ass5: More draft 18

superconductivity_assignment5_kvkempen.tex

@@ -132,16 +132,37 @@ Conventional superconductors we can describe using BCS theory or some extension
 Others we do not yet have a theory for.
 Some are type-I, others type-II.
 What they do have in common, is that they can be characterized by some key quantities.
-Starting with the coherence length $\xi$ and the penetration depth $ \lambda$, we also have their critical temperatures $T_c$ and can relate them to the energy band gap around the Fermi surface in BCS theory.
+Starting with macroscopic ones, we have the critical temperature $T_c$ and critical field(s) $H_c$.
+The microscopic behavior is described by three characteristic lengths\cite[p.62]{annett}:
+the coherence length $\xi$ of the Cooper pairs, the penetration depth $\lambda$ of the external field, and the mean free path $\ell$ of the electrons.
+These quantities are related to the energy band gap around the Fermi surface in BCS theory.
+A nice table summarizing these quantities can be found in \cite[table 10.1, p. 191]{waldram}.
 In this essay, we will take a look at what the penetration depth can tell us about the superconducting energy gap, and will go into measuring the penetration depth.
 
-In the weak coupling limit of BCS theory, the coherence length can be related to the gap parameter\cite[chapter 9]{waldram} as
+The penetration depth $\lambda$ is determined by the superfluid density $n_s$ in the two fluid model.
+As $n_s$ can be related to energy gap $\Delta$\cite[ch. 7]{annett}, so can thus $\lambda$ be.
+It is not the maximum value of $\Delta$ we are after, but more its variation in momentum space.
+Results on the relation between the nodes of the energy gap and the type of superconducting wave we deal with can be seen in figure \ref{fig:waves}.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=.8\textwidth]{Lecture-9-slides-for-printing-slide-8-wave-types.pdf}
+	\caption{Different types of superconductor waves have different node patterns. The figure is from the slides of lecture 9 by Alix McCollam.}
+	\label{fig:waves}
+\end{figure}
+
+In the weak coupling limit of BCS theory, the coherence length can be related to the gap parameter\cite[ch. 9]{waldram} as
 \[
 	\xi_{BCS} = \frac{\hbar v_F}{\pi\Delta},
 \]
 for $v_F$ the Fermi velocity.
-Furthermore, it has the physical interpretation of the size of the separation in a Cooper pair\cite[p. 62]{annett}.
+BCS describes s-wave superconductors.
+Around the Fermi sphere, we see a constant band gap.
+For other types, this is not the case: there is gap anisotropy.
 
+But now the question is how we can measure this gap anisotropy.
+A hard requirement, is that the probe should be sensitive to the direction of the electron momenta\cite[p.207]{waldram}, for which there are multiple methods.
+We will focus on using $\lambda(T)$ measurements using tunnel diode oscillators (TDO)\cite{ozcan}, as that technique is used in the provided paper.
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