Some major updates to the methods section.

Author: jvdoorn

chapters/conventions.tex

@@ -1,15 +1,18 @@
 \chapter{Method}
 \label{chap:method}
+The design of the setup roughly consists of two parts: the nanofabrication of the chip and the experimental setup. We first present a schematic overview of the on-chip design and provide a description of the nanofabrication process. We then describe the experimental setup that includes the external Helmholtz coils and the optical readout.
 
-\section{Sample fabrication}
-\label{sec:sample-fabrication}
-A \qty{9}{\mm} by \qty{5}{\mm} undoped \ce{Si} wafer of \qty{500}{\um} thick is used as a substrate. The substrate is spincoated at \qty{2000}{\rpm} with positive resist AR-P 662.06 and baked at \qty{150}{\celsius} for \qty{3}{\min}. This step is then repeated in order to coat 2 layers in total resulting in a total thickness of \qty{1}{\um}. A lithography step is performed using the Raith 100 EBPG exposing the resist to \qty{400}{\micro\coulomb\per\square\cm}. The resist is developed using a 1:3 mixture of MIBK and isopropanol for \qty{45}{\s}, the development is stopped using isopropanol. The Z-407 sputtering machine deposits a \qty{5}{\nm} \ce{MoGe} sticking layer followed by a \qty{500}{\nm} \ce{Au} layer. The lift-off is performed in acetone.
+\section{On-chip design}
+\label{sec:on-chip-design}
+The design of the chip is based on the proposed on-chip design by \textcite{perdriat}. The design consists of two coplanar loops. In the theory chapter we assumed these loops to be perfectly symmetric and infinitely thin. In reality this is ofcourse not the case and the design needs to be optimized. An important consideration is how the current is distributed in the finite sized tracks and how this affects the magnetic potential. A schematic of the loop design and the distribution of the currents (for $\xi=2$) is shown in \autoref{fig:schematic-tracks}. A finite element simulation in COMSOL is used to model the gradient of the magnetic field. The simulation is done in 3D using a stationary dc-analysis. The curvatures will be presented in \autoref{chap:results}. The thickness of the tracks is \qty{500}{\nano\meter}. Refer to \autoref{tab:sample-dimensions} for a summary of the dimensions of the sample.
 
-Inside of the inner loop of the coil, we mill a hole of roughly \qty{15}{\um} deep with a diameter of \qty{100}{\um} in the \ce{Si} substrate using a \ce{Ga+} focussed ion beam (Aquilos 2 Cryo-FIB). A similar hole is also made in a microscope coverslip. Using a micromanipulator a \ce{NdFeB} particle of \qty{12}{\um} is placed inside the \ce{Si} hole\footnote{The easiest way to do so is by sticking the particle to the bottom of the needle and then scraping the particle off on the sides of the \ce{Si} hole.} and the coverslip is placed on top of the \ce{Si} substrate. The holes are carefully aligned by moving the coverslip using a micromanipulator. The coverslip is then glued to the \ce{Si} substrate using an epoxy.
-
-The particle is magnetised by putting the whole sample in a magnetic field of approximately \qty{1.3}{\tesla} for several minutes at room temperature and pressure.
+\begin{SCfigure}
+    \centering
+    \includegraphics{figures/sample/schematic_tracks.pdf}
+    \caption{The design of the on-chip magnetic Paul trap. The dashed lines serve as a guide to the eye. The colormap indicates the current density through the tracks for a current ratio of $\xi = 2$. It can be seen that a majority of the current is concentrated in the inside bends. The extremely high and low currents in the sharp bends are an artefact of the simulation and are not physical. As a reference for the dimensions of the design refer to \autoref{tab:sample-dimensions}.}
+    \label{fig:schematic-tracks}
+\end{SCfigure}
 
-A summary of the dimensions of the sample is shown in \autoref{tab:sample-dimensions}. For a schematic illustration of the sample see \autoref{fig:sample-dimensions}.
 \begin{SCtable}
     \centering
     \begin{tabular}{lcc}
@@ -25,71 +28,40 @@ \section{Sample fabrication}
         \ce{Si}/Glass hole depth & & \qty{15}{\um} \\
         \bottomrule
     \end{tabular}
-    \caption{Dimensions of the sample. The radii of the inner and outer loops are their inner radius. Refer to \autoref{fig:sample-dimensions} for further clarification.}
+    \caption{Dimensions of the sample. The radii of the inner and outer loops are their inner radius. Refer to \autoref{fig:schematic-tracks} for further clarification.}
     \label{tab:sample-dimensions}
 \end{SCtable}
 
+\subsection{Nanofabrication}
+\label{subsec:nanofabrication}
+A \qty{9}{\mm} by \qty{5}{\mm} undoped \ce{Si} wafer of \qty{500}{\um} thick is used as a substrate. The substrate is spincoated at \qty{2000}{\rpm} with positive resist AR-P 662.06 and baked at \qty{150}{\celsius} for \qty{3}{\min}. This step is then repeated in order to coat 2 layers in total resulting in a total thickness of \qty{1}{\um}. A lithography step is performed using the Raith 100 EBPG exposing the resist to \qty{400}{\micro\coulomb\per\square\cm}. The resist is developed using a 1:3 mixture of MIBK and isopropanol for \qty{45}{\s}, the development is stopped using isopropanol. The Z-407 sputtering machine deposits a \qty{5}{\nm} \ce{MoSi} sticking layer followed by a \qty{500}{\nm} \ce{Au} layer. The lift-off is performed in acetone and anisole.
+
+Inside of the inner loop of the coil, we mill a hole of roughly \qty{15}{\um} deep with a diameter of \qty{100}{\um} in the \ce{Si} substrate using a \ce{Ga+} focussed ion beam (Aquilos 2 Cryo-FIB). A similar hole is also made in a microscope coverslip. To aid this process a very thin (\qty{5}{\nano\meter}) \ce{Pt} layer is deposited on the glass. Using a micromanipulator a \ce{NdFeB} particle of \qty{12}{\um} is placed inside the \ce{Si} hole. The easiest way to do so is by sticking the particle to the bottom of the needle and then scraping the particle off on the sides of the \ce{Si} hole. The coverslip is placed on top of the \ce{Si} substrate. The holes are carefully aligned by moving the coverslip using a micromanipulator. The coverslip is then glued to the \ce{Si} substrate using an epoxy. The coverslip creates a enclosed environment for the particle to move in. This prevents the particle from escaping the trap. The particle is magnetised by putting the whole sample in a magnetic field of approximately \qty{1.3}{\tesla} for several minutes at room temperature and pressure. \autoref{fig:optical-microscope-image-sample} shows an optical microscope image of the sample.
+
 \begin{figure}
     \centering
-    \begin{subfigure}[c]{.5\textwidth}
-        \centering
-        \includegraphics{figures/sample/trap_optical_microscope.pdf}
-    \end{subfigure}%
-    \begin{subfigure}[c]{.5\textwidth}
-        \centering
-        \includegraphics{figures/sample/trap_geometry.pdf}
-    \end{subfigure}%
-    \caption{An optical microscope image of the sample (\textbf{left}). A couple features can be seen in the image 1) in the bottom left is a discoloration due to the epoxy 2) the dark streak in the bottom right is the edge of the cover glass 3) the dark discoloration is the \ce{Pt} layer of the glass. In the center of the trap you can clearly see the \ce{NdFeB} particle. Furthermore, you can see that their is a small misalignment between the hole in the \ce{Si} and hole in the cover glass. On the \textbf{right} you see a schematic of the sample. A definition of the symbols and their numerical value can be found in \autoref{tab:sample-dimensions}.}
-    \label{fig:sample-dimensions}
+    \includegraphics{figures/sample/trap_optical_microscope.pdf}
+    \caption{An optical microscope image of the sample. A couple features can be seen in the image 1) in the bottom left is a discoloration due to the epoxy 2) the dark streak in the bottom right is the edge of the cover glass 3) the dark discoloration is the \ce{Pt} layer of the glass. In the center of the trap you can clearly see the \ce{NdFeB} particle. Furthermore, you can see that their is a small misalignment between the hole in the \ce{Si} and hole in the cover glass.}
+    \label{fig:optical-microscope-image-sample}
 \end{figure}
 
-\begin{SCfigure}
-    \centering
-    \includegraphics{figures/data/current_through_tracks.pdf}
-    \caption{The distribution of current through the tracks. Blue (red) areas indicate a low (high) current density. The simulation was done in COMSOL using a stationary dc-analysis. The current ratio is $\xi = 2$.}
-    \label{fig:current-distribution}
-\end{SCfigure}
-
-An important design consideration is the distribution of current through the tracks. Simulations in COMSOL show that the current is concentrated in the inside bends. In our case this is beneficial and allows us to get close to the $\xi = 2$ ratio. The current distribution is shown in \autoref{fig:current-distribution}. The extremely high and low currents in the sharp bends are an artefact of the simulation and are not physical.
-
 \section{Experimental setup}
-The sample is mounted on a printed circuit board (PCB) with a thick (\qty{1}{\mm}) copper baseplate. The top of the sample is exactly flush with the top of the PCB. Additionally the pads for the wirebonds on the sample and the PCB are aligned and directly next to each other. This allows for very short wirebonds to be used. This significantly reduces the resistance of the wirebonds and thus the maximum current that can be applied to the sample. The wirebonds are made of \qty{25}{\um} thick \ce{Pt}.
-
-The PCB is then placed inbetween two Helmholtz coils with a diameter of \qty{30}{\mm} and 825 windings. They Helmholtz coils provide the uniform magnetic field to align the sample and the gradient magnetic field to control the vertical position of the particle. A dc current on the order of \qty{1}{\ampere} is sent through the coils by two power supplies (Tenma 72-2540).
+The sample is mounted on a printed circuit board (PCB) with a thick (\qty{1}{\mm}) copper baseplate. The top of the sample is exactly flush with the top of the PCB. Additionally the pads for the wirebonds on the sample and the PCB are aligned and directly next to each other. This allows for very short wirebonds to which significantly reduces the resistance and thus the maximum current that can be applied to the sample. The wirebonds are made of \qty{25}{\um} thick \ce{Al}.
 
-A microscope objective with a $10\times$ magnification and large working distance is used to provide a visual image of the sample. Additionally a laser (\qty{635}{\nm}, \qty{1}{\milli\watt}) is coupled in. The reflection of the laser is then imaged on a photodiode (ThorLabs PDA36A2). The photodiode is connected to a lock-in amplifier (Zurich Instruments MFLI) allowing for the detection of the motion of the particle. In addition to the photodiode, we can use the camera feed to analyse the motion of the particle. This is done using Python and OpenCV using the CSRT tracker. This tracker is quite robust at the cost of being computationally expensive. As such it cannot be used in real-time.
+The PCB is then placed inbetween two Helmholtz coils with a diameter of \qty{30}{\mm} and 825 windings. These are the same coils as used in our previous work\cite{eli,mart}. The Helmholtz coils provide the uniform magnetic field to align the sample and the gradient magnetic field to control the vertical position of the particle. A dc current on the order of \qty{0.5}{\ampere} is sent through the coils by two power supplies (Tenma 72-2540).
 
-\section{Simulations}
-There are two assumptions in the theory: infinitely thin tracks; and perfectly symmetric loops. Because of this the ideal current ratio will be different in practice. A simulation in COMSOL is used to guide our choice of the current ratio. The simulation is performed in 3D using a stationary dc-analysis.
+A microscope objective with a $10\times$ magnification and large working distance is used to provide a visual image of the sample. Additionally a laser (\qty{635}{\nm}, \qty{1}{\milli\watt}) is coupled in. The reflection of the laser is then imaged on a photodiode (ThorLabs PDA36A2). The photodiode is connected to a lock-in amplifier (Zurich Instruments MFLI) allowing for the detection of the motion of the particle. In addition to the photodiode, we can use the camera feed to analyse the motion of the particle.
 
-\subsection{Q-factor}
-\label{subsec:q-factor}
-In addition to this we estimate our effective Q-factor. A finite Q-factor is caused by damping. Below we discuss several sources of damping.
+TODO: schematic of setup
 
-\subsubsection{Gas damping}
-\label{subsubsec:gas-damping}
-Gas damping on levitated micrometer sized particles has been studied by \citeauthor{millen}. For pressures above \qty{1E-6}{\milli\bar} and $K_n \ll 1$ stachastic forces dominate the damping rate. The damping rate in this case is given by \autoref{eqn:gas-damping-rate}.
+\subsection{Laser analysis}
+TODO
 
+\subsection{Camera analysis}
+To analyse the camera feed we use a flat-field correction without accounting for the dark frame image. The flat-field is obtained by taking 20 images of the sample when the particle is not trapped and rapdily moving around. The flat-field is then calculated by taking the average of these images. The following transormation is then applied to the frames of the camera feed:
 \begin{equation}
-    \frac{\Gamma_{\text {gas}}}{2 \pi}=3 \mu_v \frac{a}{m} \frac{0.619}{0.619+K_n} \left( 1+c_K \right)
-    \label{eqn:gas-damping-rate}
+    C = \frac{m \cdot R}{F}
+    \tag{flat-field correction}
+    \label{eq:flat-field-correction}
 \end{equation}
-
-$K_n$ is the Knudsen number defined as $K_n = \bar{l}/a$, $\bar{l}$ the mean free path of air molecules, $\mu_v$ the gas viscosity, $a$ the diameter of the particle, $m$ the mass of the particle and the constant $c_K = 0.31 K_n / \left(0.785 + 1.152 K_n + K_n^2 \right)$. A calculation for the gas damping is shown in \autoref{fig:gas-damping}.
-
-\begin{SCfigure}
-    \centering
-    \includegraphics{figures/data/gas_damping.pdf}
-    \caption{The gas damping predicted for our particle. The damping rate is calculated using \autoref{eqn:gas-damping-rate}. The calculations have been performed for $T=\qty{300}{\kelvin}$.}
-    \label{fig:gas-damping}
-\end{SCfigure}
-
-\subsubsection{Inductive damping}
-\label{subsubsec:inductive-damping}
-The enclosed flux in the loops will change as the particle moves. A changing flux induces a current in the loops which is a dissipative process. Using COMSOL we make an estimate of the dissipation by calculating the induced current in the loops.
-
-TODO: we still need to do this simulation, but I think it will be very similar to the Eddy current damping? Need to check what COMSOL does exactly.
-
-\subsubsection{Eddy current damping}
-\label{subsubsec:eddy-current-damping}
-Eddy currents are small `whirlpools' of current that are induced by a changing magnetic field. Due to the resistance of the conductor this will lead to dissipation of energy. We consider the case that the moving particle creates a changinging magnetic field which induces the Eddy currents in either the tracks or the residual \ce{Ga} around the trap due to the FIB milling. To estimate the dissipation we perform a simulation in COMSOL where we use the estimated maximum velocity of the particle to include a Lorentz transformation in the magnetic field. The simulation is done in 3D using a stationary dc-analysis.
+where $C$ is the corrected image, $m$ is the average value of the flat-field, $R$ is the raw image and $F$ is the flat-field. The camera feed is processed in grayscale. Due to the flat field correction we get a very clear indication of the position of the particle. This allows us to fit a 2D Gaussian to the particle and determine its position based on the center of the Gaussian.

chapters/method.tex

@@ -116,7 +116,7 @@ \section{Q-factor dependence on pressure}
 \begin{figure}
     \centering
     \includegraphics{figures/data/q_factor_pressure_dependence.pdf}
-    \caption{The pressure dependence of the Q-factor for the $x$-, $y$- and $z$-mode. The dashed lines are a prediction based on the theory from \autoref{subsec:q-factor}. The experimental Q-factors are determined using the ringdown method. The \zmode has not been observed experimentally. The \xmode and \ymode were driven sinusoidally at their respective resonance frequencies.}
+    \caption{The pressure dependence of the Q-factor for the $x$-, $y$- and $z$-mode. The dashed lines are a prediction based on the theory from \autoref{sec:damping}. The experimental Q-factors are determined using the ringdown method. The \zmode has not been observed experimentally. The \xmode and \ymode were driven sinusoidally at their respective resonance frequencies.}
     \label{fig:q-factor-pressure-dependence}
 \end{figure}