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ass1: Elaborate on difference type-I and type-II

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superconductivity_assignment1_kvkempen.tex
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%\usepackage[nottoc,numbib]{tocbibind}
\usepackage{float}
\usepackage{mathtools}
\usepackage{svg}
\title{Superconductivity - Assignment 1}
\author{
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\section{Difference between 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.
Using figure \ref{fig:typevs}, we will go through their differences.
\begin{figure}
\label{fig:type-i-vs-ii}
\includesvg[width=\textwidth]{Lecture-2-slides-for-printing-type-i-vs-ii.svg}
\label{fig:typevs}
\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.}
\end{figure}
Type-I superconductors (sc) are described by BCS theory.
An example of a type-I sc is lead (Pb).
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,
such that the net field is zero.
This is called the Meissner effect.
The perfect diamagnetism is quantized as having susceptibility $\chi = -1$.
There is, however, a limit to how large a field can be completely expelled.
This is called the critical field $B_c$.
If $B_c$ is exceeded, the superconductivity breaks down, thus the material will cease to expell the field,
the diagmagnetism drops to $\chi = 0$.
This effect is seen in the bottom-left plot in figure \ref{fig:typevs}.
The relation between these two critical values is given as
\[
B_c(T) = B_c(0) \left[ 1 - \frac{T}{T_c}^2 \right].
\]
An example of this curve is plotted in the top-left of figure \ref{fig:typevs}.
Two phases can be distinguished:
the Meissner (or superconducting) state under the graph,
and the normal state outside it.
The transition between these states is a first-order phase transition due to the discontinuity in the magnetization $M = \frac{dF}{dB}$, the first derivative of the free energy to the applied field.
Then there are type-II sc, for example, niobium (Nb).
Instead of only having a Meissner or sc phase, they have another phase:
the vortex state.
Below $T_c$ and lower critical field $B_{c1}(T)$, the type-II sc is in the Meissner state,
and the material thus completely cancels the externally applied field.
Above the upper critical field $B_{c2}(T)$, the material is in the normal state.
When the field is between these two critical fields, $B_{c1}(T) < B < B_{c2}(T)$,
the material is in the vortex state.
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.
These vortices are normal conducting regions and allow magnetic flux to pass.
Type-I superconductors are thought to be described by BCS theory.
They are phonon-mediated.
They have different phase diagrams from type-II superconductors,

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