chap05.tex

% !TeX spellcheck = <en-US>
\chapter{Evaporations and measurement}
\section{Evaporation configuration}

To optimize the penumbra of Pb islands on a \ce{Si} sample, a series of evaporations were performed. The \ce{Si}(111) sample was prepared and cleaned according to the process described in Section \ref{sec:sample_prep}. The cleanliness of both the sample and mask was confirmed optically before they were inserted into the Load Lock.

Five evaporations were conducted to assess the edge sharpness of the evaporated dots at different distances. The measurements began at a distance of $25 \pm 5$ {\textmu}m from the sample. The approach curve to full contact was recorded, and the first evaporation was performed at this point of full contact. The approach curve is shown in Figure \ref{fig:evaporation_approach_curve}.

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{img/Evaporation/Approach_Curve_Field01.pdf}
    \caption{The approach curve measured for field 1 until full contact.}
    \label{fig:evaporation_approach_curve}
\end{figure}

The 3 capacitance sensors appear heavily correlated and the uncertainty on C2 and C3 is an order of magnitude larger than the step in $dC$. For this reason C1 was primarily used for alignment. C2 and C3 were recorded but not utilized. The other evaporations were performed by retracting the mask $1000$ steps and approaching. 

The subsequent evaporations were performed by retracting the mask $1000$ steps and then approaching the sample. Four additional evaporations were conducted at different lateral positions on the sample. Each evaporation consisted of a $9 \times 9$ field of $3$ {\textmu}m Pb circles, as previously shown in Figure \ref{fig:mask_aligner_nomenclature_capacitances_mask}. Each field was evaporated at different mask-sample distances, as described by the approach curve. The evaporations were performed with the following stop conditions:

\begin{itemize}
	\item Field 1: $1$ {\textmu}m distance to sample (Full contact)
	\item Field 2: $16$ {\textmu}m distance to sample (First Contact)
	\item Field 3: $16$ {\textmu}m (Shortly before first contact at stop condition $0.12$ pF)
	\item Field 4: $4$ {\textmu}m (Second Contact)
	\item Field 5: $1$ {\textmu}m (Full Contact)
\end{itemize}

The parameters used for the evaporator are shown in Appendix \ref{app:evaporation}. The turbomolecular pump was by mistake not turned off during evaporation. \\

\begin{figure}[H]
    \centering
		\begin{subfigure}{0.45\linewidth}
		\centering
		\includegraphics[width=0.9\linewidth]{img/Evaporation/SampleImage.pdf}
		\caption{}
		\label{fig:Evaporation_diagramm_sample_img}
	\end{subfigure}
	\begin{subfigure}{0.45\linewidth}
		\centering
		\includegraphics[width=0.9\linewidth]{img/Evaporation/Mask01_Aspect.png}
		\caption{}
		\label{fig:Evaporation_diagramm_mask_img}
	\end{subfigure}
    
    \caption{(\subref{fig:Evaporation_diagramm_sample_img}) diagram showing the Evaporation performed on the sample. Red squares represent the positions of the evaporated fields. The number shows the order of evaporations. Distances are measured using an optical microscope. Fields are at a $10^\circ$ angle with respect to the sample holder. (\subref{fig:Evaporation_diagramm_mask_img}) microscope image of the mask taken before evaporation. The mask holder was aligned with respect to the camera view.}
    \label{fig:Evaporation_diagramm}
\end{figure}

After each evaporation the sample was moved laterally by $5000$ steps. Initially, the movement was in the -x direction, and after the third evaporation, the direction was reversed to +x. The final positions of the fields on the sample are shown in Figure \ref{fig:Evaporation_diagramm_sample_img}. \\
The fields were found to be angled at approximately 10° with respect to the sample edge. This misalignment is attributed to a slight deviation in the mask's positioning on the mask holder, as evident in Figure \ref{fig:Evaporation_diagramm_mask_img}.

\section{Contamination}
The entire surface of the sample is contaminated with small particles, approximately $50$ nm in height and $10$ nm in diameter. These contaminants are not visible under an optical microscope. Although the sample was cleaned, it was only inspected optically after cleaning, so it is unclear whether the contaminants were present after cleaning or were deposited later.

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.495\linewidth}
		\centering
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Contamination.png}
		\caption{}
		\label{fig:evaporation_contamination_img}
	\end{subfigure}
	\begin{subfigure}{0.495\linewidth}
		\centering
    	\includegraphics[width=\linewidth]{img/Evaporation/Contamination.pdf}
		\caption{}
		\label{fig:evaporation_contamination_anal}
	\end{subfigure}
    \caption{(\subref{fig:evaporation_contamination_img}) AFM image of field $3$ without any grain removal applied. Data was obtained on multiple different spots on the sample. (\subref{fig:evaporation_contamination_anal}) line cuts obtained from contamination particles. \textcolor{tab_red}{Red} and \textcolor{tab_green}{green} lines show the average height and width of the contamination particles obtained from peak fits.}
    \label{fig:evaporation_contamination}
\end{figure}

The data in Figure \ref{fig:evaporation_contamination} shows that the particles are up to $\approx 40$ nm in height and with an average height of $24 \pm 10$ nm. The particle's average width is $40 \pm 10$ nm. Height and width were obtained by fitting flattened Gaussian functions to the particles line cuts and extracting $2\sigma$ as well as the height of the peak. The distribution of particles across the sample surface is isotropic.

In addition, the sample was contaminated with larger particles possibly from long exposure at atmospheric conditions\footnote{Additionally, contamination could originate from vacuum bakeout. The system was heated to $>100$°C for several days on $2$ separate occasions.}. The size of these larger particles was determined to be in the order of $\mathcal{O}(100 \text{nm})$ using SEM and on the order of $\mathit{O}(10)$ {\textmu}m in diameter. \\
As the sample was only inspected optically before being inserted into the UHV chamber, it is possible that the small particle contamination was not detected. To avoid this issue in the future, it is recommended that the sample be examined for contaminants using AFM before being inserted into the chamber.

\section{Penumbra}

AFM measurements reveal that the dots are not circular (Fig. \ref{fig:penumbra_tilt_sigmas}). This deviation from circularity could be attributed to a tilted mask (as illustrated in Figure \ref{fig:penumbra_explanation_tilt}). Due to the elliptical aberration visible on the dots, two different penumbra widths were analyzed, denoted as $\sigma_s$ and $\sigma_l$. Both of these widths are defined along the major axis of the elliptical aberration. Additionally, the angle of tilt and the semi-major and semi-minor axes of the ellipse were measured. An example of how these measurements were taken can be seen in Figure \ref{fig:penumbra_tilt_sigmas}. \\

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.49\linewidth}
	\includegraphics[width=0.95\linewidth]{img/Plots/Background/IllustrationSigmas.pdf}
	\caption{}
	\label{fig:penumbra_tilt_sigmas}
	\end{subfigure}
	\begin{subfigure}{0.49\linewidth}
	\includegraphics[width=0.95\linewidth]{img/Evaporation/Field3_TR.pdf}
	\caption{}
	\label{fig:Evaporation_diagramm_field}
	\end{subfigure}
    
    \caption{(\subref{fig:penumbra_tilt_sigmas}) AFM image of evaporated \ce{Pb} dot illustrating the penumbral widths used for evaporation analysis $\sigma_s$ and $\sigma_l$, depicted in \textcolor{tab_red}{red}, and the major axis of the tilt \textcolor{tab_green}{(green)}.  $\sigma_s$ is drawn larger than actually measured, to aid visibility. The \textcolor{tab_blue}{blue} lines are the major $a$ and minor $b$ axis of the ellipse formed on the evaporated dot. Inset shows the same image in the phase data. The data is from Evaporation 5. (\subref{fig:Evaporation_diagramm_field}) AFM image of the top right part of field $3$. Grains were reduced using post-processing. Black circles show the dots chosen for further examination on this particular field.}
    \label{fig:penumbra_tilt_sigmas_and_field_show}
\end{figure}

Each field was studied by taking low resolution measurements of the lower left and the upper right side of the field. A few \ce{Pb} dots, representative of the edges and the center of the field were chosen for high resolution imaging. An example of this is shown in Figure \ref{fig:Evaporation_diagramm_field}. The dot visualized on the left of the image is near the center of the whole field, as the image shows only a partial field. \\

The data is filtered by masking the contamination of the \ce{Si} sample. This approach was effective because the dots' height is approximately $3$ nm, whereas the contamination particles are significantly taller, at around $50$ nm. The area under the mask is interpolated in order to remove most of the particles. \\
Line cuts close to the line along which the tilt of the dots points were obtained. By fitting a Gaussian falloff to the slopes of the line cut, the penumbra width is measured. The fit function is:

\begin{equation}
 f(x, b, h, \mu, \sigma_s, \sigma_l, r) = \begin{cases} 
      b + h * \exp(\frac{(x - (\mu - r))^2}{2\sigma_s^2}) & x\leq \mu - r\\
      b + h & \mu - r\leq x\leq \mu + r \\
      b + h * \exp(\frac{(x - (\mu + r))^2}{2\sigma_l^2}) & \mu + r \leq x 
   \end{cases}
\end{equation}

Where $r$ is the radius of the dot, $b$ is an offset from $0$, $\mu$ is the midpoint of the dot, $h$ is the height of the dot and $\sigma_s$ and $\sigma_l$ are the two different penumbras. This fit function allows the determination of the height, radius and penumbra of each dot.

An example is shown in Figure \ref{fig:evaporation_analysis}. In the example, the elliptical shape of the dot, induced by a tilt, can be easily seen in both image and line cut.
 
\begin{figure}[H]
    \centering
	\begin{subfigure}{0.45\linewidth}
		\centering
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Field5_top_demo01.png}
    	\caption{}
	\end{subfigure}
	\begin{subfigure}{0.45\linewidth}
		\centering
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Field5_top_demo02.png}
    	\caption{}
	\end{subfigure}
	\begin{subfigure}{0.6\linewidth}
		\centering
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/TopField5Fit.pdf}
    	\caption{}
	\end{subfigure}
	\caption{Example of the analysis conducted on each of the recorded dots for a single line cut. (a) raw AFM data before cleaning with a large amount of very bright contaminant particles. (b) cleaned image. The black lines in (b) show how multiple line cuts were obtained on a single image to obtain values for $\sigma_s$ and $\sigma_l$. The fit parameters are the two different penumbra widths induced by the tilt $\sigma_s$ and $\sigma_l$ for a single line cut. (c) line cut data from one line as an example. This line cut was obtained from \textcolor{tab_green}{(green)} line in (b). }
    \label{fig:evaporation_analysis}
\end{figure}

This process was performed for every recorded dot and with multiple line cuts near the tilt line, for both trace and retrace images. This gives multiple data points for each image for $\sigma_s$ and $\sigma_l$. The final values for each dot were then obtained by taking the mean and standard deviation for all line cuts. Contamination grains were avoided when drawing line cuts to increase data accuracy.

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.495\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/sigmas_summary.pdf}
    	\caption{}
		\label{fig:evaporation_measured_penumbra_sigs}
	\end{subfigure}
	\begin{subfigure}{0.495\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/sigmal_summary.pdf}
    	\caption{}
		\label{fig:evaporation_measured_penumbra_sigl}
	\end{subfigure}
	\begin{subfigure}{0.495\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/heights_summary.pdf}
    	\caption{}
		\label{fig:evaporation_measured_penumbra_height}
	\end{subfigure}
	\begin{subfigure}{0.495\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/umbra_summary.pdf}
    	\caption{}
		\label{fig:evaporation_measured_penumbra_circle_r}
	\end{subfigure}
	\caption{Data obtained from the previously described method for each of the 5 evaporations, one dot each from the center, the left, the right, the bottom and the top. (\subref{fig:evaporation_measured_penumbra_sigs})smaller penumbra $\sigma_s$.  (\subref{fig:evaporation_measured_penumbra_sigl}) larger penumbra $\sigma_l$. (\subref{fig:evaporation_measured_penumbra_height}) height of the dot. (\subref{fig:evaporation_measured_penumbra_circle_r}) diameter of the circle.}
    \label{fig:evaporation_measured_penumbra}
\end{figure}
Figure \ref{fig:evaporation_measured_penumbra} presents the results obtained from the analysis of \ce{Pb} dots in each field.
For $\sigma_s$, most of the data falls below the $100$ nm threshold, with the majority of fields exhibiting a penumbra of approximately $50$ nm. This suggests that very sharp evaporation patterns can be achieved. 
It is expected that fields $1$ and $5$ should show similar results, with smaller penumbras than the other fields, since they were evaporated at the lowest distance. However, this is not the case. Field $5$ exhibits some of the smallest penumbras, but they are more comparable to field $3$ than field $1$. Field $4$, which was evaporated at the point of second contact, unexpectedly shows the largest penumbras. \\

Fields $2$ and $4$ have the largest uncertainties, likely due to noisier data. This noise is attributed to the age of the AFM tip, which failed shortly after these measurements were taken. Unfortunately, new measurements could not be performed in time.\\
The differences in penumbra width between the top, bottom, right, left, and center of the fields are within the measurement uncertainty, indicating no significant variation across the field. \\

The height of the dots (Figure \ref{fig:evaporation_measured_penumbra_height}) is spread around a mean value of $2.6 \pm 0.3$ nm and shows deviation from the expected $5$ nm expected from flux. \\

The diameter of the \ce{Pb} dots is expected to decrease with subsequent evaporation due to clogging of the mask. This trend is mirrored in the data. The average diameter of evaporation decreases from $3.02 \pm 0.04$ {\textmu}m for field $1$ to $2.947 \pm 0.008$ {\textmu}m for field $5$. From a linear regression a decrease in diameter of $0.017 \pm 0.004$ {\textmu}m per evaporation is determined. \\

The eccentricity of the dot's outer shape was determined by measuring the diameter of multiple line cuts on the circle via fit and comparing measurements of perpendicular line cuts. The resulting eccentricity was as in the weighted mean $0.2 \pm 0.1$, which suggest that the dots are circular within measurement accuracy. The outer dot shape is not affected by the tilting effects.\\

The larger penumbra data (Figure \ref{fig:evaporation_measured_penumbra_sigl}) indicates no clear pattern within each field, except for possibly a reduction in penumbra for the bottom and center dots. This might be explained by different dots being chosen for each analysis. In the following, the penumbra and direction of tilt will be treated more thoroughly. \\

\section{Tilt and deformation}

All evaporated dots exhibited elongation of the circle, even when the mask was in full contact with the sample. If this elongation were due to misalignment between the entire mask and the sample, one would expect the direction of the tilt to be uniform across the sample. Additionally, the size of $sigma_l$ would be expected to decrease along the direction of the tilt.\\

To investigate this, the direction of the angle of the major axis was measured (as shown in the example in Figure \ref{fig:evaporation_tilts_example}) and recorded for all fields (Figure \ref{fig:evaporation_tilts_all}). The results, presented in Figure \ref{fig:evaporation_tilts_all}, reveal that the direction of the tilt points outward for dots located on the edge of the sample. This suggests that the mask itself is bent towards the edges, rather than being misaligned with the sample. \\

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.495\linewidth}
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Field3Angle.pdf}
    	\caption{}
		\label{fig:evaporation_tilts_example}
	\end{subfigure}
	\begin{subfigure}{0.495\linewidth}
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/FieldsAngle.pdf}
    	\caption{}
		\label{fig:evaporation_tilts_all}
	\end{subfigure}
	\caption{(\subref{fig:evaporation_tilts_example}) image of the reconstruction of the tilt angle for Field 3. (\subref{fig:evaporation_tilts_all}) the same for all fields. For fields 1, 4, 5 the full field scans were performed at low resolution and due to this the direction of the tilt could not be determined from the images. The only dots drawn in this case are the high resolution AFM scans of single dots.}
    \label{fig:evaporation_tilts}
\end{figure}

The smallest minor axis found in the AFM data was $2.15 \pm 0.08$ {\textmu}m compared to the $3.01\pm 0.05$ {\textmu}m of the evaporated circle. This would imply a tilt from one side of the dot on the mask to the other of $44 \pm 9 ^\circ$. This corresponds to a difference in mask sample distance of $2.08 \pm 0.31$ {\textmu}m. This tilt and distance are too large to be plausible. Another effect is likely at play in addition to some bending. \\

\begin{figure}[H]
    \centering
    \begin{subfigure}{0.49\linewidth}
    	\centering
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/SEM/SEM_Mask_cropped.pdf}
    	\caption{}
    	\label{fig:evaporation_SEM_mask}
    \end{subfigure}
    \begin{subfigure}{0.483\linewidth}
    	\centering
    	\includegraphics[width=0.96\linewidth]{img/Evaporation/SEM/ShowingClog.pdf}
    	\caption{}
    	\label{fig:evaporation_SEM_analysis_clog}
    \end{subfigure}

	\caption{(\subref{fig:evaporation_SEM_mask}) SEM image of the mask. The inset shows another image of the same mask. The image of the mask was very unstable due to heavy charging effects. (\subref{fig:evaporation_SEM_analysis_clog}) example of the clogging noticed on $4$ of the mask holes.}
    \label{fig:evaporation_SEM}
\end{figure}

To check whether the Mask was undamaged during the evaporation, the mask was examined via SEM. The resulting images can be seen in Figure \ref{fig:evaporation_SEM}. 

The image of the mask (Figure \ref{fig:evaporation_SEM_mask}) reveals no visible damage to the mask. The white areas in the image are charging artifacts, which were not stable across multiple images. The mask appears to be bending, but this is not a real deformation. Instead, it is an artifact caused by charging effects and the inherent fish-eye distortion that occurs in SEM images at high magnification.

\begin{figure}[H]
    \centering
	
	\begin{subfigure}{0.49\linewidth}
		\centering
    	\includegraphics[width=\linewidth]{img/Evaporation/SEM/SEM_CloggingOverlay.png}
    	\caption{}
		\label{fig:evaporation_SEM_analysis_clog_overlay}
	\end{subfigure}
	\begin{subfigure}{0.482\linewidth}
		\centering
		\includegraphics[width=0.95\linewidth]{img/Evaporation/SEM/SEM_Probe_01_cropped.png}
		\caption{}
		\label{fig:evaporation_SEM_sample}
	\end{subfigure}
	
	\caption{(\subref{fig:evaporation_SEM_analysis_clog_overlay}) tilt direction from \ref{fig:evaporation_tilts} overlayed over the SEM image of the mask after it was rotated to match the fields. (\subref{fig:evaporation_SEM_sample}) SEM images of field 2 on the sample.}
    \label{fig:evaporation_SEM_analysis}
\end{figure}

An example of this clogging in the SEM image is shown in Figure \ref{fig:evaporation_SEM_analysis_clog}
To further check if the clogging artifacts correspond to the directions of tilt shown in Figure \ref{fig:evaporation_tilts} the directions are overlayed in Figure \ref{fig:evaporation_SEM_analysis_clog_overlay}. The directions correspond to the direction shown in the SEM image, except for some outliers. It also points outwards. However, for many points, the clogging is not clearly visible in the image, making it difficult to draw a strong conclusion from the SEM image alone.\\

The evaporation of field $2$ shown in Figure \ref{fig:evaporation_SEM_sample} shows the elliptical tilt also visible in the AFM images. The elliptical part of the dot shows different value in the SEM image, which is an indicator, that the conductivity is different for that part of the dot. \\

This data suggests multiple possible hypothesis for this elliptical dot shape. It could be that the mask deformed during evaporation or is permanently deformed. Additionally, a displacement of the mask due to vibration could cause elliptical artifacts. 
Similar effects were previously observed, when turbo pumps were in operation during evaporation~\cite{Simon}


\section{Simulation} \label{sec:simulation}
\subsection{Overview and principle}
To gain a deeper understanding of the various hypotheses surrounding the tilted evaporation dots, a simple evaporation simulation program was developed. The simulation is based on ray tracing and was written using the open-source Godot game engine. The choice of Godot was motivated by its native support for ray collision detection, which enabled the rapid implementation of a ray tracing simulation. \\

Objects in the Godot game engine are moved, rotated and scaled with a $3 \times 4$ matrix called a "transform" matrix. This matrix performs rotations via their quaternion representation, which is a way to represent $3$-dimensional rotations as a $4$ component complex number. Modifying the transform matrix directly is possible, but would be very unintuitive and cumbersome, so the engine allows modification of the component's displacement and scale via $3$D vectors. The components of the displacement vector will be called x, y and z. The rotation can be modified via Euler angles. Internally the Euler angles are called x, y and z as well, based on the axis they rotate around. To avoid confusion the angles will be referred to as $\alpha$, $\beta$ and $\gamma$, where $\alpha$ rotates around the x-axis, $\beta$ around the y-axis and $\gamma$ around the z-axis. \\

%The simulation works as follows:
%At a time $0$ and at a distance $L$ from the sample a random point inside a circle is generated. This represents the aperture of the crucible. From it a ray is cast to a point behind the sample. The point behind the mask is chosen such that the ray casts in a cone with opening angle $\phi$. The ray is then checked for collision with a mask hole, which is represented by a cylinder with very small height. If collision with the mask hole is determined, the position at which the sample is hit is determined. Otherwise the ray is discarded. This position is then recorded in an array. It is structured like an image, spanning a user defined area around the middle of the sample with user specified resolution in pixels. For each element in the array, the amount of hits it has received is stored. This step is repeated many times in a single time step. \\

The model used in this simulation makes several assumptions:

\begin{itemize}
	\item Molecules travel in straight paths.
	\item Deposition occurs immediately upon impact with the sample with a sticking factor of $1$.
	\item No diffusion of particles occurs after deposition.
	\item Particles are assumed to be smaller than one pixel in the final image, effectively treating them as point-like objects.
\end{itemize} 

The simulation operates as follows:
\begin{enumerate}
\item At time $0$ and at a distance $L$ from the sample, a random point is generated within a circle, representing the aperture of the crucible.
\item From this point, a ray is cast towards a point behind the sample, such that the ray forms a cone with an opening angle $\phi$.
\item The ray is then checked for collision with a mask hole, which is modeled as a cylinder with a very small height.
\item If a collision with the mask hole is detected, the position where the sample is hit is determined. Otherwise, the ray is discarded.
\item The hit position is recorded in an array, which is structured like an image, covering a user-defined area around the center of the sample with a specified resolution in pixels.
\item For each element in the array, the number of hits it has received is stored.
\item This process is repeated multiple times within a single time step.
\end{enumerate}
This simulation allows for the modeling of the evaporation process and the resulting deposition pattern on the sample. \\

%\begin{figure}[H]
%    \centering
%	\includegraphics[width=0.6\linewidth]{img/Evaporation/Sim/GodotCoordinate.png}
%	\caption{Diagram depicting the coordinate system Godot uses. The order of rotation for the Euler angles is $\alpha$, $\beta$ and $\gamma$.}
%    \label{fig:evaporation_simulation_godotcoords}
%\end{figure

In order to simulate vibration effects, the cylinder collider of the mask can be moved and rotated in a periodic manner. The rotation, position and oscillation period are parameters given by the user. After each time step, the collider is moved to a new position, which is determined by interpolating between the start and end positions and rotations. The interpolation parameter is determined with the function $|\sin(\frac{t}{T})|$, where $T$ is the period of the oscillation in time steps and $t$ is the current time step. This allows for the simulation of $3$D vibrations. It does not take into account possible bending of the mask, since the colliders are stiff rigid bodies, but using rotation, bending can be locally approximated. \\

After a user specified time has passed, the amount of hits on each pixel is saved into a file and the image can then be displayed using a python script. For a more detailed look at the different parameters the simulation provides see the Appendix \ref{sec:appendix_raycast}.\\

\subsection{Results}

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.45\linewidth}
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Sim/Field3_right.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_first_compare_AFM}
	\end{subfigure}
	\begin{subfigure}{0.47\linewidth}
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Sim/Field3_right_sim_simple.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_first_compare_SIM}
	\end{subfigure}
	\caption{(a) a recorded AFM image, colors are for easier identification. (b) a simulated evaporation with parameters obtained from measurement in the AFM image. }
    \label{fig:evaporation_simulation_first_compare}
\end{figure}

An image of a simple simulation for an oscillating mask dot with parameters obtained from the AFM measurement can be seen in Figure \ref{fig:evaporation_simulation_first_compare_SIM}. The parameters for the amplitude of the oscillation were extracted from the AFM image shown in Figure \ref{fig:evaporation_simulation_first_compare_AFM}. Vibrations were assumed to be harmonic during the deposition and different sticking factors of \ce{Pb}-\ce{Si} and \ce{Pb}-\ce{Pb} were not considered. The oscillation was modeled with a displacement of $0.143$ {\textmu}m in x and $-0.358$ {\textmu}m in z direction and a tilt of $-41.12^\circ$ in $\alpha$, $10^\circ$ in $\beta$ and $31^\circ$ in $\gamma$. \\

A local deformation of approximately $45^\circ$ at a single hole site would result in significant strain on the mask. The observed tilt is likely the outcome of a combination of x-y displacement and bending of the mask. If the tilt were solely caused by vibrations, the mask would oscillate between two positions, resulting in an elliptical overlap of the two extreme positions.

However, if there is an additional displacement component in the z-direction, a smaller circle would form on top of the flat mask position. It is probable that the effect observed at the edge is an overlap of both the bending of the mask, which gives it an angle, and an additional contribution from displacement in both the x-y and z directions. A simulation of this phenomenon is shown in Figure \ref{fig:evaporation_simulation_overlap}.

\begin{figure}[H]
    \centering
	\includegraphics[width=0.5\linewidth]{img/Evaporation/Sim/OverlapCircles.pdf}
	\caption{Simulation showing the effect of only x-y vibration on the resulting evaporation. White circles show the extreme positions of the circular mask. }
    \label{fig:evaporation_simulation_overlap}
\end{figure}

The amplitude of displacement in the example in Figure \ref{fig:evaporation_simulation_first_compare_SIM} is $\approx 0.4$ {\textmu}m, this is in line with the peak to peak amplitude of an active turbomolecular pump given by $1$ {\textmu}m, obtained in the PhD thesis of Priyamvada Bhaskar~\cite{Bhaskar}. While the lateral dot shift seen in Fig \ref{fig:evaporation_simulation_first_compare} is refleceted in this specific simulation, it does not represent the exact shape of the deformation of the dot. For instance, the elliptical penumbra (highlighted in red in Figure \ref{fig:evaporation_simulation_first_compare_AFM}) appears rough in the AFM image, but its height is uniform. In contrast, the simulation shows a penumbra that gradually decreases in height. Furthermore, the lower edge of the elliptical shape visible in the AFM dot below the circle (\textcolor{tab_cyan}{cyan} in Figure \ref{fig:evaporation_simulation_first_compare_AFM}) is invisible in the AFM image, while it is very pronounced in the simulated image. The lower edge is sharp in the AFM image $61 \pm 9$ nm while it is smeared out in the simulated image. Parameters were obtained as described in Figure \ref{fig:penumbra_tilt_sigmas}.\\

The different roughness of circle and ellipse might suggest different possible reasons. First it could be a chronological effect where the circle is deposited first, and the ellipse is deposited second. Another possibility is that the vibration causes the displacement and bending of the mask in a pattern that is anharmonic, which causes the extreme points of the oscillation to be preferred. In order to investigate possible sources of this effect, the simulation was amended. \\

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.325\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/Sim/Field3_right_sim_simple.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_sharpness_stick_simple}
	\end{subfigure}
	\begin{subfigure}{0.325\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/Sim/Field3_right_sim_simple_initial.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_sharpness_stick_initial}
	\end{subfigure}
	\begin{subfigure}{0.325\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/Sim/Field3_right_sim_simple_power.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_sharpness_stick_power}
	\end{subfigure}
	\caption{(\subref{fig:evaporation_simulation_sharpness_stick_simple}) Comparison of the evaporation with harmonic oscillation. (\subref{fig:evaporation_simulation_sharpness_stick_initial}) initial phase with no elliptical oscillation and then drift to the elliptical shape. (\subref{fig:evaporation_simulation_sharpness_stick_power})an anharmonic oscillation with $\sin(\frac{t}{T} + \phi)^{20}$ . The parameters of the ellipse are the same as in Figure \ref{fig:evaporation_simulation_first_compare}.}
    \label{fig:evaporation_simulation_sharpness}
\end{figure}

Another possible reason is a chronology of events where the growth happens first on the outer circle and then on the elliptical shape, as previously discussed. This was modeled as an initial phase of user defined time $t_0$ where the mask was not oscillated. \\
The effect of this can be observed in Figure \ref{fig:evaporation_simulation_sharpness_stick_initial}.
Compared to the simpler model (Figure \ref{fig:evaporation_simulation_sharpness_stick_simple}), this result is more similar to the AFM measurement. \\

Another possibility is that the oscillation is not harmonic. Instead of using the standard oscillation function $\sin(\frac{t}{T} + \phi)$, where $t$ is the current time, $T$ is the oscillation period, and $\phi$ is the phase shift, the oscillation is parametrized as $\sin(\frac{t}{T} + \phi)^p$, where $p$ is the oscillation power. The resulting image is shown in Figure \ref{fig:evaporation_simulation_sharpness_stick_power}. \\

In the AFM image the surface of the outer is rougher than the surface of the inner circle. On average, the roughness is $1.7 \pm 0.4$ times higher. This could be due to particles first forming larger grains, which is common for PVD~\cite{grain_growth}. With larger layer height this effect typically becomes less visible. \\
It is unlikely that the vibrations causing the deformation and tilt are highly anharmonic. However, due to the growth of thin films occurring near grain boundaries, the actual growth of Pb on Si is concentrated at the extreme positions of the oscillation. \\

%The grain growth can be modeled in the simulation by penalizing deposition for pixels, where no material has been deposited previously. The probability to deposit on an empty surface is a user controlled parameter called "first\_layer\_depo\_prob". It specifies the probability with which a particle hitting the sample is deposited, when no material has previously been deposited on the relevant pixel. \\

The grain growth can be modeled in the simulation by penalizing deposition on pixels where no material has been deposited previously. This is achieved through a user-controlled parameter called "first\_layer\_depo\_prob", which represents the probability of a particle being deposited on an empty surface. Specifically, it defines the likelihood that a particle hitting the sample will be deposited on a pixel where no material has been deposited before.

\begin{figure}[H]
    \centering
	\begin{subfigure}{0.325\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/Sim/Field3_right_sim_simple_power.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_rejection_prev}
	\end{subfigure}
	\begin{subfigure}{0.325\linewidth}
    	\includegraphics[width=\linewidth]{img/Evaporation/Sim/Field3_right_sim_simple_rejection.pdf}
    	\caption{}
		\label{fig:evaporation_simulation_rejection_after}
	\end{subfigure}
	\begin{subfigure}{0.32\linewidth}
    	\includegraphics[width=0.95\linewidth]{img/Evaporation/Sim/Field3_right.png}
    	\caption{}
		\label{fig:evaporation_simulation_rejection_comparison}
	\end{subfigure}
	\caption{(\subref{fig:evaporation_simulation_rejection_prev}) simulated evaporation dots without rejection. (\subref{fig:evaporation_simulation_rejection_prev}) with (\subref{fig:evaporation_simulation_rejection_after}) $90$ \% probability to reject a deposition, when no previous deposition happened on the target pixel. (\subref{fig:evaporation_simulation_rejection_comparison}) the AFM image from which the parameters were obtained. The parameters of the ellipse are the same as in Figure \ref{fig:evaporation_simulation_first_compare}.}
    \label{fig:evaporation_simulation_rejection}
\end{figure}

The results of adding this penalty for initial deposition are shown in Figure \ref{fig:evaporation_simulation_rejection_after}. Compared to the previous simulation step in Figure \ref{fig:evaporation_simulation_rejection_prev} the roughness of the dot increased while the height has decreased. The outer tail of the ellipse disappears nearly completely. This version matches the deposition in the actual AFM image more closely, but crucially the decreased roughness of the elliptical part of the dot is not mirrored in the simulation. \\

Particles impinging on the surface will typically diffuse to a nearby large nucleation site. The simulation does not take this effect into account. This can be implemented by having pixels interact with neighboring ones.\\

Apart from this the simulation image matches the one given by the AFM measurement. This shows that vibrations bending the hole pattern of the mask in combination with a displacement are a plausible explanation for the abberant penumbra of the measured dots. \\
%
%\begin{figure}[H]
%    \centering
%	\includegraphics[width=0.9\linewidth]{img/Evaporation/Sim/Field3_right_sim_progression.pdf}
%	\caption{Image of final simulation with parameters given in Figure \ref{fig:evaporation_simulation_first_compare} and an anharmonic oscillation with a power of $20$. The image is very grainy due to a low amount of rays cast.}
%    \label{fig:evaporation_simulation_progression}
%\end{figure}

%The simulation software allows for taking in progress images at specified time intervals. An example for the previously discussed case can be seen in Figure \ref{fig:evaporation_simulation_progression}. With this, the chronology of events can be made visible more easily and visualizations could be created. \\

\subsection{Software improvements}
The simulation is accurate with respect to geometrical configuration of the Mask Aligner setup, but it assumes each particle hitting the surface either sticks to it or is rejected. This is a reasonable approximation as it follows the linear behavior of the Knudsen equation (Eq. \ref{eq:hertz_knudsen}), but it currently does not take into account grain size and diffusion of particles. \\

The current way of implementing the simulation using Godot allowed for very quick implementation and bug fixing, but lacks in performance. Each ray is cast sequentially on the CPU and significant overhead is caused by the game engine computing things necessary for games, but unnecessary for the purposes of this simple simulation. This causes the render time of each image to be in the minute to hour range for images of higher resolutions. \\

In order to improve performance, a dedicated ray tracing engine with support for threading could give significant performance improvements. Parallel deployment on the \textbf{G}raphics \textbf{P}rocessing \textbf{U}nit (GPU) using \textbf{A}pplication \textbf{P}rogramming \textbf{I}nterfaces (APIs) like for example CUDA or OpenCL could improve this further. This would most likely shorten generation times by several orders of magnitude. \\

Godot uses its own units for length measurement, which are stored as $32$-bit floating point numbers. For this reason the numbers had to be converted manually from real world units. This was time-consuming, and it can potentially cause floating point rounding issues. With a dedicated ray casting engine, real world units could be used and the accuracy of the simulation could be improved by using higher precision floating point numbers. \\

\subsection{Final Remark}

The results of the simulation show that a x-y-z vibrational displacement with a component of vibrational "bending" simulated as a strong tilt can explain the shape of the penumbra obtained in the AFM, and that its peak to peak amplitude is within the expected range for this system. It shows that the sharper penumbra edge ($\approx 60$ nm) is the penumbra that one would obtain had there been no vibrational influence on the experiment. This shows that the Mask Aligner is capable of creating sharp edges. \\