(\frac{I_{ph}}{I_{0}}+1 \right)
\end{align}
\end{subequations}
As can be seen from Equation \ref{eqn:VOC}, the $V\textsubscript{OC}$ mainly depends on the saturation current, because it varies much more (by orders of magnitude) than the photo-generated current. $I\textsubscript{0}$, in turn, depends on the recombination and therefore $V\textsubscript{OC}$ is a measure of the amount of recombination in the solar cell. It is one of the most important parameters of a solar cell, since it carries information of how much power eventually can be produced and how much will be lost to recombination.
Only when $V\in (0, V\textsubscript{OC})$ will the solar cell create power. Otherwise, it will consume power. The point at which the solar cell produces the maximum amount of power is called the \textit{Maximum Power Point} $(I\textsubscript{MP}, V\textsubscript{MP})$.
The variables introduced so far are contained in the \textit{Fill Factor}:
\begin{equation}
\label{eqn:FF}
FF = \frac{I_{MP}\cdot V_{MP}}{I_{SC}\cdot V_{OC}}
\end{equation}
The fill factor is a measure of how efficiently the solar cell produces power, compared to the power $P\textsubscript{in}$ of the irradiance it receives. The \textit{conversion efficiency} $\eta$ of the solar cell is then measured under standard (ideal) test conditions, yielding $\eta\textsubscript{STC}$ and under normal operating (realistic) cell temperature conditions, yielding $\eta\textsubscript{NOCT}$\footnote{Under NOCT conditions, the cell is tested as if there was no load, so the voltage will be the $V\textsubscript{OC}$ and since energy is not dissipated in the load, this will cause more heating and therefore a higher temperature than under standard test conditions.}:
\begin{equation}
\label{eqn:conversion-efficiency}
\eta = \frac{I_{MP}\cdot V_{MP}}{P_{in}} = \frac{I_{SC}\cdot V_{OC}\cdot FF}{P_{in}}
\end{equation}
The V\textsubscript{OC} could be thought of as proportional to $E_{g}$, since this is the maximum electric energy an electron can have in the semiconductor, but because of the Urbach energy, $e\cdot V_{OC}$ will be lowered by a subtraction term. This has to do with the fact that the space charge region is not exactly sharp which causes certain effects under strong illumination.\cite{crandall-1996} I\textsubscript{SC} decreases with the band gap energy.\footnote{Why the V\textsubscript{OC} increases with E\textsubscript{g}, whereas I\textsubscript{SC} decreases, can also be explained this way: With higher E\textsubscript{g} more energy is needed to excite an electron-hole pair, so this will limit the current, but the V\textsubscript{OC} is measured under forward bias condition, where the external electric field diminishes the potential barrier across the junction more than the higher E\textsubscript{g} increases it.} Thus, in CIGSS solar cells, V\textsubscript{OC} and E\textsubscript{g} have an affine linear relationship. In CIGS solar cells, for reasons that are not understood yet\footnote{Models suggest that it has to do with an unfavourable band alignment at the n-CIGS and p-CdS interface. See figure \ref{fig:CIGS-layer-structure}. Models cited in: Weinhardt et. al (Ibid.)}, this cannot be said as easily.\cite{weinhardt-2017}

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{Figures/Fig8.9_I-V characteristics_mirrored.jpg}
\caption{I-V characteristic. The lower black line shows the current due to thermal-generation and diffusion. It approaches $I\textsubscript{0}$ under reverse bias voltage, as can only be seen barely, and otherwise shows an exponential increase. The upper black line is the illuminated I-V curve, i.e. the dark I-V curve shifted by the photo-generation current (blue line). The product of I and V yields the power output, which has a maximum at the Maximum Power Point ($J\textsubscript{MP}$,$V\textsubscript{MP}$). Edited from this image source: \cite[p.99]{smets-2016}}
\label{fig:I-V-characteristics}
\end{figure}

If one takes into account the non-ideality of the solar cell, new variables, such as for shunt and ohmic (parallel) resistance are introduced, and there are different models as well as different methods to account for this and to determine these variables.\cite{cotfas-2013}

\subsection{CIGS solar cells}
\label{ssec:cigs-solar-cells}
Copper indium gallium diselenide (CIGS) belongs to the second generation of solar cells. In contrast to older materials, it is cheaper to produce and can be deposited on flexible materials which allows to build thinner solar cells.\footnote{For c-Si, all of the light is absorbed within a few micrometers, but the absorber is about \SI{180}{\micro\meter} thick for mechanical support.\cite{ramanujam-2017}} Thereby, the energy payback time ($\sim$ 1 year) is less than that of c-Si ($\sim$ 2 years) now. CIGS was an improvement of the \textit{copper indium diselenide} (CIS: CuInSe\textsubscript{2}) material which was first synthesized in 1953, by Harry Hahn et. al. It received major attention in the 80s, when efficiencies of over 9\% were reported. Gallium and Sulfide (for CIGSS) were added then.\cite{ramanujam-2017}
\\
Chemically, CIGS is a solid solution of CIS and \textit{copper gallium diselenide} (CGS: CuGaSe\textsubscript{2}) with the chemical formula CuIn\textsubscript{x}Ga\textsubscript{1-x}Se\textsubscript{2}, where $x\in [0,1]$ is a measure of the indium/gallium ratio. Varying x from 0 to 1 allowed Xin Cui et. al to \textit{tune} the band gap from 1.08 to 1.53 eV\footnote{If sulfur is present as well, the Se:S ratio y can be tuned as well, allowing for a band gap range of 1.0 to 1.7 eV.\cite[p.192]{smets-2016}}.\cite{cui-2015} Tuning the band gap with gallium allows to better match the solar spectrum ($\lambda_{red}\approx$ 1.77 eV)\cite{ramanujam-2017} and a higher gallium to indium ratio coincides with a lower charge collection efficiency, except for cells that might have a larger amount of defects and inhomogeneities.\cite{pyrlik-2020-bachelor} CIGS has a chalcopyrite structure which means that its tetrahedrally bonded. It has intrinsic copper defects which is part of the reason why it is intrinsically p-doped.\footnote{The physical description of this and other defects in CIGS solar cells is still pending more ongoing research.} CIGS is a direct band gap material and therefore has a higher absorption coefficient which means that an excited charge carrier will recombine earlier. This necessitates a thinner material and allows to classify CIGS as a \textit{thin film technology}.
\begin{figure}
\centering
  \begin{subfigure}{0.4\linewidth}
    \centering\includegraphics[width=0.7\linewidth]{Figures/Fig13.22a_layer_structure.png}
    \caption{Layer structure of CIGS solar cell. The p-n junction is also referred to as a \textit{buried junction}, because the separation of electron-hole pairs happens one layer above. Image source: \cite[p.193]{smets-2016}}
    \label{fig:CIGS-layer-structure}
  \end{subfigure}
\hspace{1cm}
\centering
  \begin{subfigure}{0.4\linewidth}
    \centering\includegraphics[width=0.7\linewidth]{Figures/Fig13.22b_band_diagram.png}
    \caption{Band diagram of CIGS solar cell. Image source: \cite[p.193]{smets-2016}}
    \label{fig:CIGS-band-diagram}
  \end{subfigure}
\end{figure}
Figure \ref{fig:CIGS-layer-structure} shows the different layers that a CIGS solar cell is composed of. The substrate is glass in this case, and then a Molybdenum layer of about 500 nm thickness serves as an electric back contact. Molybdenum is chosen over other electrically conductive materials, not only due to its high conductivity, but also because it is chemically and mechanically stable during the CIGS growth (\textit{selenization}).\cite{ong-2018} Next, a p-type absorber layer of about \SI{2}{\micro\meter} and a thin n-type layer (for example with a high Indium/Gallium ratio) are deposited. The n-type CIGS prevents the absorption of holes. Next, another thin layer, consisting of Cadmium Sulfide, with around 50 nm thickness, is deposited. The CdS serves as a buffer layer to shield the absorber surface during the further layer deposition process and also to smoothen the band offset across the heterojunction (see Figure \ref{fig:CIGS-band-diagram}).\footnote{This CdS layer has been much thicker in the past, before it was replaced with the subsequent i-ZnO and Al-ZnO layers.\cite{ramanujam-2017}}\cite{rana-2017} This aims at better utilizing the solar spectrum, because the CdS acts as a \textit{window layer} that transmits low-energy photons and absorbs high-energy photons. A non-optimal CdS can cause severe reductions to the V\textsubscript{OC}.\cite{weinhardt-2017} \textit{Shockley Read Hall} (SRH) recombination becomes dominant at the n-CIGS/CdS interface, whereas for the n-type CIGS it is dominant in the bulk region. An intrinsic zinc oxide (i-ZnO) layer is then followed by an aluminium-doped layer (Al-ZnO) which is effectively n-doped. The i-ZnO is highly resistive to prevent aluminium from the Al-ZnO layer to diffuse through shunting paths into the CIGS absorber. layer.\cite{ramanujam-2017} The Al-ZnO is transparent to light and acts as the \textit{Transparent Conductive Oxide} (TCO). As seen in Figure \ref{fig:CIGS-band-diagram}, with its band gap energy of 3.3 eV, no visible photon will be absorbed there.
The photo-generated electron-hole pairs are separated at the CdS-CIGS interface, to be collected at the Al-ZnO.
\section{Experiment}
\subsection{Experimental environment}
This work analyzes data which has been acquired amongst several other measurements that were led by Dr. Smart in November 2018 at Deutsches Elektronen-Synchrotron (DESY) in Hamburg, Germany. There, special interest is taken in the nanoscale structure of solar cell probes. The forelast station at the P06 beamline at DESY's particle accelerator PETRA III offers such a unique opportunity for examining X-ray beam induced currents (XBIC)\footnote{XBIC was first applied to solar cells in 2000 by Hieslmaier et. al.\cite{hieslmair-2000-csscmp}(Cited in: \cite{stuckelberger-2017-jmr})}. It is positioned in the \textit{microhutch} where the experiments are conducted. It receives a hard X-ray beam from the optics hutch, about 40 to 60 meters from the microhutch, with an energy that can be continuously tuned between 6 and 18 keV when using the channel-cut crystal monochromator (CCM). The prefocusing system of the optics hutch is based on beryllium compound refractive lenses (Be CRL) and is used to further adapt the photon flux and coherence properties for the micro- and nanohutch.\cite{schropp-2020-jac} Usually, the aim with this setup is to produce X-ray beams with a FWHM below 50 nm, but by using corrective phase plates, the FWHM lies around 100 nm and diffraction-limited X-rays are achieved.\cite{seiboth-2017-ncomm} This setup is often used when the microhutch is operated in conjunction with the nanohutch (for Ptychography\footnote{In the nanohutch, resolving probes below 10 nm is the current goal, and to focus a 50 nm FWHM X-ray beam necessitates advanced X-ray optics which cumulated in the ptychographic nano-analytical microscope (PtyNAMi).}).
The microhutch's microscope focuses the X-ray beam with Kirkpatrick-Baez (KB) mirrors onto the probe that is fixed at the stage. The stage then is able to move steps with a size of 50 nm.
Furthermore, the P06 employs many different techniques at the same time to get a much better imaging representation of probes with complex composition such as CIGS solar cells, that have multiple layers (compare subsection \ref{ssec:cigs-solar-cells}). This \textit{multi-modal} approach is a unique possibility that the P06 can provide and has successfully combined so far XBIC with X-ray fluorescence (XRF)\cite{west-2017-jsr}\cite{west-2017-nanoen}, X-ray excited optical luminescence (XEOL)\footnote{There are currently ongoing endeavours to allow for a portable XEOL setup.}, X-ray diffraction (XRD), X-ray beam induced voltage (XBIC)\cite{stuckelberger-2017-pvsc-xbiv} and ptychography (which revealed grain boundaries for the first time in 2018: \cite{stuckelberger-2018-mimi}).\cite{ossig-2019-jove} This vast combination of different techniques under hard X-ray illumination allows for point-by-point correlation in 3D space, however still limited by the necessity of thorough 3D simulations to complement the measurements, increased perturbations due to X-ray beam induced temperature, bias voltage, bias light and locally generated electric fields and, most importantly, the sheer amount of data that needs to processed and still have pending reconstruction algorithms, possibly being alleviated by machine learning in the future.\cite{stuckelberger-2017-jmr}
The hard X-ray beam allows for depper penetration depth, revealing more of the nanoscale ingredients.

\subsection{Experimental setup (Nov 2018)}
\label{ssec:experimental-setup}
For this experiment, the X-ray beam was focused to 105 $\times$ 108 nm (vertical $\times$ horizontal) at FWHM with an energy per photon of 15.25 keV. Its direction defines the \textit{downstream} X-axis, and the axis parallel/perpendicular to the ground defines then the Y-/Z- axis respectively. The probe was mounted on a stage that was moved, amongst others, by a PI piezo motor. The movement is a so-called \textit{snake scan}, where the stage moves 200 steps, each 50 nm, in the Y direction, then simultaneously 1 step in the Y and Z direction, and then reverses the Y movement. One step has a \textit{dwell time} of 9.7 ms on average.\footnote{The scans were not optimized for simultaneous XRF measurements which require about ten times as many photons for good signal/noise ratio. As the chopper halves the amount of photons, it is moved out of the beam for XRF-optimized scans.} A scan during one step is here referred to as a \textit{pixel}. Thereby, a 201 $\times$ 200 grid can be extracted.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{Figures/Cablingoverview29012019hellblau.png}
\caption{Setup scheme of the experiment with focus on the signal paths. The current is converted into a voltage and pre-amplified. Then it is split up into two signal paths. One path is lock-in amplified, the other is not. Both paths then lead into the voltage-to-frequency converters (V2F) and are separated into positive and negative signals where the data acquirement unit (DAQ) picks them up. Image source: \cite[p.9]{ossig-2019-jove}}
\label{fig:setup-of-experiment}
\end{figure}
Figure \ref{fig:setup-of-experiment} shows the setup of the experiment. The beam is first modulated by a chopper with 8.015 kHz and then focused with KB mirrors onto the device under test (DUT), i.e. CIGS probe. The probe was provided by Empa. The front contact (n-type, compare subsection \ref{ssec:cigs-solar-cells}) was grounded, because the X-ray photons have enough energy to eject charge carriers, close to surface, from the side facing upstream. These are of interest for total electron yield (TEY) measurements, but not for XBIC, since a replacement current will be caused, mainly by the majority charge carriers there that then find minority charge carriers to recombine with.\cite{stuckelberger-2017-jmr}  The voltage to frequency converters (V2f) do not accept negative voltages, therefore an inverter is used to also capture negative signals. The frequency can be calculated back to the solar cell current, as described by Ossig et. al (compare subsection \ref{ssec:data-preparation}).\cite{ossig-2019-jove}
\subsubsection{Pre-amplification and Lock-in amplification}
\label{sssec:pa-and-lia}
Usually, XBIC is measured under short-circuit condition. However, a voltage series was conducted here; alternating from negative to positive bias voltage to avoid hysteresis. The bias voltage was supplied by the Stanford Research Systems SR570 current-sensitive pre-amplifier which converts the current into a voltage, with an amplification factor of 10\textsuperscript{6} V/A. Its signal was both dispensed to the lock-in amplifier as well as directly to the DAQ. In the latter case, the signal is referred to as the \textit{direct signal}.
\\
Lock-in amplification is a technique that especially plays to its strengths when an original AC signal needs to measured which is smaller than the amount of noise in the input signal. It improves the signal/noise ratio by orders of magnitude, decreases the sensitivity to measurement artifacts, enables applying bias voltage and measuring in environments with light noise (aka \textit{bias light}).\cite{stuckelberger-2018-mimi}
For this high-precision experiment, a UHFLI dual-phase lock-in amplifier (LIA) by Zurich Instruments that is certified to only have 4 $nV/\sqrt{Hz}$ noise was used with an amplification factor 1 V/V. Figure \ref{fig:lia-scheme} sketches its working principle. Detailed explanations are found in \cite{zurich-instruments-2016} and \cite{meade-1983}. The steps described in the following can also be achieved by analog means, however as this adds noise, they are performed digitally in high-precision instruments such as the UHFLI LIA. The probe needed to be small, because increasing the size would increase the voltage-induced (DC) current which is regarded as noise by the LIA (compare \cite{stuckelberger-2017-pvsc-xbiv}. First, the input (AC) signal $V_{s}(t)$ is split. The reference signal $V_{r}(t)$ is a square wave modulation supplied by the optical chopper. Since there can be a delay between the phase of the carrier and the reference signal, V\textsubscript{r} is also shifted by $\phi = \frac{\pi}{2}$ which later reveals $\phi$. For demodulation, both signals are then multiplied by the input signal and averaged over a certain time span that can be specified by the number of modulation cycles/time. The calculation is simplified when representing V\textsubscript{s} and V\textsubscript{r} as complex numbers\footnote{If one does not prefer to calculate with complex numbers, the in-phase and quadrature components can also be calculated by separately multiplying with $sin(\omega_{r}t)$ and $cos(\omega_{r}t)$. Terms will then cancel out during averaging due to the orthogonality of sine and cosine. $R$ is the root-mean-square of the amplitude, hence the prefactor $\sqrt{2}R$.}:
\begin{align}
\label{eqn:lock-in-multiplication}
    \left[V_{s}\right]\cdot \left[V_{r}\right] &= \left[\sqrt{2}R\cdot cos(\omega_{s}t + \phi)\right]\cdot \left[\sqrt{2}cos(\omega_{r}t) - i\sqrt{2}sin(\omega_{r}t)\right] \nonumber \\
                    &= \left[\frac{R}{\sqrt{2}}e^{+i(\omega_{s}t+\phi)} + \frac{R}{\sqrt{2}}e^{-i(\omega_{s}t+\phi)}\right]\cdot \left[\sqrt{2}e^{-i\omega_{r}t}\right] \nonumber \\
                    &= R\left[e^{i[(\omega_{s}-\omega_{r})t+\phi]}+e^{-i[(\omega_{s}+\omega_{r})t+\phi]}\right]
\end{align}
When $\omega_{r} \approx \omega_{s}$, the first term in equation \ref{eqn:lock-in-multiplication} will become time-independent and the second term is harmonic and therefore can be time-averaged to zero. The product of V\textsubscript{s} and V\textsubscript{r} then simplifies to
\begin{equation}
\label{eqn:lock-in-final-product}
    V_{s}\cdot V_{r} = R\cdot e^{i\phi}.
\end{equation}
The real part of equation \ref{eqn:lock-in-final-product} is called the \textit{in-phase component} and often denoted by $X$, whereas the imaginary part is called the \textit{quadrature component Y}. R then gives the root-mean-square amplitude of the original signal for the modulation cycle (i.e. time interval over which has been averaged). The time-averaging electronically corresponds to connecting an RC low-pass filter of n-th order that specifies the roll-off behaviour. Averaging over longer periods will yield a trade-off between less noise and a slower response of R. The modulation frequency should be set below the frequency of the noise, as the low-pass filter cuts off frequencies that are higher than the reference signal. With square waves, the demodulation needs to be performed at several frequencies, since in the frequency spectrum they correspond to several peaks at the fundamental and higher harmonics frequencies. Lastly, it should be remarked that LIA itself is prone to noise. For example, all electrical instruments in the experiments should be grounded to the same point to avoid ground loops.

\begin{figure}[!ht]
\centering
\includegraphics[width=0.6\textwidth]{Figures/lockin_primer_fig_02_dual_phase_corrected.png}
\caption{The modulated input signal V\textsubscript{s} is (digitally) split and multiplied with the reference signal V\textsubscript{r} and a $\frac{\pi}{2}$ phase-shifted reference signal respectively. The result is fed to a low-pass filter with a specified cut-off frequency at which the signal power (\textasciitilde amplitude\textsuperscript{2}) is diminished by half.  Edited from this image source: \cite[p.2]{zurich-instruments-2016}}
\label{fig:lia-scheme}
\end{figure}

\section{Data preparation}
\label{ssec:data-preparation}
The data gathered from the DAQ has the HDF5 file format. It contains several dimensions of data, of which this analysis uses:
\begin{itemize}
 \item Y position: Fast scan, encoded
 \item Z position: Slow scan, encoded
 \item Frequency counts from the V2f converter: amplitude R
 \item Frequency counts from the V2f converter: Negative direct signal
 \item Frequency counts from the V2f converter: Positive direct signal
 \item QBPM counts: normalized photon flux
 \item Dwell time: seconds of exposure until the stage moves (according to a snake scan, as described in subsection \ref{ssec:experimental-setup})
\end{itemize}

The DAQ data can be converted to a current by following the formula described by Christina Ossig \cite[p.4]{ossig-2019-jove}. Subsequently, one needs to normalize each pixel to the photon flux which is composed of the QBPM count and exposure time. The QBPM (\textit{Quadrupole Beam Position Monitor}) does not measure the absolute number of photons, but instead a current proportional to it. Therefore, all XBIC measurements have to be seen relative to the probe and measurement condition. It is not possible to compute from this data a quantity such as the EQE (\textit{External Quantum Efficiency}), nor is it possible to interpolate how much current or voltage the tested solar cell would produce if a larger area was illuminated with the same overall photon energy, because the physical processes are much different when a small part of the cell is illuminated by an X-Ray, compared to operation under normal sunlight illumination.
Next, the normalized current is interpolated from a 201 $\times$ 200 px to a so-called \textit{supergrid} with 1000 $\times$ 1000 px. This allows for subpixel translations during image registration and more precise application of the Gaussian filter (see subsection \ref{sssec:image-registration}). Its nearest-neighbour method accounts for deviations of the dialed positions from the encoded positions (compare \cite{pyrlik-2020-pvsc}).

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{Figures/AllXBICinput.png}
\caption{All channels from the DAQ: a) lock-in -10 to 0V, b) lock-in 0 to +10V, c) lock-in R, d) direct signal -10 to 0V, e) direct signal 0 to +10V. Each column represents a bias in the voltage series -1V, -500mV, -200mV, -100mV, 0mV (no bias), +100mV, +200mV, +500mV and +600mV.}
\label{fig:AllXBICinput}
\end{figure}

If this is done, one can plot the input from the DAQ as shown in Figure \ref{fig:AllXBICinput}. Each column represents one scan. From left to right, the voltage series -1 V, -500 mV, -200 mV, -100 mV, 0 mV (no bias), +100 mV, +200 mV, +500 mV and +600 mV is shown. As mentioned in subsubsection \ref{sssec:pa-and-lia}, this order does not correspond to the order in time during the experiment, to avoid hysteresis. The first two rows show the channel which records the in-phase X and quadrature component Y of the lock-in signal which is separated by the intervals -10 to 0 and 0 to +10V, because X and Y can have both positive and negative values. As can be seen from the overview, mainly the Y channel contains a signal, because the phase shift remains mainly constant. If the phase shift was not relatively stable, lock-in amplification would not work. The third row contains the R amplitude of the lock-in signal and is the most valuable data for this analysis. The last two rows contain the direct signals from the pre-amplifier (compare Figure \ref{fig:setup-of-experiment}), whereas the fourth contains the data from -10 to 0V and the fifth contains the data from 0V to +10V. Here, only data is recorded in either one of those and the other contains noise, because the current can only be positive or negative. The voltage acceptance range here actually represents if the current is negative or positive. In the case of the acceptance range for the lock-in data, the different intervals only serve to separate X and Y and have no meaning whatsoever regarding whether the current is positive or negative.
Furthermore, one can see in the figure that sensible data is recorded only in the range from -200 to +200 mV. This is because larger voltages lead to higher currents which were beyond the sensitivity of the pre-amplifier. This analysis can only use the scans from -200 to +200 mV.
The profile of the X-Ray beam is not quadratic like the pixels, but has a Gaussian profile which leads to overlapping for each scanned pixel. To account for this oversampling, a Gaussian filter with a kernel size of 33 pixels is applied.\footnote{The height and width of the Gaussian filter kernel size must be odd, but can have different sizes, to properly define a middle point of the filter matrix.}

\subsection{Image Registration and Gaussian Filter}
\label{sssec:image-registration}
During such high-precision measurements, it is inevitable that the scans do not overlap precisely, due to unaccounted beam shift, sample or stage movement. From this the need for \textit{image registration} arises. The openCV for Python library offers the solution with its \texttt{findtransformEEC} function which calculates six affine transformation coefficients. The Gaussian filter helps to separate characteristic features from noise. The border values of the current grids where the filter size would exceed the grid are set to NaN.\footnote{One must be careful here that the Gaussian filter does not reflect the border values which would intensify the random distribution of border values. To be safe, a border of 1 px width can be manually set to NaN. Thereby, the filter will only be applied where the filter size does not exceed the grid.} The six coefficients are applied to each pixel (Y,Z) by the openCV \texttt{warpAffine} function as following:

\begin{gather}
\begin{pmatrix}
Y' \\
Z' \\
\end{pmatrix} =
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{bmatrix} \cdot
\begin{pmatrix}
Y \\
Z \\
\end{pmatrix} +
\begin{bmatrix}
b_{1} \\
b_{2} \\
\end{bmatrix}
\end{gather}

a\textsubscript{11} to a\textsubscript{22} are the coefficients for rotation and shearing. b\textsubscript{1} and b\textsubscript{2} are the coefficients for translation. For this data, no shearing or rotation, but only translation needs to be corrected, thus a\textsubscript{11} = a\textsubscript{22} = 1 and a\textsubscript{12} = a\textsubscript{21} = 0, whereas b\textsubscript{1} and b\textsubscript{2} contain the pixel values for translation in Y and Z direction.
\texttt{warpAffine} applies the coefficients to the supergrid by shifting it b\textsubscript{1} pixels in Y and b\textsubscript{2} pixels in Z position and inserts NaN values at the vacant pixels.
For more details about the effect of different Gaussian filter sizes on the correlation coefficients and about affine transformations see \cite{pyrlik-2020-pvsc}. At last, all pixel positions (Y,Z), where any of the scans is NaN, all other scans at (Y,Z) will be set to NaN as well.

\begin{figure}
    \centering

\begin{minipage}[t]{0.4\textwidth}
\centering \textbf{a)} \\
\includegraphics[width=\textwidth]{Figures/singleplots/Scan185.png}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{0.4\textwidth}
\centering \textbf{b)} \\
\includegraphics[width=\textwidth]{Figures/singleplots/Scan185_direct.png}
\end{minipage}

\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan183.png}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan183_direct.png}
\end{minipage}

\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan181.png}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan181_direct.png}
\end{minipage}

\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan182.png}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan182_direct.png}
\end{minipage}

\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan186.png}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Figures/singleplots/Scan186_direct.png}
\end{minipage}

\caption{The data used for this analysis: a) XBIC from the lock-in amplifier, b) voltage-induced current from the direct pre-amplifier signal. For the unbiased scan \#181, the same NaN border is added.}
\label{fig:lia-and-pa-XBICplot}

\end{figure}

If this has been done, the XBIC data from the lock-in and the direct pre-amplifier channel can be plotted, as seen in Figure \ref{fig:lia-and-pa-XBICplot}. The unbiased scan was taken as a reference in image registration for calculating the transformation coefficients, because there one can expect the least artefacts due to biasing. The artefacts were filtered out by the lock-in amplification, but can well be observed in the direct signal. One also notices that the lock-in signal exhibits a relatively small change of currents, whereas the direct signal changes from large positive to large negative values. The main difference between those two is that the lock-in signal filters out the voltage-induced current that is mainly seen in the direct signal plots.

\section{Results and Discussion}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.7\textwidth]{Figures/violin_currents.png}
\caption{Current distribution for each bias, -200 to +200mV from left to right. Violins are scaled by the number of counts and counts are taken from Seaborn's kernel density estimator.}
\label{fig:violin_currents}
\end{figure}

Before actually looking at the spatially-resolved IV curves, one can already read from Figure \ref{fig:violin_currents} that apparently the distribution of currents seems to be only slightly dependent on the bias voltage. Also, the distribution in itself seems appears to be almost unchanged. This leads to the question whether the pixels of the solar cell that are high-performing remain their status as high-performing and if the same is valid for the weaker performing pixels.

\begin{figure}
\centering
  \begin{subfigure}{0.18\linewidth}
    \includegraphics[width=\linewidth]{Figures/MedianGroups1.png}
    \label{fig:MedianGroups1}
  \end{subfigure}
\centering
  \begin{subfigure}{0.18\linewidth}
    \includegraphics[width=\linewidth]{Figures/MedianGroups2.png}
    \label{fig:MedianGroups2}
  \end{subfigure}
 \centering
  \begin{subfigure}{0.18\linewidth}
    \includegraphics[width=\linewidth]{Figures/MedianGroups3.png}
    \label{fig:MedianGroups2}
  \end{subfigure}
 \centering
  \begin{subfigure}{0.18\linewidth}
    \includegraphics[width=\linewidth]{Figures/MedianGroups4.png}
    \label{fig:MedianGroups2}
  \end{subfigure}
 \centering
  \begin{subfigure}{0.18\linewidth}
    \includegraphics[width=\linewidth]{Figures/MedianGroups5.png}
    \label{fig:MedianGroups2}
  \end{subfigure}
 \caption{The values above the median are grouped in group 1 and the values below the median in group 2. As an example, the plot was made for the scan with -200mV. Visually, there is no difference to the plots of the other scans. The Pearson correlation coefficient lies between 0.91 and 0.94.}
\end{figure}

High-/low-performing groups are defined as being above/below the median of the respective scan respectively. When the plots as in Figure \ref{fig:median_groups} are compared successively, a Pearson correlation coefficient between 0.91 and 0.94 is yielded. In other words, 81.1\% of the examined pixels display the behaviour that they don't change their allocation to either group 1 or 2 at all. The other 18.9\% are situated at the edges of group 1 and 2. This demonstrates the voltage-independency of the charge collection efficiency.

\begin{figure}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[trim={0 0 34.5cm 0},clip, width=\linewidth]{Figures/plotIV.png}
    \caption{In the background, a sample of 64 IV curves, uniformly spread across the current grid, and in bold, the mean IV curve for \textit{all} pixels are displayed.}
    \label{fig:plotIV}
  \end{subfigure}
\hspace{0.6cm}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[trim={0 0 34.5cm 0},clip, width=\linewidth]{Figures/plotIV_direct.png}
    \caption{The same display style as for Figure \ref{fig:plotIV}, but for the direct signal. The single curves show less relative deviation from the  average curve.}
    \label{fig:plotIV_direct}
  \end{subfigure}
\end{figure}

Each pixel has its own IV curve which differ both in their slopes from one bias to another, as well as in their absolute value. Figures \ref{fig:plotIV} and \ref{fig:plotIV_direct} show the mean IV curves for all pixels, including exemplary single IV curves. In Figure \ref{fig:plotIV_direct}, which shows the direct signal, it can be seen that the curve differs from the exponential curve of the Shockley equation (Equation \ref{eqn:shockley}. This is due to the fact that the X-rays strongly increase the conductivity of the solar cell, perhaps introducing shunts to the solar cell (compare \cite{stuckelberger-2017-pvsc-xbiv}).
However, Figure \ref{fig:plotIV} neither represents the Shockley equation. Fitting the curve to Equation \ref{eqn:shockley} was not possible.

A rather open research question is how to classify all of the IV curves. To answer this, the unsupervised machine learning k-means clustering algorithm can be used. As an input it takes data points with the same number of (any) dimension and randomly places k points, so-called \textit{centroids}. Each data point will be assigned to the centroid where the Euclidian distance is minimized. Then each centroid will be replaced by a new one that corresponds to the mean of all of its data points. This algorithm proceeds until it converges.

\begin{figure}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[width=1.2\linewidth]{Figures/ClusterPlots/ClusterCurves_2-k_1-kmeans-input_False-normalize-samples.png}
    \caption{In the background, a sample of 26/38 IV curves for group 1/2, uniformly spread across the current grid, and in bold, the mean IV curve for \textit{all} pixels of the respective group are displayed. K-means input: absolute values.}
    \label{fig:plotClusterCurves1}
  \end{subfigure}
\hspace{0.6cm}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[width=\linewidth]{Figures/ClusterPlots/ClusterPosition_2-k_1-kmeans-input_False-normalize-samples.png}
    \caption{Group 1 and 2 for the absolute values as k-means input.}
    \label{fig:plotClusterPosition1}
  \end{subfigure}
\end{figure}

Different derivations of the original XBIC data can be used as an input for kmeans clustering. If the IV curves with their five absolute values are taken as an input for k-means clustering, the result in Figure \ref{fig:plotClusterPosition1} resembles the original XBIC map from Figure \ref{fig:lia-and-pa-XBICplot}. This is clearly because the relative changes of the I(V) values are roughly about ten times smaller compared to the absolute offset. However, using the slope as a relative measure did not result in any meaningful clustering: The mean IV curves did not show any significant difference.

Generally, combining different slopes as an input for the clustering didn't yield much different results as when using a one-dimensional input, such as the total slope from the current at -200 mV to +200 mV. For this, the slope and intercept were fitted by minimizing the sum of squared distances to the fitted line of all five points.

\begin{figure}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[width=1.2\linewidth]{Figures/ClusterPlots/ClusterCurves_2-k_8-kmeans-input_False-normalize-samples.png}
    \caption{In the background, a sample of 24/40 IV curves for group 1/2, uniformly spread across the current grid, and in bold, the mean IV curve for \textit{all} pixels of the respective group are displayed. K-means input: fitted line for all five points.}
    \label{fig:plotClusterCurves8}
  \end{subfigure}
\hspace{0.6cm}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[width=\linewidth]{Figures/ClusterPlots/ClusterPosition_2-k_8-kmeans-input_False-normalize-samples.png}
    \caption{Group 1 and 2 for the fitted line from -200mV to +200mV values as k-means input.}
    \label{fig:plotClusterPosition8}
  \end{subfigure}
\end{figure}

In Figure \ref{fig:plotClusterCurves8}, it can be seen that those curves who show a smaller slope (group 1) also have a higher absolute value. Figure \ref{fig:plotClusterPosition8} therefore resembles the current map.

Taking a single slope from one (I,V) point to its next, does not yield any significant difference in the mean IV curves for each group, except for the slope from +100 to +200 mV.

\begin{figure}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[trim={0 0 0 1.5cm},clip, width=1.2\linewidth]{Figures/ClusterPlots/ClusterCurves_2-k_7-kmeans-input_False-normalize-samples.png}
    \caption{In the background, a sample of 24/40 IV curves for group 1/2, uniformly spread across the current grid, and in bold, the mean IV curve for \textit{all} pixels of the respective group are displayed. K-means input: slope from I(+100mV) to I(+200mV).}
    \label{fig:plotClusterCurves8}
  \end{subfigure}
\hspace{0.6cm}
\centering
  \begin{subfigure}{0.46\linewidth}
    \includegraphics[trim={0 0 0 1.7cm},clip, width=\linewidth]{Figures/ClusterPlots/ClusterPosition_2-k_7-kmeans-input_False-normalize-samples.png}
    \caption{Group 1 and 2 for the  slope from I(+100mV) to I(+200mV) as k-means input.}
    \label{fig:plotClusterPosition8}
  \end{subfigure}
\end{figure}

Interestingly, the curves are clearly shifted, although the clustering only received the relative slope as an input. To reconfirm that, a violin plot can be made that shows

\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{Figures/violin_groups_7.png}
\caption{The current distribution for each scan, separated into group 1 and 2 for clustering with the }
\label{fig:}
\end{figure}

\section{Conclusions}


\section*{References}
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\section*{Acknowledgements}
thank you all guys!

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\section*{Erklärung}
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Hiermit erkläre ich, dass ich die vorliegende Arbeit selbstständig angefertigt habe. Es wurden nur die in der Arbeit ausdrücklich benannten Quellen und Hilfsmittel benutzt. Wörtlich oder sinngemäß übernommenes
Gedankengut habe ich als solches kenntlich gemacht.
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Hamburg, \today & & \\
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Ort, Datum& & Engel Reims\\
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\newpage
\begin{appendices}
\section{Appendix A}
\label{app:pyth}

\end{appendices}
\pagebreak

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