Minor patches everywhere.

chapters/introduction/main.tex

@@ -1,8 +1,8 @@
 % !TEX root = ../../thesis.tex
 \chapter{Introduction}
-Josephson junctions have a wide variety of applications such as qubits\cite{placeNewMaterialPlatform2021,pechenezhskiySuperconductingQuasichargeQubit2020}; superconducting electronics through Josephson diodes\cite{zhangReconfigurableMagneticfieldfreeSuperconducting2023a,ciacciaGateTunableJosephson2023}; and microscopic imaging techniques\cite{clarkeSQUIDHandbook2004,rogSQUIDontipMagneticMicroscopy2022,pranceSensitivityDCSQUID2023}. The behaviour of a Josephson junction is governed by its current-phase relation (CPR). Probing the CPR can lead to new insights and applications. Par example by measuring the CPR it is possible to if the junction's behaviour is ballistic or diffusive\cite{muraniBallisticEdgeStates2017,endresCurrentPhaseRelation2023,kayyalhaHighlySkewedCurrent2020}; it has shown the existence of $0$-$\pi$ and $\varphi_0$ junctions\cite{frolovMeasurementCurrentPhaseRelation2004,muraniBallisticEdgeStates2017,strambiniJosephsonPhaseBattery2020,szombatiJosephsonPh0junctionNanowire2016}; as well as non-$2\pi$ periodic CPRs\cite{endresCurrentPhaseRelation2023}.
+Josephson junctions have a wide variety of applications such as qubits\cite{placeNewMaterialPlatform2021,pechenezhskiySuperconductingQuasichargeQubit2020}; superconducting electronics through Josephson diodes\cite{zhangReconfigurableMagneticfieldfreeSuperconducting2023a,ciacciaGateTunableJosephson2023}; and microscopic imaging techniques\cite{clarkeSQUIDHandbook2004,rogSQUIDontipMagneticMicroscopy2022,pranceSensitivityDCSQUID2023}. The behaviour of a Josephson junction is governed by its current-phase relation (CPR). Probing the CPR can lead to new insights and applications. Par example by measuring the CPR it is possible to if the junction's behaviour is ballistic or diffusive\cite{muraniBallisticEdgeStates2017,endresCurrentPhaseRelation2023,kayyalhaHighlySkewedCurrent2020} and it has shown the existence of $0$-$\pi$ and $\varphi_0$ junctions\cite{frolovMeasurementCurrentPhaseRelation2004,muraniBallisticEdgeStates2017,strambiniJosephsonPhaseBattery2020,szombatiJosephsonPh0junctionNanowire2016} as well as non-$2\pi$ periodic CPRs\cite{endresCurrentPhaseRelation2023}.
 
-In our group there is an additional interest in the CPR of homogenous \ce{Sr2RuO4} rings. Recent work by Lahabi \textit{et al.} provides evidence for the existence of chiral domain walls in these rings that act as Josephson junctions.\cite{lahabiSpintripletSupercurrentsOdd2018} As such, homogenous \ce{Sr2RuO4} rings show dc-SQUID like behaviour without the presence of constrictions, grain boundaries or an interface with a different material. More definitive proof for chiral domain walls could be found by measuring the Josephson energy.\cite{lahabiSpintripletSupercurrentsOdd2018,sigristRoleDomainWalls1999} The most elegant way to determine the Josephson energy is to measure the CPR.
+In our group there is an additional interest in the CPR of homogenous \ce{Sr2RuO4} rings. Recent work by Lahabi \textit{et al.} provides evidence for the existence of chiral domain walls in these rings that act as Josephson junctions.\cite{lahabiSpintripletSupercurrentsOdd2018} As such, homogenous \ce{Sr2RuO4} rings show dc-SQUID like behaviour without the presence of constrictions, grain boundaries or an interface with a different material. Definitive proof for chiral domain walls can be found by measuring the Josephson energy.\cite{lahabiSpintripletSupercurrentsOdd2018,sigristRoleDomainWalls1999} The most elegant way to determine the Josephson energy is by measuring the CPR.
 
 Two of the key benefits of our method is that it directly measures the full CPR and only needs a simple analysis. Details on the analysis can be found in Section~\ref{sec:analysis-method}. An alternative method uses a strongly asymmetric dc-SQUID where the junction with a much smaller critical current dominates the behaviour of the dc-SQUID.\cite{muraniBallisticEdgeStates2017,dellaroccaMeasurementCurrentPhaseRelation2007} A downside of this method is that it is much more difficult to produce the samples. Microwave measurements have been used to determine the skewness of the CPR but cannot probe the CPR directly.\cite{schmidtProbingCurrentphaseRelation2020}
 

chapters/method/main.tex

@@ -10,7 +10,7 @@ \chapter{Method}
 	\begin{circuitikz}
 		% Main loop with single Josephson Junction and an inductor for clarity.
 		\draw (0,0) to [short, *-, i=$I_t$] (2,0)
-		to [josephsonjunction, i=$I_s$, l_=$JJ$] (2, -2)
+		to [josephsonjunction, i=$I_s$] (2, -2)
 		to [short, -*, i=$I_t$] (0, -2);
 		\draw (2,0) to [short, i=$I_l$] (4, 0)
 		to [inductor, l_=$L_l$] (4, -2)
@@ -27,17 +27,17 @@ \chapter{Method}
 		\node[] at (6,-1) {$\Phi_s$};
 	\end{circuitikz}
 
-	\caption{Schematic depiction of the system. The left loop is inductively coupled to the dc-SQUID on the right. This is illustrated by $L_l$ and $L_s$. The junction itself has an inductance $L_{JJ}$. The current $I_t$ is controlled externally. The flux through the two loops is denoted by $\Phi_l$ and $\Phi_s$. The junction under study is part of the left loop. Please note that the four contacts used for the dc-SQUID readout are not shown.}
+	\caption{Schematic depiction of the system. The left loop is inductively coupled to the dc-SQUID on the right. This is illustrated by $L_l$ and $L_s$. The junction under study is part of the left loop and has an inductance $L_{JJ}$. The current $I_t$ is controlled externally. The flux through the two loops is denoted by $\Phi_l$ and $\Phi_s$. Please note that the four contacts used for the dc-SQUID readout are not shown.}
 	\label{fig:schematic-setup}
 \end{figure}
 
 \section{Analysis method}
 \label{sec:analysis-method}
-Our method requires little to no analysis, which immediately highlights one of the key benefits of this method. The measurements are performed by passing a current through the junction's loop $I_t$. Constrained by flux quantization and the CPR of the junction it will distribute the current between the loop and the junction:
+Our method requires little to no analysis, which highlights one of the key benefits of this method. The measurements are performed by passing a current $I_t$ through the junction's loop. Constrained by flux quantisation and the CPR of the junction it will distribute the current between the loop and the junction:
 \begin{equation}
 	I_t = I_s + I_l
 \end{equation}
-This can be seen by applying Ohm's law to the circuit in Figure~\ref{fig:schematic-setup}. By simultaneously measuring $V_s$ we are able to determine $\Phi_s$. Using the dc-SQUID's flux we can determine both $\gamma$ and $\Phi_l$. More details on this can be found in Section~\ref{sec:flux-phase-relation} and Section~\ref{sec:magnetic-coupling}.
+This can be seen by applying Ohm's law to the circuit in Figure~\ref{fig:schematic-setup}. By simultaneously measuring $V_s$ it is possible to determine $\Phi_s \propto \Phi_l \propto \gamma$. More details on this can be found in Section~\ref{sec:flux-phase-relation} and Section~\ref{sec:magnetic-coupling}.
 \begin{equation}
 	I_l = \frac{\Phi_s}{M} \qquad \Phi_l = \frac{L_l}{M}\Phi_s
 \end{equation}
@@ -116,7 +116,7 @@ \subsection{Magnetic coupling}
 Again the (mutual) inductance can be estimated numerically in SuperScreen.\cite{bishop-vanhornSuperScreenOpensourcePackage2022} The numerical simulation does not take into account the possibility of magnetic lensing.\cite{prigozhin3DSimulationSuperconducting2018} As such the mutual inductance might be larger in practice. This means that the dc-SQUID might react more sensitively to changes in the junction's loop.
 
 \section{Sample geometries}
-The diameter of the dc-SQUID is chosen such that the periodicity of the SQUID interference pattern is on the order of a few \unit{\milli\tesla}. This means the effective diameter should be around \qtyrange{1}{2}{\micro\meter}. Furthermore, the width of the loop together with the thickness of the superconductor determine the geometric factor $\tilde{j}$, they are chosen such that the figure of merit is sufficiently small. In practice this means that the width is around \qty{0.3}{\micro\meter} and the thickness around \qty{100}{\nano\meter}. Details on this can be found on a per sample basis later as well as details on the type of junction.
+The diameter of the dc-SQUID is chosen such that the periodicity of the SQUID interference pattern is on the order of a few \unit{\milli\tesla}. This means the effective diameter should be around \qtyrange{1}{2}{\micro\meter}. Furthermore, the width of the loop together with the thickness of the superconductor determine the geometric factor $\tilde{j}$, they are chosen such that the figure of merit is sufficiently small. In practice this means that the width is around \qty{0.3}{\micro\meter} and the thickness around \qty{100}{\nano\meter}. Details on this can be found on a per sample basis in Chapter~\ref{chapter:samples}.
 
 \section{Sample fabrication}
 \label{sec:method-sample-fabrication}

chapters/samples/CP1.2H/fabrication.tex

@@ -1,5 +1,5 @@
 % !TEX root = ../../../thesis.tex
-This sample uses lateral constriction junctions (ScS)\footnote{Also called a Dayem bridge, see~\cite{likharevSuperconductingWeakLinks1979}.} fabricated by FIB milling. The advantage of these junctions is that the behaviour of the depends strongly on $\xi$.\cite{likharevSuperconductingWeakLinks1979} As such, due to the temperature dependence of $\xi$ we can tune the junction behaviour. The size of the constriction should ideally be around $3\xi$.\cite{likharevSuperconductingWeakLinks1979}
+This sample uses lateral constriction junctions (ScS)\footnote{Also called a Dayem bridge, see~\cite{likharevSuperconductingWeakLinks1979}.} fabricated by FIB milling. The advantage of these junctions is that the behaviour of the depends strongly on $\xi$. As such, due to the temperature dependence of $\xi$ we can tune the junction behaviour. The size of the constriction should ideally be around $3\xi$.\cite{likharevSuperconductingWeakLinks1979}
 
 Our method benefits from a small $\lambda$, as can be seen in our figure of merit (Section~\ref{sec:figure-of-merit}). As such we used \ce{Nb} as our superconductor\footnote{Other candidates were \ce{NbTi} and \ce{MoGe}, they however have a much larger penetration depth.}. At \qty{0}{\kelvin} \ce{Nb} has $\lambda = \qty{47}{\nano\meter}$ and $\xi = \qty{38}{\nano\meter}$.\cite{maxfieldSuperconductingPenetrationDepth1965}
 

chapters/samples/CP1.2H/results.tex

@@ -1,5 +1,5 @@
 % !TEX root = ../../../thesis.tex
-The measurements were performed in a \qty{1.6}{\kelvin} cryostat equipped with a \qty{7}{\tesla} magnetic field. While cooling down, a 4-point measurement was used to determine the resistance of the dc-SQUID as a function of temperature. The RT-curve is shown in Figure~\ref{fig:CP1.2H-SQUID-RT}. We note that the sample becomes superconducting at \qty{8}{\kelvin} and has a quite sharp transition. The lower critical temperature compared to pure \ce{Nb} ($T_c=\qty{9.2}{\kelvin}$\cite{maxfieldSuperconductingPenetrationDepth1965}) is expected when using sputtering methods.
+The measurements were performed in a \qty{1.6}{\kelvin} cryostat equipped with a \qty{7}{\tesla} magnetic field. While cooling down, a 4-point measurement was used to determine the resistance of the dc-SQUID as a function of temperature. The RT-curve is shown in Figure~\ref{fig:CP1.2H-SQUID-RT}. The sample becomes superconducting at \qty{8}{\kelvin} and has a quite sharp transition. The lower critical temperature compared to pure \ce{Nb} ($T_c=\qty{9.2}{\kelvin}$\cite{maxfieldSuperconductingPenetrationDepth1965}) is expected when using sputtering methods.
 
 \begin{figure}[ht!]
 	\centering
@@ -19,7 +19,7 @@
 	\label{fig:CP1.2H-SQUID-critical-current-temperature-dependence}
 \end{figure}
 
-Figure~\ref{fig:CP1.2H-SQUID-SQI} shows the interference pattern of the dc-SQUID at \qty{7.6}{\kelvin}. The periodicity of the interference pattern is between \qtyrange{3}{3.5}{\milli\tesla}. This means the effective area of our dc-SQUID should be between \qtyrange{0.6}{0.7}{\square\micro\meter} which corresponds to a diameter between \qtyrange{0.87}{0.94}{\micro\meter}. However, we know from the SEM images (Figure~\ref{fig:CP1.2H-SEM-images}) that the diameter of our effective area must be between \qtyrange{1.2}{1.6}{\micro\meter} (a periodicity between \qtyrange{1}{1.8}{\milli\tesla}). This larger periodicity suggests that less flux is present in the dc-SQUID.
+Figure~\ref{fig:CP1.2H-SQUID-SQI} shows the interference pattern of the dc-SQUID at \qty{7.6}{\kelvin}. The periodicity of the interference pattern is between \qtyrange{3}{3.5}{\milli\tesla}. This means the effective area of our dc-SQUID should be between \qtyrange{0.6}{0.7}{\square\micro\meter} which corresponds to a diameter between \qtyrange{0.87}{0.94}{\micro\meter}. However, the SEM images (Figure~\ref{fig:CP1.2H-SEM-images}) show that the diameter of our effective area must be between \qtyrange{1.2}{1.6}{\micro\meter} (a periodicity between \qtyrange{1}{1.8}{\milli\tesla}). This larger periodicity suggests that less flux is present in the dc-SQUID.
 
 \begin{figure}[ht!]
 	\centering
@@ -28,8 +28,8 @@
 	\label{fig:CP1.2H-SQUID-SQI}
 \end{figure}
 
-Flux lensing\cite{prigozhin3DSimulationSuperconducting2018} does not provide an explanation. It would cause a higher flux in the dc-SQUID by focussing the magnetic field and thus a smaller periodicity. We furthermore trust the scale in the SEM images to be correct. In our view this leaves an incorrect field readout as explanation. During our use of the cryostat we later noticed that one of the two power supplies of the magnet was turned off. The current supplies operate in parallel, as such only half the current could be delivered effectively. If the magnet does not independently measure how much current is delivered then this would explain the factor two difference. As such we believe that this caused the incorrect field (readout). Due to time constraints however we have not been able to definitively test this hypothesis.
+Flux lensing\cite{prigozhin3DSimulationSuperconducting2018} does not provide an explanation. It would cause a higher flux in the dc-SQUID by focussing the magnetic field and thus a smaller periodicity. Furthermore, we trust the scale in the SEM images to be correct. In our view this leaves an field (readout) anomaly as explanation. During our use of the cryostat we later noticed that one of the two power supplies of the magnet was turned off. The current supplies operate in parallel, as such only half the current could be delivered effectively. If the magnet does not independently measure how much current is delivered then this would explain the factor two difference. As such we believe that this caused the incorrect field (readout). This hypothesis is further supported by the fact that the magnet was not able to produce more than \qty{4}{\tesla} fields. Due to time constraints however we have not been able to definitively test this hypothesis.
 
-Furthermore, the highest sensitivity of the dc-SQUID (in the linear regime) is \qtyrange{18}{21}{\micro\volt\per\milli\tesla} (or \qtyrange{36}{42}{\micro\volt\per\milli\tesla} if the field readout was indeed off by a factor two). This is not very good compared to dc-SQUIDs produced in our group earlier. Par example \cite{rogSQUIDontipMagneticMicroscopy2022} reported a dc-SQUID with similar geometries\footnote{The junctions were however SNS junctions and not constriction junctions as a layer of \ce{Ag} was present.} with a sensitivity around \qty{105}{\micro\volt\per\milli\tesla}. Improving the sensitivity for our current device further would be difficult because we are already near the critical current where we go through the bulk transition. Changing the temperature would be an option, however since taking a SQUID interference pattern takes a long time it was not viable due to the limited measurement time. Measurements at a constant bias current were attempted but did not work due to software limitations.
+Furthermore, the highest sensitivity of the dc-SQUID (in the linear regime) is \qtyrange{18}{21}{\micro\volt\per\milli\tesla} (or \qtyrange{36}{42}{\micro\volt\per\milli\tesla} if the field (readout) was indeed off by a factor two). Compared to dc-SQUIDs produced in our group earlier this is not good. Par example \cite{rogSQUIDontipMagneticMicroscopy2022} reported a dc-SQUID with similar geometries\footnote{The junctions were however SNS junctions and not constriction junctions as a layer of \ce{Ag} was present.} and a sensitivity around \qty{105}{\micro\volt\per\milli\tesla}. Improving the sensitivity of our current device further would be difficult because we are already near the critical current where we go through the bulk transition. Changing the temperature would be an option, however since taking a SQUID interference pattern takes a long time it was not viable due to the limited measurement time. Measurements at a constant bias current were attempted but did not work due to software limitations.
 
-Finally, an attempt was made to measure the CPR at \qty{7.6}{\kelvin}\footnote{Due to the poor dc-SQUID performance we did not expect this to be extremely fruitful. However it is a quick measurement and would not hurt.}. To do so we biased the dc-SQUID at \qty{200}{\micro\ampere} and sweeped a current through the junction's loop from \qtyrange{-250}{250}{\micro\ampere} and simultaneously measured the voltage over the dc-SQUID. The IV-curve we measured looked suspiciously much like the IV-curve of the dc-SQUID at \qty{7.6}{\kelvin}. This does not make sense and the measurement probably failed for the following two reasons. Firstly, the bias current through the dc-SQUID was too high, pushing it into its normal regime. This can be seen in Figure~\ref{fig:CP1.2H-SQUID-critical-current-temperature-dependence}. Secondly, there most likely was a short between some of the connections. The first issue could have been fixed easily, but was only noted too late. The second issue was not investigated further due to time constraints. Even if we had done so we likely would not have been able to fix it in a timely manner. More people in our group reported issues with this specific batch of \ce{Si} wafers. As such it appears to be a likely culprit.
+Finally, an attempt was made to measure the CPR at \qty{7.6}{\kelvin}\footnote{Due to the poor dc-SQUID performance we did not expect this to be extremely fruitful. However it is a quick measurement and would not hurt.}. To do so we biased the dc-SQUID at \qty{200}{\micro\ampere} and sweeped a current through the junction's loop from \qtyrange{-250}{250}{\micro\ampere} and simultaneously measured the voltage over the dc-SQUID. The measured IV-curve looked very similar to the IV-curve of the dc-SQUID at \qty{7.6}{\kelvin}. This does not make sense and the measurement probably failed for the following two reasons. Firstly, the bias current through the dc-SQUID was too high, pushing it into its normal regime. This can be seen in Figure~\ref{fig:CP1.2H-SQUID-critical-current-temperature-dependence}. Secondly, there most likely was a short between some of the connections. The first issue could have been fixed easily, but was only noted too late. The second issue was not investigated further due to time constraints. Even if we had done so we likely would not have been able to fix it in a timely manner. More people in our group reported issues with this specific batch of \ce{Si} wafers. As such it appears to be a likely culprit.