ass5: The points before Ozcan seem to be done

references.bib

@@ -136,3 +136,33 @@ Results from previous neutron measurements are found to be consistent with the X
 	year = {2003},
 	pages = {412--418},
 }
+
+@article{giannetta_london_2021,
+	title = {London {Penetration} {Depth} {Measurements} {Using} {Tunnel} {Diode} {Resonators}},
+	issn = {1573-7357},
+	url = {https://doi.org/10.1007/s10909-021-02626-3},
+	doi = {10.1007/s10909-021-02626-3},
+	abstract = {The London penetration depth \$\${\textbackslash}lambda \$\$is the basic length scale for electromagnetic behavior in a superconductor. Precise measurements of \$\${\textbackslash}lambda \$\$as a function of temperature, field and impurity scattering have been instrumental in revealing the nature of the order parameter and pairing interactions in a variety of superconductors discovered over the past decades. Here we recount our development of the tunnel-diode resonator technique to measure \$\${\textbackslash}lambda \$\$as function of temperature and field in small single crystal samples. We discuss the principles and applications of this technique to study unconventional superconductivity in the copper oxides and other materials such as iron-based superconductors. The technique has now been employed by several groups world-wide as a precision measurement tool for the exploration of new superconductors.},
+	language = {en},
+	urldate = {2022-05-18},
+	journal = {Journal of Low Temperature Physics},
+	author = {Giannetta, Russell and Carrington, Antony and Prozorov, Ruslan},
+	month = oct,
+	year = {2021},
+}
+
+@article{hardy_precision_1993,
+	title = {Precision measurements of the temperature dependence of {\textbackslash}ensuremath\{{\textbackslash}lambda\} in \$\{{\textbackslash}mathrm\{{YBa}\}\}\_\{2\}\$\$\{{\textbackslash}mathrm\{{Cu}\}\}\_\{3\}\$\$\{{\textbackslash}mathrm\{{O}\}\}\_\{6.95\}\$: {Strong} evidence for nodes in the gap function},
+	volume = {70},
+	shorttitle = {Precision measurements of the temperature dependence of {\textbackslash}ensuremath\{{\textbackslash}lambda\} in \$\{{\textbackslash}mathrm\{{YBa}\}\}\_\{2\}\$\$\{{\textbackslash}mathrm\{{Cu}\}\}\_\{3\}\$\$\{{\textbackslash}mathrm\{{O}\}\}\_\{6.95\}\$},
+	url = {https://link.aps.org/doi/10.1103/PhysRevLett.70.3999},
+	doi = {10.1103/PhysRevLett.70.3999},
+	abstract = {A miniature superconducting resonator operating at 1.3 K and 900 MHz has been used to measure the change in λ(T) from 1.3 K to Tc in very high quality single crystals of YBa2Cu3O6.95. The data, which have a resolution of 1-2 Å, show a strong linear term extending from approximately 3 to 25 K. We believe the strong linear dependence to be characteristic of the pure system and that its apparent absence in thin films and some crystals is due to the presence of defects.},
+	number = {25},
+	urldate = {2022-05-18},
+	journal = {Physical Review Letters},
+	author = {Hardy, W. N. and Bonn, D. A. and Morgan, D. C. and Liang, Ruixing and Zhang, Kuan},
+	month = jun,
+	year = {1993},
+	pages = {3999--4002},
+}

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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