ass2: Finalize assignment 2, hand it in

superconductivity_assignment2_kvkempen.tex

@@ -31,6 +31,7 @@
 \usepackage{amsmath}
 \usepackage{todonotes}
 \setuptodonotes{inline}
+\usepackage{mhchem}
 
 \newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
 
@@ -218,6 +219,8 @@ If you apply an external magnetic field, this thus means that a superconductor w
 This generated current is called a supercurrent.
 Superconductivity 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 zero resistance property follows from the perfect diagmagnetism.
+It is impossible for the material to let these supercurrents flow indefinitely with resistance, as heat would be generated.
 
 The class of superconductors we have a model for, is the class of conventional superconductors.
 In this class, there are two types, called type-I and type-II superconductors.
@@ -234,7 +237,51 @@ This passing through is done by creating normally conducting channels throughout
 This fixed amount is a multiple of the flux quantum $\Phi_0$.
 The material generates current around these channels in accordance to the Maxwell-Amp\`ere law, conforming to the let through magnetic field inside the vortex and cancelling the field on the outside the vortex.
 
+There are lots of applications for both the perfect diagmagnetism and the zero resistivity.
+There is even a Wiki about them: \url{https://en.wikipedia.org/wiki/Technological\_applications\_of\_superconductivity}.
+What is most notable about these applications, is that maintaining a temperature below the critical temperature is the biggest challenge.
+A real breakthrough for superconductivity would be the discovery of room-temperature superconductors at atmospheric pressure, or materials close to that.
+Currently, the highest $T_c$ material we know is carbonaceous sulfur hydride (\ce{CH8S}) with $T_c = \SI{15}{\degreeCelsius}$ but at a pressure of a whopping $\SI{267}{\giga\pascal}$.
+At atmospheric pressure, the highest $T_c$ material known is a cuprate \cite{dai-synthesis-1995} \ce{HgBa2Ca2Cu3O_{8+\delta}} at $T_c = \SI{135}{\kelvin}$.
+The quest for this breakthrough is intensely researched, although most is experimental.
+The clue is that most of the high $T_c$ materials that are being discovered, are unconventional superconductors.
+As there is no theory for them (yet), the search is mostly educational guessing.
+By trying to find patterns in the previously high $T_c$ materials, similar materials are studied to see if they also exhibit superconductivity.
+One of the patterns is that superconductivity in cuprates is high $T_c$.
+We'll focus on these materials in the following.
 
+Currently, most hopeful candidates are cuprates.
+These materials are made of layers of copper oxides (\ce{CuO2}) alternated with layers of other metal oxides.
+The copper oxide layers are the superconductive layers, and the other metal oxides are used as charge reservoirs, doping electrons (or holes) into the copper oxide layers.
+Due to the geometry of these materials, there is anisotropy in the resistivity of the material.
+Parallel to the layers, superconductivity takes place in the copper oxide layers.
+Perpendicular to the layers, this is not the case.
+
+The behaviour of the material can be tuned by tuning the doping, thus the other metal oxides as mentioned before.
+A typical phase diagram as function of the doping can be seen in figure \ref{fig:cuprate-phase}.
+The material can be steered from being antiferrimagnetic to superconductive by increasing doping.
+
+\begin{figure}
+	\centering
+	\includegraphics{cuprate-phase.pdf}
+	\caption{For high $T_c$ superconducting cuprates, a typical phase diagram as function of doping looks like this.\cite{chubukov} }
+	\label{fig:cuprate-phase}
+\end{figure}
+
+As can be seen, there is an optimal doping fraction for achieving highest $T_c$.
+Aiming for this doping yields the desired material.
+
+Now the question is what direction to search for.
+The timeline in figure \ref{fig:timeline} might give a direction for the most promising types of cuprates to look into.
+It could be, however, that other types have higher $T_c$.
+A lot of creativity is therefore needed to find them.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=\textwidth]{Timeline-of-Superconductivity-from-1900-to-2015.pdf}
+	\caption{The last century, a lot of research has been done in the direction of cuprate superconductivity. Pia Jensen Ray made this overview for his master thesis.\cite{ray-2016} The different paths are different types of cuprates. Please see his thesis for the meaning of the labels. On the right side, an idea of the temperature is givin by comparing it to common cooling agents.}
+	\label{fig:timeline}
+\end{figure}
 
 ---
 
@@ -243,10 +290,6 @@ The start was a good recap of the breakthroughs relevant to conventional superco
 but in pages 61--63, the theory is worked through a little quickly.
 I might reread it some times.
 
-
-
-\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.
 In the mixed or vortex state, superconductors let through a number of finite flux quanta $\Phi_0$.