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Minor updates to results.
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jvdoorn committed Jun 26, 2023
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2 changes: 1 addition & 1 deletion chapters/samples/CP2.6B/results.tex
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In an attempt to fix the `noise' caused by the feedback loop, we turned off the magnet. Unfortunately, it never turned back on as the magnet controller was broken. A downside of this, is that we were also no longer able to read out the value of the field. We decided to perform a CPR measurement again by using a constant bias of \qty{380}{\micro\ampere} for the dc-SQUID, sweeping a current through the junction loop and then measuring the voltage over the dc-SQUID. It should be mentioned that during these measurements there appeared to be a lot of drift causing measurement to differ widely. We suspect that an external magnetic field was responsible for this.

Figure~\ref{fig:CP2.6B_SQUID_voltage_over_total_current} shows the raw data from one of the CPR measurements. We note a clear periodicity in the signal of around \qty{200}{\micro\ampere} and a somewhat linear trend on top of that. We present this specific measurement because it has no jumps in the dc-SQUID voltage, has a sufficiently small sample spacing, covers a wide enough range to see several periods and measured both positive and negative currents. Different sample spacings did not affect measured periodicity. For transparency we have included a few other data sets in Appendix~\ref{app:CP2.6B-data}.
Figure~\ref{fig:CP2.6B_SQUID_voltage_over_total_current} shows the raw data from one of the CPR measurements. We note a clear periodicity in the signal of around \qty{200}{\micro\ampere} and a somewhat linear trend on top of that. We present this specific measurement because it has no jumps in the dc-SQUID voltage, has a sufficiently small sample spacing, covers a wide enough range to see several periods and measured both positive and negative currents. Different sample spacings did not affect measured periodicity.

\begin{figure}[ht!]
\centering
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8 changes: 6 additions & 2 deletions chapters/samples/CP2.6B_revisited/results.tex
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Previously the CPR was measured at \qty{3}{\kelvin}. An increase in temperature degrades the sensitivity of our dc-SQUID and the critical current of the junction loop. We note that our measurement was approximately \qty{200}{\micro\ampere} periodic. In order to seem multiple periods it is preferable to have a critical junction around \qty{400}{\micro\ampere}. This allows us to see around 4 to 5 periods. Through trial and error we found the highest usable temperature to be \qty{3.4}{\kelvin}.

Similarly to last time we then measured SQIs for several bias currents. A bias current of \qtyrange{300}{350}{\micro\ampere} was used. They are shown in Figure~\ref{fig:CP2.6B_revisited_SQIs}. Whilst the SQIs are far from perfect, they are usable. The curved background is due to the magnetic field dependence of the critical current of the bulk.Near zero field the periodicity as well as the horizontal offset is reproducible. The amplitude of the oscillations is stable as well. Since the junction loop will only create a small flux this should be sufficient. Around zero field we can approximate the oscillations as sinusoidal. After fitting it gives a sensitivity of \qtylist{491.37;189.73;326.49;410.49;419.73}{\micro\volt\per\fluxquantum} for \qtylist{2.8;3.0;3.2;3.4;3.6}{\kelvin} respectively. These are comparable to previous results. Please note that the first sensitivity is significantly higher due to a higher bias current.
Similarly to last time we then measured SQIs for several bias currents. A bias current of \qtyrange{300}{350}{\micro\ampere} was used. They are shown in Figure~\ref{fig:CP2.6B_revisited_SQIs}. Whilst the SQIs are far from perfect, they are usable. The curved background is due to the magnetic field dependence of the critical current of the bulk. Near zero field the periodicity as well as the horizontal offset is reproducible. The amplitude of the oscillations is stable as well. Since the junction loop will only create a small flux this should be sufficient. Around zero field we can approximate the oscillations as sinusoidal. After fitting it gives a sensitivity of \qtylist{491.37;189.73;326.49;410.49;419.73}{\micro\volt\per\fluxquantum} for \qtylist{2.8;3.0;3.2;3.4;3.6}{\kelvin} respectively. These are comparable to previous results. Please note that the first sensitivity is significantly higher due to a higher bias current.

\begin{figure}[ht!]
\centering
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\label{fig:CP2.6B_revisited_SQIs}
\end{figure}

After the calibrations the current-phase relations were measured. They are shown in Figure~\ref{fig:CP2.6B_revisited_CPRs}. The amplitude of the oscillations appears to decrease as the temperature increases. Except for the measurement at \qty{3.6}{\kelvin} where it increases again. We suspect that this is due to a human error. A likely explanation is that a different bias current was used for the dc-SQUID compared to the bias current used for the calibration. The change in amplitude corresponds to a change in critical current. Ignoring the measurement at \qty{3.6}{\kelvin}, the critical current goes from \qtyrange{90}{45}{\micro\ampere} between \qtyrange{2.8}{3.4}{\kelvin}. Besides the change in critical current we also note that the asymmetry changes. Both these changes are in line with theoretical models \cite{likharevSuperconductingWeakLinks1979}.
After the calibrations the current-phase relations were measured. They are shown in Figure~\ref{fig:CP2.6B_revisited_CPRs}. The amplitude of the oscillations appears to decrease as the temperature increases. Except for the measurement at \qty{3.6}{\kelvin} where it increases again. We suspect that this is due to a human error. A likely explanation is that a different bias current was used for the dc-SQUID compared to the bias current used for the calibration.

Ignoring the measurement at \qty{3.6}{\kelvin}, the critical current goes from \qtyrange{90}{45}{\micro\ampere} between \qtyrange{2.8}{3.4}{\kelvin}. Besides the change in critical current we also note that the asymmetry changes. Both these changes are in line with theoretical predictions\cite{likharevSuperconductingWeakLinks1979}. From the slope in our data we determined the mutual inductance to be 0.4 times the simulated value in Table~\ref{tab:CP2.6B-geometries}. For the loop inductance the value is 3.5 times the simulated value. These deviations might be explained by the fact that the simulation does not take the layer of \ce{Cu} into account. Additionally these values do not match what we previously found for this sample. Possibly this is because our calibration was not optimal.

\begin{figure}[ht!]
\centering
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\label{fig:CP2.6B_revisited_CPRs}
\end{figure}

Having to fit the mutual and loop inductance is one of the weaknesses of our method. In the linear regime of the dc-SQUID extracting the mutual inductance should be trivial. The loop inductance however is determined by the periodicity in the data. Whilst possible to compare the loop inductance to a simulated value, a factor two difference already changes a $2\pi$-periodic CPR to a $4\pi$-periodic current-phase relation. To overcome this, a reference loop without any junction could be added.

Previously we thought the steep curve was caused by a multivalued measurements. However, with these new results we think we wrong mainly because we see such a gradual transition.

Clearly the results show a few artefacts most notably at \qtylist{2.8;3.0;3.4}{\kelvin}. We attribute this to the fact that we are not in the linear regime of our dc-SQUID. We are confident that if the dc-SQUID would be biased in the linear regime that these artefacts would disappear. Additionally because we did not control our dc-SQUID's bias, it is not possible to say what point $\gamma = 0$. As such we have artificially centred the curves such that $\gamma = 0$ for $I_s=0$.

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