243 lines
13 KiB
TeX
Executable File
243 lines
13 KiB
TeX
Executable File
\documentclass[a4paper, 11pt]{article}
|
|
\usepackage[utf8]{inputenc}
|
|
|
|
\usepackage[
|
|
a4paper,
|
|
headheight = 20pt,
|
|
margin = 1in,
|
|
tmargin = \dimexpr 1in - 10pt \relax
|
|
]{geometry}
|
|
|
|
\usepackage{fancyhdr} % for headers and footers
|
|
\usepackage{graphicx} % for including figures
|
|
\usepackage{booktabs} % for professional tables
|
|
\setlength{\headheight}{14pt}
|
|
|
|
|
|
\fancypagestyle{plain}{
|
|
\fancyhf{}
|
|
\fancyhead[L]{\sffamily Radboud University Nijmegen}
|
|
\fancyhead[R]{\sffamily Superconductivity (NWI-NM117), Q3+Q4}
|
|
\fancyfoot[R]{\sffamily\bfseries\thepage}
|
|
\renewcommand{\headrulewidth}{0.5pt}
|
|
\renewcommand{\footrulewidth}{0.5pt}
|
|
}
|
|
\pagestyle{fancy}
|
|
|
|
\usepackage{siunitx}
|
|
\usepackage{hyperref}
|
|
\usepackage{float}
|
|
\usepackage{mathtools}
|
|
\usepackage{amsmath}
|
|
\usepackage{todonotes}
|
|
\setuptodonotes{inline}
|
|
\usepackage{mhchem}
|
|
\usepackage{listings}
|
|
\usepackage{subcaption}
|
|
|
|
\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
|
|
|
|
\title{Superconductivity - Assignment 5}
|
|
\author{
|
|
Kees van Kempen (s4853628)\\
|
|
\texttt{k.vankempen@student.science.ru.nl}
|
|
}
|
|
\AtBeginDocument{\maketitle}
|
|
|
|
% Start from 8
|
|
\setcounter{section}{16}
|
|
|
|
\begin{document}
|
|
|
|
\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 band gap energy $\Delta(k)$ is a useful order parameter for superconductivity.
|
|
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 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}.
|
|
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}
|
|
\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 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
|
|
\[
|
|
\lambda_L(T) = \sqrt{\frac{m_e^*}{\mu_0e^2n_s(T)}}.
|
|
\]
|
|
If $n_s(T)$ can be related to energy gap $\Delta$, so can $\lambda$, and luckily we can.
|
|
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.
|
|
This in turn implies a linearly increasing relation to $\lambda(T) = \lambda(0) + cT$ for some constant $c$.
|
|
|
|
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. 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.
|
|
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.
|
|
|
|
In \cite{ozcan}, heavy-fermion superconductor \ce{CeCoIn5} is investigated using the TDO technique to measure its penetration depth.
|
|
It is an unconventional superconductor, and the question is what type of wave-symmetry it exhibits.
|
|
The paper found a non-linear $\lambda(T)$ relation.
|
|
See figure \ref{fig:linear-lambda} for their results.
|
|
They do conclude that the material is in a $d_{x^2-y^2}$ superconductor ground state.
|
|
I would expect there to be no nodes in the band gap energy, in this case, which however is the case.
|
|
The authors also seem puzzled at the beginning.
|
|
They suspect strong-scattering impurities to alter the $\lambda(T)$ relation.
|
|
To exclude this possibility, they checked a couple of possible explanations.
|
|
Purity was checked and impurity content was determined to be a factor 100 smaller than the deviation in $\lambda(T)$ would imply.
|
|
Other theories were also ruled out, on impossibility of far-fetchedness.
|
|
They conclude by proposing non-Fermi-liquid renormalisation in both the normal and superconducting state of \ce{CeCoIn5} to take place, yielding the well fitting relation as seen in the inset of figure \ref{fig:linear-lambda}.
|
|
This means that their would be quantum criticality in the superconducting state, i.e. a phase transition at zero temperature.
|
|
That would be exotic.
|
|
In conclusion, the behavior of \ce{CeCoIn5} was not explained with certainty at the point this paper was published (2003), although quantum criticality was a possibility.
|
|
However, years later (2014), further research supports their hypothesis \cite{paglione_quantum_2016}.
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=.6\textwidth]{ozcan-linear.png}
|
|
\caption{For \ce{CeCoIn5}, $\lambda(T) \propto T^2/(T-T^*)$ is plotted in the main plot. In the inset, the concluding hypothesis of the authors \cite{ozcan} is presented, i.e. $\lambda(T) \propto T^{1.5}$.}
|
|
\label{fig:linear-lambda}
|
|
\end{figure}
|
|
|
|
\bibliographystyle{vancouver}
|
|
\bibliography{references.bib}
|
|
|
|
\end{document}
|