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ass1: Add references, add note on uncertainty

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Kees van Kempen
2022-02-18 08:49:40 +01:00
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superconductivity_assignment1_kvkempen.tex
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@ -114,11 +114,13 @@ No idea yet.
In the realm of conventional superconductors, we have type-I and type-II superconductors. In the realm of conventional superconductors, we have type-I and type-II superconductors.
Both types are mediated by electron-phonon coupling, but there are quite some differences. Both types are mediated by electron-phonon coupling, but there are quite some differences.
Using figure \ref{fig:typevs}, we will go through their differences. Using figure \ref{fig:typevs}, we will go through their differences.
The material covered is mostly from the Solid State Physics course by Steffen Wiedmann and the Superconductivity course by Alix McCollam, with some statistical physics knowledge due to Mikhail Katsnelson's Advanced Statistical Physics.
Holes in my knowledge were mostly filled by \cite{waldram}.
\begin{figure} \begin{figure}
\label{fig:typevs} \label{fig:typevs}
\includegraphics[width=\textwidth]{Lecture-2-slides-for-printing-type-i-vs-ii.pdf} \includegraphics[width=\textwidth]{Lecture-2-slides-for-printing-type-i-vs-ii.pdf}
\caption{Figure is borrowed from the presentation of week two of the Superconductivity course by Alix McCollam.} \caption{On the left, the behaviour of a type-I superconductor is displayed, on the right the type-II. The top graphs show the critical fields $H_c$ over the temperature of the sc. The bottom graphs show the magnetization of the materials over the temperature. Phase transitions for both types are clearly visible. The figure is borrowed from the presentation of week two of the Superconductivity course by Alix McCollam in 2022.}
\end{figure} \end{figure}
Type-I superconductors (sc) are described by BCS theory. Type-I superconductors (sc) are described by BCS theory.
@ -126,7 +128,7 @@ An example of a type-I sc is lead (Pb).
Below a critical temperature $T_c$, these materials exhibit perfect diamagnetism. Below a critical temperature $T_c$, these materials exhibit perfect diamagnetism.
Inside the sc, a magnetic field is generated to expell the externally applied field, Inside the sc, a magnetic field is generated to expell the externally applied field,
such that the net field is zero. such that the net field is zero.
This is called the Meissner effect. This is called the Meissner-Ochsenfeld effect.
The perfect diamagnetism is quantized as having susceptibility $\chi = -1$. The perfect diamagnetism is quantized as having susceptibility $\chi = -1$.
There is, however, a limit to how large a field can be completely expelled. There is, however, a limit to how large a field can be completely expelled.
This is called the critical field $H_c$. This is called the critical field $H_c$.
@ -151,18 +153,19 @@ and the material thus completely cancels the externally applied field.
Above the upper critical field $H_{c2}(T)$, the material is in the normal state. Above the upper critical field $H_{c2}(T)$, the material is in the normal state.
When the field is between these two critical fields, $H_{c1}(T) < H < H_{c2}(T)$, When the field is between these two critical fields, $H_{c1}(T) < H < H_{c2}(T)$,
the material is in the vortex state. the material is in the vortex state.
See the top-right plot in figure \ref{fig:typevs}.
In the vortex state, the externally applied magnetic field is not completely expelled. In the vortex state, the externally applied magnetic field is not completely expelled.
Instead, the material consists of normal and sc regions, the former called vortices. Instead, the material consists of normal and sc regions, the former called vortices.
These vortices are normal conducting regions and allow magnetic flux to pass. These vortices are normal conducting regions and allow magnetic flux to pass.
The flux let through by the vortices is quantized by flux quanta,
and more flux is let through by allowing more quanta through the material.
Type-I superconductors are thought to be described by BCS theory. In practice, this means that more vortices appear,
They are phonon-mediated. which is energetically more favorable than having vortices with more flux.
They have different phase diagrams from type-II superconductors, As can be seen in the magnetization graph on the bottom-right of figure \ref{fig:typevs},
with two critical fields but one critical temperature. the magnetization is continuous over the temperature, thus no first-order phase transition is observed.
There is a superconducting phase and a vortex phase. A second-order phase transition, however, does take place at $H_{c1}$,
and is due to a discontinuity in $\chi = \frac{d^2F}{dH^2}$.
Something about quantized flux by the Meissner-Ochsenfeld effect. The transition from the vortex to the normal state is also of second order.\footnote{Sources on this seem hard to find. The magnetization graph seems continuous, so a second-order transition seems plausible. By describing the transition as the disappearance of the order parameter $M$, it can be seen that there indeed is a true phase transition.}
\bibliographystyle{vancouver} \bibliographystyle{vancouver}
\bibliography{references.bib} \bibliography{references.bib}