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\title{Superconductivity - Assignment 5}
\author{
	Kees van Kempen (s4853628)\\
	\texttt{k.vankempen@student.science.ru.nl}
}
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\section{$T_c$ upper limit in BCS}
In BCS theory, the formation of Cooper pairs is mediated by phonons.
There is a phonon-electron interaction quantified by the dimensionless quantity 
\[
	\lambda := Vg(\epsilon_F)
\]
with $V$ Cooper's approximate potential and $g(\epsilon_F)$ the density of states near the Fermi surface for the electrons.
A thorough discussion can be found in Annett's book \cite[chapter 6]{annett} and in the slides of week 6 of this course.
The binding energy of the Cooper pairs (i.e. the energy gain of forming these pairs) is
\[
	 -E = 2\hbar\omega_De^{-1/\lambda} =: \Delta_0,
\]
which is also called the gap parameter $\Delta_0$ at zero temperature for BCS.
In the weak coupling limit of the BCS theory, the case we have considered so far, it is assumed that $\lambda << 1$.
It should be noted that this weak limit also means that the gap is smaller than the thermal energy of the highest excited energy phonon, which corresponds to the Debye temperature
\[
	\Delta < k_B\Theta_D.
\]
It is when this assumption breaks down, BCS does not work and we find an upper limit to the critical temperature $T_c$.
We will look at a way to express the critical temperature in terms we can derive, and then look at the values that maximize this critical temperature whilst still following BCS theory.

From the derivation of the BCS coherent state, this gap parameter at finite temperature is found.
There is a temperature dependence $\Delta(T)$ as in figure \ref{fig:gap-T}.
\begin{figure}
	\centering
	\includegraphics[width=.4\textwidth]{Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf}
	\caption{By taking the gap parameter to zero, we find the critical temperature. Figure from the slides of lecture 7.}
	\label{fig:gap-T}
\end{figure}
For larger temperatures, thermal energy is increased, and less energy is required to break up Cooper pairs, thus degrading the superconductivity.
This puts a limit $T_c$.

We will mostly follow the derivation by Waldram \cite[paragraph 7.9, mostly p.128--130]{waldram}.
The superconducting state breaks down at high temperature, at which also $\Delta$ vanishes so that the gap parameter is a good order parameter for the state.

Let's consider the gap parameter
\[
	\Delta_{\vec{k}} = -\sum_{\vec{k'}}(1-2f_{\vec{k'}})u_{\vec{k'}}v_{\vec{k}}V_{\vec{k'}\vec{k}},
\]
with $u$ and $v$ occupation functions for the BCS state, $f$ the Fermi occupation number, and $V$ the potential between the states.
Minimizing $\Delta_{\vec{k}}$ and taking that $V_{\vec{k'}\vec{k}} = -V$ is constant gives us a self-consistent relation for the gap parameter.
We also recognize that the states that we sum over all all those states such that they have smaller energy than the highest excited phonon.
\[
	\Delta_{\vec{k}} = V\sum_{\epsilon_{\vec{k'}}}(1-2f_{\vec{k'}})\frac{\Delta_{\vec{k'}}}{2E_{\vec{k'}}}.
\]
Now the right-hand side is independent of $\vec{k}$ but does contain $\Delta_{\vec{k'}}$.
We can thus conclude that the gap parameter should be constant over all states $\vec{k}$!
That means we can divide both sides by it, giving us
\[
	1 = V\sum_{\epsilon_{\vec{k'}}}(1-2f_{\vec{k'}})\frac{1}{2E_{\vec{k'}}}.
\]
Converting the equation to an integral, and substituting in $f(E) = [\exp{(E/(k_BT))}+1]^{-1}$ and $E = \sqrt{\epsilon^2 + \Delta(T)^2}$ yields
\[
	1 = 2g(\epsilon_F)V\int_0^{k_B\Theta_D}\frac{1-2[\exp{(E/(k_BT))}+1]^{-1}}{2\sqrt{\epsilon^2 + \Delta(T)^2}} \textup{d}\epsilon.
\]

I believe Waldram that one could find that
\[
	T_c = 1.14\Theta_D\exp{(-1/(g(\epsilon_F)V))} = 1.14\Theta_D\exp{(-1/(\lambda)}
\]
from this nice equation.

As limiting value, we take $\lambda = 0.3$, as was posed as a reasonable limit for the weak coupling by Alix in lecture 7,
although Waldram \cite{waldram} thinks it is more like $\lambda \approx 0.4$.

For metals, Waldram thinks $\Theta_D \leq \SI{300}{\kelvin}$ is a good limit.
This leads to our final maximum
\[
	T_c \leq 1.14 \cdot 300 \cdot \exp{(-1/0.4)} \approx \SI{28}{\kelvin}.
	%https://www.wolframalpha.com/input?i=1.14*300*e%5E%28-1%2F.4%29
\]
(Using $\lambda = 0.3$ yields an even lower $T_c \leq \SI{12}{\kelvin}$.)

For larger $T_c$ values, larger binding of Cooper pairs would be needed to overcome the thermal energy.
This means our assumption of weak coupling breaks down, making most of the derivation invalid without further arguments.

\clearpage

\section{Penetration depth $\lambda$ and measuring it}
There are many species of superconductors.
Conventional superconductors we can describe using BCS theory or some extension of it.
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 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.

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.
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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