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ass5-final
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PhysRevLett.70.3999-d-wave.png
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60
ass3-12-a-weak-junction.py
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60
ass3-12-a-weak-junction.py
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@ -0,0 +1,60 @@
|
||||
#!/bin/env python3
|
||||
|
||||
import numpy as np
|
||||
from scipy.integrate import odeint
|
||||
from matplotlib import pyplot as plt
|
||||
import pandas as pd
|
||||
|
||||
def phi_dot(phi, t, I_DC, I_RF):
|
||||
R = 10.e-3 #Ohm
|
||||
I_J = 1.e-3 #A
|
||||
omega_RF = 2*np.pi*.96e9 #rad/s
|
||||
hbar = 1.0545718e-34 #m^2kg/s
|
||||
e = 1.60217662e-19 #C
|
||||
|
||||
return 2*e*R/hbar*( I_DC + I_RF*np.cos(omega_RF*t) - I_J*(np.sin(phi)) )
|
||||
# I attempted to solve it without the constants, as I suspected overflows
|
||||
# were occurring. The next line did not improve the result.
|
||||
#return R*( I_DC + I_RF*np.sin(omega_RF*t) - I_J*(np.sin(phi)) )
|
||||
|
||||
# We need an initial value to phi
|
||||
phi_0 = 0
|
||||
|
||||
# Let's try it for a lot of periods
|
||||
N_points = 10000
|
||||
t = np.linspace(0, 100, N_points)
|
||||
|
||||
hbar = 1.0545718e-34 #m^2kg/s
|
||||
e = 1.60217662e-19 #C
|
||||
|
||||
df = pd.DataFrame(columns=['I_DC','I_RF','V_DC_bar'])
|
||||
|
||||
# For testing:
|
||||
#phi = odeint(phi_dot, phi_0, t, (.5e-3, .5e-3))[:, 0]
|
||||
#for I_DC in [1e-4, .5e-3, 1.e-3, 1.5e-3, 2.e-3, 2.5e-3]:
|
||||
|
||||
for I_DC in np.arange(0, 1e-3, 1e-5):
|
||||
for I_RF in [0., .5e-3, 2.e-3]:
|
||||
# The individual solutions for phi do seem sane, at least, the ones
|
||||
# I inspected.
|
||||
phi = odeint(phi_dot, phi_0, t, (I_DC, I_RF))
|
||||
# I initially thought to average over the tail to look at the asymptotic behaviour.
|
||||
#N_asymp = N_points//2
|
||||
#V_DC_bar = hbar/(2*e)*np.mean(phi[N_asymp:]/t[N_asymp:])
|
||||
# Then I choose to just take the last point to see if that gave better results.
|
||||
V_DC_bar = hbar/(2*e)*phi[-1]/t[-1]
|
||||
print("For I_DC =", I_DC, "\t I_RF = ", I_RF, "\twe find V_DC_bar =", V_DC_bar)
|
||||
df = df.append({'I_DC': I_DC, 'I_RF': I_RF, 'V_DC_bar': V_DC_bar}, ignore_index = True)
|
||||
|
||||
## Plotting the thing
|
||||
plt.figure()
|
||||
|
||||
plt.xlabel("$\\overline{V_{DC}}$")
|
||||
plt.ylabel("$I_{DC}$")
|
||||
|
||||
for I_RF in df.I_RF.unique():
|
||||
x, y = df[df.I_RF == I_RF][["V_DC_bar", "I_DC"]].to_numpy().T
|
||||
plt.plot(x, y, label="$I_{RF} = " + str(I_RF) + "$")
|
||||
|
||||
plt.legend()
|
||||
plt.show()
|
||||
BIN
lec4-vortexlattice.png
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|
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3
makefile
3
makefile
@ -9,3 +9,6 @@ ass2:
|
||||
|
||||
ass3:
|
||||
latexmk -xelatex superconductivity_assignment3_kvkempen
|
||||
|
||||
ass5:
|
||||
latexmk -xelatex superconductivity_assignment5_kvkempen
|
||||
|
||||
BIN
ozcan-linear.png
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|
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74
references.bib
Normal file → Executable file
74
references.bib
Normal file → Executable file
@ -108,3 +108,77 @@ Staging is observed for all four samples at low temperatures with X-ray measurem
|
||||
The temperature dependent phase transitions for both the Bmab structure and the staging reflections are investigated between 5 and 300 K. The critical exponents for the Bmab reflections are found to be significantly lower than results from similar materials in literature, although with transition temperatures consistent with literature for comparable sample compositions. It is found that the critical exponents for the staging reflections increase for increasing doping while the transition temperatures decrease, both consistent with results on the isostructural La2NiO4+y.
|
||||
|
||||
Results from previous neutron measurements are found to be consistent with the X-ray measurements in this work, and measured reciprocal space maps from this work show a large variety of other superstructure reflections which will be interesting to investigate in the future.}, publisher={figshare}, author={Ray, Pia Jensen}, year={2016}, month={Feb} }
|
||||
|
||||
@book{annett,
|
||||
edition = {1},
|
||||
series = {Oxford Master Series in Condensed Matter Physics},
|
||||
title = {Superconductivity, {Superfluids}, and {Condensates}},
|
||||
volume = {5},
|
||||
isbn = {978-0-19-850756-7 0-19-850756-9},
|
||||
url = {http://physics.ut.ac.ir/~zadeh/teachings/Adv_Super/SC_Annett.pdf},
|
||||
language = {English},
|
||||
publisher = {Oxford University Press},
|
||||
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},
|
||||
}
|
||||
|
||||
@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},
|
||||
}
|
||||
|
||||
@article{paglione_quantum_2016,
|
||||
title = {Quantum {Critical} {Quasiparticle} {Scattering} within the {Superconducting} {State} of {CeCoIn} 5},
|
||||
volume = {117},
|
||||
issn = {0031-9007, 1079-7114},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevLett.117.016601},
|
||||
doi = {10.1103/PhysRevLett.117.016601},
|
||||
language = {en},
|
||||
number = {1},
|
||||
urldate = {2022-05-18},
|
||||
journal = {Physical Review Letters},
|
||||
author = {Paglione, Johnpierre and Tanatar, M. A. and Reid, J.-Ph. and Shakeripour, H. and Petrovic, C. and Taillefer, Louis},
|
||||
month = jun,
|
||||
year = {2016},
|
||||
pages = {016601},
|
||||
}
|
||||
|
||||
Binary file not shown.
@ -32,6 +32,7 @@
|
||||
\usepackage{todonotes}
|
||||
\setuptodonotes{inline}
|
||||
\usepackage{mhchem}
|
||||
\usepackage{listings}
|
||||
|
||||
\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
|
||||
|
||||
@ -57,31 +58,119 @@ $\kappa = \frac{\lambda}{\xi} = 34 > \frac{1}{\sqrt{2}}$,
|
||||
which means we are indeed dealing with a type-II superconductor.
|
||||
As $B_{c1} < B_E < B_{c2}$, the cylinder is in the vortex state.
|
||||
From the previous set of assignments, we know what the currents in the cylinder look like.
|
||||
From free energy considerations, we have found in lecture 4 that for type-II superconductors, it is favorable to allow flux quanta inside the superconductor in this vortex state.
|
||||
In this derivation, the contribution of one flux quantum is considered, but the consideration holds for many vortices, until they start to interact and repel eachother.
|
||||
At that point, the vortex-vortex interaction orders the vortices in a lattice.
|
||||
When the vortex cores start to overlap, there are no superconducting regions left, thus the material enters the normal conducting state.\footnote{I wanted to paint a complete picture althought it is not needed to answer the question.}
|
||||
Minimizing the free energy over the flux shows the energy is lowered for determined thresholds $B_{c1} < B_E < B_{c2}$.
|
||||
|
||||
The average field inside the cylinder is gives as
|
||||
Let's start with the result from said free energy considerations.
|
||||
The average field inside the cylinder is given by the following self-consisting equation as
|
||||
\[
|
||||
\langle \vec{B} \rangle = \frac{1}{V_{\text{cylinder}}} \int_{\text{cylinder}} \vec{B}(\vec{r}) d\vec{r} .
|
||||
B = B_E - \frac{\phi_0}{8\pi\lambda^2}\ln{\frac{\phi_0}{4\exp{(1)}\xi^2B}}.
|
||||
\]
|
||||
Plugging in the values for \ce{Nb3Sn}, $B_E = \SI{1}{\tesla}$, and $\phi_0 = \SI{2.0678}{\weber}$, $B$ is found as $B = \SI{.986}{\tesla} \approx B_E$ by intersection.
|
||||
%https://www.wolframalpha.com/input?i=B+%3D+1+-+%282.0678*10%5E-15%29%2F%288*pi*%28124*10%5E-9%29%5E2%29+*+ln%282.0678*10%5E-15%2F%284*exp%281%29*%283.65*10%5E-9%29%5E2+*+B%29%29
|
||||
This is in the range as provided in the assignment ($B = \SI[separate-uncertainty]{.981\pm.019}{\tesla}$).
|
||||
|
||||
To determine this $\vec{B}$ inside the material, we first need to know how many vortices there are.
|
||||
We assume that every vortex lets through only one flux quantum $\Phi_0$,
|
||||
and that the vortices will arange themselves as far as possible from eachother.
|
||||
If their distance then is large enough to assume there is no overlap between regions of finite $\vec{B}$ around them,
|
||||
we can calculate the average field by just summing over the quanta and lastly over the field that penetrates the material in the outside of the cylinder.
|
||||
For this latter calculation, we can use the field for a type-I superconductor.
|
||||
To investigate the inhomogenity of the field inside the cylinder, we look at the gradient $\nabla B$ inside the material.
|
||||
As we assume a vortex lattice that fully fills the cross section of the cylinder,
|
||||
and we assume that the fields due to each vortex die out quickly enough to not overlap,
|
||||
it suffices to calculate the gradient over just one vortex.
|
||||
These assumptions coincide with slide 15 of lecture 4, from which I took figure \ref{fig:lec4-vortexlattice}.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\label{fig:lec4-vortexlattice}
|
||||
\includegraphics[width=.4\textwidth]{lec4-vortexlattice.png}
|
||||
\caption{The vortices are arranged in a lattice to maximize their distance, as this lowers their repulsive interaction and thus the energy.}
|
||||
\end{figure}
|
||||
|
||||
On slide 19 from the same week, we find an expression $B(r)$ for the field at distance $r$ from the vortex core as
|
||||
\[
|
||||
B = \frac{\phi_0}{2\pi\lambda^2} K_0(r/\lambda) = B_0 K_0(r/\lambda),
|
||||
\]
|
||||
where $K_0$ is the modified Bessel function of the second kind.
|
||||
For small $r$ (i.e. $r << \lambda$), we can approximate this and find that
|
||||
\[
|
||||
K_0 \propto - \ln{(r/\lambda)},
|
||||
\]
|
||||
and notice a singularity at $r = 0$.
|
||||
For the gradient we thus find
|
||||
\[
|
||||
\nabla B \propto \pfrac{K_0}{r}(r/\lambda) \propto \pfrac{-\ln{(r/\lambda)}}{r} = -\lambda/r.
|
||||
\]
|
||||
The size of the supercurrent density has the same relation, $J_S \propto 1/r$.
|
||||
|
||||
\section{Superconducting wire}
|
||||
\textbf{(a)}
|
||||
The voltage $U = \SI{1.5e-5}{\volt}$ across the wire of length $\ell = \SI{.08}{\meter}$ induces a current $J_t$. % through the resistive wire with unknown resistivity $\rho$ according to Ohm's law.
|
||||
Due to the presence of the magnetic field $B = \SI{5}{\tesla}$, if the vortices move with velocity $v_L$, a Lorentz force $f_L$ per vortex acts on the vortices.
|
||||
This results in a power input $P_L = f_Lv_L = J_tBv_L$ per vortex.
|
||||
%$\epsilon = Bv_L$
|
||||
This power should come from the current induced by the voltage, thus $P_L = \epsilon J_t = \frac{U}{\ell}J_t$.
|
||||
Equating these expressions and rewriting yields
|
||||
\[
|
||||
v_L = \frac{U}{B\ell} = \SI{3.75e5}{\meter\per\second}.
|
||||
% https://www.wolframalpha.com/input?i=1.5*10%5E-5+%2F+%285*+.08%29
|
||||
\]
|
||||
|
||||
\textbf{(b)}
|
||||
The vortices are aranged in a lattice with separation $r_{sep} = \sqrt{\frac{\Phi_0}{B}}$.
|
||||
They move along the wire with velocity $v_L$ as determined above.
|
||||
The expected frequency is then given by their velocity over the separation, as that is the period of the changing fields due to the vortices:
|
||||
\[
|
||||
f = \frac{v_L}{r_{sep}} = \frac{U}{B\ell}\sqrt{\frac{B}{\Phi_0}} = \frac{U}{\ell\sqrt{B\Phi_0}} = \SI{1.84}{\kilo\hertz},
|
||||
% https://www.wolframalpha.com/input?i=1.5*10%5E-5+%2F+%28.08%29+%2Fsqrt%285+*+2.067*10%5E%28-15%29%29
|
||||
\]
|
||||
where we used that $\Phi_0 = \SI{2.067e-15}{\volt\second}$.
|
||||
This is very close to what is written in the assignment, but not precisely the same, so maybe I used a different value for $\Phi_0$.
|
||||
|
||||
\section{Fine type-II superconducting wire}
|
||||
|
||||
\section{Critical currents}
|
||||
\textbf{(a)}
|
||||
Silsbee's rule states that the supercurrents through the wire must not generate magnetic fields in excess of $B_c$ at the surface of the wire.
|
||||
We assume that the supercurrent is maximal at the surface with a maximum value of $J_{max}$, and that the supercurrent decays linearly from the surface to zero at a penetration depth $\lambda$ deep.
|
||||
We thus find a relation for the supercurrent as function of the cylindrical radius $r$ as
|
||||
\[
|
||||
J_s(r) = \frac{J_{max}}{\lambda} \left[ r - R + \lambda \right].
|
||||
\]
|
||||
|
||||
Now we can use the Maxwell-Amp\`ere law to find this value for $J_{max}$.
|
||||
\[
|
||||
\oint \vec{B}\cdot d\vec{\ell} = \mu_r\mu_0\iint\vec{J_s}\cdot d\vec{S}
|
||||
\]
|
||||
Using $\vec{B} = \vec{B_c}$, $\mu_r = 1$ as we're calculating the field outside the sc, $J_s = J_s(r)$, the area over which $d\vec{S}$ runs to be the small ring from $r = R - \lambda$ to $r = R$, and the path along which $d\vec{\ell}$ runs to be the loop $2\pi R$ along the surface of the wire.
|
||||
This gives us
|
||||
\[
|
||||
2\pi R B_c = \mu_0 \int_{\phi = 0}^{2\pi}\int_{r=R-\lambda}^R \frac{J_{max}}{\lambda}\left[ r - R + \lambda \right] rdr d\phi.
|
||||
\]
|
||||
Solving the integral over $\phi$ results in
|
||||
\[
|
||||
R B_c = \mu_0 \int_{r=R-\lambda}^R \frac{J_{max}}{\lambda}\left[ r - R + \lambda \right] rdr = \frac{\mu_0J_{max}}{\lambda} \left[ \frac{r^3}{3} - \frac{(R + \lambda)r^2}{2} \right]_{r = R-\lambda}^R.
|
||||
\]
|
||||
Solving for $J_{max}$, this yields the beautiful expression
|
||||
\[
|
||||
J_{max} = \frac{6B_c\lambda R}{\mu \left[ 4\lambda^3 - 9\lambda^2R + 3\lambda R^2- 3\lambda R + 3R^3 -3R^2 \right]}.
|
||||
% https://www.wolframalpha.com/input?i=R*B+%3D+m*x%2Fl*%28%28R%5E3-%28R-l%29%5E3%29%2F3+-+%28R%2Bl%29*%28R+-+%28R-l%29%5E2%29%2F2%29
|
||||
\]
|
||||
|
||||
\textbf{(b)}
|
||||
|
||||
|
||||
\section{A weak junction}
|
||||
See the code in appendix \ref{appendix:program-task-12}.
|
||||
|
||||
It unfortunately does not seem to produce any useful results.
|
||||
In the code, I left many comments as it is mostly in a debugging state.
|
||||
|
||||
\bibliographystyle{vancouver}
|
||||
\bibliography{references.bib}
|
||||
|
||||
%\appendix
|
||||
\appendix
|
||||
\section{Program to task 12}
|
||||
\label{appendix:program-task-12}
|
||||
\lstinputlisting[language=python,breaklines=true]{ass3-12-a-weak-junction.py}
|
||||
|
||||
\end{document}
|
||||
|
||||
BIN
superconductivity_assignment4_kvkempen.pdf
Normal file
BIN
superconductivity_assignment4_kvkempen.pdf
Normal file
Binary file not shown.
BIN
superconductivity_assignment5_kvkempen.pdf
Executable file
BIN
superconductivity_assignment5_kvkempen.pdf
Executable file
Binary file not shown.
242
superconductivity_assignment5_kvkempen.tex
Executable file
242
superconductivity_assignment5_kvkempen.tex
Executable file
@ -0,0 +1,242 @@
|
||||
\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}
|
||||
Reference in New Issue
Block a user