A Study of Narrowband Noise Characteristics Associated with
Vortex Motion in High Temperature Superconductors
Thomas J Bullard III
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Physics
Uwe C. Täuber, Chair
Beate Schmittmann
Tetsuro Mizutani
Jimmy Ritter
Dick Zallen
May 11, 2005
Blacksburg, Virginia
Keywords: nonequilibrium Monte Carlo, superconductor, vortex, washboard noise
Copyright 2005, Thomas J Bullard III
A Study of Narrowband Noise Characteristics Associated with Vortex Motion in High
Temperature Superconductors
Thomas J Bullard III
ABSTRACT
Vortex motion plays an important role in the transport properties of high Tc superconductors. In the
presence of a sufficiently large applied current vortices will drift creating an ohmic resistance in the
material, while defects in the material will tend to inhibit their motion. Some types of material defects
are more effective at pinning then others, and therefore, above the depinning threshold, may effect the
motion of vortices differently. To investigate their motion, voltage noise generated by moving vortices is
studied for different material defect types using a nonequilibrium Metropolis Monte Carlo simulation.
The current-voltage (I-V) characteristics obtained from the simulation for various vortex densities and
defect types show features similar to those obtained in experiments. The power spectra generated for
point and columnar disorder are then compared for increasing vortex density. Above, but near the
depinning threshold, broadband noise associated with plastic vortex flow is observed for columnar
defects at low vortex densities, while for higher densities a triangular lattice is obtained along with a
washboard signal and higher harmonics. For point defects a washboard signal with higher harmonics is
always observed in the region investigated. These results suggest that power spectra for both point and
columnar defects are qualitatively similar for higher vortex densities (larger magnetic fields). A second
comparison is made by observing, on the one hand, the power spectra for finite linear defects increasing
in length and, on the other hand, increasing point defect strength. Power spectra and structure factor
results are very similar for these results as well. Both show a trend from an ordered to a disordered
system with a washboard peak first increasing and then decreasing in power with increasing pinning
efficiency. For both defect types the power spectrum is eventually dominated by broadband noise
indicating the approach to the pinned glassy phases.
Acknowledgements
I would like to thank my advisor, Uwe Täuber, for his support through this process. Without his wisdom
and patience none of this would have been possible. Uwe’s friendship made this work bearable in the tough
times and joyous in the good times. I would also like to thank Beate Schmittmann for her willingness to
answer an research questions I might have.
I would like to thank Jayajit Das for ”spearheading” this work. Jayajit wrote a large portion of the
simulation, and his insights into the underlying physics concerning this research were useful throughout
the entire research process. I also want to thank him for his willingness to answer any questions I had.
I would like to acknowledge George Daquila who has been a great help programming sections into
the simulation as well as getting the code ready for System X. Some of the fruits of that work presented
here.
I owe a debt of gratitude to Chris Thomas for guiding me through all of the administrative hoops, as
well as Roger Link who kept me up and running on the computer network.
Finally, I would like to thank my family for being a constant support and encouragement through
these six years.
iii
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Approach to Modeling Vortices in NESS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Quantized Vortices in Superconductors
2.1 Vortex Structure . . . . . . . . . . . . .
2.2 The London Approximation . . . . . . .
2.3 Equilibrium Vortex Phases . . . . . . .
2.4 Nonequilibrium Vortex Phases . . . . .
1
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3 Voltage Noise in Type II Superconductors
3.1 Early Experiments Relating Resistance and Voltage Noise to Vortex Motion . . . . . . . .
3.2 Washboard Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Effects of Disorder on Vortex Motion and Shape . . . . . . . . . . . . . . . . . . . . . . .
19
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4 The Model
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5 Results
5.1 Vortex Lattice Orientation in Equilibrium . . . . . . . . . . . . . . . . .
5.2 I-V Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Narrow Band Noise Characteristics . . . . . . . . . . . . . . . . . . . . .
5.3.1 Randomly Distributed Columnar Defects . . . . . . . . . . . . .
5.3.2 Point Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Variable Columnar Defect Length versus Point Pinning Strength
5.4 Finite Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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29
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36
36
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6 Summary and Conclusion
65
A Obtaining the Power of Individual Peaks in the Power Spectrum
68
iv
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
5.1
The origin of surface energy for a type I and type II superconductor. a) The penetration
depth and coherence length at the boundary. b) The contributions to the free energy. c)
The total free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic flux versus applied magnetic field is plotted for type I and type II materials.
Compared to type I materials the transition to the normal state is delayed for type II
materials by allowing magnetic field to penetrate starting at Hc1 . For type II materials
the transition to the normal state occurs at Hc2 . . . . . . . . . . . . . . . . . . . . . . . .
The mixed state in an applied magnetic field of strength greater than Hc1 . a) A lattice of cores with encircling currents. b) The variation with position of concentration of
superelectrons. c) The variation of the flux density. d) A schematic of a vortex. . . . . .
Generation of e.m.f. by flux flow in the mixed state . . . . . . . . . . . . . . . . . . . . . .
Typical v-f characteristic for T = 0 and T > 0. Three main regions are distinguished:
the creep regime, characterized by thermally activated motion well below the threshold,
the critical regime, characterized by plastic flow near the threshold, and the large velocity
region well above the threshold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Snapshot of a perfect (nondisordered) lattice moving along the x direction (b) By
superimposing images at different times one would see that at T = 0 the particles follow
perfectly straight lines. (For T > 0 the channels remain straight, but increase in width.)
(c) In the presence of weak disorder the particles are elastically coupled between channels
and the lattice remains topologically ordered. This is known as the moving Bragg glass.
(d) For stronger disorder the positions of the particles may decouple. This is the moving
transverse glass. (e) Superposition of snapshots from (c) or (d) show particles traveling
through elastic channels. (f) For d=3 vortex channels are ”sheets” that can form an
anisotropic type of smectic layering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
8
10
12
15
18
Depinning from a columnar defect via a ”half-loop” and ”double kink” excitation in the
direction of the drive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The solid line represents a possible configuration for a flux-line moving through point
disorder. The dotted line represents a configuration at a later time in the presence of a
force per unit length fL . All arrows lie parallel to the upper and lower planes. Note the
arrows are not necessarily parallel to the direction of the drive. . . . . . . . . . . . . . . .
23
A triangular
lattice composed of 25 points (left) does not ”fit” in the rectangular system
√
(2 : 3 aspect ratio) with periodic boundary conditions. The top row is not at the nearest
neighbor locations for the bottom row (located by a circle with an x through it). On the
other hand, the triangular lattice composed of 16 points (right) does preserve the triangular
lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
v
23
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Coordination number
histogram measured at equilibrium for the system size aspect ratio
√
(a) 1:1 (b) 2 : 3. (c) Density correlation plot of a typical square vortex lattice found in
(a). The square lattice is a more readily available vortex configuration for the system with
a square aspect ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Orientation√angle histogram measured at equilibrium for the system size aspect ratio (a)
1:1 (b) 2 : 3. c) Orientation angle is measured from the horizontal axis (the long axis of
the rectangular system) to the closest lattice vector. Lattices oriented at approximately
±5◦ as shown in d) satisfy a chiral wrapping about the periodic boundary conditions of
the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity vs Force curves plotted for random columnar and point defects for various vortex
densities. For random defects the depinning threshold decreases with increasing density.
The same occurs for point defects. This can be seen most easily in the inset in (b). The
force per unit length is reported in units of ²0 and the velocity in units of b0 /M CS where b0
is the pin radius and M CS stands for Monte Carlo steps. The following symbols represent
densities reported as the number of lines in an area of size 150 √23 b0 × 150b0 . N − 144,
¥ − 100, F − 64, ° − 36, × − 16. Data points have been connected as a guide for the eye.
I-V curves for interacting (N) and noninteracting (¥) vortices. Note the curves cross at a
force approximately equal to .1. Vortex motion is limited by the repulsion of neighbors,
hence a more quickly saturating I-V curve for interacting vortices. The force per unit
length is reported in units of ²0 and the velocity in units of b0 /M CS. . . . . . . . . . . . .
Washboard frequency versus number of vortices in the system. Results show good agreement between the measured and calculated values. Vortex density is reported as the
number of vortices per unit area. ω is reported in units of rad/MCS. Error bars for the
measured values are obtained from the full width at half-maximum of the washboard peaks
in Fig 5.7. N - measured, ¥ - calculated . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity power spectra measured in the x-direction and diffraction plots are displayed for
increasing vortex density in the presence of columnar defects. For 16 lines a diffraction
plot associated with an isotropic liquid is obtained along with a broadband signal in the
velocity power spectrum. By 36 lines two peaks appear located transverse to the drive
direction in the diffraction plot accompanied with a drop in broadband noise in the power
spectrum. The peaks in the diffraction plot indicate the forming of parallel plastic channels
running along the drive direction. At 64 lines the central peak in the diffraction plot is
surrounded by 6 other peaks. The peaks with an x-component are lower than those with
only a y-component indicating greater order in the y-direction. In the power spectrum a
large washboard frequency peak appears as well as higher harmonics. As the density is
increased to 100 and then 144 lines the lattice shows complete macroscopic order, and the
peaks in the power spectrum drop in power and decrease in width. ω is reported in units
of rad/MCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power spectra measured in the y-direction for increasing vortex density. Pinning centers are
randomly distributed columns. The power spectrum developes in a manner qualitatively
similar to the results in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Delaunay triangulation plots for 16 (a) and 64 lines (b) with random columnar defects.
Topological defects have been marked. A disordered structure is obtained for a ”snapshot”
of 16 lines, while a triangular array free of topological defects is obtained for 64 lines. A
”time exposure” for 16 and 64 lines has been plotted in (c) and (d) to give a sense of the
motion. For 16 lines, trajectories cross suggestive of plastic flow, while for 64 lines parallel
elastic channels have formed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
31
32
33
34
35
40
42
43
5.10 Components of the average radius of gyration for columnar defects. The radius of gyration decreases with increasing vortex density. At higher densities vortices are squeezed,
preventing stretching. A larger deviation in the average stretching value is observed at
64 lines, and might help explain the larger width of the corresponding narrowband peak.
Length is reported in units of the pin radius b0 . N - x-component, ¥ - y-component . . .
5.11 Velocity power spectra measured in the x-direction and diffraction plots for increasing
vortex density with point defects. Six-fold coordination of the central peak is observed for
all plots. For 16 lines the Bragg peaks decrease with increasing wave number. By 36 lines
the system shows all peaks with equal height in the diffraction plot and only remnants of
broadband noise in the power spectrum remain. . . . . . . . . . . . . . . . . . . . . . . .
5.12 Delaunay triangulation for 16 and 36 lines with random point defects. Topological defects
have been marked. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Power spectra measured in the y-direction for increasing vortex density. Pinning centers are
randomly distributed point pins. The power spectrum develops in a manner qualitatively
similar to the results in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 Components of the average radius of gyration for point defects. The radius of gyration decreases with increasing vortex density. At higher densities vortices are squeezed, preventing
stretching. N - x-component, ¥ - y-component . . . . . . . . . . . . . . . . . . . . . . . .
5.15 Power spectra and diffraction plots for increasing columnar defect length. Results reveal
the system evolves from order to disorder as the length increases as shown in the diffraction
plots. Likewise the washboard signal decreases as the broadband noise grows. . . . . . .
5.16 Results for pinning strength increased from its original value, p0 to 1.5p0 . These results
are qualitatively similar to results obtained by increasing columnar defect length in Fig 5.15.
5.17 Radius of gyration (¨ - x-component, F - y-component) for increasing columnar defect
length and increasing pinning strength. The data for increasing length is fit to an exponential for both components, while the data for increasing pinning strength is best fit to a
quadratic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.18 Structure factor and power spectra plots for systems with an extended cutoff length in the
presence of columnar defects. The cutoff length is double the length used in Fig 5.7(a)
and Fig 5.7(b). Unlike the results for a short cutoff length, peaks do not appear in the
structure factor plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.19 Structure factor and power spectra plots for systems with an extended cutoff length in the
presence of point pins. The cutoff length is double the length used in Fig 5.11(a) through
Fig 5.11(c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 Power spectra and structure factor plots for 64 lines of a length 3 times that of all other
runs. The results for (a) are very similar to those of the same density for shorter lines
for columnar defects shown in Fig 5.7(c). For point defects the power drops noticeably
compared to Fig 5.11(c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 A typical power spectrum plot used to demonstrate how the area is found under a peak .
vii
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48
49
51
52
56
58
59
61
62
63
69
List of Tables
5.1
5.2
5.3
Point Defects: Power of Largest Three Peaks . . . . . . . . . . . . . . . . . . . . . . . . .
Columnar Defects: Power of Largest Three Peaks . . . . . . . . . . . . . . . . . . . . . . .
Point Defects: Power of Largest Three Peaks for an Extended Cutoff Length . . . . . . .
viii
46
49
64
Chapter 1
Introduction
1.1
Motivation
The phenomenon of superconductivity has held great promise for many technical applications ever since
its discovery in 1911 by Heike Kamerlingh Onnes [2]. The two hallmarks of superconductivity, perfect
conductivity and perfect diamagnetism, are manifested when certain materials (e.g. most metals) are
brought below some critical temperature and magnetic field. While such properties present great energy
saving potential the critical parameters are very often prohibitive. For instance, mercury transitions to
the superconducting state below a critical temperature of 4 K and a critical field of .04 Tesla rendering
many applications unfeasible and scientific investigation difficult.
In 1986 a new class of superconducting materials was discovered by Bednorz and Müller [27] rekindling
interest in the field. Labeled ”high temperature”, some of these superconducting ceramic oxide materials
possess critical temperatures that extend beyond the boiling point of liquid nitrogen (77 K). Furthermore,
these materials fall into a previously known class of materials (type II) that allowed quantized amounts
of magnetic flux to penetrate into the superconductor delaying the transition to the normal state.
However, limitations remain. First, these materials are brittle and cannot be easily drawn into wire
making their application difficult. Second, as with all type II materials penetrating magnetic flux lines
(also known as vortices due to the current encircling each flux quantum) experience a Lorentz force in
the presence of an external current resulting in a drift velocity. The motion of the magnetic field yields
an electric field across their normal cores resulting in Ohmic power dissipation.
Despite these challenges there have been some successes. Among small scale devices the best known
is the SQUID magnetometer. (SQUID is an acronym for Superconducting Quantum Interference Device.) A SQUID is composed of a superconducting ring separated at certain points by a ”weak link”
known as a Josephson junction. Superconducting electrons tunnel through these links and the associated
superconducting wave function passes through the junctions as though it were passing through parallel
slits resulting in a modulating current. The phase difference of the wave function passing through the
two junctions is related to the amount of magnetic flux passing through the ring. One period of current
1
variation corresponds to the penetration of one quantum of flux through the center of the ring enabling
the magnetometer to detect magnetic fields 100 billion times smaller than the Earth’s field.
Other small scale devices that have reached commercial availability include passive radio and microwave frequency filters for wide band communication and radar. These filters have the advantage
of very low noise and much higher selectivity enabling desired frequencies to be passed and undesired
frequencies to be blocked in high congestion applications such as cellular telephone systems.
Among large scale applications transmission of large currents and generation of strong magnetic fields
are of primary interest. Superconducting power transmission cables carrying up to five times the current
of a copper cable of the same diameter are coming into commercial use in some cities. Other large
scale applications include powerful magnets used in particle accelerators and magnetic resonance imaging
(MRI), as well as electric generators and motors which are far more efficient and about half the size of
their conventional counterparts. For an extended review of superconductor applications see [60].
In large scale applications the transport of current without energy loss in magnetic fields is highly
desirable; therefore, the motion of vortices is of particular importance. As previously mentioned, in the
presence of applied currents vortices tend to move and dissipate power making superconducting materials
much less useful for application. Naturally occurring pinning sites such as oxygen vacancies and twin
grain boundaries tend to delay the onset of motion, and more effective centers such as columnar defects
can be introduced into the material through heavy ion irradiation. However, as an applied current or
magnetic field is increased vortices will depin putting an upper limit on the ability of applications to
transport current without loss.
It is the goal of this study to investigate the effects of different pinning structures on the velocity
noise spectrum of the moving vortices. In experiments this spectrum is obtainable by measuring the
voltage (proportional to the average velocity) across the material in question. It is hoped that voltage
noise might help to characterize the relevant type of pinning centers in the material.
1.2
Approach to Modeling Vortices in NESS
The dynamical properties of a classical many-body system are governed by Newton’s equations of motion.
Its temporal evolution may be completely described if the initial conditions of each particle and the total
potential energy of the system are known. However, a fully atomistic description is rarely possible for
macroscopic systems, and regardless is not the aim of this study. Rather a coarse-grained picture of the
motion of vortices is sufficient, where a subset of the system’s degrees of freedom is considered. Therefore,
a stochastic approach has been employed where the degrees of freedom that are left out of the description
(i.e. those with shorter time and length scales then the relevant degrees of freedom) act as a heat bath
inducing transitions among the relevant degrees of freedom [58].
Such an approach is well established for systems in thermal equilibrium [1]. The probability of a
system, at fixed temperature T, being in a particular configuration C is proportional to the Boltzmann
factor e−βH(C) where β = 1/kB T and H(C) is the Hamiltonian for configuration C. If one can obtain
the Hamiltonian for each configuration one may then calculate averages of various macroscopic quantities
2
associated with the system.
The two advantages of describing systems in equilibrium are 1) that the distribution of configuration
probabilities is time-independent, and 2) all that is required to obtain the distribution of configuration
probabilities are the Hamiltonians for the microscopic configurations. Unfortunately the vast majority of
systems in nature are not in equilibrium. In this case, probabilities are usually time-dependent, and the
systems energy can no longer be described by a configuration dependent Hamiltonian making the problem
intractable. The approach then is to study nonequilibrium systems that have settled into a steady state.
While still non-Hamiltonian, the probability distributions are once again time-independent.
For the current system of interest, a nonequilibrium steady state is a good description of vortices
driven by a constant applied current and will be the direction our model will follow. More formally, we
consider a generic stochastic process that generates a series of system configurations C1 → C2 → ...Cn ,
where the probability of the system being in a new configuration is based on prior configurations:
P (Cn |Cn−1 , Cn−2 , ...C1 ).
(1.1)
For the special case where the probability of a new configuration is dependent upon only the immediately
preceding configuration we may interpret the probability as a transition rate to move from Ci to Cj :
W (Ci → Cj ) = P (Cj |Ci ).
(1.2)
For such transition rates the sequence of configurations is known as a Markov chain, and may be interpreted as the dynamical evolution of the system where time is associated with the sequential labels of
the states [26].
Since total probability is conserved a continuity or master equation may be used to describe the
time-dependent probability of being in configuration Cj
dP (Cj , t) X
=
[W (Ci → Cj )P (Ci , t) − W (Cj → Ci )P (Cj , t)].
dt
(1.3)
Ci
The two terms in the sum represent ”gain” and ”loss” terms increasing or decreasing the probability of
being in configuration Cj as time progresses [49].
The specific choice of rates for the master equation determines both the dynamics of the model,
and, with appropriate boundary conditions, the steady state distribution P ∗ (C) = P (C, t → ∞) at
fixed temperature T . To obtain these rates, thermal equilibrium is first considered where P ∗ (C) =
Peq (C) ∼ e−βH . For the model to tend towards the equilibrium distribution the condition of detailed
balance is imposed. Detailed balance requires that each pair of terms in the master equation cancel out
dP
guaranteeing dteq = 0. By plugging the equilibrium distribution into the master equation and imposing
detailed balance the ratio of the transition rates is obtained,
Peq (Cj )
W (Ci → Cj )
=
= e−β∆H ,
W (Cj → Ci )
Peq (Ci )
3
(1.4)
where ∆H = H(Cj ) − H(Ci ). One choice of rates that satisfy this ratio is given by the well known
Metropolis rates which are implemented as follows: a system configuration is generated and its energy is
compared to that of the previous one. If the change in energy is negative the new configuration is kept.
If ∆H is positive the new configuration is kept if e−β∆H ≤ r where r is a randomly generated number
in the unit interval. The implementation of the rates may be simply expressed as min[1, e−β∆H ]. It is
important to note that the Metropolis rates generate a sequence of configurations that may be interpreted
as a Markov chain since the probability of acceptance of a new configuration is only dependent upon the
difference in energy between itself and the prior configuration allowing us to interpret the results as a
possible time evolution of the system.
To achieve a non-equilibrium steady state a driving field F is introduced. Such a field favors motion
along its direction while suppressing motion against it and leaving transverse motion unaffected. The
field is included by adding a work term to the Hamiltonian [23]:
W (Cj → Ci ) = e−β(∆H+lF )
(1.5)
where l specifies displacement in the system. Note that Eq.(1.5) retains the form of Eq.(1.4), a ”local
detailed balance” condition, enabling one to still define a ”local temperature” for the model. Also note
that in the limit as F → 0 the rates return to the thermal equilibrium result obtained in Eq.(1.4).
Finally, since we are interested in non-equilibrium steady states, periodic boundary conditions are
chosen to exclude the dynamics from being derivable from a global Hamiltonian. Combined with the
driving field a non-trivial global current may be induced while maintaining translational invariance.
In conclusion, it should be kept in mind that the description of a dynamical process via Monte Carlo
methods is a phenomenological modeling of what one thinks are the essential steps of a coarse grained
description, and therefore is most representative of reality when one has a good separation of time scales.
In fact, for a very good separation of time scales it is possible to reduce the Liouville equation to a
Markovian master equation such as that written in Eq.(1.3) [58].
1.3
Overview
In the following chapter a phenomenological description of the pertinent structure and behavior of superconducting vortices is given as well as an overview of their nonequilibrium phases. In chapter 3 a brief
history is given on the measurement of voltage noise in type II superconductors as well as a discussion of
possible effects of disorder on the related velocity fluctuations. A description of the model is presented
in chapter 4. In chapter 5 the results are presented, and chapter 6 provides a summary and conclusion
to the results.
4
Chapter 2
Quantized Vortices in
Superconductors
The microscopic theory of superconductivity was presented in 1957 by Bardeen, Cooper, and Schrieffer
[10], and has had good success in explaining superconducting phenomena. Central to this theory is the
phonon mediated pairing of electrons of opposite spin and momenta forming a new bound state known as
a Cooper pair. With a net spin of zero, Cooper pairs behave as bosons making the Fermi surface unstable
and condensing into a new bosonic ground state. In this new state (separated from the excited state by
an energy gap proportional to the pairing energy) Cooper pairs behave collectively, acting as a single
quantum mechanical entity, and may be represented by a coherent wave function extending over the
entire volume of the superconductor. Consequently, the astounding (and useful) properties of materials
in the superconducting state may be described as macroscopic quantum mechanical phenomena.
The phenomenon central to this study is the existence of quantized magnetic vortices. However, we
are interested in the ”bulk” motion of vortices rather than their quantum mechanical origins. Therefore,
the following sections are meant to provide physical intuition for their origin and fundamental properties
rather than a strict quantum mechanical derivation.
2.1
Vortex Structure
To describe the origin and structure of vortices it is first necessary to discuss two important characteristic
length scales common to all superconductors. The first may be obtained from the London equation that
describes the bulk magnetic properties of a superconductor:
B=−
4πλ2
∇ × J,
c
5
(2.1)
where λ (the first characteristic length of interest) is known as the penetration depth. The second London
equation describing perfect conductivity is
E=
4πλ2 ∂
J
c2 ∂t
(2.2)
and is included for completeness. Combining Eq.(2.1) with the Maxwell equation
∇×B =
4πJ
c
∇2 B =
B
.
λ2
(2.3)
yields
The solution of this equation reveals that magnetic fields are exponentially screened from the interior
of the material with characteristic length λ regardless of time dependence unlike a ”perfect conductor”.
This superconducting screening is known as the Meissner effect [21].
The magnetic fields impinging on the material are screened by electrical currents set up inside the
superconductor; however, the behavior of the electrons that make up the screening currents is obviously
different from the behavior in a material that is not superconducting. In the framework of the two-fluid
model [3], below the transition temperature the conduction electrons can be segregated into two classes,
those behaving as ”superconducting electrons” (i.e. those that have formed Cooper pairs), and those
behaving as normal electrons. From Ginzburg-Landau theory a pseudowavefunction Ψ(r) acting as a
2
complex order parameter maybe introduced where |Ψ(r)| is equal to the density of the superconducting
electrons ns (r). The value of Ψ(r) (and hence the density of super-electrons) will take on a value lower
than its maximum in the presence of a magnetic field, but will return to a maximum value deep inside
the bulk of the material where all magnetic fields are screened. The characteristic distance over which
the wave function returns to its maximum value is known as the coherence length ξ [17].
Having a basic understanding of what these two characteristic lengths represent we now consider the
free energy of the system. Due to the Meissner effect, a superconductor displays a perfect diamagnetic
response. Therefore, in the presence of an applied magnetic field Ha the free energy of a superconductor
will increase by an amount Ha2 /8π [17]. However, above an applied field strength Hc it becomes energetically unfavorable for a material to remain in the superconducting state; it will transition to the normal
state allowing the applied field to penetrate. Depending on the geometry of the surface, a magnetic field
below Hc may effectively be above the critical field in certain areas of the materials, and as a result part of
the material may return to the normal state creating interfaces between the normal and superconducting
sections. For this ”intermediate state” to be in equilibrium the free energy density on either side of the
interface must be equal. As shown in Fig. 2.1 (a) and (b), as the magnetic field from the normal region
penetrates into the superconducting region it is screened exponentially with characteristic length λ, and
the free energy per unit volume associated with this diamagnetic response increases by a value of Hc2 /8π
6
Type I
Normal
Type II
Superconducting
Normal
Number of
Superelectrons
Magnetic
flux density
Magnetic
flux density
ξ
Superconducting
Number of
Superelectrons
ξ
a)
λ
Free energy
density
b)
g
λ
Hc
__
8π
n
g
2
ξ
λ
Free energy
density
2
λ
Hc
__
8π
n
2
ξ
Free energy
density
c)
g
2
Hc
__
8π
Hc
__
8π
Free energy
density
g
n
n
Figure 2.1: The origin of surface energy for a type I and type II superconductor. a) The penetration
depth and coherence length at the boundary. b) The contributions to the free energy. c) The total free
energy
in response to the field strength in the normal region. On the other hand, the density of superconducting
electrons increases with length ξ while their contribution lowers the free energy density due to their more
favorable thermodynamic ground state. Since the intermediate state is a stable state the two regions
must be in equilibrium, and therefore the free energy increase from the magnetic field is canceled by the
condensation energy of the superconducting ground state.
However, the characteristic lengths tend to differ, and therefore so do the gradients of the free energy
contributions. Hence the energy contributions tend not to cancel at the interface as shown in Fig. 2.1
c). For instance, if the coherence length is longer than the penetration depth the total free energy is
positive at the boundary and the interface surface area must be minimized to minimize the total free
energy. On the other hand, for materials with a negative surface energy it turns out that at a field
strength Hc1 below Hc it is energetically favorable for magnetic field to penetrate if the interface surface
area is maximized. Such materials are classified as type II materials while materials with a positive
surface energy are classified as type I. The exact breakpoint in Ginzburg-Landau theory between these
√
two regimes is when κ = λ/ξ = 1/ 2, and the surface energy of the interface may be approximated as
Hc2 (ξ − λ)/8π. The magnetic flux versus applied magnetic field is plotted for type I and II materials in
Fig 2.2.
To maximize negative surface energy normal regions subdivide in type II materials increasing the
7
Flux Penetration
B
Type II
Type I
Hc1
Hc
Hc2
H
Figure 2.2: Magnetic flux versus applied magnetic field is plotted for type I and type II materials.
Compared to type I materials the transition to the normal state is delayed for type II materials by
allowing magnetic field to penetrate starting at Hc1 . For type II materials the transition to the normal
state occurs at Hc2 .
interface surface area until a magnetic quantum limit is reached. Threading through the material, centered
on each normal region (also known as a core), is a single quantum of flux Φ0 = hc/e∗ where e∗ = 2e
is the charge of the Cooper pair. To demonstrate this quantization we consider the total magnetic flux
through a region in the material:
Z
0
Φ =
(B +
4πλ2
∇ × J ) · dS,
c
(2.4)
where the first term is the magnetic flux through an area due to the applied field, and the second term
is the flux due to the superconducting currents. Eq.(2.4) may be rewritten as
I
Φ0 =
(A +
m∗ c
v) · dl,
e∗
(2.5)
where m∗ = 2me is the mass of the Cooper pair. The contour of integration encloses any region where
magnetic field penetrates the material. The integrand may be identified as the canonical momentum
associated with the motion of a charged particle, and may be rewritten as
c
Φ =~ ∗
e
0
I
∇φ · dl,
where φ is the phase of the superconducting wavefunction.
8
(2.6)
Due to the Aharanov-Bohm effect, as an electron traverses a path in the presence of a magnetic vector
potential its wave function will pick up an additional phase. For the superconducting wave function to
remain single valued as the closed loop is traversed φ must change by integral multiples of 2π leading to
flux quantization:
~
c
e∗
I
∇φ · dl = nΦ0 .
(2.7)
From an historic point of view, F. London was the first to recognize that superconductivity was a macroscopic quantum phenomenon, and guided by this insight suggested that Eq.(2.6) must conform with the
Bohr-Sommerfeld quantization rule for the (quasi-classical) motion of an electron [6].
From Maxwell’s equation the velocity of the currents associated with the quantized magnetic flux may
be obtained. By rewriting J in terms of v in Eq.(2.3) and substituting for the second term in Eq.(2.5)
we have
I
(A + λ2 ∇ × B) · dl = Φ0 .
(2.8)
The contour of integration is chosen to be at a distance from the flux axis where ns is a constant, and hence
the second term dominates allowing the first term to be set to zero. As shown in Fig. 2.3 the orientation
of the magnetic flux is chosen to be in the ẑ direction, and hence has only a radial dependence; therefore,
∇ × B = dB(r)
dr θ̂. By performing the integral it is found that
dB(r)
Φ0
=
θ̂.
dr
2πrλ2
(2.9)
By substituting Eq.(2.9) into Eq.(2.3) and noting that the magnitude of superconducting current density
is ns evs and λ2 = mc2 /4πns e2 the radially dependent current velocity profile is obtained:
vs =
~
θ̂.
2mr
(2.10)
Eq.(2.9) may also be integrated to obtain an approximate solution for the magnetic field:
B(r) =
Φ0
λ
ln .
2
2πλ
r
(2.11)
The encircling supercurrents allow only a quantized amount of flux to enter the normal region and
screen the flux from the bulk of the material. As a result the quantum of magnetic flux is spatially
distributed such that it takes a maximum value at the center of the normal region and at large distances
decays exponentially into the superconducting bulk with characteristic length λ. (Eq.(2.11) describes the
magnetic flux for λ À r À ξ. A better approximation for B(r) will be obtained in the next chapter.)
Likewise, at the center of the core the superconducting electron density is zero and increases to a maximum
value with characteristic length ξ. The quantum of magnetic flux, the normal core, and the encircling
9
J
Ha
a)
Φ
0
ns
b)
d)
2ξ
J
x
B
c)
2λ
z
x
r
Figure 2.3: The mixed state in an applied magnetic field of strength greater than Hc1 . a) A lattice of
cores with encircling currents. b) The variation with position of concentration of superelectrons. c) The
variation of the flux density. d) A schematic of a vortex.
supercurrents are the elements that comprise what is known as a vortex [67].
As the magnetic field is increased the vortex cores are packed more tightly together, and eventually at
the upper critical field Hc2 the separation distance becomes of the order of ξ (i.e. the cores overlap). At
this upper critical field the superconductor returns to the normal state. Due to the partial flux penetration, the diamagnetic energy cost of screening the magnetic field is less than in a type I superconductor,
√
so Hc2 is greater than Hc , and it turns out that Hc2 = 2κHc .
If a vortex is in the presence of a current it will experience a repulsive force
f = J (r) ×
Φ0
,
c
(2.12)
where f is the force per unit length on the vortex, J (r) is the current density at r, and Φ0 is parallel to
the applied field. As a result, parallel vortices will repel each other and tend to arrange themselves into
a triangular lattice (known as an Abrikosov lattice) allowing for greatest separation of nearest neighbors
as shown in Fig 2.3. Of course most materials are not perfect, and therefore locations in the material
exist where no superconductivity occurs. These material defects can occur naturally as oxygen vacancies
or twin grain boundaries, or they can be artificially introduced through heavy ion irradiation creating
columnar defects. By overlapping vortex normal cores with non-superconducting locations the cost in
free energy is minimized and as a result some of the vortices may be pinned to these sites resulting in
what may be described as a vortex glass rather than a regular triangular lattice.
If a transport current is applied the vortices will experience a Lorentz force, perpendicular to the
10
direction of the applied current. However, the vortices may remain stationary if the Lorentz force is
less than the pinning strength. Furthermore, despite not sitting at a pinning site, vortices may still be
motionless. Due to repulsion, the vortex lattice has a certain amount of rigidity, so that if only a few
vortices are pinned the whole vortex structure is immobilized. Far above a critical force fc the entire
structure depins, and rather than accelerating the vortices maintain an average velocity while dissipating
power. While the precise mechanism for this dissipation is not fully understood one possibility is that as
magnetic flux moves an e.m.f is induced drawing current across the normal core. Regardless, it may be
supposed that the vortices dissipative motion is due to some viscous drag and the work required to keep
the vortices in motion comes from the applied current resulting in a voltage difference between the ends
of the specimen. Since the voltage drop is proportional to the average velocity of the moving vortices this
allows measurement of their bulk motion [17]. A simple diagram measuring this voltage drop is shown
in Fig. 2.4.
In more general terms superconducting vortices fall into a larger class of structures in ordered system
known as topological defects. In the phase transition to the superconducting state the macroscopic
wave function plays the part of the order parameter. At temperatures above the superconducting state
individual electrons possess wave functions with random relative phases. As the system goes through
the phase transition to the superconducting state the electrons form Cooper pairs, and enter the Bose
ground state. In passing through this transition the local gauge symmetry is spontaneously broken and
the Cooper pairs in the ground state can be described by a single wave function with a single coherent
phase. However, the ground state energy for the system is phase independent, resulting in a degeneracy
between thermodynamic states. In general when a system is taken below its critical temperature to an
ordered state, and a continuous symmetry is broken there is a competition between degenerate states as
the system equilibrates. This competition is relieved by topological defects [64].
Some examples of defects include dislocations and grain boundaries in periodic crystals, and vortices
in superfluid helium and type II superconductors. Topological defects play a large role in the properties
of the material since they represent a break in the regularity of the ordered state. For example, the
mechanical properties of solids are influenced by the number of defects in the material. Likewise, as has
been explained above, the transport properties in superconductors are dependent upon the existence of
vortices.
2.2
The London Approximation
For many type II materials λ is much larger than ξ making the normal to superconducting interface very
diffuse and therefore the radius of the normal core difficult to define. Furthermore, the thickness of the
material through which the magnetic field penetrates is much longer than either of the characteristic
lengths. Therefore, it is convenient to treat the vortex as a one-dimensional line as outlined by Tinkham
[21]. It is first noted that |Ψ(r)|2 reaches its maximum value in approximately a distance ξ (as shown for
type II materials in Fig 2.1); therefore, we take the limit ξ → 0 allowing Ψ(r) to be treated as a constant
everywhere. In this case the London equations govern the magnetic fields and currents. To account for
11
V
d
J
J
Figure 2.4: Generation of e.m.f. by flux flow in the mixed state
the presence of the core a source term is added to Eq.(2.1)
B+
4πλ2
∇ × J = ẑΦ0 δ2 (r),
c
(2.13)
where ẑ is a unit vector along the vortex and δ2 (r) is a two-dimensional δ function at the location of the
core. Combining Eq.(2.13) with Eq.(2.3) we obtain
λ2 ∇ × (∇ × B) + B = ẑΦ0 δ2 (r).
(2.14)
Since ∇ · B = 0 this can be written as
∇2 B −
Φ0
B
= − 2 ẑΦ0 δ2 (r).
λ2
λ
(2.15)
This equation has the exact solution
µ ¶
Φ0
r
B(r) =
K0
.
2πλ2
λ
(2.16)
Here, K0 is a modified Bessel function of zeroth order, and can be described qualitatively as diverging
logarithmically as r → 0 and decreasing exponentially for long distances. Of course in reality the magnetic
flux is finite at the center and our approximate description of the magnetic flux starts breaking down at
r < ξ where the superconducting electron density starts dropping to zero.
The free energy per unit length may also be calculated when κ = λ/ξ À 1. Neglecting the core region
the primary contributions are the field energy and the kinetic energy of the currents
1
²1 =
8π
Z
(B 2 + λ2 |∇ × B|2 )dS.
12
(2.17)
This can be transformed by a vector product rule and Gauss’s theorem to
1
²1 =
8π
Z
λ2
B · (B + λ ∇ × ∇ × B)dS +
8π
2
I
(B × (∇ × B)) · dl.
(2.18)
Substituting Eq.(2.14) into the first integrand we have
1
²1 =
8π
Z
λ2
|B|Φ0 δ2 (r)dS +
8π
I
(B × (∇ × B)) · dl.
(2.19)
The first term contributes nothing since we avoid the core in the integration. However, it is interesting
to note that if the first term were included the result would contain B(r → 0) which is logarithmically
divergent due to the approximations leading to Eq.(2.16). To evaluate the second term the integration
path is broken up into three parts: a circle at a very large radius, which makes a vanishingly small
contribution due to the exponential decay of B(r) at large r, two counter-traversing radial paths from
the outer circle to an inner circle of very small radius (canceling each other), and finally a path around
the inner circle which makes the only nonvanishing contribution [55]. Taking advantage of the geometry
and orientation of the flux-line we have
·
¸
λ2
dB
²1 =
B
2πr ,
(2.20)
8π
dr
ξ
where ξ has been chosen for the inner loop radius. This corresponds to the physically reasonable assumption that the field divergence is removed at the coherence length (i.e. B(ξ) ≈ B(0)). Substituting
Eq.(2.9) into the previous equation gives
²1 =
Φ0
B(ξ).
8π
(2.21)
Substituting in Eq.(2.11) we finally obtain
µ
²1 ≈
Φ0
4πλ
¶2
ln κ.
(2.22)
The interaction energy between two parallel vortices may be obtained in the same manner. Since the
medium is linear, superposition may be used
B(r) = B 1 (r) + B 2 (r) = [B(|r − r 1 |) + B(|r − r 2 |)]ẑ,
(2.23)
where r 1 and r 2 are the locations of the two cores. The energy may be calculated by substituting the
total magnetic flux into Eq.(2.17) to obtain the increase in free energy
∆F
=
Φ0
[B1 (r 1 ) + B1 ((r2 ) + B2 ((r1 ) + B2 ((r2 )]
8π
13
=
Φ0
Φ0
B1 (r 1 ) +
B1 (r 2 ).
4π
4π
The first term is the sum of the two individual line energies, while the second is the interaction energy.
Substituting Eq.(2.16) into the second term the final form for the interaction energy is obtained.
F12 =
µ
¶
Φ20
r12
.
K
0
8π 2 λ2
λ
(2.24)
The interaction energy is repulsive for the usual case in which the flux penetrates in the same direction
for both vortices.
Within the type II class of materials exists a subgroup known as high temperature superconductors.
As previously mentioned, many of these materials have critical temperatures above 77K making them
useful for applications. The common structural element in these materials are copper oxide planes
which are thought to dominate the superconducting properties. A convenient model, first introduced by
Lawrence and Doniach [18] to investigate the consequences of a layered superconducting structure, is to
consider a stacked array of two-dimensional superconductors coupled together by Josephson tunneling
between adjacent layers. Different masses are introduced into the model to reflect the different modes
of charge transport both in the planes (mab ) and perpendicular to them (mc ). (Here, we have taken
the conventional crystallographic notation with z as the coordinate along ĉ and x, y as coordinates in
the ab planes.) The mass anisotropy causes the coherence length and the penetration depth to also be
anisotropic with the following relationship.
µ
Γ≡
mc
mab
¶1/2
=
λc
ξab
=
.
λab
ξc
(2.25)
Typical anisotropy values for high Tc materials range between 7 for YBCO and 150 for BSCCO. A mass
anisotropy greater than 1 reflects the fact that currents prefer to circulate in the planes. As a result
penetrating magnetic flux is nearly perpendicular as it crosses each plane resulting in a stack of ”pancake
vortices” along each fluxline. Depending on the interplane spacing the pancakes are coupled by either
order parameter tunneling (also known as Josephson coupling) or magnetic interaction depending on
the magnitude of the anisotropy. Of course in the limit Γ → 0 magnetic interaction dominates while
Josephson dominated coupling is estimated to occur when Γ ≤ λab /s where s is the interplane spacing.
For the model developed for this study, Josephson coupling is assumed to be the dominant interaction.
This may be modeled as a simple harmonic potential [40]. A pancake model for zero Josephson coupling
has been introduced by Clem [36].
2.3
Equilibrium Vortex Phases
The mean-field prediction for the equilibrium vortex phase is the Abrikosov lattice. However, upon
closer examination it is found that the equilibrium vortex phase diagram is richly detailed. This arises
14
v
LARGE VELOCITY
CRITICAL
CREEP
T=0
T>0
Fc
f
Figure 2.5: Typical v-f characteristic for T = 0 and T > 0. Three main regions are distinguished: the
creep regime, characterized by thermally activated motion well below the threshold, the critical regime,
characterized by plastic flow near the threshold, and the large velocity region well above the threshold.
from four competing energies: thermal energy which favors a vortex liquid, vortex repulsive interaction
which favors an Abrikosov lattice, pinning potential wells which favor an amorphous or glassy solid,
and coupling energy between superconducting layers which controls vortex line formation from pancake
vortices in adjacent layers [48]. Two important phases are discussed here.
The Bragg glass [50, 52] exists in the presence of weak point disorder. This phase is characterized by
an algebraic decay of translational order and no topological defects. As a result Bragg peaks are exhibited
in the structure factor. The vortex system is no longer in the Bragg glass phase when the distance over
which translational disorder develops becomes greater than the Abrikosov vortex spacing, leading to a
new phase known as the vortex glass [35].
In the presence of parallel columnar disorder the vortex system may exist in the Bose glass phase
[34, 39]. Unlike point defects the pinning force for columnar damage tracks adds coherently over an
extended length resulting in a more effective pinning mechanism. For magnetic fields applied sufficiently
parallel to the defects, and for sufficiently low temperatures each vortex is localized to at most a few
columnar pins resulting in zero linear resistance. This is known as the Bose glass phase due to the fact
that the localized vortices in three dimensions may be mapped to localized bosonic particles in potential
minima in two dimensions [21]. One of the many characteristics of this phase is a diverging tilt modulus
due to the correlated pinning. Associated with this is a transverse Meissner effect where vortices remain
pinned rather than align with an applied magnetic field with a component perpendicular to the columnar
defects.
15
2.4
Nonequilibrium Vortex Phases
When studying the motion of vortices in the presence of disorder and external drive one important
relationship to determine is between the average velocity and the applied force (also known as the I-V
characteristics since the applied current and induced voltage are proportional to the force and average
velocity respectively). For T = 0 and for an applied force below a critical threshold the vortex structure
is pinned in some metastable state. As the applied force is increased through a critical value the structure
depins and moves with an average velocity. Vortices fall into a more general group of elastic lines whose
depinning behavior is described by the power law v ∼ (f − fc )β [51]. As the force is further increased the
relationship becomes linear, with disorder and dissipation resulting in an effective viscosity. For T > 0
the v-f characteristics ”smoothen out” in the vicinity of the critical force as shown in Fig. 2.5. Typical
experimental results showing this temperature dependence are shown in [43].
For T > 0 the v-f curve can be broken into three regimes. Below the pinning threshold vortices
may occasionally move via thermal activation. In this regime flux lines jump from one pinning point
to another seeking out lower energy configurations. This is accomplished either by vortices depinning
through half-loop excitations and wandering or by the development of kinks in the flux lines that search
out and then attach to favorable pinning sites. If the vortex interaction distance is much larger than the
vortex spacing, hopping to new sites will occur in bundles [12], [41].
Close to the critical force in the presence of strong disorder, depinning is observed to proceed via the
flowing of vortices through plastic channels [24]. These ”rivers” of vortices form, freeze, and reappear
at different locations in the sample flowing around ”islands” of temporarily trapped vortices resulting in
incoherent motion. Such behavior has been observed experimentally [47] and in 2D numerical simulations
[30], [31]. Plastic flow is characterized by 1/f α noise in the power spectrum as shown in recent 3d
numerical work [69]. For very weak pinning it is predicted that a Bragg glass will depin without entering
the plastic phase, but will rather move as a coherent structure known as a moving Bragg glass [52].
Well above the critical force it has been observed that vortices are more translationally ordered than
at low velocities [44]. One might expect that at high enough drive the vortices would form a moving
Abrikosov lattice since the force from the disorder on each vortex varies rapidly and would therefore be
less effective [45]. However, it has recently been shown by Le Doussal and Giamarchi [52] that some
modes of the disorder are not effected by the motion even at large velocities. As a result the vortices
exist in what has been labeled the moving glass phase which displays the following generic properties.
First, as vortices move, transverse displacements are pinned into preferred time-independent configurations resulting in static 2d channels or paths. These channels are a result of a subtle competition
between elastic energy, disorder, and dissipation. Simply stated they are the easiest paths for the vortices. These channels deviate from the the paths of a perfect lattice as shown in Fig. 2.6. It is important
to note that these channels are fundamentally different from those used to describe plastic motion between pinned islands. Unlike plastic flow, in the moving glass vortices are aligned almost parallel and are
coupled along the y-direction; hence, the channels are labeled ”elastic channels”.
Second, associated with elastic channels is the existence of transverse barriers. Once channels are
16
established it is energetically unfavorable for them to reorient. Therefore, below some critical force
applied perpendicular to and in the plane of the motion the glass is pinned in the transverse direction.
The moving glass will depin as the transverse force is increased above a critical value. Unlike the
longitudinal critical force a transverse critical force does not exist for a single moving vortex in a shortrange correlated random potential. It does exist, however, if the disorder is sufficiently correlated along
the direction of motion. (such as a tin roof potential, constant in the x-direction and periodic in the
y-direction). In a similar manner, once a pattern of elastic channels is established vortices are sufficiently
correlated that barriers exist preventing the channels from reorienting.
There are a few possible regimes for coupling of vortices in between elastic channels. For weak point
disorder or large velocities relative deformations grow only logarithmically with distance; hence, the
vortex structure maintains quasi-long-range order corresponding to complete elastic coupling between
vortices. This is known as a moving Bragg glass and is characterized by algebraically divergent peaks in
the structure factor at small reciprocal lattice vectors. By increasing the disorder the system transitions
to the moving transverse glass characterized by decoupling of the channels while the periodicity along
the transverse direction is maintained. Here divergent Bragg peaks with nonzero reciprocal-lattice vector
components along the direction of motion are lost while peaks in the transverse direction remain.
Finally, a moving Bose glass is formed in the presence of columnar disorder. Due to the cooperative
effects of the correlated disorder the structure factor will resemble that of the moving transverse glass.
Furthermore, like the equilibrium Bose glass the moving Bose glass possesses a diverging tilt modulus
due to a localization effect arising from the columnar pins. This novel feature combined with a transverse
critical force results in a critical transverse applied magnetic field. Below this critical field moving vortices
will tend to stay aligned with the columnar disorder, while above the critical field the vortices will align
with the applied field. This phenomenon is known as the dynamical transverse Meissner effect and can
be used as an unambiguous signature of the moving Bose glass [66].
17
a)
b)
c)
e)
d)
f)
F
y
x
Figure 2.6: (a) Snapshot of a perfect (nondisordered) lattice moving along the x direction (b) By superimposing images at different times one would see that at T = 0 the particles follow perfectly straight lines.
(For T > 0 the channels remain straight, but increase in width.) (c) In the presence of weak disorder the
particles are elastically coupled between channels and the lattice remains topologically ordered. This is
known as the moving Bragg glass. (d) For stronger disorder the positions of the particles may decouple.
This is the moving transverse glass. (e) Superposition of snapshots from (c) or (d) show particles traveling
through elastic channels. (f) For d=3 vortex channels are ”sheets” that can form an anisotropic type of
smectic layering.
18
Chapter 3
Voltage Noise in Type II
Superconductors
3.1
Early Experiments Relating Resistance and Voltage Noise
to Vortex Motion
As mentioned in the introduction, this study is concerned with the measurement of voltage noise in type
II superconductors in the mixed state and its relationship to the motion of vortices. The development
of the theory relating the two was inspired by a number of different experimental and theoretical results
obtained in the late 1950’s and early 1960’s. Mendelssohn and Moore [4] were the first to suggest
that magnetic fields might penetrate certain superconducting materials forming a mesh of magnetic
filaments; however, a much more complete model of this behavior was given by Abrikosov[9]. Noting
that the Ginzburg-Landau theory of superconductivity [7] predicted a group of materials with a negative
surface energy at the normal to superconducting interfaces, he solved the Ginzburg-Landau equation for
a superconductor in a magnetic field perpendicular to its surface and obtained a spatially periodic square
lattice solution for the superconducting order parameter. (Later work done by Kleiner and coworkers
showed that a triangular array was the more favorable solution [14].) He was able to show that at places
where the order parameter went to zero the magnetic field in the material would match the external
field, while at places where the order parameter was a maximum the penetrating field was zero. Noting
that superconducting current would curl around these filaments, he suggested that these structures were
analogous to the vortices predicted by Onsager [5] and Feynman [8] in liquid helium. He called the state
in which these vortices existed the ”mixed state” and labeled materials that behaved this way ”type II”
materials; materials with a positive energy interface he labeled ”type I”. Using these observations he was
able to estimate a number of important quantities such as the line tension and interaction energy of the
superconducting vortices.
Using the picture of penetrating flux filaments Anderson addressed critical currents observed in type
19
II materials by developing a model for their motion. This ”flux-creep” model [11] described bundles
of magnetic flux filaments hopping via thermal activation between local free energy minima near the
critical current. The model turned out to be quite accurate matching the behavior of the temperature
dependence of the critical Lorentz force observed in experiments. This, along with Abrikosov’s results
opened the door to the study of vortex motion in type II materials.
Inspired by Anderson’s ideas, Kim, Hempstead, and Strnad noted that if flux lines were to move,
they would have to move dissipatively; a fact that might explain previously observed resistivity in type
II materials above the critical current. More specifically, a moving magnetic flux line would induce an
electric field leading to power dissipation. They found that this idea was supported in a number of their
experiments. For example, working with strips of type II materials exposed to perpendicular magnetic
fields, they were able to show that the resistivity of the superconducting material was proportional to the
magnetic flux [12]. For applied currents close to the critical current they found that their results agreed
well with the ”flux creep” model put forward by Anderson, while for larger currents the motion could
be be described as ”flux flow”. In this regime the Lorentz force on the vortices dominated effects due to
pinning or thermal fluctuations resulting in the vortices moving en masse. Measurement of the resistivity
revealed a linear relationship between the voltage and the current in this regime suggesting dissipative
flow for the vortices [13].
In an independent experiment performed by Giaever [15] a dc transformer was constructed out of two
superconducting films separated by a thin insulating layer. A current was passed through one layer (the
primary) and increased until a voltage drop was measured across the material. An equal voltage was then
measured in the secondary film even though no current passed through it. If the current through the
primary was increased to the point where it returned to the normal state no voltage would be measured
across the secondary. These results strongly suggested the interpretation that flux filaments created in
the primary and driven by the applied current were being dragged through the secondary resulting in the
same voltage drop.
Using the recently developed theories of flux line motion van Ooijen and van Gurp [16] reasoned that
vortices moving across a conductor of width W at velocity v = cE/B should generate voltage pulses of
duration τ = W/v yielding a ”shot noise” spectrum which cuts off above ω = 1/τ , and whose amplitude is
a measure of the amount of flux moving in each independent, discrete entity. Their experiments confirmed
their prediction; however, they observed that except for the highest currents and fields the moving entities
typically carried as many as 1000 flux quanta. This was presumably due to pinning which was expected
to cause flux lines to move in bundles of vortices as predicted by Anderson. These results suggested that
the concept of flux motion was correct, and confirmed the previously held suspicion that defects in real
material samples could considerably complicate the idealized picture.
3.2
Washboard Noise
An important and distinct feature of the moving Bragg glass is the existence of a sharp frequency
associated with the periodicity of the lattice. A uniformly driven periodic structure of average velocity
20
< v > and lattice constant a oriented in the direction of the drive moving through a random distribution
of pinning sites will acquire an ac component ω/2π =< v > /a. Such a phenomenon is not unique to
type II superconductors, and is in fact well-known in charge- and spin-density wave systems [33]. The
washboard frequency was first observed in superconductors by Fiory [19] who applied both an ac and dc
current to an aluminum film in the mixed state. The frequency of the ac component was held constant
while the dc current was increased changing the dc velocity of the vortices. When the time for the
Abrikosov lattice to travel one lattice constant corresponded to integer or half integer multiples of the
period of the ac current a drop in the resistance was observed. In a similar experiment using YBCO
performed by Harris et al.[46] it was observed that when the ac frequency was increased the minima in
the resistance shifted linearly to higher dc drive. These results suggested that the minima were due to
interference of the ac current frequency and a washboard frequency.
Direct observation of washboard noise has been observed in the creep regime by various experiments.
Troyanovsky and coworkers [56] have observed washboard motion using N bSe2 crystals in the mixed
state. Vortices were able to be imaged by Scanning Tunneling Microscopy (STM) due to modulation
of the tunneling conductance associated with the order parameter variation of the vortex lattice. The
sampling frequency of the STM was high enough that it was possible to observe the motion of the vortices
in the creep and plastic regimes by compiling a sequence of images into a movie. For columnar defects
plastic motion was observed for the highest applied magnetic fields. For point defects a moving vortex
lattice was observed for a number of different applied magnetic fields. By tracking the velocity of a number
of individual vortices a frequency corresponding to the washboard frequency was measured. However, for
these experiments the limited scanning area allowed on the order of 20 vortices to be imaged at one time.
It was not clear from these experiments whether the washboard signal was obtainable for larger systems.
Macroscopic observation of the washboard frequency in the creep regime has been obtained by Togawa
et al. [57]. They measured the voltage noise spectrum of BSCCO crystals in the mixed state subject
to a constant current for various applied magnetic field strengths. For low magnetic fields broad-band
noise (BBN) was observed and attributed to plastic vortex flow. As the applied magnetic field was
increased BBN reached a maximum and then decreased while narrow-band noise (NBN) corresponding
to the washboard frequency appeared. As the magnetic field was further increased the NBN increased
in frequency while decreasing in height and sharpness. The increase in frequency was due to a tighter
packing of the vortices and hence a shorter lattice constant, while the decrease in height and increase in
width is still not well understood.
Washboard noise has also been observed in numerical simulations. Olson et al. [53] performed 2d
molecular dynamics simulations of vortices driven through a system of random columnar defects. By
varying the drive strength they observed that the number of regimes available to the system above the
plastic flow phase depended on the vortex-vortex interaction strength. For weak vortex interactions only
smectic ordering was observed above depinning regardless of drive strength. This regime was indicated
by peaks in the structure factor located transverse to the drive direction. For midrange vortex interaction
values the smectic phase was followed at higher drive by a transition regime from uncoupled to coupled
channel flow. For strong vortex interactions the vortices immediately entered this transition regime
21
above depinning avoiding the smectic phase. For higher drive, systems with both mid and high vortex
interaction strength entered a coupled channel regime indicated by sixfold coordination of all vortices.
In this regime washboard noise was observed. Furthermore, it was noted that the washboard signal
decreased as the system size was increased. It is believed that this was due to multiple domains forming
in the vortex lattice resulting in decoherence of the noise signal. 2d simulations using Langevin dynamics
and varying drive and pinning strength have been performed [61] with similar washboard noise results.
Gotcheva et al. [70] have brought into question the validity of using driven diffusive Metropolis Monte
Carlo dynamics in simulations for vortices in 2d superconducting networks. The far from equilibrium
steady states obtained using Metropolis Monte Carlo and continuous time Monte Carlo dynamical rules
were compared as a function of temperature and driving force. Results differed dramatically depending
on which dynamical rule was used. The Metropolis algorithm yielded a spatially disordered moving
steady state for much of the temperature-force phase diagram. The argument put forward by the authors
was that this is due to intrinsic randomness contained in the Metropolis Monte Carlo updating rules.
Continuous time Monte Carlo rules preserved order (for finite size systems) over much of the driven phase
diagram. To what extent these results can be extend to 3d off-lattice simulations is unclear.
Using a 3d frustrated anisotropic X-Y model Chen and Hu [65] investigated the first-order transition
from the moving Bragg glass to the moving smectic. In the moving Bragg glass phase NBN corresponding
to the washboard frequency was observed along with higher harmonics for various driving strengths at
T = 0. For T > 0 the washboard peak remained while the harmonics disappeared. It was also observed
that the moving Bragg glass became a moving liquid at high drive, and hence the washboard signal
was destroyed. The authors believed that this was due to thermally activated vortex loops inducing
dislocations in the Bragg glass. This may be an explanation for the temporal decoherence observed in
the previously mentioned experiment performed by Togawa and coworkers [57].
3.3
Effects of Disorder on Vortex Motion and Shape
As previously pointed out, the type of disorder in superconducting materials plays a large role in how
vortices behave. In equilibrium, in the absence of disorder, at zero temperature vortices will arrange into
an Abrikosov lattice. At higher temperatures thermal fluctuations alone will lead to flux line wandering
and an entangled vortex liquid at the appropriate applied magnetic field strength [29]. The inclusion
of uncorrelated point disorder further increases flux-line wandering [28]. At lower temperatures, in the
presence of strong point disorder vortices are pinned and stretch across a number sites resulting in no
order in the system. This is known as the vortex glass [35]. On the other hand, in the presence of weak
point disorder repulsive interactions dominate, straightening vortices and arranging them into a Bragg
glass [52].
By contrast, correlated disorder (i.e. columnar defects) promotes localization of the vortex. At low
temperatures it is energetically favorable for flux-lines directed parallel to columnar disorder to be isolated
to one or two columnar pins. This is the aforementioned Bose glass phase [34], [39]. In this phase vortices
are fairly straight and upright like in the Bragg glass, but randomly distributed and pinned like the vortex
22
Double Kink
fL
z
r
fL
Half−Loop
Figure 3.1: Depinning from a columnar defect via a ”half-loop” and ”double kink” excitation in the
direction of the drive.
f
L
f
L
Figure 3.2: The solid line represents a possible configuration for a flux-line moving through point disorder.
The dotted line represents a configuration at a later time in the presence of a force per unit length fL .
All arrows lie parallel to the upper and lower planes. Note the arrows are not necessarily parallel to the
direction of the drive.
23
glass. A nice discussion of the competition between columnar and point defects on the shape of vortices
is given by Arsenin et al. [42].
Unlike equilibrium, for driven systems columnar defects can cause fluxlines to bend and stretch.
Vortices in the presence of an external current less than the depinning current, will hop from one columnar
pinning site to the next via thermally activated motion [11]. Depending on the strength of the drive this
hopping is done by vortices deforming and stretching, temporarily trading elastic deformation energy for
a lower overall energy configuration. For the lowest applied currents, vortices will send out ”double kinks”
to adjacent pinning sites or ”super double-kinks” across many pinning sites locating new energetically
favorable location [41] (analogous to Mott variable range hopping of electrons between localized states
in semiconductors [25]). Once the ”tongue” connects to a compatible energy site it moves outward and
spreads to the new site. For higher drives (still below the critical current) vortices will depin via a half
loop excitation. In the presence of the external drive the loop will grow until the flux-line is completely
free and will wander to a new pinning site. These depinning modes are shown in Fig 3.1.
Well above the depinning current remnants of these half-loop excitations and double kinks might still
exist. At high driving values these remnant excitations would occur predominantly in the direction of
the drive, while other excitations would still be suppressed by the localizing effect of the columnar defect.
Vortices moving in the presence of columnar defects at high drive has been identified as the ”moving
Bose glass” phase. It is predicted that this phase will possess smectic order due to the effectiveness of
the columnar pins [52].
For point defects one can imagine vortices in the ”flux creep” regime wandering by attaching to
random pins via kinks in all directions with a tendency to move in the direction of the drive as portrayed
in Fig 3.2. As the drive increases to a strength well above the critical force vortices will arrange into
elastic channels running along the direction of the drive. Depending on the strength of the point disorder
the channels will either be coupled or uncoupled. The former is the moving Bragg glass; the latter is the
”moving transverse glass”.
As mentioned above, in the diffraction plot for the Bragg glass the central peak is surrounded by six
other peaks, while for the moving transverse glass only two peaks located transverse to the drive remain.
It is assumed the moving Bose glass and the moving transverse glass will have a similar structure factor
plots due to the dominant role of the disorder [50], [52]. For the moving Bragg glass vortices would be
fairly straight and upright, although we might expect more wandering transverse to the drive than for
the moving Bose glass, again due to the localization effects of the columnar defects. For the moving
transverse glass we would expect vortices to be bent and stretched along the direction of the drive, while
transverse to the drive we would expect results similar to the Bragg glass.
If the type of defect in the system effects the shape of nonequilibrium vortices as suggested, namely
columnar defects suppressing transverse fluctuations while point defects encourage them, this effect might
be observable in the velocity noise spectrum. Specifically, at a particular drive strength and vortex density
washboard noise might be completely wiped out for point defects due to transverse fluctuations, but not
columnar defects. With this information it may then be possible to identify the dominant type of defect
in a high Tc sample from the noise power spectrum. While it is possible that such signatures exist in the
24
broadband spectrum as power laws, such results would be hard to distinguish for a finite-size simulation.
A preliminary investigation of the broadband noise done for a single flux line showed that, in addition
to finite size effects, the spectrum is dominated by Brownian (1/f 2 ) noise [63]. Therefore, we propose to
investigate the effects of disorder on the narrowband washboard signal.
25
Chapter 4
The Model
For this numerical study vortices are considered in the London approximation (λ À ξ) as elastic lines.
The total free energy associated with the tension of NL lines is
EL


NL Z L
 dr i (z) 2
²1 X

dz 
 dz  ,
2 i=1 0
=
(4.1)
¡ ¢ −2
Γ
where r i (z) describes the configuration of the ith flux line. The line stiffness is given by ²1 = ²0 ln λξab
ab
where λab is the in-plane London penetration depth, and ξab is the superconducting coherence length,
¡ φ0 ¢2
both in the ab plane. ²0 = 4πλ
is the energy scale for the vortex interactions, and φ0 = hc
2e is
ab
 dri (z) 2
 ¿ Γ−1 where
the magnetic flux quantum. The stretching energy of Eq.(4.1) holds as long as 
dz
Mz
denotes the effective mass ratio for the elastic line. For this study high Tc materials are
Γ2 = M
⊥
considered for which Γ À 1. In the simulation each flux line is modeled by Np points located at (r i , zi ).
Each point is confined to a constant zi (a separate ab plane) and interacts with its nearest neighbor via
a simple harmonic potential.
The total interaction potential between different vortices is
Eint =
NL Z
X
i=j
L
¡
¢
V |r i (z) − r j (z)| dz,
0
where
V (r) = 2²0 K0
³ r ´
.
λab
(4.2)
Here, K0 is a modified Bessel function of zeroth order, and can be described qualitatively as diverging
logarithmically as r → 0 and decreasing exponentially for long distances. Interactions between vortices
occur within the z planes (i.e. no cross-plane interaction: This approximation is valid as long as the
L
requirements for Eq.(4.1) are satisfied.) For the simulation, vortex interactions are cut off at min( L2x , 2y )
26
where Lx and Ly are the dimensions in the x and y directions. To minimize the effects of the cut-off
λab has been decreased to prevent numerical artifacts observed in the simulation. With a severe cutoff the system becomes trapped in artificial local minima: each line becomes trapped by its neighbors’
step-function potentials created by the cut-off.
Defects in the system are modeled by a distribution of cylindrical square wells. The free energy
contribution to the system of flux-lines by the defects is given by
ED =
NL Z
X
L
VD (r j (z))dz,
0
j=1
where
VD (r j (z)) =
ND
X
(p)
U Θ(rd − |r j (z) − r k |).
k=1
(p)
Here rd is the pin radius in the ab plane, Θ denotes the Heaviside step function, r k indicates the location
of the k th pin, and U denotes the well depth per unit length.
Finally, in the presence of an external current the vortices experience a force per unit length f L =
1
c ẑ × J , hence a work term is introduced:
W =−
NL Z
X
i=1
0
L
f L · r i (z)dz.
The total free energy of the system can then be written as
Etot = EL + Eint + ED + W.
(4.3)
For this simulation the applied magnetic field is assumed parallel to the c axis (oriented parallel to
ẑ); therefore, at t = 0 straight lines are placed vertically in a system of size Lx × Ly × Lz with periodic
boundary conditions in all directions. We have found that the initial configurations of the lines do not
(p)
affect the steady state. Defect centers at positions r k , are also distributed throughout the system, either
randomly or correlated parallel to the c axis to model columnar defects. The state of the system is then
updated according to Monte Carlo Metropolis rates, where for each trial a point on one of the lines is
randomly selected, moved, and ∆Etot is calculated. This updated position is then accepted if ∆Etot < 0
or if some random number r < exp(−∆Etot /kB T ) where 0 < r < 1. When the number of times this
procedure is repeated is equal to the number of points on a line multiplied by the total number of lines
in the system, this constitutes a single Monte Carlo step (MCS) and serves as the unit of time in the
simulation. The maximum distance for a move is limited to b20 where b0 is the radius of a defect. This is
done to help guarantee interaction with all appropriate defects.
Parameter values used in the simulation correspond to typical high Tc materials, and are reported in
27
units of b0 and interaction energy scale ²0 . The parameters ξab , ²1 , U, and Γ2 are chosen to be .5b0 , .25²0 ,
.0075²0 , and 16 respectively. λab is assigned a value of 16b0 which is about 1/3 of typical high Tc values.
As previously mentioned this was done to minimize artifacts due to the interaction cut-off. The average
separation distance between randomly distributed defects in each ab plane is taken to be 15b0 .
Since this study is primarily concerned with the effect of defect correlations in the dynamics the
√
−1
temperature is chosen such that T /T ∗ < 1, where T ∗ = kB
²U b0 is the temperature above which
entropic corrections due to thermal fluctuations become relevant for pinned flux lines. Here, thermally
induced bending and wandering of the flux lines are largely suppressed, and the results can be interpreted
in terms of low-temperature kinetics. kB T has been given a value of .004²0 . The average or center of
mass (CM) velocity is then calculated as
v cm =
Nv
1 X
r cmi (t + τ ) − r cmi (t)
,
Nv i=1
τ
(4.4)
where r cmi is the center of mass position of the ith vortex, and τ is the time interval between measurements. τ is set to 30 MCS, and the simulation is then run for 105 MCS to arrive at a steady state. Data
is then collected for the next 2.5 × 105 MCS.
From the collected data, power spectra and vortex lattice structure plots are obtained for various
vortex densities and defect configurations. Vortex densities are reported as a count of the number of
vortices in a unit system of size 150 √23 b0 ×150b0 . Both types of plots are averaged over defect distributions.
The velocity power spectra are further averaged (preceding the averaging over defect distributions) by
breaking the data into overlapping segments, windowed to minimize spectral leakage, and averaged over
each Fourier transformed section as described in reference [32]. The method used for obtaining the power
of individual peaks is described in the Appendix.
The structure factor (diffraction) plot is obtained in the following manner. Five ”snapshots” of vortex
positions are taken at regular intervals during a run. For each snapshot the vortices are sliced into
individual layers (one for each point making up a vortex). The density correlation plot is found for each
layer and the results are averaged together along with the results from the other snapshots resulting in a
time averaged 2d density correlation plot. This result is Fourier transformed yielding the structure factor
plot.
Delaunay triangulation and ”time exposure” plots have also been included. Both are plots of a
particular (or successive) snapshot (or snapshots) of a particular vortex lattice layer. No averaging has
been done for these plots.
28
Chapter 5
Results
5.1
Vortex Lattice Orientation in Equilibrium
As a preliminary comparison with expected results a square number of vortices were initially put into a
system free of pinning sites with an x-y aspect ratio of 1:1. The vortices were observed to arrange into
a number of different lattice configurations as shown in Fig 5.2(a). While a number of runs ended with
configurations where vortices were primarily six-fold coordinated (as expected), a nontrivial number of
runs ended in other configurations. The prevalence of a four-fold square lattice arrangement, having a
configuration energy slightly above that of the Abrikosov lattice [21], suggests that the square system size
aspect ratio increases the energy of the Abrikosov lattice arrangement by forcing the lattice to deform to
satisfy boundary conditions. The vortex system has been observed to enter four different arrangements:
the lattice may stretch or compress to maintain six-fold coordination, topological defects may form in
the lattice, the vortices may arrange into a square lattice, or some combination of the first three.
Different lattice orientations have also been observed as shown in Fig 5.3(a). Vortex configurations
find local energy minima by either aligning a lattice vector parallel to the systems horizontal axis, aligning
the lattice vector by 30◦ to the horizontal, or by arranging such that the lattice twists about the system
at a chiral angle satisfying periodic boundary conditions as shown in Fig 5.3(d). It has been observed in
experiments [20] that the vortex lattice will reorient itself in the presence of defects such that the principle
lattice vector points in the direction of an applied drive; however, this is not observed in our simulations.
We believe that reorientation is too energetically expensive, and therefore the system is locked into the
initial orientation found in equilibrium. Whether reorientation is observed while simultaneously annealing
and driving the system, or slowly increasing the drive, bringing the system through plastic flow is still to
be investigated.
√
To avoid these difficulties the system’s x-y aspect ratio has been chosen to be 2 : 3 to accommodate
a triangular lattice. The vortices are then placed in the system prearranged in a triangular lattice.
Choosing this aspect ratio allows an even square number of vortices to ”fit” in the system while arranged
in a triangular lattice. Odd square numbers of vortices do not fit in the rectangular system as shown
29
Figure
√ 5.1: A triangular lattice composed of 25 points (left) does not ”fit” in the rectangular system
(2 : 3 aspect ratio) with periodic boundary conditions. The top row is not at the nearest neighbor
locations for the bottom row (located by a circle with an x through it). On the other hand, the triangular
lattice composed of 16 points (right) does preserve the triangular lattice.
in Fig 5.1. Results for runs with initially randomly distributed vortices are shown in Fig 5.2(b), and
Fig 5.3(b). Compared to the square system the number of runs with six-fold coordinated vortices increases
noticeably as does the number of systems with the principle axis oriented parallel to the horizontal axis.
5.2
I-V Characteristics
The average velocity of the vortices versus applied force for systems with randomly distributed columnar
and point defects has been plotted for various flux line densities in Fig 5.4. In addition to averaging
over vortices the velocity is averaged over time and defect distributions. For both graphs the number of
pinning sites is equal.
Pinning is observed for both columnar and point defects up to a critical depinning force, and approximately linear behavior is observed in a small region after depinning. The depinning threshold is higher
for columnar defects indicating a more effective pinning structure for vortices. This is expected since the
force from the pinning sites adds coherently over the length of the vortex. The inset for Fig 5.4(b) is
included to show the region of the curve associated with depinning for point defects. Results for the most
and least dense systems are shown.
The results reveal that denser systems depin sooner for both types of pinning structures. This is due to
the fact that as the magnetic flux density increases the vortex interaction energy for each vortex becomes
greater. As a result it becomes more energetically favorable for a vortex to remain at a particular point in
a particular lattice configuration (either moving or nonmoving) than trapped at a pinning site. We find
that these results are qualitatively comparable to experimental results for high temperature materials
[37], [68], [43].
The velocity versus force curves generated by the simulation are observed to cross. This is especially
clear for point defects. The reason for this is not fully understood. However, for higher drive the
30
% Coordination Number
% Coordination Number
100
80
60
40
20
0
2
4
6
8
10
100
80
60
40
20
0
Coordination Number
2
4
6
8
10
Coordination Number
(a) Square Aspect Ratio
(b) Rectangular Aspect Ratio
45
0
15
15
0
45
(c) Density Correlation Plot
Figure 5.2: √
Coordination number histogram measured at equilibrium for the system size aspect ratio (a)
1:1 (b) 2 : 3. (c) Density correlation plot of a typical square vortex lattice found in (a). The square
lattice is a more readily available vortex configuration for the system with a square aspect ratio.
31
20
15
15
# of Runs
# of Runs
20
10
5
10
5
-30-25-20-15-10 -5 0
-30-25-20-15-10 -5 0
5 10 15 20 25 30
Orientation Angle
5 10 15 20 25 30
Orientation Angle
(a) Square Aspect Ratio
(b) Rectangular Aspect Ratio
30
25
20
15
10
α
5
0
5
0
(c) Measured Angle
10
15
20
25
30
(d) Twist Angle
Figure 5.3:
√ Orientation angle histogram measured at equilibrium for the system size aspect ratio (a) 1:1
(b) 2 : 3. c) Orientation angle is measured from the horizontal axis (the long axis of the rectangular
system) to the closest lattice vector. Lattices oriented at approximately ±5◦ as shown in d) satisfy a
chiral wrapping about the periodic boundary conditions of the system.
32
0.008
<v>
0.006
0.004
0.002
0
0
0.01
0.02
0.03
0.04
0.05
f
(a) Random Columnar Defects
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0.01
<v>
0.008
0.002 0.004 0.006 0.008 0.01
0.006
0.004
0.002
0
0
0.01
0.02
0.03
0.04
0.05
f
(b) Random Point Defects
Figure 5.4: Velocity vs Force curves plotted for random columnar and point defects for various vortex
densities. For random defects the depinning threshold decreases with increasing density. The same occurs
for point defects. This can be seen most easily in the inset in (b). The force per unit length is reported
in units of ²0 and the velocity in units of b0 /M CS where b0 is the pin radius and M CS stands for Monte
Carlo steps. The following symbols represent densities reported as the number of lines in an area of size
150 √23 b0 × 150b0 . N − 144, ¥ − 100, F − 64, ° − 36, × − 16. Data points have been connected as a guide
for the eye.
33
0.05
<v>
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
f
Figure 5.5: I-V curves for interacting (N) and noninteracting (¥) vortices. Note the curves cross at a
force approximately equal to .1. Vortex motion is limited by the repulsion of neighbors, hence a more
quickly saturating I-V curve for interacting vortices. The force per unit length is reported in units of ²0
and the velocity in units of b0 /M CS.
34
0.014
0.014
0.013
0.012
0.01
Ω
Ω
0.012
0.011
0.01
0.008
0.009
0.006
0.008
0
20
40
60
80
100
120
140
80
vortex density
100
120
140
vortex density
(a) Random Point Defects
(b) Random Columnar Defects
Figure 5.6: Washboard frequency versus number of vortices in the system. Results show good agreement
between the measured and calculated values. Vortex density is reported as the number of vortices per
unit area. ω is reported in units of rad/MCS. Error bars for the measured values are obtained from the
full width at half-maximum of the washboard peaks in Fig 5.7. N - measured, ¥ - calculated
velocity versus force curves are observed to change slope and level off as demonstrated in Fig 5.5. Curves
for systems with interacting and noninteracting vortices have been plotted. While linear behavior (i.e.
overdamped motion) is desired for this simulation we find that the velocity saturates at high applied
drive values. We believe this is due to a limitation present in the Monte Carlo simulation. As the force
is increased ”moves” made by the vortex in the direction of the drive must also increase proportionally
to maintain a linear relationship. However, while the number of moves opposite the direction of the
drive decreases and moves in the direction of drive increases (more easily overcoming the elastic energy
of the vortex and the potential energy of the pinning sites) the number of ”small” moves in the direction
of the drive remains statistically the same restricting the velocity. As an example consider a system
with infinite drive where no moves opposite the direction of the drive occur. To maintain a proportional
velocity all moves must be ”infinite” (obviously the finite size of a system in a real simulation restricts
this possibility); however, ”smaller” moves are still likely resulting in a value less than infinite.
This limitation is exacerbated by further restrictions placed on the motion of the vortex and damping
of vortex fluctuations by physical factors. First, updates in the position of the points on the vortex
have been limited to distances less than one fourth of the defect radius helping to guarantee interaction
with the defects. This will obviously put an upper limit on velocities. Second, vortices are stiffened
due to repulsion by nearest neighbors suppressing large moves. This is demonstrated in Fig 5.5. While
the interacting system depins earlier as explained above, the system also saturates sooner than in the
noninteracting case.
35
5.3
Narrow Band Noise Characteristics
As previously mentioned a characteristic frequency known as the washboard frequency is expected in the
velocity power spectra for a vortex lattice moving through a random distribution of pinning sites. The
washboard frequency is calculated in the following manner: the measured average velocity hvi (averaged
over time and defects) is divided by the vortex crystal lattice constant in the direction of the applied
drive. Due to the aspect ratio of the system this distance is obtained by dividing the system length in
the direction of the drive Lx by the square root of the number of vortices in the system.
√
ω = 2πhvi
N
Lx
(5.1)
To investigate this phenomenon the power spectrum and vortex structure factor has been obtained
for increasing vortex density as shown in Fig 5.11 and Fig 5.7 . Peaks are observed as well as higher
harmonics in the power spectra for both point and columnar defects. The harmonics are always located at
integer multiples of the fundamental frequency. The fundamental frequency is observed to increase with
increasing vortex density, and in Fig 5.6 the measured frequency is plotted versus the number of vortices
per unit system size along with the predicted washboard frequency for both point and columnar disorder.
We obtain good agreement between the measured and predicted values. Error bars for the measured
frequency are approximated by measuring the full width at half maximum of the fundamental peak while
the uncertainty from the calculated value is obtained from the standard deviation of the average velocity.
5.3.1
Randomly Distributed Columnar Defects
Power spectral density and structure factor plots for random columnar defects are displayed in Fig 5.7.
With the average spacing between defects set to 15b0 the number of columns in a unit area (150 √23 b0 ×
150b0 ) is 115. For a density of 16 lines per unit area (a filling fraction of ∼ 17 ) only broadband noise is
observed. The diffraction pattern shows a ring typically associated with an isotropic liquid; the radius
of the ring corresponds to the inverse of the spacing between vortices. A typical Delaunay triangulation
plot of a snapshot of a particular run for 16 lines is shown in Fig 5.9(a). Here, each vertex represents a
randomly chosen set of points (one point per vortex) all in the same plane. The plot shows that a number
of topological defects exist in the vortex system. A ”time exposure” of the positions of one of the layers
of the vortices shows trajectories reminiscent of ”braided rivers” observed in plastic flow [54]. Vortices’
paths appear to move and cross within winding channels. For 36 lines, in addition to the isotropic ring,
two small peaks emerge at locations perpendicular to the drive direction. Though not sufficient, this
result is suggestive of the predicted moving transverse glass mentioned above.
As the system is increased from 36 to 64 lines Bragg peaks in the diffraction plot appear, surrounding
the main peak in Fig 5.7(c). These peaks correspond to the wave vectors of the Bragg planes that make
up the vortex lattice. Stronger peaks located perpendicular to the direction of the drive are observed. In
the Delaunay plot (Fig 5.9(b)) topological defects in the vortex lattice disappear, and parallel planes of
vortices travel in comparatively straight parallel channels (Fig 5.9(d)). At densities of 100 and 144 lines
36
the structure plot reveals a well ordered array of vortices.
The results for random columnar defects are a good demonstration of the competing energies in the
system. Random pinning sites favor a random vortex distribution while vortex repulsion favors a regular
array. For a density of 16 lines the structure factor displays a random vortex configuration suggesting
that the lattice structure is dominated by the random pinning sites. Individual vortices are pinned for
periods of time that are long compared to the time it takes for the vortex lattice to move one lattice
constant, tearing apart the vortex structure. At a density of 36 lines vortex repulsion begins to separate
vortices into parallel channels resulting in spatial periodicity in the y-direction, and by 64 lines the system
of vortices is dominated by the repulsive vortex potential. At this stage the vortex structure heals when
a vortex is freed from a pin and can be thought of as an elastic manifold [22]. Distinct Bragg peaks
emerge in the diffraction plot, but peaks possessing an x-component are weaker suggesting that planes
running parallel to the drive are more well ordered than those with a component oriented perpendicular
to the drive. This anisotropy observed at 36 and 64 lines is a result of the drive in the x-direction. The
vortices that make up the planes oriented perpendicular to and traveling in the direction of the drive will
be deformed by the pinning sites. Vortices that make up these Bragg planes will become temporarily
trapped forcing the vortex to come out of alignment with the other vortices in the plane. Depending on
the density of the vortices, at some point nearest neighbors will force the defect out of the pin and back
into the lattice position. Since there is no drive in the y-direction planes perpendicular to the y-direction
are not driven into the defects in such a way that alignment in the x-direction is affected. As the density
is increased to 100 and then 144 lines the repulsive energy between the vortices increases and further
dominates in both directions resulting in a symmetric six-fold structure.
Information about the structure of the lattice carries over into the power spectrum plot. A broadband
signal is measured for the system with isotropically distributed vortices. Similar broadband noise has been
observed in a number of studies and is associated with incoherent motion such as observed in Fig 5.9(c)
[54], [69], [57]. As the vortices begin to arrange into a lattice a hint of a washboard peak appears. A
slight bump corresponding to this peak is observed at 36 lines. At 64 lines a strong narrowband signal
appears. As the vortex density is increased the power of the peak decreases as shown in Table 5.2 as
well as the width. A decrease in the width of the peak indicates a greater temporal coherence between
vortices. This is not totally surprising; as the density of the vortices is increased each vortex is more
tightly held in its place in the lattice and stiffened by its nearest neighbors. For similar reasons remnants
of the broadband noise continues to flatten as the density is increased. The decrease in power of the
fundamental is also due to the increase in stiffness of the lattice structure as the density is increased. As
a vortex passes through a pinning site it will be less effected in a more dense system for the same reasons
just mentioned resulting in a smaller velocity fluctuation, and hence a smaller power output.
In Table 5.2 the ratio of the power of the 1st and 2nd harmonic with respect to the 3rd are also
recorded for power measured in x-direction. The ratios decrease as the density increases indicating a
change in the shape of the velocity time curve. The ratios seem to approach results similar to point
defects (Table5.1). The exact shape of the wave remains to be investigated.
A narrow band signal is also measured in the y-direction for densities of 64 through 144 lines. Plots of
37
the noise are shown in Fig 5.8. The power of the fundamental and the ratio of the harmonics is reported
in Table 5.2. The frequency of the fundamental peak and higher harmonics is identical to the frequencies
measured in the x-direction. While the power is much lower in the y-direction (anywhere from 10 to 40
times lower) the ratio of the peaks are somewhat similar. Unlike the power of the fundamental measured
in the x-direction, the y-direction narrowband power does not follow the same decreasing trend. Since
there is no drive in the y-direction the motion might be described as a ”transverse wiggle” occurring at
the same frequency as the washboard motion. A likely explanation is that as a vortex becomes trapped in
a pinning potential fluctuation transverse to the motion are suppressed until the vortex departs resulting
in periodic behavior of the component of the velocity transverse to the direction of the drive.
To obtain information about the shape of the vortices moving through columnar defects the radius
of gyration, averaged over time and defect configurations, has been obtained. The components of the
radius of gyration in the x- and y-direction versus vortex density are plotted in Fig 5.10. As the density
is increased the radius of gyration decreases indicating that the vortices are straightened with higher
density. This is expected since, as the density increases the stronger repulsion of nearest neighbors would
cause the vortices to stiffen, and hence, counteract stretching. The data also shows an anisotropy in the
stretching of the vortices. The magnitude of the stretching is greater along the direction of the drive
than perpendicular to it with the x- and y-components approaching each other at large densities. For low
density systems we would expect the competition between the drive and the pinning potential to result
in flexible vortices depinning in sections, with some parts of the vortex leaving the columnar pin while
other parts remain trapped like the ”double kinks” and ”half-loops” mentioned above. In the presence
of the drive the free sections move forward stretching the vortex until either the elastic energy of the
vortex is greater than the drive stopping the stretching, or the pinned sections depin. Since there is no
drive in the y-direction the y-component of the radius of gyration is smaller. However, the same trend is
observed for the y-component of the radius of gyration. As previously shown in the time-exposure plot
for 16 lines (Fig 5.9(c)) vortices do not always move parallel to the drive, but in winding channels (i.e.
plastic flow). Hence, we might expect the y-component of the radius of gyration to be larger at lower
densities. At high densities the magnitude of the x-component approaches the y-component. Since the
vortices are stiffer, rather than stretching, depinning tends toward an ”all or nothing” process. If one
point of a stiff vortex depins all of the other points will tend to depin trying to stay in line.
The standard deviation of the x-component of the radius of gyration for 64 lines is larger than other
uncertainties obtained for columnar defects. While this does not completely account for the spread in
the width of the washboard peak at 64 lines it is conceivable that it does play a part in the temporal
decoherence. As a hypothetical row of vortices with an average separation distance travels in the direction of the drive (x-direction), each vortex will collide with the same sequence of defects resulting in a
characteristic frequency. If the vortices are stretched to various degrees in the drive direction the time
over which they contact the defects will differ resulting in a spread in the frequency. It is not clear, at
this point, as to whether the stretching observed here is a result unique to correlated disorder.
38
5 ´ 10-6
S HΩL
4 ´ 10-6
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) 16 lines
5 ´ 10-6
S HΩL
4 ´ 10-6
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) 36 lines
5 ´ 10-6
S HΩL
4 ´ 10-6
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
(c) 64 lines
39
0.04
0.05
0.06
5 ´ 10-6
S HΩL
4 ´ 10-6
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(d) 100 lines
5 ´ 10-6
S HΩL
4 ´ 10-6
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(e) 144 lines
Figure 5.7: Velocity power spectra measured in the x-direction and diffraction plots are displayed for
increasing vortex density in the presence of columnar defects. For 16 lines a diffraction plot associated
with an isotropic liquid is obtained along with a broadband signal in the velocity power spectrum. By
36 lines two peaks appear located transverse to the drive direction in the diffraction plot accompanied
with a drop in broadband noise in the power spectrum. The peaks in the diffraction plot indicate the
forming of parallel plastic channels running along the drive direction. At 64 lines the central peak in
the diffraction plot is surrounded by 6 other peaks. The peaks with an x-component are lower than
those with only a y-component indicating greater order in the y-direction. In the power spectrum a large
washboard frequency peak appears as well as higher harmonics. As the density is increased to 100 and
then 144 lines the lattice shows complete macroscopic order, and the peaks in the power spectrum drop
in power and decrease in width. ω is reported in units of rad/MCS.
40
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
1 ´ 10-7
0.01
0.02
0.03
Ω
0.04
0.05
0.06
0.04
0.05
0.06
0.04
0.05
0.06
(a) 16 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
1 ´ 10-7
0.01
0.02
0.03
Ω
(b) 36 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
1 ´ 10-7
0.01
0.02
0.03
Ω
(c) 64 lines
41
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
1 ´ 10-7
0.01
0.02
0.03
Ω
0.04
0.05
0.06
0.04
0.05
0.06
(d) 100 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
1 ´ 10-7
0.01
0.02
0.03
Ω
(e) 144 lines
Figure 5.8: Power spectra measured in the y-direction for increasing vortex density. Pinning centers are
randomly distributed columns. The power spectrum developes in a manner qualitatively similar to the
results in the x-direction.
42
30
30
25
25
20
20
15
15
10
10
5
5
5
10
15
20
25
30
5
10
(a) 16 lines
30
25
25
20
20
15
15
10
10
5
5
10
15
20
20
25
30
25
30
(b) 64 lines
30
5
15
25
5
30
(c) 16 lines
10
15
20
(d) 64 lines
Figure 5.9: Delaunay triangulation plots for 16 (a) and 64 lines (b) with random columnar defects.
Topological defects have been marked. A disordered structure is obtained for a ”snapshot” of 16 lines,
while a triangular array free of topological defects is obtained for 64 lines. A ”time exposure” for 16 and
64 lines has been plotted in (c) and (d) to give a sense of the motion. For 16 lines, trajectories cross
suggestive of plastic flow, while for 64 lines parallel elastic channels have formed.
43
radius of gyration
4
3
2
1
0
0
20
40
60
80
100
120
140
vortex density
Figure 5.10: Components of the average radius of gyration for columnar defects. The radius of gyration
decreases with increasing vortex density. At higher densities vortices are squeezed, preventing stretching.
A larger deviation in the average stretching value is observed at 64 lines, and might help explain the
larger width of the corresponding narrowband peak. Length is reported in units of the pin radius b0 . N
- x-component, ¥ - y-component
44
5.3.2
Point Defects
The results for point defects show many qualitative similarities to columnar defects; however, the diffraction plot associated with an isotropic liquid is never observed at the lowest densities, but rather Bragg
peaks. The Bragg peaks surrounding the central peak in the structure factor are observed to increase in
height as the density is increased indicating a greater degree of order in the system. For the lowest density
system (Fig 5.11(a)) the magnitude of the peaks in the direction of the drive drop off more quickly than
those perpendicular to the drive (like the results for columnar defects) suggesting that vortices are not
as well coupled perpendicular to the drive as they are along the direction of the drive. This is further
shown by the Delaunay triangulation plot shown in Fig 5.12(a). The vortices (located at each vertex) are
well aligned forming parallel planes along the direction of drive (x-direction), while not as well aligned
perpendicular to the drive. At this density topological defect pairs are oriented perpendicular to the drive
allowing parallel planes to slide by each other in the x-direction. Compared to 16 lines the Delaunay plot
for 36 lines shows greater alignment in the y-direction due to the disappearance of topological defects in
the x-y plane. The reason for this is similar to that for columnar defects. The defects introduce shear
between planes running parallel in the x-direction resulting in an anisotropy in the coupling of the Bragg
planes. As the density increases the vortex repulsion becomes the dominant energy in the system, and
the coupling between parallel channels increases.
In the power spectrum for 16 lines (Fig 5.11(a)) the narrowband signal sits on a small broadband signal.
As the vortex density increases the broadband signal decreases as well as the power of the washboard
peak from 64 to 144 lines. The width of the peaks are smaller than those observed for columnar defects,
and unlike columnar defects, the width of the fundamental peak does not decrease. The power of the
fundamental for different vortex densities is shown in Table 5.1. For point defects there is a significant
decrease in the ratios from 36 to 64 lines. A second change at 144 lines is also statistically significant.
Comparing these results to columnar defects, we see that at 144 lines the ratios for columnar defects is
comparable to those of point defects at higher densities.
A narrow band signal is also measured in the y-direction for point defects. In the cases where
peaks are resolvable the fundamental peak and higher harmonics are located at frequencies identical to
the frequencies measured in the x-direction. Results are shown in Fig 5.13. The measured power and
harmonic ratios are reported in Table 5.1. The ratio of the power of the 1st and 2nd harmonic with
respect to the 3rd are also recorded for power measured in x-direction. Ratios for 16, 64, and 100 lines
are similar suggesting a similar velocity-time trace shape. For 144 lines the ratio is different. At this
point it is not clear as to what dictates these ratios.
The radius of gyration versus vortex density for point defects in plotted in Fig 5.14. The behavior
for both components is similar to that of columnar defects. A larger radius of gyration is observed for
low density systems and decreases for both components as vortex density is increased. Unlike columnar
defects, the pinning force for uncorrelated point defects does not add coherently over the length of the
vortex. While the random distribution of point defects might promote flux-line wandering, stretching of
the vortices at depinning is not as severe. Comparing results to columnar defects, for identical densities,
45
Table 5.1: Point Defects: Power of Largest Three Peaks
vortex number
runs
16 lines
44
36 lines
44
64 lines
276
100 lines
88
144 lines
88
Sx (ω) × 10−7
8.9 ± .9
6.0 ± .5
2.1 ± .2
6.2 ± .6
2.8 ± .4
1.3 ± .2
11.5 ± .4
6.7 ± .2
2.8 ± .1
6.1 ± .4
3.4 ± .2
1.5 ± .1
3.1 ± .3
2.4 ± .2
.58 ± .05
ratio
4.7 ± .8
3.2 ± .5
1
4.9 ± .8
2.2 ± .4
1
4 ± .2
2.4 ± .1
1
4.1 ± .4
2.3 ± .2
1
5.3 ± .5
4.2 ± .5
1
Sy (ω) × 10−7
NA
NA
NA
.34 ± .06
.14 ± .04
.13 ± .05
.98 ± .04
.5 ± .03
.16 ± .02
.72 ± .06
.28 ± .06
NA
.65 ± .05
.28 ± .03
.05 ± .02
ratio
NA
NA
NA
3±1
1.1 ± .5
1
6.0 ± .8
3 ± .5
1
NA
NA
NA
12 ± 5
5±2
1
the magnitude of both components of the radius of gyration are smaller for point defects. As the density
is increased both components for point defects approach the same value for the radius of gyration as do
the curves for columnar defects.
5.3.3
Variable Columnar Defect Length versus Point Pinning Strength
To further compare the effects of point and columnar defects on the velocity power spectrum, results
are obtained for various columnar defect lengths at a constant vortex density of 16 lines. Each set of
results are obtained by averaging runs over random distributions of defects of a particular length. The
number of pinning sites is held constant throughout the entire set of runs. Defect lengths vary from a
single pinning site (random point defects) to a length of 10 pinning sites, and the points that make up
each column are contiguous. Power spectra and diffraction plot results are shown in Fig 5.15.
As the defect length is increased the order in the system decreases. Comparing results for columnar and
point defects from the previous two sections we see that pins correlated along the z-direction are a more
effective pinning arrangement than randomly distributed point defects. Therefore, it is not surprising
that as the length of the columnar defect is increased it becomes a more effective pinning structure as it
grows destroying the periodicity of the vortex lattice. As was observed in the diffraction plots for both
point and columnar defects, peaks with a component along the drive direction are weaker than those
perpendicular to the drive. As the length of the pinning structures increase the peaks decrease, and by
length 5 any peak with an x-component has disappeared, while peaks perpendicular to the drive remain.
This result is reminiscent of the predicted moving Bose/transverse glass [52].
46
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) 16 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) 36 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
(c) 64 lines
47
0.04
0.05
0.06
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(d) 100 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(e) 144 lines
Figure 5.11: Velocity power spectra measured in the x-direction and diffraction plots for increasing vortex
density with point defects. Six-fold coordination of the central peak is observed for all plots. For 16 lines
the Bragg peaks decrease with increasing wave number. By 36 lines the system shows all peaks with
equal height in the diffraction plot and only remnants of broadband noise in the power spectrum remain.
48
50
30
45
25
40
20
35
15
30
10
25
5
5
10
15
20
25
30
5
(a) 16 lines
10
15
20
25
30
(b) 36 lines
Figure 5.12: Delaunay triangulation for 16 and 36 lines with random point defects. Topological defects
have been marked.
Table 5.2: Columnar Defects: Power of Largest Three Peaks
vortex number
runs
64 lines
115
100 lines
44
144 lines
44
Sx (ω) × 10−7
330 ± 20
67 ± 4
9.8 ± .6
200 ± 20
62 ± 6
11 ± 1
110 ± 20
42 ± 5
13 ± 2
49
ratio
34 ± 3
6.8 ± .6
1
19 ± 2
5.6 ± .8
1
8±1
3.2 ± .5
1
Sy (ω) × 10−7
14 ± 1
3.5 ± .3
1.1 ± .2
15 ± 2
4.5 ± .4
1 ± .2
11 ± 1
4.0 ± .5
.6 ± .3
ratio
12 ± 2
3.1 ± .6
1
14 ± 3
4.3 ± .8
1
19 ± 9
7±3
1
7 ´ 10-8
6 ´ 10-8
S HΩL
5 ´ 10-8
4 ´ 10-8
3 ´ 10-8
2 ´ 10-8
1 ´ 10-8
0.01
0.02
0.03
Ω
0.04
0.05
0.06
0.04
0.05
0.06
0.04
0.05
0.06
(a) 16 lines
7 ´ 10-8
6 ´ 10-8
S HΩL
5 ´ 10-8
4 ´ 10-8
3 ´ 10-8
2 ´ 10-8
1 ´ 10-8
0.01
0.02
0.03
Ω
(b) 36 lines
7 ´ 10-8
6 ´ 10-8
S HΩL
5 ´ 10-8
4 ´ 10-8
3 ´ 10-8
2 ´ 10-8
1 ´ 10-8
0.01
0.02
0.03
Ω
(c) 64 lines
50
7 ´ 10-8
6 ´ 10-8
S HΩL
5 ´ 10-8
4 ´ 10-8
3 ´ 10-8
2 ´ 10-8
1 ´ 10-8
0.01
0.02
0.03
Ω
0.04
0.05
0.06
0.04
0.05
0.06
(d) 100 lines
7 ´ 10-8
6 ´ 10-8
S HΩL
5 ´ 10-8
4 ´ 10-8
3 ´ 10-8
2 ´ 10-8
1 ´ 10-8
0.01
0.02
0.03
Ω
(e) 144 lines
Figure 5.13: Power spectra measured in the y-direction for increasing vortex density. Pinning centers are
randomly distributed point pins. The power spectrum develops in a manner qualitatively similar to the
results in the x-direction.
51
radius of gyration
2
1.5
1
0.5
0
0
20
40
60
80
100
120
140
vortex density
Figure 5.14: Components of the average radius of gyration for point defects. The radius of gyration
decreases with increasing vortex density. At higher densities vortices are squeezed, preventing stretching.
N - x-component, ¥ - y-component
52
In the power spectrum we note that the power of the washboard peak (located at ω ≈ .005rad/M CS)
increases as the length of the columnar defect increases from length 1 to length 3, while the intensities
of the harmonics located at higher multiples of the washboard frequency decrease. In the presence of
more effective pinning sites, as long as the lattice maintains its structure, one would expect the velocity
to vary more, and the resulting amplitude of the a typical velocity-time trace of the vortex lattice would
increase resulting in a larger narrowband power signal. However, by length 4 this trend reverses, and by
length 5 the signal is almost indistinguishable from other peaks that have appeared in the noise. In the
corresponding diffraction pattern all Bragg peaks with an x-component have disappeared by length 5,
revealing that the spatial periodicity in the x-direction is completely wiped out. From these results we
infer that the defect length for which maximum narrowband power is observed is the length above which
correlated disorder becomes effective enough to begin to overcome the restoring elasticity of the lattice.
Above this point columnar defects do more to tear the lattice than vary the velocity, resulting in a drop
in temporal periodicity. The system is transitioning to a moving plastic phase.
The broadband noise in the spectrum also increases with increasing defect length. As shown before,
this is associated with incoherent motion of the vortices, and since the effectiveness of the pins is increasing
one expects the vortex lattice, for this particular vortex density, to eventually be torn apart. By length
10, a washboard peak that is the same height as other unidentified peaks that have arisen in the spectrum
is observed sitting on a broadband spectrum, as well as an isotropic ring in the structure factor plot.
The origin of these unidentified peaks in the power spectra is unclear. They are first observed at length
1 surrounding the washboard peak. Their symmetric location suggests they are harmonics associated with
the washboard peak. As the length is increased to 2, a second peak corresponding to the system size
appears. This peak may be an indicator of some type of deformation in the vortex lattice with a time
constant much larger than the period of the time it takes for the lattice to travel the system length. With
periodic boundary conditions the deformation would repeatedly travel over the same defect distribution
resulting in a frequency corresponding to the system length. A similar interpretation has been given
to narrowband signals observed in measuring vortex density fluctuations in anisotropic superconductors
[59, 62]. In these experiments the origin of the density fluctuation was attributed to macroscopic defects
in the material, namely the edge of the sample. Such a defect might create extended topological defects
in the vortex structure. Although, there are no such macroscopic defects in the simulation, such an
interpretation is not inconsistent with larger columnar defects. Like the washboard peak the systemlength peak increase and then decreases in size. The decrease may also be attributed to vortex structure
no longer preserving its shape. It also has a number of unidentified peaks associated with it, and as the
defect length increases so does the power of these peaks. By length 10 the amplitude of the system-length
peak is of the same height as the unidentified peaks.
As a comparison of the pinning effectiveness of defect length versus pinning strength, similar sets of
runs were performed at the same vortex density for various pinning strengths. The depth of the pinning
potential well was increased up to 1.5 times its original value. Results were obtained by averaging over
random distributions of point defects. The number of pinning sites was held constant as before; only the
pinning strength was changed. Results are shown in Fig 5.16. The results are qualitatively very similar to
53
increasing columnar defect length. In fact, the only qualitative difference is the lack of peaks surrounding
the washboard and system-length peak.
While the power spectra are qualitatively similar, the growth of the radius of gyration is not. As shown
in Fig 5.17 the growth of the radius of gyration of the vortices is quantitatively different. The result for
columnar disorder is best fit to an exponential function, while the data for point defects fits a quadratic.
The reason for the quantitative difference is not fully understood, but some physical observations can be
made. First we consider the increased point pinning strength case with an applied drive strong enough
to depin a vortex from a point pin set to the initial pinning strength. The degree to which a single
vortex will stretch (transverse to the z-direction) is determined (in this simulation) by the elasticity of
the vortex, the strength of the pin (i.e. the depth of the pinning potential), and the strength of the drive.
If a vortex is trapped on a single pinning point it will stretch until the applied drive and the elastic ”pull”
from its nearest neighbors are greater than the pinning potential. The stronger the pin, the longer it
will take (on average) for the vortex to depin. Therefore, the stronger the pin, the greater the stretching
in the x-y plane resulting in the increasing behavior observed in Fig 5.17(b). However, eventually the
pinning threshold is reached. At this point the vortex will stretch until the elastic restoring force is equal
to the applied drive resulting in no motion and a maximum radius of gyration. Therefore we expect the
curve shown in Fig 5.17(b) to eventually become convex and flatten at a higher pinning strength.
The same competition between pinning, elasticity, and applied drive occurs in the presence of columnar
defects of increasing length. However, rather than the strength of the pin, the number of points on the
vortex that are pinned increases. This means that there is a drive strength above which the vortex will
depin, even if the defect length spans the height of the sample. (This is observed in the high density
cases for increasing vortex density in chapter 5.3.1.) This means that as the vortex depins via half loops
and double kinks, these vortex excitations will only develop so far before the vortex is free from the pin.
The stretching contest between the columnar pin and the drive is not competitive enough to stretch the
vortex to its maximum length unlike the strong pinning point. Therefore, the radius-of-gyration curve
tends towards a saturation value as shown in Fig 5.17(a).
To sum up, we expect saturation for both radius-of-gyration curves, but for different reasons and
at different vortex velocities. In the case of increasing point defect strength, the pinning strength will
eventually win out, and saturation will occur because of the eventual stalemate between the elastic
stretching and applied drive. This saturation would occur as the vortex velocity approaches zero. For
columnar defects saturation occurs because the columnar defect can never win against appropriately high
applied drive, and the stretching can only go so far before the vortex will depin rather than stretch. This
guarantees a final vortex system velocity above zero (albeit lower due to the increased effectiveness of the
pin), and we would expect this saturation to occur as the velocity levels off to its final average value.
5.4
Finite Size Effects
To investigate the effect of the cutoff on spatial correlations, results have been obtained for an interaction
cutoff length twice as long as that used to obtain results for Fig 5.7 and Fig 5.11. To obtain these results
54
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) length 1
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) length 2
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
(c) length 3
55
0.04
0.05
0.06
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(d) length 4
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(e) length 5
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
Ky
4 ´ 10-7
2 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(f) length 10
Figure 5.15: Power spectra and diffraction plots for increasing columnar defect length. Results reveal the
system evolves from order to disorder as the length increases as shown in the diffraction plots. Likewise
the washboard signal decreases as the broadband noise grows.
56
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) p0
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) 1.1 p0
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
(c) 1.2 p0
57
0.04
0.05
0.06
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(d) 1.3 p0
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(e) 1.4 p0
1.4 ´ 10-6
1.2 ´ 10-6
S HΩL
1 ´ 10-6
8 ´ 10-7
6 ´ 10-7
4 ´ 10-7
Ky
-7
2 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(f) 1.5 p0
Figure 5.16: Results for pinning strength increased from its original value, p0 to 1.5p0 . These results are
qualitatively similar to results obtained by increasing columnar defect length in Fig 5.15.
58
5
4
3
2
1
0
0
2
4
6
8
10
(a) increasing columnar defect length
8
6
4
2
0
0.8
1
1.2
1.4
1.6
(b) increasing pinning strength
Figure 5.17: Radius of gyration (¨ - x-component, F - y-component) for increasing columnar defect
length and increasing pinning strength. The data for increasing length is fit to an exponential for both
components, while the data for increasing pinning strength is best fit to a quadratic.
59
system are is increased by a factor of four. Results are plotted in Fig 5.18 and Fig 5.19. The results for
columnar defects show no spatially periodic order in the system for all results obtained. By comparison,
for a short cutoff length, transverse peaks appear in the structure factor plot at a density of 36 lines, and
by 64 lines six-fold coordination of the center peak is observed. For point defects the structure factor
plots for a long cutoff show six peaks surrounding the central peak; however, for 36 and 64 lines the peaks
with an x-component are lower than those observed at similar densities for a short cutoff as shown in
Fig 5.11.
The long-cutoff power spectra results for columnar disorder show a drop in the broadband noise as
vortex density is increased. A similar trend is observed with a short cutoff length as the system changes
from plastic flow to what appears to be a moving smectic (possibly a moving bose glass as predicted in
the literature [52]). Whether an ordered phase like that observed for the shorter cutoff in Fig 5.7(e) is
obtainable is still to be seen. For point defects the power output of the fundamental peak decreases with
increasing density as it does for a short cutoff length. However, the length of the cutoff effects the ratio
of the harmonics in the power spectrum. For a long cutoff length the ratio of the first and second peak to
the third is larger for the longer cutoff as shown in Table 5.3. The reason for this is not yet understood.
It is predicted that, in the presence of point defects, deformations in the Bragg glass grow logarithmically with distance, and likewise the height of the Bragg peaks decay algebraically [52] suggesting that
the high degree of order observed in our results at high densities and short cutoff lengths may not be
physical (if the Bragg glass is indeed what is being observed in the results for point defects). Results
from the longer cutoff length suggest that spatial correlations have been attenuated by the cutoff length
preventing the growth of disorder in the vortex lattice. A similar effect is observed for columnar defects.
However, the results also show an evolution similar to the short cutoff length as the density is increased
suggesting that the majority of results might be recovered at higher densities. However computational
resources are limited. With an increased total area for an extended cutoff length, the simulation run-time
for high density systems interacting with point defects becomes extremely long. One possible way of
saving time may be to parallelize the code.
The effect of the vortex length on the power spectrum is also examined. Results for longer vortices
(3 times the length of the vortices shown in all other results) at a density of 64 lines have been included
in Fig 5.20(a) and Fig 5.20(b). Qualitatively, the results are quite similar to the shorter vortex length;
however, in the presence of point defects, compared to the results shown in Fig 5.11(c), the narrowband
power is lower. On the other hand, the drop in power is not observed for columnar defects (Compare
Fig 5.20(a) to Fig 5.7(c)). The additional length of the vortices and the lack of spatial correlation in the
z-direction of the point defects further demonstrates the difference in effective pinning between point and
columnar defects. For columnar defects the lengths of both the defect and the vortex span the height of
the sample just as they did for the shorter system; hence a similar effect on the motion and the power
spectrum. For point defects, as the length of the vortex is increased the effect of a point defect on the
longer vortex decreases resulting in a smaller power output.
60
4 ´ 10-6
S HΩL
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) 16 lines
4 ´ 10-6
3.5 ´ 10-6
S HΩL
3 ´ 10-6
2.5 ´ 10-6
2 ´ 10-6
1.5 ´ 10-6
Ky
1 ´ 10-6
-7
5 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) 36 lines
4 ´ 10-6
3.5 ´ 10-6
S HΩL
3 ´ 10-6
2.5 ´ 10-6
2 ´ 10-6
1.5 ´ 10-6
Ky
1 ´ 10-6
-7
5 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(c) 64 lines
Figure 5.18: Structure factor and power spectra plots for systems with an extended cutoff length in the
presence of columnar defects. The cutoff length is double the length used in Fig 5.7(a) and Fig 5.7(b).
Unlike the results for a short cutoff length, peaks do not appear in the structure factor plot.
61
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) 16 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
1 ´ 10-7
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) 36 lines
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
-7
1 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(c) 64 lines
Figure 5.19: Structure factor and power spectra plots for systems with an extended cutoff length in the
presence of point pins. The cutoff length is double the length used in Fig 5.11(a) through Fig 5.11(c).
62
5 ´ 10-6
S HΩL
4 ´ 10-6
3 ´ 10-6
2 ´ 10-6
Ky
1 ´ 10-6
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(a) columnar defects
7 ´ 10-7
6 ´ 10-7
S HΩL
5 ´ 10-7
4 ´ 10-7
3 ´ 10-7
2 ´ 10-7
Ky
-7
1 ´ 10
Kx
0.01
0.02
0.03
Ω
0.04
0.05
0.06
(b) point defects
Figure 5.20: Power spectra and structure factor plots for 64 lines of a length 3 times that of all other
runs. The results for (a) are very similar to those of the same density for shorter lines for columnar
defects shown in Fig 5.7(c). For point defects the power drops noticeably compared to Fig 5.11(c).
63
Table 5.3: Point Defects: Power of Largest Three Peaks for an Extended Cutoff Length
vortex number
runs
16 lines
44
36 lines
44
64 lines
22
Sx (ω) × 10−7
11.6 ± .8
5.2 ± .4
1.4 ± .2
5.8 ± .4
2.7 ± .3
.62 ± .06
4.6 ± .6
1.3 ± .1
.4 ± .06
64
ratio
8.4 ± 1
3.8 ± .5
1
9.4 ± 1.2
4.3 ± .6
1
11 ± 2
3.2 ± .6
1
Sy (ω) × 10−7
NA
NA
NA
.18 ± .04
.06 ± .03
NA
.22 ± .03
.07 ± .02
NA
ratio
NA
NA
NA
NA
NA
NA
NA
NA
NA
Chapter 6
Summary and Conclusion
To summarize, a nonequilibrium Monte Carlo simulation code has been developed to study the effects of
disorder on the velocity power spectrum of vortices driven in the steady state. Specifically, the evolution of
the washboard frequency signal has been measured for increasing vortex density, increasing defect length,
and increasing point pinning strength. For increasing vortex density, results for point and columnar
defects were compared. The velocity power spectrum was also related to the average radius of gyration
of the vortices, and their 2d structure factor plots.
Concerning the velocity versus force curves, results were qualitatively quite similar to I-V curves
obtained in experiments. Changes in the curves due to pinning site type and vortex density occur as
expected. Vortices take on velocities greater than zero at lower critical forces for higher density and less
correlated disorder. The depinning threshold for point defects is much lower than for columnar defects,
and while results close to depinning have been obtained for the highest and lowest vortex densities, a
more thorough investigation of the I-V curves in that region is required. As a reference for future study,
it is to be kept in mind that the velocity versus force curves are observed to saturate at high drive due
to a limitation in the simulation algorithm. This puts an upper limit on the validity of these curves.
The diffraction plots for increasing vortex density show very similar results for point and columnar
defects. As the vortex density is increased for systems with both defect types order is observed to
increase in the diffraction plot. For columnar defects the vortex system is observed to change from that
of a liquid, to a smectic, to a triangular array with smaller peaks along the drive direction, and finally
a triangular lattice. For point defects only the triangular array is observed in the region studied with
peaks located along the drive direction increasing with greater vortex density. We observe that the plots
at low vortex densities for point defects appear qualitatively similar to the results for columnar defects
at higher densities.
Likewise the power spectra for both types of defects also show similarities. Specifically, if a diffraction
plot shows order in the x-direction, a narrowband signal corresponding to the washboard signal is observed
for both defect types. Harmonics are also observed at multiples of the washboard frequency for both types
of disorder. However, the ratio of the harmonics are different. For columnar defects the ratios of the
65
power of the first and second harmonic to the third are larger than for point defects. As the density is
increased the ratio decreases for columnar defects. Whether the ratios for columnar defects arrive at the
ratios obtained for point defects at higher vortex densities is still to be seen.
Another difference observed between power spectra results for point and columnar defects is the spread
of the fundamental frequency observed only for columnar defects, with the width of the washboard peak
decreasing as density increases. It is not clear whether this is a unique effect on the vortex shape
caused by the columnar defects. Similar results might be seen for point defects at different vortex
densities. Enhanced point pinning strength is not observed to increase the width of the signal at low
densities. On the other hand, the washboard peak is observed to widen in the presence of point defects as
magnetic density is increased in the experiment performed by Togawa et al. [57]. It would be interesting
to investigate lower density systems for point defects. The width of the washboard peak should be
investigated for point defects at the same point as was done for columnar disorder; namely when the
system transitions from a structure factor with two peaks located perpendicular to the drive to six peaks
surrounding the central peak.
Comparisons of our results to current experiments that have observed washboard noise ([57, 56])
should be done cautiously. The washboard peak in these experiments was reported to be in the ”coherent
flux creep” regime, while all runs performed in our work are best described as being in either in the
plastic or flux flow regimes. In the coherent flux creep regime the system is at a density where the vortex
lattice preserves its structure as it hops from one local energetic minimum to the next. This suggests
that a fruitful area of focus for our simulation may be in this regime.
The radius of gyration was also obtained for point and columnar defects in the x- and y-directions.
The general trend (as expected) is for the radius of gyration to decrease as the density of the vortices
is increased. A large uncertainty in the radius of gyration is observed in the x-direction for 64 lines in
the presence of correlated disorder. This might help to explain the spread in the washboard peak. If
the radius of gyration is varying dramatically (compared to point defects) from one disorder distribution
to the next this would result in a greater temporal decoherence of the vortices, and hence a spread in
the washboard peak width. Results for the radius of gyration also reveal that transverse fluctuations do
not appear to play a large role in changing the shape of the vortices. The transverse component rarely
exceeds the radius of a pinning site except for the lowest densities.
Changing disorder from point-like to columnar-like shows similar results to increasing point defect
pinning strength. Similar results are obtained in both the diffraction and power spectrum plots. It would
be interesting to repeat this comparison for a higher vortex density. In this case the vortex lattice would
be preserved over an extended ”defect-effectiveness” range minimizing broadband noise and allowing for
a better comparison of the evolution of the narrowband peak. Different effects on the vortices due to
adjusting these two different pinning mechanisms can be seen by comparing the growth of the radius
of gyration in the x-direction. While results seem to correspond to physical intuition, a precise theory
explaining the quantitative growth of the radius of gyration for both situations needs to be developed.
Data obtained for an extended cut-off length suggest that the current length may be too small.
Disorder is observed for higher densities revealing that lattice deformations propagate farther for a longer
66
cutoff length. This suggests that the highly ordered results observed at high vortex densities (100 to 144
vortices) may not be physical. Simulations of larger systems, at higher densities, with a longer cutoff
length need to be performed to investigate how much order survives.
Extending the cutoff length has implications for the harmonics observed in the power spectra. It
is observed that the ratios change for point defects depending on the cutoff length. For a short cutoff
length, in the presence of columnar defects, as the vortex density is increased the ratio of the harmonics
decreases. If narrowband noise survives in the Renascence of columnar defects with a long cutoff length,
it is assumed that a final set of ratios will be reached such as for point defects. Obviously these final
values should be cutoff length independent.
It is not clear whether the washboard peak will be resolvable in future experiments performed on high
Tc materials in a similar regime to the one investigated in this study. While the Bragg glass is predicted
to survive at high drive, the effects of the point disorder may be so small on the bulk motion of the vortex
lattice that the peak may not be observed above background noise. Our results obtained for increasing
vortex length seem to suggest this, but further study is required.
One area of future investigation would be to pursue indicators to help identify moving vortex phases
in the model. Various results obtained throughout this study have been identified in other work as
indicators of nonequilibrium vortex phases mentioned in section 2.4, but further verification is required.
One example of this would be to measure the dynamical transverse Meissner effect by simply tilting the
columnar defects while the system is being driven, and then measure the average angle of the vortices
as a function of the tilt angle. The behavior of the vortices would then help to identify whether or not
the system was in the moving Bose glass phase. Likewise, measurement of the transverse depinning of
systems moving in elastic channels would be a useful step in identifying the moving Bragg glass.
The narrowband signature of other defect types in superconductors is another area of future investigation. Examples include splayed columnar pins, periodic columnar pins, and twin grain boundaries.
For periodic pins and grain boundaries one can imagine the vortex lattice interacting with the underlying defects differently depending on the relative orientation of the one with respect to the other. The
washboard signal may then show dependence on the orientation of the sample. Similarly, it might be
interesting to investigate the effects of the degree of tilt of the splayed columns on the washboard noise.
67
Appendix A
Obtaining the Power of Individual
Peaks in the Power Spectrum
To obtain the power spectral density estimate of the velocity v(t) of the vortex system, the velocity
is sampled at N discrete times during the simulation. The discrete Fourier transform of the sampled
velocity-time trace is then computed.
Vk =
N
−1
X
vj e2πijk/N
(A.1)
j=0
where k = 0, ..., N − 1. The power spectral density estimate is then defined at N/2+1 frequencies as
S0
=
1
|V0 |2
N2
Sk
=
1
(|Vk |2 + |VN −k |2 )
N2
SN/2
=
1
|VN/2 |2
N2
(A.2)
In this case k = 0, 1, ..., N2 − 1.
The total power for a particular peak is found by summing the area under the peak.
S = Sm + Sm+1 + Sm+2 ... + Sm+n
where m is the first bin of the peak and m + n is the last. Bins m and m + n are chosen by finding the
first ”significant” minima in the power spectral density plot in the neighborhood of the peak as shown in
68
S( ω )
ωm
ω m+ n
ω
Figure A.1: A typical power spectrum plot used to demonstrate how the area is found under a peak
69
Fig A.1. The area of the noise on which the peak ”sits” (shown by the dotted lines) is also calculated.
A=
1
n(Sm+1 + Sm+n )
2
A is subtracted from S to yield the power of the peak. This step implies two assumptions: first, the
broadband signal is uncorrelated to the narrowband peaks, and second, the broadband noise possesses a
fairly linear slope underneath the peaks. As to the first assumption, this is admittedly a guess and has
not been investigated. However, for higher density systems the broadband noise disappears while the
peaks remain ”sitting” on a flat noise background, so the majority of measurements avoid this potential
problem. Concerning the second assumption, the broadband signal appears to vary smoothly for low
density systems (ex. Fig 5.7(a)), and since the width of the peaks are fairly small compared to the
change in slope of the noise, the assumption is probably a good one.
The uncertainty in the power for each peak is calculated using the standard equation for error propagation.
sµ
σS
=
=
p
∂S
σm
∂Sm
¶2
µ
+
∂S
σm+1
∂Sm+1
¶2
µ
+ ...
∂S
σm+n
∂Sm+n
¶2
σm 2 + σm+1 2 + ... + σm+n 2
The uncertainty in the area underneath the peaks is
σA =
1 p 2
n σm + σm+n 2
2
and finally the uncertainty of the power of the peak is
√
σ=
σS 2 + σA 2
√
M
where M is the number of trials over which the power spectral results have been averaged.
70
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