ass5: The points before Ozcan seem to be done

superconductivity_assignment5_kvkempen.tex

@@ -33,6 +33,7 @@
 \setuptodonotes{inline}
 \usepackage{mhchem}
 \usepackage{listings}
+\usepackage{subcaption}
 
 \newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
 
@@ -165,11 +166,54 @@ In the weak coupling limit of BCS theory, around the Fermi sphere, we see a cons
 There thus are no nodes.
 BCS describes s-wave superconductors.
 For other types, this is not the case: there is gap anisotropy.
-A result like in figure \ref{fig:waves} can thus tell us what kind of superconductor we see.
+A result like in figure \ref{fig:waves} can thus tell us what kind of superconductor we see. Looking at $\lambda(T)$, we find plots as in figure \ref{fig:sd}.
+
+\begin{figure}
+	\centering
+	\begin{subfigure}{.45\textwidth}
+		\centering
+		\includegraphics[width=\linewidth]{PhysRevLett.70.3999-s-wave.png}
+		\caption{For superconductors without nodes (s-wave, BCS), there is a constant gap energy, resulting in $\lambda(T) \propto [n_s(0)(1-\alpha\exp{(\frac{\Delta}{k_BT})})]^{-1/2}$.}
+		\label{fig:s}
+	\end{subfigure}
+	\begin{subfigure}{.45\textwidth}
+		\centering
+		\includegraphics[width=\linewidth]{PhysRevLett.70.3999-d-wave.png}
+		\caption{For superconductors with line nodes, such as d-wave and some p-wave, $\lambda(T) \propto T$ is observed as was expected.}
+		\label{fig:d}
+	\end{subfigure}
+	\caption{Both figures are from \cite{hardy_precision_1993}.}
+	\label{fig:sd}
+\end{figure}
 
 But now the question is how we can measure this gap anisotropy in practice.
-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.
+To image the complete $k$-dependence of the gap, it is required that the probe is sensitive to the direction of the electron momenta\cite[p.207]{waldram}, for which there are multiple methods.
+A direct way would be to use ARPES, as that directly probes the band gap energy and is angular resolved, thus yielding a $k$-dependent measurement.
+However, we want to take a look at a different approach.
 We will focus on using $\lambda(T)$ measurements using tunnel diode oscillators (TDO)\cite{ozcan}, as that technique is used in the provided paper, and we just discussed the relation between $\lambda$ and the band gap.
+Do note that angular information will not be obtained this way.
+
+A thorough discussion about $\lambda$ measurements using a TDO is presented in \cite{giannetta_london_2021}.
+The idea is to measure the resonant frequency of an $LC$-circuit which inductance $L$ changes as function of the penetration depth.
+A piece of superconductor material is inserted in the coil of the $LC$-circuit, preferably a slab, cylinder or sphere, as these yield exact results to the London equations that are used for determining the dependence.
+The $LC$-circuit is turned on by some AC signal.
+This in turn induces an alternating magnetic field $H$ inside the coil.
+Following the London equations, this induces a magnetic moment $m$ inside the superconductor sample that is linear to the field and depends on the geometry of the sample, thus $m = C(\lambda) H$.
+This magnetic moment in its turn affects the inductance of the coil, resulting in a resonant frequency change
+\[
+	\delta f = f(\textup{with sample}) - f(\textup{without sample}) = Gm = GC(\lambda)H,
+\]
+with $G$ the effective volume of the coil.
+As determining the geometry and field directions for $C$ is quite error prone and hard due to the smallness of the quantities, they are usually not determined.
+They are, however, kept constant, and $\lambda$ is what is varied by changing the temperature such that we can easily write
+\[
+	\Delta \lambda = \lambda(T) - \lambda(0).
+\]
+With knowledge about $\lambda(0)$ from other sources, $\lambda(T)$ is determined by determining $\Delta \lambda$ from $\delta f$.
+Now the superfluid density can be determined.
+
+
+
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