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ass2: The essay is hard.

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superconductivity_assignment2_kvkempen.tex
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@ -199,16 +199,44 @@ We apply a gauge transformation as follows.
\end{enumerate}
\section{Type II superconductors and the vortex lattice}
\begin{em}
In 2003, Alexei Abrikosov was one of the winners of the Nobel Prize in Physics ``for pioneering contributions to the theory of superconductors and superfluids''.
For this occasion, he gave a lecture called ``Type II superconductors and the vortex lattice''\cite{abrikosov}
explaining the discoveries that led to the understanding of conventional superconductors.
To get started, let me first explain what superconductors are.
% Begin copy from philosophy
Superconductors are characterized by perfect diamagnetism and zero resistance.
Perfect diamagnetism is the ability by superconductors to have a net zero magnetic field inside.
If you apply an external magnetic field, this thus means that a superconductor will let a current flow on its inside to generate a field to counteract this external field $\vec{H}$.
This generated current is called a supercurrent.
This is, however, a phase of the material.
Superconductors only have these properties below a certain temperature, its critical temperature $T_c$, and can only expel a maximum external magnetic field, its critical magnetic field $B_c(T)$, which is a function of the temperature.
The class of superconductors we have a model for, is the class of conventional superconductors, which are explained by a theory called BCS (and some extensions).
In this class, there are two types, called type-I and type-II superconductors.
% End copy from philosophy
In type-I superconductors, there is only one phase in which the superconductor material exhibits perfect diamagnetism:
when the externally applied magnetic field $H < B_c(T)$ and $T < T_c$.
In type-II superconductors, there are two phases distinct from the normal conducting state.
One is the superconducting state which behaves as in type-I superconductors, with critical field $B_{c1}(T)$.
This state is reached for $T < T_c$ and $B_E < B_{c1}(T)$.
The other state is a mixed state that allows some flux to pass through the material.
This passing through is done by creating normally conducting channels throughout the material where a fixed amount of flux can pass through.
This fixed amount is a multiple of the flux quantum $\Phi_0$.
The material generates current around these channels cancelling the field on the inside of the superconducting part of the material.
---
The nobel prize lecture by Abriskosov \cite{abrikosov} was really interesting.
The start was a good recap of the breakthroughs relevant to conventional superconductivity,\footnote{Why is that every story on superconductivity includes KGB captivity?}
The start was a good recap of the breakthroughs relevant to conventional superconductivity,\footnote{Why is it that every story on superconductivity includes KGB captivity?}
but in pages 61--63, the theory is worked through a little quickly.
I might reread it some times.
\end{em}
\todo{The essay so far is just a draft. Choosing a topic was hard.}
\todo{The essay so far is just a draft. Choosing a topic was hard. As we are to aim at bachelor students not knowing sc, I thought a proper introduction was appropriate.}
\section{Currents inside type-II superconducting cylinder}
For $B_{c1} < B_E < B_{c2}$, the cylinder of type-II superconductor material is in the mixed state.