Merge development into master

.github/workflows/latex.yml

@@ -10,17 +10,12 @@ jobs:
   compile:
     runs-on: ubuntu-latest
     permissions: write-all
-    
+
     steps:
       - name: Fetch
         uses: actions/checkout@v2
         with:
           fetch-depth: 0
-      - name: Compile
-        uses: dante-ev/latex-action@latest
-        with:
-          root_file: thesis.tex
-          args: -lualatex -shell-escape -latexoption=-file-line-error -latexoption=-interaction=nonstopmode
       - name: Versioning
         uses: anothrNick/github-tag-action@master
         id: version
@@ -29,7 +24,15 @@ jobs:
           WITH_V: true
           RELEASE_BRANCHES: main
           DEFAULT_BUMP: ${{ github.ref == 'refs/heads/main' && 'minor' || 'patch' }}
-          DRY_RUN: true  # We want to manually create the release.
+          DRY_RUN: true # We want to manually create the release.
+      - run: |
+          echo "${{ steps.version.outputs.new_tag }}" > version
+          cat version
+      - name: Compile
+        uses: dante-ev/latex-action@latest
+        with:
+          root_file: thesis.tex
+          args: -lualatex
       - name: Release
         uses: "marvinpinto/action-automatic-releases@latest"
         with:

.gitignore

@@ -304,4 +304,7 @@ TSWLatexianTemp*
 .DS_Store
 
 # Ignore main output file.
-thesis.pdf
+thesis.pdf
+
+# Ignore any SAVE-ERROR files
+*-SAVE-ERROR

.latexmkrc

@@ -0,0 +1,5 @@
+$pdf_mode = 1; # tex -> pdf
+@default_files = ('thesis.tex');
+
+$lualatex = 'lualatex -shell-escape -latexoption=-file-line-error -f';
+$pdf_previewer = 'start zathura';

.zed/tasks.json

@@ -0,0 +1,11 @@
+[
+  {
+    "label": "Compile Thesis",
+    "command": "latexmk -lualatex -pvc",
+    "use_new_terminal": false,
+    "allow_concurrent_runs": false,
+    "reveal": "always",
+    "hide": "on_success",
+    "shell": "system"
+  }
+]

chapters/conventions.tex

@@ -0,0 +1,3 @@
+\chapter*{Conventions}
+\label{chap:conventions}
+In this thesis we regularly have to project 3D structures onto a 2D plane. To clarify the x-, y- and z-directions they have been color coded. This is done consistently throughout the thesis and matches with the colors assigned to the axes by COMSOL. The x-direction is colored in \textcolor{x_axis_color}{red}, the y-direction in \textcolor{y_axis_color}{green} and the z-direction in \textcolor{blue}{blue}. Additionally an attempt has been made to optimize figures for colorblind readers based on Paul Tol's color schemes\cite{paul_tol}.

chapters/method.tex

@@ -0,0 +1,58 @@
+\chapter{Method}
+\label{chap:method}
+
+\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.
+
+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.
+
+A summary of the dimensions of the sample is shown in \autoref{tab:sample-dimensions}. For a schematic illustration of the sample see TODO.
+\begin{table}
+    \centering
+    \begin{tabular}{lcc}
+        \toprule
+        \textbf{Parameter} & \textbf{Symbol} & \textbf{Value} \\
+        \midrule
+        Track thickness & $d$ & \qty{500}{\nm} \\
+        Inner track width & $w$ & \qty{50}{\um} \\
+        Inner track width & $w$ & \qty{100}{\um} \\
+        Inner loop radius & $r_1$ & \qty{85}{\um} \\
+        Outer loop radius & $r_2$ & \qty{170}{\um} \\
+        \ce{Si}/Glass hole diameter & & \qty{100}{\um} \\
+        \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 the radii at the center of the track.}
+    \label{tab:sample-dimensions}
+\end{table}
+
+\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).
+
+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.
+
+\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 models the \ce{Si} wafer, the \ce{Au} tracks and a air box surrounding them. The simulation is performed in 3D using a stationary dc-analysis.
+
+\subsection{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.
+
+\subsubsection{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}.
+
+\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}
+\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 $c_K = 0.31 K_n / \left(0.785 + 1.152 K_n + K_n^2 \right)$.
+
+\subsubsection{Inductive damping}
+
+
+\subsubsection{Eddy current damping}

chapters/results.tex

@@ -0,0 +1,36 @@
+\chapter{Results}
+\label{chap:results}
+The datasheet of our \ce{NdFeB} particles specifies that their residual magnetic field is \qtyrange{730}{760}{\milli\tesla}. Full saturation (\qty{>95}{\percent}) is achieved when a \qty{2}{\tesla} field is applied. Our method of magnetising the particle was limited to a \qty{1.4}{\tesla} field. As such, it is likely that we did not achieve the full saturation of the particle and that the residual magnetic field is lower than specified. Assuming a roughly linear relation between the two values, we estimate the residual magnetic field of our particles to be \qtyrange{510}{530}{\milli\tesla}. The density of the particles is given as \qty{7430}{\kilo\gram\per\cubic\meter}.
+
+These values, together with the simulation of the magnetic field distribution in the trap, allows us to estimate the required driving frequency and the resulting eigen frequency.
+
+\begin{figure}
+    \centering
+    \includegraphics{figures/magnetic_field_curvature.pdf}
+    \caption{the \textbf{top} figures from left to right show $z$-component of the magnetic field evaluated along a line parallel to the $x$-, $y$- and $z$-axis respectively through the origin. The gray zones indicate the boundaryies of the trap. The \textbf{bottom} figure shows the curvature of the magnetic field for these calculations evaluated in the extrema. The simulations were performed for $i_1=\qty{0.5}{\ampere}$ and $\xi$ between \numrange{-0.5}{-5}. The curves have been labelled with the corresponding value for $\xi$.}
+\end{figure}
+
+\section{Measurements at atmospheric pressure}
+\label{sec:measurements-at-atmospheric-pressure}
+At atmospheric pressure we determined the dependence of $\omega_{x,y}$ on $\Omega$ and $i_1$. In these measurements the ratio $\xi$ was kept constant. The spectra are obtained using the lock-in amplifier connected to the photodiode. Due to the low Q-factor at atmospheric pressure, it is not possible to tell the $x$- and $y$-modes apart. The results are shown in \autoref{fig:xy-mode-dependence-1bar}. The $z$-mode is not visible in these measurements. In addition to this the dependence of the rotational mode on $B_0$ is shown in \autoref{fig:rotational-mode-dependence-1bar}. We note three curves with a apparently linear dependence on $B_0$. The two additional curves are modulations of the resonance peak with the trapping frequency. A fourth curve can be seen in the bottom right. Its origin is unclear. A fit has not been made due to the sparsity of the data. $\omega_\alpha$ has not been observed.
+
+\begin{figure}
+    \centering
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics{figures/xy_mode_dependence_on_driving_frequency.pdf}
+        %\label{fig:xy-mode-dependence-on-driving-frequency-1bar}
+    \end{subfigure}
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics{figures/xy_mode_dependence_on_inner_current.pdf}
+        %\label{fig:xy-mode-dependence-on-inner-current-1bar}
+    \end{subfigure}
+    \caption{The dependence of $\omega_{x,y}$ on $\Omega$ (\textbf{left}) and $i_1$ (\textbf{right}) at atmospheric pressure. The dashed lines are a fit following the theory of $\omega_{x,y}(\Omega) \sim 1 / \Omega$ and $\omega_{x,y}(i_1) \sim i_1$. The corresponding prefactors are $2\pi \cdot \qty{377.33170211685874\pm9.030914540546724}{\kilo\hertz\per\hertz}$ and \qty{626.5282946695123\pm20.984120764687987}{\hertz\per\ampere}. The normal value for $\Omega = 2\pi \cdot \qty{2.5}{\kilo\hertz}$ and $i_1 = \qty{200}{\milli\ampere}$ when they are not part of the sweep, the ratio $\chi = 2$ is always kept constant.}
+    \label{fig:xy-mode-dependence-1bar}
+\end{figure}
+
+\begin{figure}
+    \centering
+    \includegraphics{figures/rotational_mode_dependence_on_B0.pdf}
+    \caption{The dependence of $\omega_{\gamma,\tilde\beta}$ on $B_0$ at atmospheric pressure.}
+    \label{fig:rotational-mode-dependence-1bar}
+\end{figure}

chapters/theory.tex

@@ -0,0 +1,71 @@
+\chapter{Theory}
+\label{chap:theory}
+
+\section{Magnetic levitation}
+\label{sec:magnetic_levitation}
+Obtaining and then maintaining stable magnetic levitation requires three fields: a static homogenous field to align the magnet ($\vec{B_0}$); an alternating field containing the saddle point ($\vec{B_1}$); and a gradient field to counter the gravitational offset ($\vec{B_2}$). This follows from simulations and our previous experience \cite{perdriat,mart,eli}. Combining this with the magnetic moment of the levitated particle $\vec{\mu}$ allows us to derive (up to second order) the magnetic potential. Averaging over the oscillation period gives allows us to derive the associated eigenfrequencies.
+
+\subsection{Fields}
+\label{subsec:fields}
+The homogeneous field is given in \autoref{eq:homogeneous_field}. It is oriented in the $z$-axis with magnitude $B_0$. Homogenous fields can (locally) be created using Helmholtz coils.
+\begin{equation}
+    \vec{B_0}(\vec{r}, t) = \vec{B_0} = B_0 \zhat
+    \label{eq:homogeneous_field}
+\end{equation}
+
+Rotating a saddle point sufficiently fast effectively creates a (local) minimum. The field is given in \autoref{eq:saddle_point}. $B_1''$ is the curvature of $\vec{B_1}$ and $\Omega / 2\pi$ the frequency of the oscillation. The field can be created using two loops of wire with a current in opposite directions in the same plane, a so called magnetic Paul Trap. These loops are the main focus of this thesis. Given the radius of the inner loop $r_1$ we can express the curvature as $B_1'' = -\frac{9}{16}\mu_0i_1/r_1^3$ where $i_1$ is the current through the loop\footnote{This uses the requirement that $i_1/i_2 = -r_1/r_2$ and that $i_1$ and $i_2$ oppose each other (clockwise and anti-clockwise).}.
+\begin{equation}
+    \vec{B_1}(\vec{r}, t) = \frac{B_1''}{2} \begin{pmatrix}
+        -xz \\
+        +yz \\
+        z^2 - \frac{1}{2}\left(x^2 + y^2\right)
+    \end{pmatrix} \cos(\Omega t)
+    \label{eq:saddle_point}
+\end{equation}
+
+Finally we have the gradient field, given in \autoref{eq:gradient_field}. The required gradient can be expressed as $B_2' = mg/\mu$ where $m$ is the mass of the levitated particle, $g$ the gravitational acceleration and $\mu$ the magnetic moment of the particle. The gradient serves to offset the effect of gravity. As such the gradient is oriented in the $z$-axis. The gradient field can be created by using a larger current in one of the Helmholtz coils.
+\begin{equation}
+    \vec{B_2}(\vec{r}, t) = B_2' \begin{pmatrix}
+        -x / 2 \\
+        -y / 2 \\
+        z
+    \end{pmatrix}
+    \label{eq:gradient_field}
+\end{equation}
+
+\subsection{Magnetic potential}
+\label{subsec:magnetic_moment}
+Using the $zyz$ convention for the Euler angles ($\alpha$, $\beta$, $\gamma$) we can express the magnetic moment as in \autoref{eq:magnetic_moment}. In this equation $\tilde\beta = \beta - \pi/2$.
+\begin{equation}
+    \vec{\mu} = -\mu \begin{pmatrix}
+        -\cos(\alpha)\sin(\tilde\beta)\cos(\gamma) - \sin(\alpha)\sin(\gamma) \\
+        \cos(\alpha)\sin(\gamma) - \sin(\alpha)\sin(\tilde\beta)\cos(\gamma) \\
+        -\cos(\tilde\beta)\cos(\gamma)
+    \end{pmatrix}
+    \label{eq:magnetic_moment}
+\end{equation}
+
+By taking the inner product of the magnetic moment and the fields we can derive the magnetic potential. This is given in \autoref{eq:magnetic_potential}. The potential is a function of the position of the particle $\vec{r}$ and the orientation of the magnetic moment $\vec{\mu}$.
+\begin{equation}
+    E_\text{mag}(\vec{r}, \vec{\mu}) = \mu B_0 \left(\frac{\gamma^2}{2} + \frac{\tilde\beta^2}{2}\right) - \frac{\mu B_1''}{2} \left(z^2 - \frac{1}{2}\left(x^2 + y^2\right)\right)\cos(\Omega t)
+    \label{eq:magnetic_potential}
+\end{equation}
+
+\subsection{Eigenfrequencies}
+\label{subsec:eigenfrequencies}
+Starting from \autoref{eq:magnetic_potential} and averaging over the oscillation period we can derive the eigenfrequencies. They are given in \autoref{eq:eigenfrequencies}. In these equations $a$ is the radius of the levitated particle (such that $V \sim a^3$) and $\rho_m$ is the density of the particle. The eigenfrequencies associated with the orientation of the magnet depend on $B_0$, which intuitively makes sense since the orientation of the magnet is determined by the homogenous field. The eigenfrequencies associated with the position of the magnet depend on $B_1''$, which also makes sense since the alternating field is what restricts the movement of the particle.
+\begin{equation}
+    \begin{gathered}
+        \omega_\gamma = \omega_{\tilde\beta} = \sqrt{\frac{5}{2}\frac{B_0B_\text{sat}}{\mu_0 \rho_m a^2}} \\
+        \omega_z = 2\omega_x = 2\omega_y = \frac{\Omega \abs{q_z}}{2\sqrt{2}} = \frac{1}{\sqrt{2}}\frac{B_1''B_\text{sat}}{\mu_0\rho_m\Omega}
+    \end{gathered}
+    \label{eq:eigenfrequencies}
+\end{equation}
+
+The time average is valid if $\abs{q_z}$, as defined in equation \ref{eq:q-factors}, is less than or equal to $0.4$.
+\begin{equation}
+    q_z = -2q_x = -2q_y = \frac{2}{\Omega^2}\frac{B_1''B_{\text{sat}}}{\mu_0\rho_m}
+    \label{eq:q-factors}
+\end{equation}
+
+TODO: $\omega_\alpha$?