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\title{Superconductivity - Assignment 1}
\author{
	Kees van Kempen (s4853628)\\
	\texttt{k.vankempen@student.science.ru.nl}
}
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\begin{document}

\section{Electron-phonon coupling in elements}
Conventional superconductors are described by considering Cooper pairs:
pairs of electrons mediated by electron-phonon coupling.
This is usually described by BCS theory.
The hypothesis is that stronger electron-phonon coupling results in enhanced critical temperatures for the superconducting phase transition.
In order to investigate this, we need a way to determine the electron-phonon coupling strength.
We will attempt to do this by looking at the room temperature resistivity of superconducting elements.

For metals, we have the following familiar relation for resistivity $\rho$ over temperature $T$.

\begin{equation}
	\rho = \underbrace{\rho_0}_\text{{impurities}} + \underbrace{aT^2}_{\text{electron-electron coupling}} + \underbrace{bT^5}_{\text{electron-phonon coupling}}
\end{equation}

At $T = 0$, only resistivity due to impurities and lattice defects is left in the material.
Then, at low temperatures, electron-electron coupling increases resistivity.
The effect that is the largest at room temperature, is due to electron-phonon interaction, due to the fifth power in temperature.
The constants $a$ and $b$ differ from material to material.
If the hypothesis is correct, an increasing trend of critical temperature $T_c$ over room temperature resistivity $\rho_{300K}$ should be observed.

For a collection of superconducting elements, this relation is plotted in figure \ref{fig:scelements}.
The data on critical temperatures $T_c$ and (approximately) room temperature resistivity $\rho_{\SI{300}{\kelvin}}$ is from various sources, as can be found in the table in appendix \ref{appendix:scelements}.

\begin{figure}%[H]
	\label{fig:scelements}
	\includegraphics[width=\textwidth]{sc_elements.pdf}
	\caption{In this plot of the critical temperature $T_c$ versus the room temperature resistivity $\rho_{300K}$ for elemental superconductors, not one clear relation can be distinguished. For most elements, resistivity is taken at room temperature $T = \SI{300}{\kelvin}$. If it was unavailable in consulted references, the value at the temperature closest to \SI{300}{\kelvin} was chosen. See the table in appendix \ref{appendix:scelements} for the raw data including their source. The mess in the left bottom corner was hard to filter out. A log-log plot was attempted and improved separation, but obscured the observed two branches in this linear plot.}
\end{figure}

Looking at the plot, there is no obvious positive trend between $T_c$ and $\rho_{300K}$.
As a way to quantize this (lack of) correlation, we can take a look at the Pearson correlation coefficient: $r = 0.165415$.
% I used df.corr() to calculate $r$.
Pearson's $r$ is a measure of linear correlation.
If $|r| = 1$, there is a perfectly linear relation.
The lower $|r|$ is, the less correlated the points are.
The sign of $r$ gives the direction of the trend.
This slightly positive value found for the superconducting elements suggests a slightly positive but uncertain correlation.
As the relation between electron-phonon coupling and resistivity is well-established, it seems reasonable to conclude that there is no unambiguous relation between $T_c$ and $\rho_{300K}$.
There seem to be other factors we are missing in this analysis.

Looking at the plot, however, it would be too easy to conclude there is no relation between these quantities at all.
There do seem to be two branches with approximate linear correlation.
Distinguishing these two groups roughly along the line from the origin under Re, we find $r = 0.69739$ and $r = 0.621341$ respectively above and below the line.

Another missing factor is the comparison to non-superconductor elemental metals.
Although they do not really experience a phase transition to a superconducting state, so they do not have a finite $T_c$ associated to them, they could be plotted having $T_c = 0$.
I chose to exclude them from the plot, as they would only clutter it further, and there is no real relation visible.

Further distinction could be in superconductor type.
Most of the plotted elements are type-I superconductors, but vanadium, for example, is type-II.
Vanadium does, however, behave similar to the rest of the elements.

\section{Exam question electrodynamics in superconductors}
No idea yet.

\section{Difference between type-I and type-II superconductors}
Type-I superconductors are thought to be described by BCS theory.
They are phonon-mediated.
They have different phase diagrams from type-II superconductors,
with two critical fields but one critical temperature.
There is a superconducting phase and a vortex phase.

Something about quantized flux by the Meissner-Ochsenfeld effect.

\bibliographystyle{vancouver}
\bibliography{references.bib}

\appendix
\section{Superconducting elements}
\label{appendix:scelements}
\input{sc_elements.tex}

\end{document}