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ass5: More draft 18

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Kees van Kempen
2022-05-18 21:04:05 +02:00
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15
references.bib
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@ -121,3 +121,18 @@ Results from previous neutron measurements are found to be consistent with the X
author = {Annett, James F.},
year = {2004},
}
@article{ozcan,
title = {London penetration depth measurements of the heavy-fermion superconductor {CeCoIn} $_{\textrm{5}}$ near a magnetic quantum critical point},
volume = {62},
issn = {0295-5075, 1286-4854},
url = {https://iopscience.iop.org/article/10.1209/epl/i2003-00411-9},
doi = {10.1209/epl/i2003-00411-9},
number = {3},
urldate = {2022-05-18},
journal = {Europhysics Letters (EPL)},
author = {Özcan, S and Broun, D. M and Morgan, B and Haselwimmer, R. K. W and Sarrao, J. L and Kamal, Saeid and Bidinosti, C. P and Turner, P. J and Raudsepp, M and Waldram, J. R},
month = may,
year = {2003},
pages = {412--418},
}

27
superconductivity_assignment5_kvkempen.tex
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@ -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.
\bibliographystyle{vancouver}
\bibliography{references.bib}