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ass5: Now it might be a story

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superconductivity_assignment5_kvkempen.tex
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@ -139,10 +139,12 @@ These quantities are related to the energy band gap around the Fermi surface in
A nice table summarizing these quantities can be found in \cite[table 10.1, p. 191]{waldram}. 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 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.
The penetration depth $\lambda$ is determined by the superfluid density $n_s$ in the two fluid model. The band gap energy $\Delta(k)$ is a useful order parameter for superconductivity.
As $n_s$ can be related to energy gap $\Delta$\cite[ch. 7]{annett}, so can thus $\lambda$ be. It can tell us a lot about what kind of superconductor we are dealing with.
It is not the maximum value of $\Delta$ we are after, but more its variation in momentum space. It is not the maximum value of $\Delta$ we are after, but its variation in momentum space, and more specifically any nodes in it.
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}. 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}.
The regions with opposite sign correspond to regions of repulsion, whereas same sign regions have attraction.
Now it is our job to connect this to $\lambda$.
\begin{figure} \begin{figure}
\centering \centering
@ -151,18 +153,23 @@ Results on the relation between the nodes of the energy gap and the type of supe
\label{fig:waves} \label{fig:waves}
\end{figure} \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 In the theory by the London theory of superconductivity, the penetration depth is related to the superfluid density $n_s$\cite[ch. 3, ch. 7.5]{annett} (of the superfluid model) as
\[ \[
\xi_{BCS} = \frac{\hbar v_F}{\pi\Delta}, \lambda_L(T) = \sqrt{\frac{m_e^*}{\mu_0e^2n_s(T)}}.
\] \]
for $v_F$ the Fermi velocity. If $n_s(T)$ can be related to energy gap $\Delta$, so can $\lambda$, and luckily we can.
BCS describes s-wave superconductors. If there is a node in $\Delta(k)$ for some $k$, it means that there will be states available for any energy we put in.
Around the Fermi sphere, we see a constant band gap. This in turn implies a linearly increasing relation to $\lambda(T) = \lambda(0) + cT$ for some constant $c$.
For other types, this is not the case: there is gap anisotropy.
But now the question is how we can measure this gap anisotropy. In the weak coupling limit of BCS theory, around the Fermi sphere, we see a constant band gap.
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.
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. 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. 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.
\bibliographystyle{vancouver} \bibliographystyle{vancouver}
\bibliography{references.bib} \bibliography{references.bib}