Dynamics of Driven Vortices in Disordered Type-IISuperconductors

Harshwardhan Chaturvedi

Dissertation submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Physics

Uwe C. Tauber, ChairGiti KhodaparastMichel Pleimling

Eric Sharpe

November 14, 2018Blacksburg, Virginia

Keywords: Type-II Superconductors, Relaxation Dynamics, Non-Equilibrium StatisticalPhysics, Magnetic Flux Lines, Glassy Systems

Copyright 2018, Harshwardhan Chaturvedi

Dynamics of Driven Vortices in Disordered Type-II Superconductors

Harshwardhan Chaturvedi

(ABSTRACT)

We numerically investigate the dynamical properties of driven magnetic flux vortices in dis-ordered type-II superconductors for a variety of temperatures, types of disorder and samplethicknesses. We do so with the aid of Langevin molecular dynamics simulations of a coarse-grained elastic line model of flux vortices in the extreme London limit. Some original findingsof this doctoral work include the discovery that flux vortices driven through random pointdisorder show simple aging following drive quenches from the moving lattice state to both thepinned glassy state (non-universal aging) and near the critical depinning region (universalaging); estimations of experimentally consistent critical scaling exponents for the continuousdepinning phase transition of vortices in three dimensions; and an estimation of the bound-ary curve separating regions of linear and non-linear electrical transport for flux lines driventhrough planar defects via novel direct measurements of vortex excitations.

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences,Division of Materials Sciences and Engineering, under Grant No. DE-FG02-09ER46613.

Dynamics of Driven Vortices in Disordered Type-II Superconductors

Harshwardhan Chaturvedi

(GENERAL AUDIENCE ABSTRACT)

The works contained in this dissertation were undertaken with the goal of better understand-ing the dynamics of driven magnetic flux lines in type-II superconductors under differentconditions of temperature, material defects and sample thickness. The investigations wereconducted with the aid of computer simulations of the flux lines which preserve physicalaspects of the system relevant to long-time dynamics while discarding irrelevant microscopicdetails. As a result of this work, we found (among other things) that when driven by electriccurrents, flux lines display very different dynamics depending on the strength of the current.When the current is weak, the material defects strongly pin the flux lines leaving them in adisordered glassy state. Sufficiently high current overpowers the defect pinning and results inthe flux lines forming into a highly ordered crystal-like structure. In the intermediate criticalcurrent regime, the competing forces become comparable resulting in very large fluctuationsof the flux lines and a critical slowing down of the flux line dynamics.

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences,Division of Materials Sciences and Engineering, under Grant No. DE-FG02-09ER46613.

Acknowledgments

I would like to thank everyone who helped me get here. On my own, without your help, Icouldn’t imagine covering an iota of the distance that I have covered thus far. Although Iwill strive to help you whenever you need it in the future, I can never truly pay you back. Iwill however consciously pay it forward.

iv

Contents

1 Introduction 1

1.1 Type-II Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Vortex motion and Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Physical Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theoretical Background 7

2.1 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Ginzburg-Landau Equations . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Emergence of Superconductivity from GL Theory . . . . . . . . . . . 8

2.1.3 London Penetration Depth and Coherence Length . . . . . . . . . . . 10

2.2 Type-II Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The Ginzburg-Landau Parameter . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Fluxoid Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4 The Abrikosov Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . 14

2.2.5 Vortex Line Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.6 Vortex Line Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.7 Vortex Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.8 Flux Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

v

2.3 Physical Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Elastic Line Model and Simulation Description 22

3.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Langevin Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Defect Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Simulation Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 Steady-State Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.2 Drive-Quench Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Measured Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Drive Quenches into the Moving and Pinned Regimes 30

4.1 Moving and Pinned Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Quenches within the Moving Regime . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Quenches from the Moving into the Pinned Regime . . . . . . . . . . . . . . 35

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Critical Scaling and Aging near the Vortex Depinning Transition 41

5.1 Critical Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Scaling Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1.2 Fc and δ from Convexities of v-T Curves . . . . . . . . . . . . . . . . 45

5.1.3 Scaling Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Critical Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Relating Static and Dynamic Exponents . . . . . . . . . . . . . . . . . . . . 50

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Flux-flow regimes in the presence of parallel twin boundaries 52

6.1 Depinning Drive Regimes and Preferred Ordering . . . . . . . . . . . . . . . 56

6.1.1 The Pinned Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

vi

6.1.2 The Liquid Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.3 The Partially-Ordered and Smectic Regimes . . . . . . . . . . . . . . 61

6.1.4 The moving-lattice regime . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Flux Line Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.3 Widely Spaced Defect Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7 Conclusions 71

Bibliography 73

vii

Chapter 1

Introduction

This dissertation contains my work on characterizing the non-equilibrium behavior of mag-netic vortices in type-II superconductors, based primarily on data from numerical simulationsof these systems. Type-II superconductors find use in a wide range of practical applications,from MRI scanners to particle accelerators. This chapter aims to provide a broad intro-duction to type-II superconductors, the mixed phase in such materials that gives rise tomagnetic vortices, the different kinds of disorder the vortices are subject to, and the conceptof physical aging.

1.1 Type-II Superconductivity

Superconductivity is the phenomenon of perfect conductivity or zero resistance. As onecan imagine, superconductivity is a desirable property in materials; such materials can beused for very practical purposes such as dissipation-free transmission of power. The perfectconductivity of superconductors was discovered by Heike Kamerlingh Onnes, assisted byGilles Holst, in 1911, when they observed the complete disappearance of electrical resistancein certain materials when cooled below a certain critical temperature (that depends on thematerial) [1]. High-currents that are free of Ohmic losses can be also used to economicallygenerate strong magnetic fields. In fact, type-II superconductors are used for exactly thispurpose in MRI machines and particle accelerators, two applications that require powerfulmagnetic fields in order to function.

Besides perfect conductivity, another interesting property of superconductors is the completeexpulsion of magnetic field from the bulk of the material when it is cooled below its criticaltemperature. This effect was discovered by Fritz Walther Meissner and Robert Ochsenfeld in1933 and is called the Meissner effect [2]. A superconductor displaying the Meissner effect issaid to be in the Meissner state. The Meissner effect does not hold for all strengths of externalmagnetic field. If the external field is strengthened beyond a critical value (that depends

1

Figure 1.1: Mean-field phase diagram of type-II superconductors as a functionof external magnetic field H and temperature T . Thermal fluctuations and spatialdisorder introduce major alterations to this diagram.

on the temperature), the Meissner effect is destroyed. The classification of superconductorsinto type-I and type-II is done on the basis of the nature of these critical fields.

Type-I superconductors are characterized by a single critical magnetic field. Below thisfield, the material is in a pure superconducting Meissner state with complete expulsion ofmagnetic field, and above this field, the material is in a normal resistive state that allowsthe penetration of magnetic field into its bulk [3]. Type-II superconductors are not quiteas simple - they are characterized by two critical magnetic fields HC1 and HC2 as shownin the mean-field phase diagram in Fig. 1.1. Type-II superconductivity was discovered byLev Shubnikov in 1935 [4, 5]. When the external field strength is below HC1, the materialis in the Meissner state, as in the case of type-I superconductors, accompanied by completeexpulsion of magnetic flux from the material. Above HC2, the material is normal-conducting,has non-zero resistance, and there is no expulsion of magnetic fields. We are interested in theregion in between - when the external field strength is higher than HC1 but less than HC2,the magnetic field partially penetrates the sample’s surface in the form of quantized tubesof flux and the material enters a mixed phase of normal and superconducting regions. Thequantized flux tubes are also known as magnetic flux lines or vortex lines. Each line consistsof a normal-conducting core carrying φ0 = hc/2e (one quantum) of flux, with each corebeing screened from the rest of the superconductor by supercurrents that circulate aroundthe core, analogous to quantized vortices in bosonic superfluids, giving rise to the termflux vortex. As the external field is made stronger, more and more flux lines penetrate the

2

sample until we reach the upper critical field HC2 and the sample gets filled with overlapping,normal-conducting cores that destroy all superconductivity in the material.

1.2 Vortex motion and Pinning

Electric current applied externally to a type-II superconductor containing magnetic vorticesexerts a Lorentz force on these vortices, inducing them to move through through the sample.The moving magnetic flux lines induce an electric field that opposes the motion-inducingelectric current, resulting in Ohmic dissipation of electrical energy. Such dissipation resultsin the material losing its most desirable property – perfect conductivity. In order to maintainthe superconducting nature of the material while it is in the mixed phase, we must find waysto hinder the current-induced motion of the vortices. Material defects have been found toact as effective pinning sites that curb flux flow [6].

Material defects (disorder) locally suppress the superconducting charge carrier density, re-sulting in the region occupied by the defects becoming normal conducting. The supercon-ducting charge density is also locally suppressed in the normal-conducting core of a vortex.There is an energy cost associated with such local suppression of charge density. If a vortexcore and a defect overlap, this cost gets paid just once, making such a configuration ener-getically favorable. Thus disorder sites in the material act as local pinning sites that attractflux lines via short-range forces.

Some forms of disorder used for pinning are uncorrelated point-like and correlated columnaror planar disorder. These can either be naturally occurring or artificially introduced inthe material. Point-like defects naturally occur in ceramic high-TC superconductors in theform of oxygen vacancies. They can be artificially introduced by irradiating the samplewith electrons [7]. Columnar defects occur naturally in the form of line dislocations. Theymay also be artificially introduced by bombarding the material with heavy ions (such asSn, Pb, or I) or by growing extended defects with MgO nanorods [8]. Planar defects canbe commonly found in the form of twin boundaries in high-Tc cuprates such as (doped)YBa2Cu3O7−x (YBCO) and La2CuO4+δ. They occur naturally as a mosaic of twins fromone of two orthogonal families [9, 10] and may also be fabricated artificially [11–13].

Vortices repel each other via long-range electromagnetic forces that arise due to the super-currents that screen the vortex cores and experience thermal fluctuations on account of ther-mally induced microscopic currents in the surrounding charge liquid. In a low-temperaturesystem free of disorder, vortices in equilibrium self-assemble into a hexagonal Abrikosov lat-tice containing long-range crystalline order. Introducing weak point disorder into the systemdestroys this crystalline order resulting in a vortex glass [11, 14–17] or Bragg glass phasepossessing quasi long-range positional order [18–23]. The introduction of columnar defectsresults in a distinct strongly pinned Bose glass phase [11,24–27] where flux lines get attachedto the entire length of the extended linear defects. The vortex line localization brought about

3

in this manner makes columnar defects significantly more effective at pinning than point de-fects; this has been experimentally verified [28]. The flux-line tension, inter-vortex repulsiveinteractions, vortex-defect attractive interactions, and thermal fluctuations have comparableenergy scales, and together give rise to complex system that demands extensive and care-ful theoretical and experimental investigation in order to be scientifically characterized andtechnologically optimized.

1.3 Experimental Methods

In order to probe and study the properties of vortex matter in type-II superconductors,several innovative experimental methods have been developed over the years. For exam-ple, Vasyukov et al. used nano-SQUIDs (superconducting quantum interference devices) inscanning probe microscopy to obtain high-resolution images of magnetic vortices with fieldstrengths as small as 50 nT [29]. Auslaender et al. used the tip of an MFM (magneticforce microscope) to drag a single vortex by its end across the surface of a YBCO sam-ple to measure the interaction of vortices with the local disorder, with the eventual aim ofstudying vortex dynamics and their (de)pinning processes when subject to different kindsof disorder [30]. Abulafia et al. used an array of microscopic Hall sensors to experimentallyinvestigate flux creep parameters in YBCO samples [31]. Small-angle neutron scattering is atechnique that can be used to directly measure the Fourier transform of the transverse wan-derting (height-height correlations) of flux lines, thereby enabling one to access the lateralfluctuations and therefore structural properties of vortices in superconductors [32].

The techniques outlined are being (and can further be) used to dynamically characterizevortices in various materials. The results of these experimental studies can be used to im-prove vortex pinning techniques and increase the technological capabilities of these materials.These techniques could also perhaps be used to study the relaxation phenomena of flux linesin disordered media, helping us better understand the behavior of these vortices far fromequilibrium and near critical points, where they transition between different states of vortexmatter, as well as the effect of different disorder types / strengths of on such phenomena.

1.4 Physical Aging

Physical aging is a phenomenon that occurs in a system when for some property of thesystem, the function governing the time dependence of this property changes when the starttime of the measurement of the property is changed; and the functions for the differentstart times collapse on to a master curve under the application of some dynamical scaling.In other words, the function depends on the waiting time following the preparation of thesystem and breaks time-translation invariance [33]. The phenomenon was first observed

4

by L. C. E. Struik in 1977, when he performed a thorough experimental study on variouspolymers and examined their relaxation properties following some treatment [34]. Struik’scareful investigation of 40 different materials revealed aging to be a general feature thatthey all had in common, and that was independent of the specific details of the material.Besides time-translation invariance, physical aging is accompanied by slow relaxation (e.g.algebraic versus exponential time dependence) of the measured observable. A more restrictivedefinition requires the two-time (observation and waiting times) to display dynamical scalingin order for the process to qualify as physical aging [35]. One should note that physicalaging is distinct from biological or chemical aging. The latter processes involve irreversiblestructural changes in the material, accompanied by permanent modifications to the chemicalcomposition and the rupture of primary atomic bonds. Physical aging involves only reversiblemicroscopic changes to the structure of the material, with no permanent physical or chemicalmodification to the material.

Aging has been observed in superconducting materials. Du et el. found evidence for agingin disordered superconducting materials when they observed that the voltage response to anexternally applied electric current in a superconducting sample depended on the durationof the current pulse [36]. Aging was also found by Papadopolou et al. when they foundaspects of aging in their measurements of zero-field cooled (ZFC) magnetization curves inBi2Sr2CaCu2O8+x [37].

In vortex matter specifically, there have been major advances in the study of aging onboth numerical and experimental fronts. Bustingorry, Cugliandolo and Domınguez employedLangevin molecular dynamics to simulate a three-dimensional model of flux lines wherethey observed aging properties in the measurements of two-time correlation functions [38,39]. Pleimling and Tauber used Monte Carlo methods to simulate an elastic line modelof vortices in type-II superconductors and study the non-equilibrium relaxation propertiesof these vortices starting from somewhat artificial initial sample conditions that consist ofrandomly distributed, perfectly straight elastic flux lines [40]. This study yielded complexaging features in the system that were subsequently recovered in later studies that utilized avery different microscopic representation of magnetic flux lines involving Langevin moleculardynamics [41, 42]. The investigation of out-of-equilibrium relaxation dynamics of type-IIsuperconductors aims to identify and differentiate between dynamical features that dependon the material parameters and those that are universal and do not depend on the microscopicdetails of the physical or numerical sample.

1.5 Overview

Chapter 2 provides the theoretical background for superconductivity, starting with Ginzburg-Landau theory, followed by deriving the Ginzburg-Landau equations and outlining the essen-tial features needed for the study of superconductivity. We then discuss Abrikosov’s solution

5

to these equations that lead to the conclusion that the minimum-energy configuration of vor-tices in a disorder-free system is a hexagonal lattice. The chapter ends with a mathematicaldescription of physical aging and dynamical scaling.

We describe our coarse-grained elastic line model of flux lines and the Langevin moleculardynamics algorithm used to simulate this model in Chapter 3. We go on to describe inthis chapter the model parameters used, the types of defects we have studied, the simulationprotocols employed for obtaining steady-state and time-dependent (post-quench) results, anddefinitions of the one- and two-time observables that we measure.

Chapter 4 delves into the relaxation of flux lines in the presence of random point disorderfollowing drive down quenches within the crystalline moving lattice regime and from themoving regime deep into the glassy pinned regime. The two families of quenches yieldcompletely different relaxation behavior post quench, with the vortices relaxing exponentiallyfast following the intra-moving-regime quenches, but slowing down dramatically followingquenches into the pinned regime such that height-autocorrelations decay algebraically withtime. The latter system is shown to exhibit simple (but non-universal) aging. This work ispublished in Ref. [43].

Chapter 5 focuses on the depinning crossover that flux lines must undergo as they transitionfrom the low-drive pinned glassy state to the high-drive moving lattice state in the presence ofpoint defects. This crossover is confirmed to be a continuous (second-order) phase transitionat zero temperature via finite-temperature scaling analyses that allow us to compute thecritical scaling exponents (β, δ, and ν) associated with the phase transition. This is followedby aging analysis of flux line relaxation following drive quenches near the critical point,revealing universal aging scaling features not seen in the case of quenches into the pinnedregime. We end with the calculation of additional dynamic and static scaling exponents(ζ, z and λC) with the aid of hyperscaling relations. We will be submitting this work forpublication in the near future.

Chapter 6 is dedicated to characterizing flux-flow regimes in a system with parallel planardefects. For a specific horizontal orientation of the system, we see the emergence of a richcollection of drive regimes not observed for point or columnar defects. Further investigationreveals this behavior to be a consequence of a preferred ordering of vortices by the anisotropicplanar defect configuration. We also perform novel direct measurements of flux-line exci-tations such as half-loops, single kinks and double kinks that appear in the system due tothe peculiar pinning characteristics of the planar defects, and use the results to compute theboundary curve that separates regions of linear and non-linear current-voltage response inthe system. This work has been accepted for publication in The European Physical JournalB [44].

We draw overall conclusions from this body of work in Chapter 7.

6

Chapter 2

Theoretical Background

We now delineate the theoretical background that underlies our numerical work. We in-troduce Ginzburg-Landau theory in Section 2.1, which is a phenomenological framework toexplain superconductivity. This section describes the important features of the theory, aswell as the key parameters that give rise to superconductivity. Section 2.2 is devoted tohighlighting the conditions that classify superconductors into type-I and type-II, as well asthe emergence of magnetic vortices in type-II superconductors. We cover in the last section(2.3) of this theoretical chapter the concept of physical aging and properties related to itthat will be useful in understanding the work presented in later chapters.

2.1 Ginzburg-Landau Theory

Vitaly Lazarevich Ginzburg and Lev Landau described in 1950 a phenomenological theorythat explains superconductivity without requiring knowledge of any microscopic propertiesof the material [45]. Seven years later, a microscopic theory of superconductivity was for-mulated and presented by John Bardeen, Leon Cooper, and John Robert Schrieffer [46,47].The BCS theory, as it is called, essentially describes superconductivity as a consequenceof Cooper pairs condensing into a bosonic state. The two theories are fundamentally dif-ferent, with Ginzburg-Landau (GL) theory having a top-down thermodynamical approach,and BCS theory having a bottom-up approach rooted in quantum mechanics. In 1989 how-ever, two years after the appearance of BCS theory, GL theory was shown by Gor’kov to bederivable in some limit of BCS theory, who also provided a microscopic interpretation to allparameters of the former [48]. The simplicity and macroscopic nature of GL theory make itvery desirable for use in our computationally intensive work, and we will therefore focus ourattention on it for the remainder of the section.

Ginzburg and Landau [45] built upon Landau’s existing theory on second-order phase transi-tions [49] to introduce a complex order parameter ψ that is related to the density of supercon-

7

ducting electrons. They argued that the free energy F of a material in the superconductingstate can be expressed in terms of powers of ψ and its gradient, using phenomenologicalparameters as expansion coefficients, in the following manner:

F = FN +

∫d3x

[α|ψ|2 +

β

2|ψ|4 +

1

2m|(−i~∇− q

c~A)ψ|2 +

| ~B|2

2µ0

]. (2.1)

Here, FN is the free energy of the material in its normal state, α and β are phenomenologicalparameters that depend on the system temperature, m = 2me and q = −2e are the massand charge respectively of the superconducting electron or Cooper pair (where me and e are

respectively the elementary mass and charge of an electron), and ~A is the electromagnetic

vector potential that gives rise to the magnetic field ~B = ~∇× ~A.

2.1.1 The Ginzburg-Landau Equations

When we minimize the free energy F (2.1) with respect to variations in the order parameterψ, we arrive at the equation of motion for ψ, the first of two Ginzburg-Landau differentialequations:

αψ + β|ψ|2ψ +1

2m

(−i~∇− q

c~A)2

ψ = 0 . (2.2)

When we minimize the free energy F with respect to variations in the vector potential ~A,we obtain the equation for the supercurrent density ~j as

~j =q

m<[ψ∗(−i~∇− q

c~A)ψ]. (2.3)

One should note that (2.2) takes a form similar to the time-independent Schrodinger equationfor a quantum particle in a magnetic field, along with a nonlinear term β|ψ|2ψ.

2.1.2 Emergence of Superconductivity from GL Theory

From (2.3), one can see that the supercurrent density becomes non-zero (and the materialbecomes superconducting) only when ψ becomes finite. Using this in (2.1), we see that the

free energy F in the absence of electromagnetic field ~A and gradients in ψ is given by

8

Figure 2.1: Ginzburg-Landau free energy F−FN as a function of order parameterψ when (a) α > 0 and (b) α < 0. (b) shows the emergence of two degenerate minimaat |ψ| = |ψ∞| when α < 0.

F − FN =

[α|ψ|2 +

β

2|ψ|4

]· V , (2.4)

where V is the volume of the sample, and as such the free energy becomes a fourth orderpolynomial in ψ (the order parameter). When the temperature-dependent parameter βis negative, the free energy gets unbounded from below and the system becomes unstableconstraining β to be positive. We are now left with two cases based on the sign of the otherterm that depends on temperature α; these are displayed in Figure 2.1.

The free energy for α > 0 is a parabola that attains minimum at ψ = 0, which correspondsto the normal state, as seen in Figure 2.1. However, when α < 0, the free energy curve takeson the shape of a Mexican hat containing minima at two points that satisfy the condition|ψ|2 = |ψ∞|2 = −α

β. The notation ψ∞ indicates that ψ approaches this value only at very

deep locations (approaching infinity) inside the superconductor. Thus the order parameteras well as the supercurrent density are non-zero only when α < 0.

If TC is the critical temperature below which the system can enter the superconducting state,α should be proportional to T − TC (at least close to TC). Expanding α about T − TC anddiscarding all but the leading term yields

9

α(T ) ≈ αC1

TC(T − TC) , (2.5)

where αC is a constant of proportionality.

Furthermore, when we expand β in a Taylor series about T = TC , we obtain

β(T ) ≈∞∑n=0

βn

(T − TCTC

)n, (2.6)

where βn are the Taylor coefficients. It becomes evident that β0 has to be dominant overthe coefficients for all the larger-order terms for the theory to be stable.

Rewriting the order parameter in the Euler form as ψ = (−αβ

+ψf )eiθ, we see that the kinetic

term in the free energy (2.1) picks up an additional term given by

− q2

2mc2

α

β| ~A|2 . (2.7)

This term consists of the the electromagnetic field ~A accompanied by constant pre-factors,and is hence considered as an effective mass that is acquired by the photon. This process isanalogous to the effective mass acquired by the Higgs field via the Anderson-Higgs mechanismin high-energy physics [50].

2.1.3 London Penetration Depth and Coherence Length

Substituting ψ as |ψ|eiθ in the free energy equation (2.1), the third term of the integrandcomes out to be

1

2m

[2(~∇|ψ|)2 +

(~∇θ − q

c~A)2

|ψ|2]. (2.8)

The first term in this expression gives us the gradient of the magnitude of the order parameterψ, while the second term gives the kinetic energy associated with ψ and the associatedsupercurrents. Under the assumption of a London gauge, the phase angle the phase angleθ is constant and the second term in (2.8) becomes q2A2|ψ|2/2mc2, which is precisely the

10

effective mass term seen in (2.7). Thus the kinetic energy density of a superconductorobtained from London’s equations is

ns2mv2

s =A2

8πλ2eff

, (2.9)

where ns = |ψ|2 is the local density of superconducting electrons, and ~vs = − q ~Amc

is the localaverage velocity in the presence of an applied field. Equating the effective mass term from(2.7) with the right-hand side of (2.9) yields the London penetration depth as

λ =

√mc2

4πq2|ψ|2. (2.10)

The London penetration depth is the characteristic length scale at which the external mag-netic field exponentially decays within the superconducting material.

The other important characteristic length required to complete the picture is the coherencelength, which can be obtained by turning off the magnetic field, i.e. by setting ~A to 0. Upondoing this, (2.2) becomes

ψ +β

α|ψ|2ψ − 2

2mα~∇2ψ = 0. (2.11)

The coherence length is the pre-factor of the third term on the right-hand side:

ξ =

√2

2m|α(t)|. (2.12)

The coherence length ξ is the characteristic length scale associated with variations in theorder parameter ψ, and is equivalent to the size of a Cooper pair in BCS theory.

2.2 Type-II Superconductors

2.2.1 The Ginzburg-Landau Parameter

The Ginzburg-Landau parameter is a dimensionless quantity that is defined as the ratiobetween the London penetration depth λ (2.10) and the coherence length ξ (2.12)

11

κ = λ/ξ . (2.13)

The value of the Ginzburg-Landau parameter determines whether a superconductor will beclassified as a type-I or type-II superconductor. If κ < 1/

√2, the material undergoes a

first-order phase transition at the critical temperature, completely losing superconductivityabove the critical field HC . Such a material is a type-I superconductor.

If κ > 1/√

2 on the other hand, the material undergoes a smooth second-order transitionfrom the superconducting to the normal state for magnetic fields above the lower critical fieldHC1. There is a negative surface energy at the phase boundaries between superconductingand normal-conducting regions when κ > 1/

√2, which facilitates the maximization of the

cumulative surface area of such phase boundaries through the creation of many normal-conducting regions inside the superconducting phase. This results in the formation of amixed phase that exists for magnetic fields above HC1, but below the upper critical fieldHC2. If the external magnetic field reaches HC2, the normal-conducting regions overlapresulting in total loss of superconductivity. A material with κ > 1/

√2 is classified as a

type-II superconductor.

2.2.2 Vortex Lines

The terms of the free energy (2.1) that depend on the order parameter are

F =

∫d3x

[|ψ|2

(α +

β

2|ψ|2

)+

1

2m|(−i~∇− q

c~A)ψ|2]. (2.14)

The first term is finite only if

|ψ(r →∞)|2 = −αβ≡ ρ2 . (2.15)

This condition requires only the magnitude of the order parameter at infinity to be fixed butnot the phase. This allows us to write down an ansatz defining the the order parameter

ψ = ρeiθ , (2.16)

where ρ ≡ |ψ| and θ is the phase of the order parameter at r →∞.

12

Outside the vortex core, supercurrents screen the magnetic field lines, resulting in the theelectromagnetic field vanishing i.e. ~A = 0. In this limit, the current density from (2.3)becomes

~j = − q

m<(ψ∗i~∇ψ) ≡ 2qh

m

ρ2

reθ . (2.17)

The above implies that supercurrents run only in the azimuthal direction, i.e. they rotateat r → ∞. This rotation of supercurrents at infinity motivates the formation of vortices intype-II superconductors.

2.2.3 Fluxoid Quantization

For a superconductor in the presence of an externally applied magnetic field, the magneticfield may be computed as

Φ =

∫~B · d~f =

∮~A · d~s . (2.18)

We may define the supercurrent velocity as m~vs = ~∇θ − qc~A. Substituting this in (2.18),

we get

Φ =cq

∮~∇θ · d~s− mc

q

∮~vs · d~s . (2.19)

The magnetic flux of a single normal-conducting region in a superconducting background isknown as a fluxoid quantum. Introduced by Fritz London in 1950 [51], a fluxoid is definedas

Φ′= Φ +

mc

q

∮~vs · d~s =

c

q

∮ (m~vs +

q

c~A)· d~s =

cq

∮~∇θ · d~s = k

hc

q. (2.20)

We can quantify the fluxoid quantum by noting that the the integral in the first term in(2.19) equals 2πk, where k takes on integer values. Using the charge of one Cooper pair|q| = 2e, we get the value of a fluxoid quantum as

Φ0 =hc

2e= 2.07× 10−15Wb . (2.21)

13

One magnetic flux quantum Φ0 is the quantity of magnetic flux carried by the normal-conducting core of a single magnetic vortex line.

2.2.4 The Abrikosov Vortex Lattice

Alexei Abrikosov in 1957 presented an approximate solution to the Ginzburg-Landau equa-tions (2.2 and 2.3) [52]. This solution is valid for field strengths near the upper critical fieldHC2, where the order parameter |ψ|2 << 1. Neglecting the nonlinear term |ψ|2ψ in (2.2)and minimizing the free energy, the linearized GL equations have solutions of the form

ψk = eikyf(x) = exp

[iky − (x− xk)2

2ξ2

], (2.22)

where k is a free parameter denoting ky and xk = kΦ0/2πH.

We choose to restrict ourselves to crystalline arrangements of vortices since these will tend tohave lower energies than a random arrangement. In order to enforce this restriction, k mustbe kn = nq, which produces a periodicity in the y direction with ∆y = 2π/q. This restrictionadditionally ensures a periodicity in the locations of the vortex cores at xn = nqΦ0/2πHequivalent to ∆x = qΦ0/2πH = Φ0/H∆y.

One can conclude from the periodicities present above in x and y that

H∆x∆y = Φ0 (2.23)

is the quantum of magnetic flux carried by each vortex lattice unit cell.

Thus, a more general solution to the linearized GL equation can be given by

ψL =∑n

Cneinqye

− (x−xk)2

2ξ2 , (2.24)

where k here is restricted to assume integer multiples of q, making the solution periodic in y.Furthermore the general solution (2.24) is periodic in x if the pre-factors Cn are chosen suchthat they are periodic in n. A triangular lattice emerges when Cn+2 = Cn and C1 = iC0.

One should note that the relative favorability of different solutions is determined by theparameter

14

βA =〈ψ4

L〉〈ψ2

L〉2, (2.25)

which is the nonlinear part of the Ginzburg-Landau free energy. It was shown by Abrikosovthat the lower the value of βA, the more favorable the corresponding solution. Numericalcalculations show that the most favorable solution is that for a triangular lattice with βA =1.16. It is worth noting that in his original paper, Abrikosov initially concluded that thelowest βA value is that for a square lattice (βA = 1.18). Kleiner et al. later corrected this byshowing that it is in fact the triangular lattice that has the most favorable value of βA [53].

An alternative means to decide the favoribility of different solutions of the Ginzburg-Landauequations is by extracting the separation a of nearest neighboring vortices in these solutions[3]. The fact that flux vortices are mutually repulsive suggests that the solution with thelargest a should be the most optimal one. This nearest neighbor distance in a triangulararray, which is when each vortex is surrounded by six equidistant vortices in a hexagonalarray, is given by

a4 =

(4

3

)1/4(Φ0

B

)1/2

= 1.075

(Φ0

B

)1/2

. (2.26)

On the other hand, the nearest neighbor distance for vortices in a square array is computedto be

a =

(Φ0

B

)1/2

. (2.27)

Since a4 > a, it is plain to see that the triangular array is more energetically favorablethan the square array. Triangular arrays have since been observed in experiment [54–56],thereby validating the theoretical analysis.

2.2.5 Vortex Line Energy

In order to obtain the free energy per unit length of a flux line or its line tension ε1, one canadd the contributions of the magnetic field energy and the kinetic energy

ε1 =1

8π

∫ [~B2 + λ2(~∇× ~B)2

]df , (2.28)

15

where the first term is contributed by the magnetic field energy and the second one arisesdue to the kinetic energy

ns

∫m

2~v2sdf =

m

2nsq2

∫~j2sdf .

Performing integration by parts for (2.28) gives

ε1 =

(Φ0

4πλ

)2

K0

(ξ

λ

), (2.29)

where K0(x) is the zeroth-order modified Bessel function.

Exploiting the logarithmic behavior of the zeroth-order modified Bessel function in (2.28) atshort distances, one can also express ε1 as

ε1 ≈

(Φ0

4πλ

)2

lnκ . (2.30)

2.2.6 Vortex Line Interaction

To study the interaction between two magnetic vortices, we may employ the concept ofsuperposition between magnetic fields, with the vortex cores being at positions ~r1 and ~r2.

~B(~r) = ~B1(~r) + ~B2(~r) = ~B(|~r − ~r1|) + ~B(|~r − ~r2|) . (2.31)

This results in an increase in the total free energy, and the increase per unit length is givenby

∆F =Φ0

8π

[B1(~r1) +B1(~r2) +B2(~r1) +B2(~r2)

]

=Φ0

8π

[2B(0) + 2B(|~r1 − ~r2|)

]= 2ε1 +

Φ0

4πB(|~r1 − ~r2|) .

(2.32)

16

We make the substitution B(r) = Φ0

2πλ2K0( r

λ) in the second term of the final result of (2.32)

in order to yield the interaction energy F12 = V (r).

V (r) = 2

(Φ0

4πλ

)2

K0

(r

λ

). (2.33)

Introducing a new energy scale ε0 = ( Φ0

4πλ)2, we obtain the final representation of the vortex-

vortex interaction as

V (r) = 2ε0K0

(r

λ

). (2.34)

As noted before, the zeroth-order modified Bessel function behaves logarithmically at shortdistances and decays as r−1/2e−r/λ at large distances. Therefore, the inter-vortex interactionin (2.34) is essentially a logarithmic repulsion that is exponentially screened at λ.

2.2.7 Vortex Motion

External currents, along with repulsive vortex-vortex interactions exert a Lorentz force oneach vortex line that is given by

~F = ~j × Φ0

cez , (2.35)

where ~j is the cumulative supercurrent density due to all the other vortices in the system.

The Lorentz force induces vortices to move within the sample with velocity ~v in a directionperpendicular to the current. This flux flow in turn induces an electric field ~E = ~B × ~v

c

antiparallel to the direction of the external current ~J . This induced field acts as a resistivevoltage, resulting in the dissipation of electric power as heat. Effectively, the motion of thenormal-conducting cores due the external current in the superconducting region produces adissipative force ~Ff = −η~v, where η is the viscous drag coefficient. The rate of energy perunit length of a flux line is

W = −~Ff · ~v = ηv2 . (2.36)

17

Figure 2.2: The phase diagram for type-II superconductors in the presence ofrandom point disorder (adapted from Ref. [57]).

2.2.8 Flux Pinning

The goal of using superconductors in technological applications is to carry large electriccurrents, often in the presence of strong magnetic fields, with little to no dissipation ofelectrical energy. Flux flow generates dissipation and should therefore be minimized. Thiscan be achieved by pinning the flux lines to defect sites. Defect sites locally suppress thesuperconducting order parameter ψ and therefore the density of superconducting Cooperpairs. Overlapping a defect with the normal-conducting core of a vortex line minimizes thefree energy of the system. Therefore defect sites exert short range attractive forces on fluxlines, thereby localizing them. The condensation energy density of a Cooper pair obtainedfrom the mean-field theory outlined above is −α2/2β. The pinning energy must thus beon the order of −V α2/2β for a defect site to be effective at pinning a core, where V is thevolume of overlap between a pinning site and a vortex core.

Pinning centers may either be naturally occurring in a sample or artificially produced. Theintroduction of any type of disorder greatly affects the phase diagrams for type-II supercon-ductors obtained via the mean-field approximation displayed above in Figure 1.1. It wasshown by Larkin and Ovchinnikov that even very weak disorder destroys the long-rangecrystalline order of the low-temperature Abrikosov vortex lattice [58]. The presence of weakpoint defects results in the formation of either a genuine disordered vortex glass phase com-

18

pletely devoid of long range crystalline order [11, 14–17] or the formation of a Bragg glassstate that retains quasi long-range positional order [18–23]. Large thermal fluctuations onthe order of the lattice constant a0 must be taken into account since they produce a meltingtransition from a vortex lattice to a vortex liquid phase [59, 60]. Another phenomenon thatmust be accounted for is that of flux creep, where thermal fluctuations cause vortex lines intype-II superconductors to jump over pinning barriers at low current densities. The ther-mally induced first-order melting transition of the flux line lattice at non-zero temperaturesis replaced by a disorder-driven continuous second-order phase transition between the frus-trated glassy low-temprature states and a fluctuating liquid phase in the presence of strongdisorder [59,61,62].

As a result, the mean-field phase diagrams are highly altered by the introduction of point-like disorder and thermal fluctuations, as is evident in Figure 2.2. This is a confirmation ofthe complexity exhibited by the vortex-matter system, and highlights the different thermo-dynamic phases that manifest due to pinning by defects and thermal fluctuations.

The presence of columnar defects leads to the emergence of a novel thermodynamic state atlow temperatures, i.e., the strongly pinned Bose glass phase that is distinct from the vortex-glass phase [11, 24–27]. In the Bose-glass phase, flux lines are localized along the entirelength of the linearly correlated columnar defects, leading to a divergence in the sample’stilt modulus in a phenomenon called the transverse Meissner effect [26, 63]. As one mayexpect, columnar defects have indeed proven more efficient at pinning than uncorrelatedpoint-like disorder on account of their extended, correlated nature [28].

2.3 Physical Aging

Physical aging is said to occur in a system when for some property of the system, thefunction governing the time dependence of this property changes when the start time forthe measurement of the property is changed, and the functions for the respective start timesdisplay dynamical scaling. Experimental results of this phenomenon were obtained by Struikthrough his work on the relaxation of polymers [34].

Struik observed the dynamics of glass-forming materials such as PVC, as they relaxed to-wards equilibrium. His experimental procedure involved preparing the system (e.g. glass-forming PVC) at a high-temperature liquid phase and suddenly lowering its temperature,i.e., quenching it to a glassy phase that is characterized by the existence of several equilib-rium states [35]. The rapid change in temperature forces the system out of equilibrium. Afterthe quench, he would then wait for a predetermined period of time known as the waiting timebefore applying a mechanical stress to the system and measuring the response of the systemto this stress. He repeated this experiment for different waiting times s. Upon analyzing thenon-equilibrium relaxation dynamics for different waiting times, Struik observed that therelaxation of the material was very slow (compared to exponential relaxation) and highly

19

sensitive to the value of the waiting time s.

Time-translation invariance is the phenomenon where a physical property of a system thatis a function of the observation time t and waiting time t effectively depends only on thetime elapsed (t− s) since the measurement of the property was begun. When the curves ofa time-translation invariant quantity for different s are plotted against t − s, they coincideor collapse on a master curve. That the response function for Struik’s polymers showed adependence on not only the measurement time t − s, but the waiting time s itself impliedthe breaking of time-translation invariance.

After noticing the breaking of time-translation invariance, Struik showed that the relaxationcurves for the different waiting times when transformed via translation along the time axisand scaling with the waiting time s, collapsed onto a master curve in a process known asdynamical scaling. In a more restrictive definition of aging, dynamical scaling is an additionalcriteria that must be fulfilled. To understand it better, let us consider the following two-timecorrelation function

C(t, s;~r) = 〈φ(t, ~r)φ(s,~0)〉 − 〈φ(t, ~r)〉〈φ(s,~0)〉 , (2.37)

where φ(t, ~r) is some order parameter at time t and position ~r, ~0 is the position vector forthe origin of the reference frame, and s is the waiting time since the quench. Furthermore,the autocorrelation function is defined as

C(t, s) ≡ C(t, s;~0) (2.38)

which can be split into a stationary part and an aging part [35]

C(t, s) = Cst(t, s) + Cage(t, s) , (2.39)

with the stationary part satisfying the condition limt→∞Cst(t, s) = 0. Further, the agingpart can be expressed in the scaling form [35]

Cage(t, s) = FC

(h(t)

h(s)

). (2.40)

The time-reparametrization function h(t) in the scaling function FC that appears in (2.40)is of the form [35]

20

h(t) = h0 exp

[1

A

t1−µ − 1

1− µ

], (2.41)

where h0 and A are constants, and µ is a free parameter. The aging part in (2.39) has beenfound by experimental analysis of glassy systems to be of the form

Cage(t, s) ≈ FC

(t

sµ

). (2.42)

The aging scaling exponent µ is used to classify aging behavior into three types: subagingwhen 0 < µ < 1, simple aging or full aging when µ = 1, and superaging when µ > 1.

In our work, we have observed full aging, characterized by µ = 1, that describes purelyrelaxational dynamics. Here, the autocorrelation function C(t, s) follows the scaling form

C(t, s) = sbfC(t/s) , (2.43)

where b is the aging scaling exponent. It is important to note that when t/s approachesinfinity, the scaling function fC follows a power law in the case of simple aging

fC(t/s) ∼ (t/s)−λC/z, (2.44)

where λC is the autocorrelation exponent, z is the dynamic exponent, and the ratio λC/z isan independent scaling exponent that needs to be evaluated.

In summary, a system is said to undergo physical aging when the relaxation of the sys-tem towards its stationary state(s) is slow (non-exponential) and shows breaking of time-translation invariance. In addition, the quantities that measure relaxation corresponding todifferent waiting times are sometimes amenable to collapse under dynamical scaling.

21

Chapter 3

Elastic Line Model and SimulationDescription

This chapter was adapted with only minor changes from the manuscripts:

H. Chaturvedi, H. Assi, U. Dobramysl, M. Pleimling, and U. C. Tauber, “Flux line relaxationkinetics following current quenches in disordered type-II superconductors” J. Stat. Mech.,vol. 2016, no. 8, p. 083301, 2016.

H. Chaturvedi, N. Galliher, U. Dobramysl, M. Pleimling, and U. C. Tauber, “Dynamicalregimes of vortex flow in type-II superconductors with parallel twin boundaries,” Eur. Phys.J. B, vol. 91, p. 294, 2018.

3.1 Model Hamiltonian

We model flux lines as mutually repulsive elastic lines [26, 64] in the extreme London limit,i.e., when the London penetration depth is much larger than the coherence length. TheHamiltonian of the system is a sum of four terms, viz. the elastic line tension energy, theattractive potential due to pinning sites, the repulsive pair interactions between vortex lineelements, and the work done by the external electric current:

H[~ri] =N∑i=1

∫ L

0

dz

[ε12

∣∣∣∣d~ri(z)

dz

∣∣∣∣2 + UD(~ri(z)) +1

2

N∑j 6=i

V (|~ri(z)− ~rj(z)|)− ~Fd · ~ri(z)

], (3.1)

where ~ri(z) represents the position vector in the xy plane of the line element of the ith fluxline (one of N), at height z.

22

The elastic line stiffness or local tilt modulus is given by ε1 ≈ Γ−2ε0 ln(λab/ξab) whereΓ−2 = Mab/Mc is the effective mass ratio or anisotropy parameter. High-TC superconductingmaterials are highly anisotropic with the crystallographic c-direction being much larger thanthe a and b-directions. This results in different effective charge carrier masses Mc and Mab

for the different directions. We assume that the magnetic field is aligned with the crystal-lographic c-direction of the material, and we assign the material properties discussed belowthe in-plane index ab.

The London penetration is λab depth and ξab is the coherence length, in the ab crystallo-graphic plane. The in-plane repulsive interaction between any two flux lines is given byV (r) = 2ε0K0(r/λab), where K0 denotes the zeroth-order modified Bessel function. It effec-tively serves as a logarithmic repulsion that is exponentially screened at the scale λab. TheLorentz force exerted on the flux lines by an external current ~j is modeled in the system asa tunable, spatially uniform drive Fd = |~j × φ0

~B/B| in the x direction, where φ0 = hc/2e isthe magnetic flux quantum..

The pinning sites are modeled as smooth potential wells, given by

UD(~r, z) = −ND∑α=1

b0

2p

[1− tanh

(5|~r − ~rα| − b0

b0

)]× δ(z − zα), (3.2)

where ND is the number of pinning sites, p ≥ 0 is the pinning potential strength, b0 is thewidth of the potential well, while ~rα and zα respectively represent the in-plane and verticalpositions of pinning site α. Each potential well is smooth at its boundary, and drops steeplyto a flat minimum −b0p/2 in its bottom.

All lengths are measured in units of b0 while energies are measured in units of ε0b0, whereε0 = (φ0/4πλab)

2 is the elastic line energy per unit length.

3.2 Langevin Molecular Dynamics

In order to simulate the dynamics of the model, we discretize the system along the z axis, i.e.,the direction of the external magnetic field, into layers, with the layer spacing correspondingto the crystal unit cell size c0 along the crystallographic c direction [64, 65]. Consequently,each elastic line is broken up into points, with each point belonging to a given line, residingin a unique layer. Any two points of the same line in neighboring layers attract each other viaan elastic force, the potential between them constituting the first term in the Hamiltonian(3.1). Points in the same layer repel each other via long-range logarithmic interactions thatare defined by the third term of the Hamiltonian. The pinning sites are also confined tothese layers perpendicular to the z axis, and are modeled as smooth potential wells (3.2).

23

The interactions between the discrete elements of the system that are described here areencapsulated in the properly discretized version of the Hamiltonian. We use this discretizedHamiltonian to obtain coupled, overdamped Langevin equations which we solve numerically:

η∂~ri,z(t)

∂t= −δH[~ri,z(t)]

δ~ri,z(t)+ ~fi,z(t). (3.3)

Here, η = φ20/2πρnc

2ξ2ab is the Bardeen-Stephen viscous drag parameter, where ρn represents

the normal-state resistivity of YBCO near TC [6,66]. We model the fast, microscopic degreesof freedom of the surrounding medium by means of thermal stochastic forcing as uncorrelatedGaussian white noise ~fi,z(t) with vanishing mean 〈~fi,z(t)〉 = 0. Furthermore, these stochasticforces obey the Einstein relation

〈~fi,z(t) · ~fj,z′(s)〉 = 4ηkBTδijδzz′δ(t− s),

which ensures that the system relaxes to thermal equilibrium with a canonical probabilitydistribution P [~ri,z] ∝ e−H[~ri,z ]/kBT in the absence of external current.

3.3 Model Parameters

We have selected our model parameters to closely match the material properties of theceramic high-TC type-II superconductor YBa2Cu3O7 (YBCO). The pinning center radiusis set to b0 = 35A; all simulation distances are measured in units of this quantity. Theinter-layer spacing in the crystallographic c direction is set to this microscopic scale, c0 = b0.The in-plane London penetration depth and superconducting coherence length are chosento be λab = 34b0 ≈ 1200A and ξab = 0.3b0 ≈ 10.5A respectively, in order to model thehigh anisotropy of YBCO, which has an effective mass anisotropy ratio Γ−2 = 1/5. The lineenergy per unit length is ε0 ≈ 1.92 ·10−6erg/cm; all simulation energies are measured in unitsof ε0b0. This effectively renders the vortex line tension energy scale to be ε1/ε0 ≈ 0.189. Thepinning potential well depth is set to p/ε0 = 0.05. The temperature in our simulations is setto around 10 K (kBT/ε0b0 ≈ 0.002 in our simulation units). The Bardeen-Stephen viscousdrag coefficient η = φ2

0/2πρnc2ξ2ab ≈ 10−10 erg · s/cm2 is set to one, where ρn ≈ 500µΩm is

the normal-state resistivity of YBCO near TC [67]. This results in the simulation time stepbeing defined by the fundamental temporal unit t0 = ηb0/ε0 ≈ 18 ps; all times are measuredin units of t0.

24

3.3.1 Defect Types

In every study in my doctoral work, we have employed one of two types of defects: point-likeor planar.

Point-like Defects

Point-like defects are modeled by distributing the smooth potential wells described describedin (3.2) throughout the system. We place 1116 pinning sites in each discretized horizontallayer of the system, using a different random distribution for each layer.

Planar Defects

In the case of planar defects, we initialize the system with two defect planes oriented per-pendicular to the direction of the drive (x direction). Each planar defect consists of columnsof point defects extending along the entire height of the system. These columns are stackedside by side along the y direction, and consecutive defects are separated by a distance of 2b0.We set up our pair of defect planes in one of two configurations – either close together, i.e.,where the planes are separated by 16b0 (∼ 5% of the system length in the x direction), or farapart with a separation of 160b0 (∼ 50% of the system length in the x direction). Besidesthe two defect planes, isolated point defects are randomly distributed throughout the systemto maintain a concentration of 1116 defects per plane. The random point defects providethe effective viscosity experienced by moving flux lines in a real physical system.

3.4 Simulation Protocol

Our system consists of N = 16 flux lines, moving in a three-dimensional space with periodicboundary conditions in the xy directions and free boundary conditions along the z direction.We set the horizontal system size to (16/

√3λab × 8λab). The particular ratio of horizontal

boundary lengths is necessary to ensure that the flux lines can equilibrate to a periodichexagonal Abrikosov lattice in the absence of quenched disorder. The system is discretizedinto L = 100 layers along the z direction.

Each simulation run starts with the 16 flux lines being perfectly straight, and distributedrandomly in the computational space containing the desired defect configuration. We thenfollow different protocols depending on whether we want to probe steady-state properties ofthe system, or dynamical time-dependent ones.

25

3.4.1 Steady-State Protocol

For steady-state simulations, after initializing the flux lines in the system, we immediatelysubject them to the desired temperature T and drive strength Fd. The lines are allowed torelax in this constant temperature-drive bath for an initial relaxation time of 100, 000t0. Atthis point, we measure various one-time observables (see Section 3.5) in the system every100 time steps, a duration larger than the correlation times in the system that range from20t0 to 45t0 depending on the strength of the applied driving force. We perform 1000 suchmeasurements and under the ergodic assumption, record their average for each observable.We simulate 10 independent realizations in this manner and perform an ensemble averageover these realizations. Between the time averaging and ensemble averaging, we thus averageeach data point over 10, 000 independent values.

3.4.2 Drive-Quench Protocol

For simulations designed to probe long-time relaxation dynamics of flux lines after subjectingthem to a drive quench, we pursue the following protocol. We begin in a similar fashionto steady-state simulations in that we immediately subject the flux lines to the desiredtemperature and initial drive strength. The lines are then left to relax beyond microscopictime scales in the temperature-drive bath. During this time, thermal fluctuations contributetowards the roughening of the lines. After this initial relaxation period that lasts 60, 000time steps, we instantaneously change (quench) the drive to the desired final drive strength.At this point, we reset the system clock t to 0. Following the drive quench, we start themeasurement of one-time quantities (see Section 3.5) and allow the system to relax for waitingtime s. After the waiting time has elapsed, we take a snapshot of the system. We then beginthe measurement of two-time quantities (see Section 3.5) which continues until the end ofthe simulation. We average our results over 1000 to 10,000 independent runs / realizationsdepending on noisiness of the data. The exact number of realizations for each result in thedissertation is specified along with the result itself.

3.5 Measured Quantities

In the course of a simulation, we measure several one- and two-time physical quantities;these are averaged over many disorder realizations and noise histories. One-time quantitiesare those that depend only on the system time t, while two-time quantities depend on both tand the waiting time s, i.e., the length of time that has elapsed after the system is quenchedor perturbed in some manner.

A one-time quantity of interest for us is the local hexatic order parameter

26

h =

⟨1

mi

mi∑j=1

cos(6θij)

⟩i,k

, (3.4)

a measure of local sixfold orientational order in the system. An h value of 0 indicates a highdegree of orientational disorder in the system while h ≈ 1 signifies a highly ordered hexagonalcrystal-like structure. In order to compute h, we first compute the mean xy positions of theflux lines in the system. This gives us a two-dimensional representation of the flux lines aspoints in a plane. For each line in this representation, we identify its nearest neighbors, i.e.,other lines that are within a cutoff distance 4/

√3λab of itself. The cutoff distance is derived

from the distance that separates neighboring lines in an ideal hexagonal Abrikosov lattice.Say line i has mi nearest neighbors; we then compute the angle θij that each neighbor jmakes with the neighbor closest to it in the clockwise direction, and subsequently calculatecos(6θij) for each j and find the mean of these cosines. This value is the measure of hexaticorder for line i with respect to its nearest neighbors. In the final expression for h statedabove, 〈. . .〉i,k represents an average over all N vortex lines i and different realizations k ofthe disorder configurations and noise histories.

Another quantity of interest is the mean radius of gyration,

rg =√〈(~ri(z)− 〈~ri〉z)2〉 , (3.5)

i.e., the standard deviation of the lateral positions ~ri(z) of the points constituting the ithflux line, averaged over all the lines. rg is a measure of overall roughness of the lines in thesystem. Here, 〈. . .〉z represents an average over all layers z, while 〈. . .〉 represents an averageover layers z of line i as well as an average over all lines and different realizations of thedisorder and the noise.

Our third observable is the mean vortex velocity

~v =

⟨d

dt~ri(z)

⟩. (3.6)

The fourth quantity we measure is the fraction of pinned line elements

fp = 〈n(r < b0)/ntotal〉k . (3.7)

Here, n(r < b0) denotes the number of line elements located at a distance r less than onepinning center radius b0 from a pinning site. ntotal is the total number of line elements inthe system. Thus, fp is the fraction of line elements in the system that are located withindistance b0 of an attractive defect site. Here, 〈. . .〉k represents an average over differentrealizations k of the disorder and the noise.

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Figure 3.1: Simulation snapshots of the system projected onto the xz plane for aside view, showing flux lines forming three different excitations: (a) half-loop, (b)single kink, and (c) double kink. The green dotted lines are flux lines, the gray dotsrepresent point pins, and the black vertical lines planar defects. The red dottedsections are those portions of the flux lines that are trapped at planar defects. Thedrive Fd is oriented along the positive x (right) direction.

In the presence of planar defects, we also measure the numbers of different flux line excitations[68] that appear in the system, viz. half-loops, single kinks, and double kinks (Figure 3.1).A flux line forms a half-loop (Figure 3.1a) when it becomes partially depinned from a defectplane and the separation between the depinned portion and the plane is smaller than theinter-planar distance. A single kink (Figure 3.1b) appears when part of a line is trapped inone defect plane while an adjacent section is trapped in the neighboring plane. A doublekink (Figure 3.1c) is similar to a half-loop but with a larger separation between the depinnedportion and the remainder of the flux line that results in the outermost portion of the half-loop being pinned to the next defect plane; this can also be viewed as a specific combinationof two single kinks and is accounted for as such in our measurements. In each simulationrun, we record the total number of each type of vortex excitation appearing in the system.The excitation numbers depend on the total number of flux lines in the system, which forus is N = 16.

The two-time quantity we measure is the normalized height autocorrelation

C(t, s) =〈(~ri,z(t)− 〈~ri,z(t)〉z)(~ri,z(s)− 〈~ri,z(s)〉z)〉

〈(~ri,z(s)− 〈~ri,z(s)〉z)2〉. (3.8)

It quantifies how the lateral positions ~ri,z of the elements of a line relative to the meanlateral line position 〈~ri,z〉z at the present time t are correlated to those relative positionsat a past time s and contains information about local transverse thermal fluctuations ofvortex line elements. It is worth noting that the term ‘height’ autocorrelation originatesfrom viewing the flux lines as fluctuating one-dimensional interfaces, the local height of

28

which corresponds to the deviation of ~ri,z from the respective line’s mean position. Weuse the height autocorrelations as a tool to investigate the existence and nature of physicalaging in our system. A system shows aging when a dynamical two-time quantitiy displaysslow relaxation and the breaking of time translation invariance [35]. Additionally, in a simpleaging scenario, the two-time quantity shows dynamical scaling and follows the general scalingform

C(t, s) = s−bfC(t/s), (3.9)

where fC is a scaling function that follows the asymptotic power law

fC(t/s) ∼ (t/s)−λC/z, (3.10)

as t→∞; b is the aging scaling exponent, λC is the autocorrelation exponent, and z is thedynamical scaling exponent.

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Chapter 4

Drive Quenches into the Moving andPinned Regimes

This chapter was adapted with only minor changes from the manuscript:

H. Chaturvedi, H. Assi, U. Dobramysl, M. Pleimling, and U. C. Tauber, “Flux line relaxationkinetics following current quenches in disordered type-II superconductors” J. Stat. Mech.,vol. 2016, no. 8, p. 083301, 2016.

We devote this chapter to an investigation of the relaxation dynamics of magnetic vortex linesin type-II superconductors following rapid changes of the external driving current within themoving regime and from the moving to the pinned regime (regimes defined in Section 4.1). Asystem of flux vortices in a sample with randomly distributed point-like defects is subjectedto an external current of appropriate strength for a sufficient period of time so as to be ina moving non-equilibrium steady state. The current is then instantaneously lowered to avalue that pertains to either the moving or pinned regime. The ensuing relaxation of the fluxlines is studied via one-time observables such as their mean velocity and radius of gyration.We have in addition measured the two-time flux line height autocorrelation function toinvestigate dynamical scaling and aging behavior in the system, which in particular emergeafter quenches into the glassy pinned state.

4.1 Moving and Pinned Regimes

In order to identify the drive ranges that correspond to states when the system of flux linesis respectively in the pinned or moving regime, we have investigated steady-state featuresof the system as a function of drive, in the manner described in Section 3.4.1. Error bars,representing the statistical error or standard deviation of the mean, are smaller than thesymbol sizes in Fig. 4.1, and therefore do not appear there. In such instances in the

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Figure 4.1: Steady-state (a) mean vortex velocity v (units of b0/t0), (b) radius ofgyration rg (units of b0), and (c) fraction of pinned line elements fp as a function ofdrive Fd (units of ε0) for a system of interacting flux lines. rg peaks at Fd ≈ 0.006ε0,where also v starts assuming non-zero values, and fp begins to decay from itspinned steady state value ∼ 0.2. Data are averaged over 10,000 independent values.Copyright (2016) by the Institute of Physics.

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dissertation where the error bars are larger than the symbol sizes, they are shown.

For zero drive, about 20% (see fp in Fig. 4.1c) of flux line elements are pinned by thepinning centers, as they have had 100, 000 time steps to move around the system exclusivelyvia thermal wandering and find point-like defects that will trap them. The absence of drivefurther increases the likelihood of the flux lines remaining relatively motionless and trappedin their pinned configurations, as seen by the zero mean velocity of the lines at Fd = 0(Fig. 4.1a). Upon introducing drive, at small values, we see an increased radius of gyrationcompared to the case with Fd = 0. This can be attributed to the relatively weak drive assistedby thermal fluctuations causing portions of the lines that are weakly pinned to break freefrom their original pins and get trapped in other nearby pins resulting in distortions of theline configurations which translates to increased line roughness and hence larger gyrationradius rg. The persistence of the pinned state under these drive conditions is supportedby the continued absence of mean line velocity v and the lack of significant change in thefraction of pinned line elements fp compared to its value (≈ 0.20) at t = 0. The radius ofgyration continues to increase with drive, until the drive is large enough to overcome theattractive forces exerted by the pins, enabling a complete depinning of the lines from thepins. This depinning point is marked by the rise of v, coinciding with a drop in rg andfp. These trends continue for the remainder of the drive values, resulting in the flux linesgetting further depinned (lower fp), moving faster (higher v) and becoming straighter (lowerrg) with increasing drive. The depinning crossover appears to occur somewhere in the driveinterval 0.004ε0 ≤ Fd ≤ 0.008ε0, the critical regime of drive. Drive values below this interval(Fd < 0.004ε0) constitute the pinned regime while those above it (Fd > 0.008ε0) constitutethe moving regime.

We have repeated these numerical operations for non-interacting flux lines and found theresults to be very similar: rg once again peaked around Fd ≈ 0.006ε0 which is also the valueat which v started assuming non-zero values, and fp began decaying from its steady initialvalue, indicating that for our purposes, the ranges for the pinned and moving regimes remainessentially unchanged for the non-interacting case.

4.2 Quenches within the Moving Regime

In a first set of numerical experiments, we quench the drive in a moving (Fd = 0.035ε0)steady-state system of vortex lines, in the presence of point-like disorder, to Fd = 0.025ε0, adrive value also in the moving regime.

For interacting lines, upon quenching, the mean velocity v of the lines drops suddenly (Fig.4.2a) due to the system being in an overdamped Langevin regime which effectively renders theelastic lines massless; the lines have no inertia and an instantaneous change in drive causes anequally abrupt change in velocity. At the moment of quench, the mean radius of gyration ofthese interacting lines starts growing (Fig. 4.2b). This is in agreement with our expectations:

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Figure 4.2: Relaxation of the (a) velocity v (units of b0/t0), (b) radius of gyrationrg (units of b0), and (c) fraction of pinned line elements fp with time (units of t0) fora system of interacting flux lines in the presence of point-like disorder, following adrive down-quench from Fd = 0.035ε0 to 0.025ε0 (moving to moving regime), withrelaxation times τv = 0, τrg = 1250t0, and τfp = 155t0, respectively. Data areaveraged over 1000 independent realizations. Copyright (2016) by the Institute ofPhysics.

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Figure 4.3: Evolution of the height autocorrelation function C(t, s) as a functionof the post-snapshot time t− s (units of t0), for waiting times s = 27t0, 29t0, 213t0,214t0, and 215t0, for systems of (a) non-interacting and (b) interacting flux lines inthe presence of point disorder, following a drive down-quench from Fd = 0.035ε0to 0.025ε0 (moving to moving regime). Time translation invariance is obeyed inboth cases for larger waiting times (s ≥ 213t0), as seen by the collapse of the cor-responding C(t, s) curves onto stationary curves that show exponential relaxation,with the interacting lines relaxing faster (τ = 693t0) than the non-interacting lines(τ = 1490t0). Data are averaged over 1000 independent realizations. Copyright(2016) by the Institute of Physics.

the reduced mean vortex velocity allows easier trapping of the lines by the pins present in thesample. This increased susceptibility to pinning coupled with thermal wandering results inthe lines assuming increasingly distorted configurations, whence their roughness is enhancedas a function of time. The growth of rg is fast (exponential) and stabilizes to a new steady-state value within a relaxation time τ = 1250t0. This exponential relaxation implies thatwhen quenching within the moving regime, the system transitions from one non-equilibriumsteady state to another quickly. The fraction of pinned line elements fp also grows rapidly(Fig. 4.2c) and reaches a new steady-state value after τ = 155t0 upon quenching the drive.This is to be expected since the lowered velocity of the lines means that a larger fractionof line elements in the system are susceptible to trapping by the pins. The free parametersfor the mathematical functions that have been fitted to the data in Fig. 4.2 and subsequentfigures were determined using the method of least squares. The time evolution of one-timephysical properties v, rg, and fp for non-interacting lines was found to be very similar tothat for the interacting case discussed above, with comparable exponential relaxation times(τv = 0, τrg = 1120t0, and τfp = 149t0). The effect of interactions on flux line dynamicsonly becomes evident when we study two-time height autocorrelations C(t, s).

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We have measured C(t, s) for waiting times s = 27t0, 29t0, 213t0, 214t0, and 215t0 as a functionof time elapsed post-quench t − s (Fig. 4.3). In both, non-interacting (Fig. 4.3a) andinteracting (Fig. 4.3b) cases, the autocorrelations for the higher waiting times (s ≥ 213t0)are observed to be time-translation invariant, i.e., they coincide and display exponentialrelaxation, with the interacting lines relaxing faster (τ = 693t0) than the non-interacting ones(τ = 1490t0). The faster relaxation in the presence of vortex interactions can be attributedto caging effects. The repulsions force the lines apart, resulting in faster depinning of the lineelements and hence straightening of the lines as well as the confining of these straightenedlines into a moving lattice. This quick straightening results in the lines becoming spatiallyuncorrelated with their initial horizontal configurations faster than in the non-interactingcase, thus explaining the faster height autocorrelation decay. Time-translation invariance isbroken, however, when we go to shorter waiting times (27t0, 29t0) for both the non-interactingand interacting cases. This is to be expected since the waiting times in question are shorterthan the relaxation time (τ = 693t0 ≈ 29.4t0), a regime where the system has not yet forgottenits initial state; therefore its relaxation behavior is dependent on when we start measuringthe autocorrelation function, i.e., it depends on the waiting time s. The observation of timetranslation invariance in the evolution of the height autocorrelation functions correspondingto higher waiting times rules out the possibility of physical aging in the system, as wasalready hinted at by the exponentially fast relaxation of the radius of gyration rg(t).

4.3 Quenches from the Moving into the Pinned Regime

For our next set of numerical experiments, we quench the drive of a system of flux lines inthe moving regime (Fd = 0.025ε0) to Fd = 0 in the pinned regime.

For the interacting lines, at the moment of quench, the velocity v, once again as in thecase of quenches within the moving regime, drops instantaneously to zero (Fig. 4.4a) asthe system enters a pinned state. The drop in velocity is accompanied by growth of theradius of gyration rg (Fig. 4.4b). This growth is very slow, however, when compared tothe exponentially fast relaxation of the radius of gyration that we observed in the case ofquenches within the moving regime. Here, the relaxation is slow enough that the radius ofgyration cannot stabilize to a steady value on the time scales we are exploring, and insteadshows a logarithmic growth with time. Initial attempts to fit the rg data to a power law bythe method of least squares yielded exponents quite close to zero. A logarithmic function wastherefore tested and found to provide a superior fit (smaller residuals) to the data than anytemporal power law. This slower logarithmic growth can be attributed to the system enteringa Bragg glass phase where the system of flux lines has access to many metastable states,each corresponding to a unique configuration. These states have a negligible mean velocitybut have similar probabilities associated with several different pinning configurations. Forinteracting lines, the growth in rg does not persist indefinitely, but terminates at a certainupper value of time t. This is a consequence of the caging effect of the repulsive vortex

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Figure 4.4: Relaxation of the (a) mean vortex velocity v (units of b0/t0), (b)radius of gyration rg (units of b0), and (c) fraction of pinned line elements fp withtime t (units of t0) for a system of interacting flux lines in the presence of point-like disorder, following a drive down-quench from Fd = 0.025ε0 to 0 (moving topinned regime). v drops instantaneously, while both rg and fp relax logarithmicallyslowly (ar = 0.05b0, af = 0.01) with t. Data are averaged over 1000 independentrealizations. Copyright (2016) by the Institute of Physics.

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interactions on the growth of the time-dependent correlation length L(t) associated withthe flux lines [69]. However this caging effect is not yet perceptible in the data shown inFig. 4.4. The interaction-induced caging effect will also affect the behavior of the two-time height autocorrelation functions at very long times. Another one-time quantity thatdisplays slow logarithmic growth post quench as the system enters the glassy pinned state isthe fraction of pinned line elements fp (Fig. 4.4c), in contrast to the fast exponential growthand stabilization of the quantity seen for quenches within the moving regime (moving-to-moving quenches). For the relaxation of one-time quantities v (τ = 0), rg (ar = 0.06b0), andfp (af = 0.01) in the interaction-free situation, as in the case of moving-to-moving quenches,we did not find remarkable qualitative differences compared to the system with interactinglines.

The two-time height autocorrelations C(t, s) for quenches into the pinned regime display slowtemporal relaxation accompanied by the breaking of time translation invariance for both non-interacting (Fig. 4.5a) and interacting lines (Fig. 4.5c). This is in contrast to the situation forquenches within the moving regime, where time translation invariance was clearly observedfor the entire period of measurement for waiting times greater than the relaxation time of thesystem. We checked the autocorrelations for dynamical scaling by testing a range of scalingexponents b in the following way. For each b under consideration, we plotted the threesbC(t, s) curves (s = 214t0, 215t0 and 216t0) against t/s. We then employed a least-squaresalgorithm to compare these functions and identified the value of b that rendered the bestdynamical scaling collapse. For the non-interacting (Fig. 4.5b) and interacting (Fig. 4.5d)cases, the algorithm yielded pairs of dynamical aging scaling exponents (b, λC/z) = (0.004,0.007) and (0.005, 0.011), respectively, for which the individual height autocorrelation curvescollapsed onto a master curve, a clear indication of physical aging in the system. The scalingonly emerges for larger t/s, when the system has had sufficient time to overcome the initiallarge fluctuations that immediately follow the quench, and to enter the aging scaling regime.

For interacting lines, the aging scaling regime will be cut short at very long times by thecaging effect of the repulsive vortex interactions (also responsible for limiting the growth ofrg) [69]. The scaling form for simple aging given in (2.44) is a special case of the more generalscaling form fC(t, s) ∼ [L(t)/L(s)]−λC . The simple aging form arises from the general casewhen L(t) grows as a simple power law of t. The algebraic growth L(t) ∼ t1/z with thedynamic scaling exponent z is limited by the interaction-induced caging effect. The agingscaling exponents b seen here are over an order of magnitude smaller than those obtained inprevious studies: one on the aging of randomly placed, interacting flux lines in the absence ofdrive [41] and another on relaxation following temperature and magnetic field quenches, alsofor randomly placed flux lines without drive [42]. In the case of drive quenches as presentedhere, we have verified that during the initial pre-quench, high-drive (Fd = 0.035) period of thesimulation, the flux lines constitute a highly correlated moving lattice. This is in contrast tothe previous studies where, on account of the absence of drive, the initial disorder dominatedstate was always random and uncorrelated. We can thus infer that the initial conditions havea significant influence on the aging scaling exponents, with a correlated initial state yielding

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Figure 4.5: Height autocorrelation function C(t, s) as a function of t−s (a, c), andscaled height autocorrelation s−bC(t, s) as a function of t/s (b, d), for systems of (a,b) non-interacting and (c, d) interacting flux lines in the presence of point disorder,following a drive down-quench from Fd = 0.025ε0 to 0ε0 (moving to pinned regime).Time translation invariance is broken (a, c) and dynamical scaling is observed (b,d) in both cases, with scaling exponents (b, λC/z) found to be (0.004, 0.007) and(0.005, 0.011), respectively, for the (b) non-interacting and (d) interacting cases.Data are averaged over 10, 000 independent realizations. Copyright (2016) by theInstitute of Physics.

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far smaller aging scaling exponents compared to an uncorrelated one.

4.4 Summary

In this chapter, we have investigated the long-time relaxation features of driven magneticflux vortices in type-II superconductors following sudden quenches of external current. Inorder to study the post-quench dynamics of these vortices in the presence of uncorrelatedpoint-like disorder, we modeled them as directed elastic lines in the presence of localizedpinning centers, and solved the associated Langevin molecular dynamics equations numer-ically. In the simulations we maintained a constant ambient temperature. The externalcurrent quenches were realized in the form of instantaneous changes in the drive, a quantityin the elastic line model that mimics the Lorentz force exerted by external current on theflux vortices. In this study, we focused on two types of drive quenches, those within themoving regime and those from the moving regime into the pinned regime.

For quenches within the moving regime, we have studied the effects of the vortex-vortexrepulsive interactions on the relaxation kinetics of the vortices by performing drive quenchesin the system with the interactions initially absent or in effect. In both cases, drive quencheswithin the moving phase result in fast exponential relaxation of the system from one non-equilibrium steady state to another, as evidenced by the rapid temporal evolution of one-timeobservables such as the mean radius of gyration of the lines and the fraction of pinned lineelements. The two-time height autocorrelation functions for different waiting times displaysimilar fast exponential relaxation as the one-time quantities, along with time translationinvariance, firmly eliminating the possibility of physical aging in the case of quenches withinthe moving regime. When turned on, the screened logarithmic repulsive interactions betweenthe flux lines significantly speed up the exponential relaxation of the height autocorrelationswith the associated relaxation time being around half that for quenches with no interactionspresent.

For our study on drive quenches from the moving to the pinned regime, in stark contrast toquenches within the moving regime, the relaxation of the system after the quench is muchslower, which is seen in the non-exponential, logarithmic time evolution of the radius ofgyration and fraction of pinned line elements. This indicates that the system fails to reacha steady state when quenched into the pinned regime on time scales that are on the orderof the simulation duration. The two-time height autocorrelations show breaking of timetranslation invariance, accompanied by dynamical scaling with t/s, evidence for aging in thesystem, as we quench it from a moving non-equilibrium steady state into a pinned, glassyone. The t/s range for which simple aging is applicable for interacting lines is bound bythe limiting of the algebraic growth of the characteristic time-dependent correlation lengthL(t), a consequence of the caging of the flux lines by the repulsive vortex interactions.Correlated initial conditions as with the moving lattice seen in the initial state in our study

39

yield markedly smaller aging scaling exponents compared to uncorrelated initial conditionssuch as those obtained in previous investigations where the flux lines were initially randomlydistributed.

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Chapter 5

Critical Scaling and Aging near theVortex Depinning Transition

This chapter contains original unpublished results that we intend to submit for publicationby Dec 2018.

We devote this chapter to the investigation of critical behavior and aging phenomena ofdriven flux lines as they transition from the pinned glassy state to the moving state. Fluxlines driven over quenched disorder are an example of a system where coherent structuresare driven through disordered media. Such systems are interesting from a theoretical pointof view since they exhibit a variety of phases resulting from the competing energy scalesassociated with the interactions, disorder, thermal fluctuations, and external drive F . Theyare also of interest from an experimental and technological standpoint since they show upin such a diverse array of physical scenarios, e.g. in magnetic flux vortices, Wigner crystals,driven colloids, charge density waves (CDWs), magnetic bubbles, magnetic domain walls,and directed polymers [70]. The non-linear dynamics of vortex motion in disordered type-IIsuperconductors have been studied extensively [6,15,16,71,72] including by the use of func-tional renormalization group methods [23, 73–75]. At low temperatures and drive strengths(below the depinning drive), flux vortices in the presence of random quenched disorder areknown to be either in a Bragg glass phase [19] characterized by quasi long-range positionalorder or in a disordered vortex glass phase [17]. In this study, we focus on the depinningtransition of flux lines as they go from the glassy pinned regime below a critical drive Fc tothe moving regime above it.

Fisher in 1985 via phenomenological arguments posited that the depinning of sliding CDWsmay be regarded as a dynamic critical phenomenon where driving force acts as the controlparameter and velocity as the order parameter [76]. Though originally proposed for CDWs,the idea has been successfully extended to several other domains. For most manifolds (andCDWs) in the elastic limit, it has been theoretically shown that depinning constitutes acontinuous phase transition at zero temperature with the mean velocity above the depinning

41

threshold showing a power law dependence on the reduced drive vT=0 ∼ fβ and at temper-atures above zero vf=0 = T 1/δ, where f is the reduced drive f = (Fd − Fc)/Fc, and β andδ are critical exponents [73, 77–82]. Outside the elastic limit, a theoretical description ofpinning is much more challenging. Marchetti et al. have proposed a coarse-grained model toaddress non-elastic dynamics in depinning where a visco-elastic coupling is used to describetopological defects or phase slips [83–87]. In the mean-field limit, the theory predicts twotypes of depinning transitions; one is continuous and belongs to the universality class ofelastic depinning, while the other is a discontinuous hysteretic one. Evidence for the firsttype has been found in experiments [88–95] and numerical studies [96–105] where a con-tinuous second-order transition has been observed. Evidence for the second type has beenfound in experiments on CDWs [106] and 3D simulations of vortices [107] in the form of adiscontinuous hysteretic transition where the flux lines jump between pinned and unpinnedstates. Hysteretic depinning in a special case of the theoretical model has been confirmedvia functional renormalization group methods [108]. Hysteretic depinning for CDWs hasbeen predicted in some phase-slip models [109,110] while other phase slip models make pre-dictions of non-hysteretic and continuous depinning [111] which is supported by numericalstudies [112, 113]. In an alternative treatment, when inertia is introduced to the equationof motion [114–116], a small inertial parameter leads to continuous depinning that breaksdown and becomes hysteretic when the magnitude of the inertia is raised beyond a certainthreshold. In periodic systems where the displacement field has two components (N = 2),numerical simulations support continuous depinning and indicate that strong disorder isaccompanied by dislocations and plasticity [102–105,117–132].

For the depinning of vortices in disordered type-II superconductors, Luo and Hu have usedmolecular dynamics simulations to study the dynamical scaling of velocity-force curves forflux lines in an three-dimensional embedded space (d = 3, N = 2), obtaining critical expo-nents β and δ in both the weak and strong pinning regimes [133]. Fily et al. have studieddepinning for two-dimensional vortex lattices (d = 2, N = 2), and determined β and δ for thescaling relation that governs the velocity-force behavior near the depinning transition [70].Di Scala et al. have computed critical scaling exponents including the growth exponent νfor the elastic depinning of vortices in two dimensions [134]. Bag et al. have recently cal-culated critical scaling exponents for experimental data they obtained for 2H-NbS2 singlecrystals [135]. For a recent review article on depinning and nonequilibrium phases in varioussystems, refer to Ref. [136].

In this chapter, we use our elastic line model (Chapter 3) to study critical behavior near thedepinning transition for vortices in the presence of weak random quenched disorder or pointdefects in a three dimensional system (d = 3) with a two-dimensional displacement vector(N = 2). We perform finite-temperature scaling on both steady-state velocity and radius ofgyration data and as a result derive the critical scaling exponents β, δ, and ν (see Section5.1.1) that characterize the depinning process as a continuous second-order phase transitionat zero temperature, obtaining good agreement with experiment for β. We probe agingdynamics in the system by quenching vortices from the high-drive moving lattice regime to

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Figure 5.1: Steady-state (a, b) velocity v (units of b0/t0) and (c, d) radius ofgyration rg (units of b0) as a function of drive Fd (units of ε0) for (a, c) interactingand (b, d) non-interacting flux lines. Each quantity is measured for temperaturesT = 0.0005, 0.0006, 0.0007, 0.0008, and 0.0009 (units of ε0b0).

the critical depinning regime (Section 4.1) and studying the ensuing two-time correlations tocompute the aging exponent b and the ratio of the autocorrelation and dynamic exponentsλc/z. Exploiting hyperscaling relations between the static and dynamic exponents, we alsoobtain estimates for λc and z separately, as well as the roughness exponent ζ.

5.1 Critical Scaling

For the purpose of obtaining the scaling exponents that characterize the critical behaviorof flux lines near the depinning transition, we focus primarily on two quantities – the meanradius of gyration rg (3.5) and the mean velocity of the lines v (3.6). We measure steady-state

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rg and v as a function of driving force Fd across the depinning transition for five temperaturesT (Figure 5.1) using the procedure outlined in Section 3.4.1. Repulsive interactions betweenvortices counter the short-range attractive forces of the defects and lower the depinningthreshold as seen by the earlier onset of v and peaking of rg for interacting vortices as seenin Figure 5.1.

5.1.1 Scaling Arguments

As per Fisher’s original argument [76], the depinning transition is a critical phenomenonthat implies the existence of critical scaling near the depinning threshold. It is useful tointroduce the reduced force

f = (Fd − Fc)/Fc , (5.1)

where Fc is the critical depinning force at zero temperature (T = 0). f plays the role of thecontrol parameter in this context, with the order parameter v emerging above the criticalpoint f = 0 (Fd = Fc). Critical scaling near f = 0 implies that

v(T = 0, f > 0) ∼ fβ (5.2)

and

v(T > 0, f = 0) ∼ T 1/δ. (5.3)

A scaling ansatz that encapsulates this behavior [70,76,133,137–139] is

v(T, f) = T 1/δS(T−1/βδf), (5.4)

where S(x) is a scaling function that satisfies the conditions

S(x→∞) ∼ xβ and S(x = 0) = const. (5.5)

Taking the limit T → 0+ in (5.4) and exploiting the asymptotic behavior of S(x) givenabove yields the scaling relation (5.2), while setting f to zero in (5.4) yields the other scalingrelation (5.3).

Additionally, we argue that the radius of gyration rg plays the role of the critical correlationlength ξ in the system that should diverge as a power law

rg(T = 0, f > 0) ∼ f−ν (5.6)

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near the critical point (f → 0+) at zero temperature, and show a power-law dependenceon temperature at f = 0. Taking these hypotheses into account while ensuring that thetemperature-scaled reduced drive (T−1/βδf) is consistent with (5.4) leads us to the followingscaling ansatz for rg:

rg(T, f) = T−ν/βδR(T−1/βδf), (5.7)

where

R(x→∞) ∼ x−ν and R(x = 0) = const. (5.8)

The first scaling relation (5.6) is obtained by taking T → 0+ in (5.7) in conjunction with(5.8), while setting f to zero gives us our second scaling relation

rg(T > 0, f = 0) ∼ T−ν/βδ. (5.9)

5.1.2 Fc and δ from Convexities of v-T Curves

Before we can check whether our data behave as predicted by the critical scaling ansatzeexpressed in (5.4) and (5.7), we must first compute the zero-temperature critical drive Fc,so that we may transform our drives Fd to reduced drives f as per (5.1). To compute Fc, weuse the relationship between v and T expressed in (5.3). As per this equation, v should goas a power law of T at f = 0 or Fd = Fc. We can compute Fc by plotting v versus T curveson a log-log scale for a range of fixed Fd values in the depinning region and identifying theFd value for which the log v vs log T curve is closest to a linear function. We performed thisanalysis on both our interacting and non-interacting velocity data. Some of the curves forthe interacting data are shown in the inset of Figure 5.2b. The curves for Fd below 0.013ε0are concave while those above are convex; the curve for Fd = 0.013ε0 is closest to a linearfunction, providing us with an estimate of Fc = 0.013± 0.0005ε0 for the interacting system.Since vFd=Fc ∼ T 1/δ, the slope of the critical curve is 1/δ. Calculating the slope, we getδ = 5.6± 0.2

Identical inflection analysis of v-T curves for the non-interacting data yields Fc = 0.015 ±0.0005ε0 and δ = 4.1 ± 0.1. Note that Fc is higher for the non-interacting system than theinteracting one, which is consistent with the behavior of the raw steady-state data (Figure5.1).

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Figure 5.2: Steady-state data from Figure 5.1 scaled using ansatze (5.4) and (5.7)with critical scaling exponents β, δ and ν. Figures (a, c) show results for the non-interacting system (β = 0.33 ± 0.03, δ = 4.1 ± 0.1, ν = 0.74 ± 0.13), while (b, d)represent the interacting system (β = 0.43± 0.04, δ = 5.6± 0.2, ν = 0.98± 0.15).Figures (a, b) show scaled velocities v for temperatures T = 0.0005, 0.0006, 0.0007,0.0008, and 0.0009 (units of ε0b0) as functions of scaled reduced drive f , and figures(c, d) show scaled radii of gyration rg for the same temperatures also as functionsof scaled f . The inset in (b) shows the velocity-temperature (v-T ) curve for thecritical drive Fc = 0.013 ± 0.0005ε0 as a solid line, dashed concave v-T curves fordrives less than Fc located below the critical curve, and dashed convex v-T curvesfor drives greater than Fc located above the critical curve. The slope of the criticalcurve is equivalent to 1/δ (see (5.3)) and yields a δ value of 5.6± 0.2.

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Source d βδ β δ ν

Luo et al. [133] 3 1.0± 0.019 0.754± 0.010 1.326± 0.018

Fily et al. [70] 2 1.73± 0.27 1.30± 0.10 1.33± 0.18

Di Scala et al. [134] 2 1.04± 0.21 0.29± 0.03 3.57± 0.64 1.04± 0.04

Bag et al. [135](experiment; averaged results)

3 1.01± 0.06 0.41± 0.02 2.47± 0.08

Our results (interacting) 3 2.41± 0.24 0.43± 0.04 5.6± 0.20 0.98± 0.15(non-interacting) 3 1.35± 0.11 0.33± 0.03 4.1± 0.10 0.74± 0.13

Table 5.1: Critical scaling exponents for vortex depinning from different studieswith displacement field of dimension N = 2 and embedding space of dimension d.All results are for interacting vortex simulations unless specified otherwise.

5.1.3 Scaling Collapse

Having estimated Fc, we compute the reduced drives f and check if v and r scale respectivelyas per (5.4) and (5.7). Using global optimization methods [140,141], we have estimated theexponents β, δ, and ν that provide optimal scaling of the temperature-dependent observ-ables, thereby facilitating their collapse onto a single master curve that is independent oftemperature as seen in Figure 5.2.

For the interacting system, the optimal values of the exponents are found to be β = 0.43±0.04, δ = 5.6± 0.2, and ν = 0.98± 0.15 (Figure 5.2b/d). Our β value shows good agreementwith experiment (Table 5.1). The product βδ has been estimated once when scaling the v-fcurves for different temperatures as βδ = 2.41± 0.24, and then again when scaling the rg-fcurves as βδ = 2.5± 0.2, with the two independent estimates agreeing within error bars.

By identical methods, we have evaluated the critical scaling exponents that yield excellentfinite-temperature scaling for the non-interacting system to be β = 0.33±0.03, δ = 4.1±0.1,and ν = 0.74 ± 0.13 (Figure 5.2a/c). The values of βδ estimated for v-f scaling and rg-fscaling are βδ = 1.35± 0.11 and βδ = 1.4± 0.1 respectively and are also in agreement.

In both the non-interacting and interacting cases, the fact that our estimates of βδ for scalingthe rg-f curves using the ansatz in (5.7) are in agreement with the values obtained by scalingthe v-f curves with the extensively verified ansatz in (5.7) [70,76,133,134,137–139] coupledwith the quality of scaling for both ansatze supports the validity of our hypothesis (5.7)that the radius of gyration rg is proportional to a characteristic length scale ξ in the systemthat diverges at the critical depinning transition point f = 0 as a power law of f at zerotemperature, and more importantly, that we are properly accessing the asymptotic criticalscaling regime in the system.

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Scheme b direct λc/z direct ζ indirect λc indirect z indirect

Non-interacting 0.56± 0.03 0.61± 0.02 0.65± 0.24

Interacting 0.29± 0.03 0.43± 0.03 0.98± 0.16 0.60± 0.09 1.40± 0.18

Table 5.2: Aging scaling exponents for relaxation of vortices following criticaldrive quenches.

5.2 Critical Aging

In addition to finite-temperature critical scaling of one-time quantities near the depinningtransition, we have studied the relaxation of the flux lines following a drive quench from themoving regime to the critical depinning regime using two-time height autocorrelations.

We begin by identifying for each temperature T , the drive strength corresponding to themaximum steady-state radius of gyration (Figure 5.1c/d). As seen from the results in thepreceding section, the gyration radius is a good proxy for correlation length in the system,and its peak value must occur at a drive strength sufficiently close to the finite-temperaturedepinning crossover point, i.e., within the critical-depinning drive regime. For the criticalquench, we prepare the system in a moving (Fd = 0.035ε0) non-equilibrium steady stateat the desired temperature T and follow the protocol in Section 3.4.2 to quench it to thedepinning crossover drive corresponding to T , following which we measure the one-timequantities of radius of gyration (3.5) and line velocity (3.6) as well as the two-time heightautocorrelation C(t, c) given by (3.8). We perform these critical quenches for five differenttemperatures: T = 0.0005, 0.0006, 0.0007, 0.0008, and 0.0009 (units of ε0b0).

When we plot the height autocorrelations C(t, s) against t − s for different waiting times(s = 26t0, 27t0, and 28t0), we see clear breaking of time-translation invariance (Figure 5.3a/b),the first indication of aging. The data are found to dynamically scale (Figure 5.3c/d) ac-cording to the full-aging ansatz C(t, s) = s−bfC(t/s) (2.43) where b is the aging exponent.Interestingly, the scaled autocorrelations collapse on a master curve that appears to be linearon a log-log scale when plotted against t/s. This implies that the scaling function fC variesalgebraically with t/s, indicating that the flux lines undergo simple aging after a criticalquench. The master curve goes as a power law (t/s)−λc/z where λc, and z are respectivelythe autocorrelation, and dynamic exponents.

We obtained excellent dynamical scaling following critical quenches (with interactions turnedon and off) for all five temperatures considered. The results for T = 0.0005ε0b0 are shown inFigure 5.3. In each interaction scheme, the values of λc/z and b were found to agree (withinerror bars) across all temperatures as seen in the insets of Figure 5.3. This further supportsthe hypothesis of vortex depinning being a critical phenomenon (at zero temperature), sinceuniversal scaling or scaling that does not depend on initial conditions or other microscopic

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Figure 5.3: Two-time height autocorrelations C(t, s) following critical quenchesat T = 0.0005ε0b0 for waiting times s = 26t0, 27t0, and 28t0 as a function of(a, b) t − s and (c, d) these autocorrelations scaled with sb as a function of t/swhere b is 0.56 ± 0.03 for (c) and 0.30 ± 0.03 for (d). Figures (a, c) and (b, d)represent respectively the non-interacting and interacting flux line systems. Thesolid black line in (c, d) shows the power law dependence of the scaled quantitiesfor T = 0.0005ε0b0 on t/s with λc/z = 0.61± 0.015 for (c) and 0.44± 0.03 for (d).The lower and upper insets in (c, d) show respectively the values for b and λc/zestimated for critical quenches at temperatures T = 0.0005, 0.0006, 0.0007, 0.0008,and 0.0009 (units of ε0b0). In each inset, the solid horizontal line represents themean value of the data points and the shaded region indicates the error of the mean.bmean = 0.56 ± 0.03 and (λc/z)mean = 0.61 ± 0.02 for (c) while bmean = 0.29 ± 0.03and (λc/z)mean = 0.43± 0.03 for (d).

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details of the system (like temperature) is a signature of a system being near a criticalpoint [142–147]. The combined results for the different temperatures are stated in Table5.2. Correlations for interacting vortices decay slower (λc/z = 0.43 ± 0.03) than they doin the non-interacting scheme (λc/z = 0.61 ± 0.02) indicating that repulsive vortex-vortexinteractions facilitate the formation of correlated vortex deformations, and slow down therelaxation of these deformations as a function of time.

5.3 Relating Static and Dynamic Exponents

Equations relating the growth exponent ν, the order parameter exponent β, the roughnessexponent ζ and the dynamic exponent z have been derived, the latter using statistical tiltsymmetry [77,78,82]:

ν = 1/(2− ζ) , (5.10)

β = (z − ζ)ν . (5.11)

One can intuit the physics behind (5.11) through dimensional analysis. The dynamic ex-ponent z governs the growth of the characteristic correlation time τ in the system at zerotemperature with the critical correlation length (rg ∼ ξ) as τ ∼ rzg . Substituting rg ∼ f−ν

from (5.6) here yields τ ∼ f−νz. Multiplying this with v ∼ fβ from (5.2) yields τv ∼ f−νz+β.Dimensionally arguing that τv ∼ rg and substituting this as well as (5.6) in the precedingexpression yields f−ν ∼ f−νz+β, from which we get the exponent relation β = (z− 1)ν. Thisis similar to (5.11), except that we must replace the dimension 1 in the expression we havederived with the roughness exponent ζ to account for the fractal nature of the flux lines. Assuch (5.11) captures the dimensional relationship between the characteristic time and lengthscales in the system near critical depinning.

Using our numerically obtained value of ν = 0.74±0.13 for non-interacting vortices in (5.10)gives the value ζ = 0.65 ± 0.24. Substituting these values of ζ, ν and our estimate of β(Table 5.1) into (5.11) gives us an estimate for the dynamic exponent z as 0.89 ± 0.24. Avalue of z < 1 is not physically realistic since it implies flux line relaxation that is faster thanballistic propagation [148]. The current errorbars do not rule out the possibility of z > 1however and performing more simulation runs would shrink the errorbars allowing for thedetermination of a more useful estimate of z for non-interacting vortices.

As with the non-interacting scheme, we have used (5.10) and (5.11) in conjunction with ournumerical results to obtain estimates ζ = 0.98± 0.16, λc = 0.60± 0.09 and z = 1.40± 0.18for the interacting scheme. We have documented these along with the result for ζ for thenon-interacting scheme in Table 5.2.

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By performing driving force quenches to different drive strengths Fd > FC near the criticaldepinning point FC , it is in principle possible to directly compute the dynamic exponent z,by computing the characteristic relaxation times τ for different Fd and using the definitionτ ∼ ξz, where ξ is the correlation length in the system. Since gyration radius rg ∼ ξ (seeSection 5.1.3), one can write down a scaling ansatz for τ (similar to the ones we used for rgand v) that governs the drive evolution of τ for different temperatures T , and independentlyobtain scaling exponents including z. This directly obtained value can be compared to thez value obtained by us via scaling exponent relations in Section 5.2, allowing us to furthertest the validity of these relations in our system.

5.4 Summary

In this study, we used our coarse-grained elastic line model of flux vortices to study theelastic depinning of vortices subject to weak random pins in three dimensions. We performedfinite-temperature scaling of one-time quantities (gyration radius and line velocity) to obtainconsistent estimates of critical exponents β, δ and ν via independent analyses of the one-timequantities, suggesting that we are properly accessing the asymptotic scaling regime in thesystem. Our estimate for ν is in agreement with the numerical result from [134] obtained viafinite-size scaling for a two dimensional vortex system. Our value for β is in good agreementwith recent experimental results [135].

We also determined aging scaling exponents b, λc and z for the relaxation of the systemfollowing drive quenches into the critical depinning regime, via the analysis of two-timeheight autocorrelations and the aid of hyperscaling relations between the static and dynamicexponents. We found evidence for universal scaling near the depinning threshold in the formof temperature independence of the aging scaling exponents indicating that we are accessingthe critical aging regime in the system, and providing further support for depinning beinga critical phenomenon. Interactions appear to repulsively cage flux lines and slow down thedecay of correlations in the system as evidenced by the smaller value of λc/z compared tothe non-interacting scheme.

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Chapter 6

Flux-flow regimes in the presence ofparallel twin boundaries

This chapter was adapted with only minor changes from the manuscript:

H. Chaturvedi, N. Galliher, U. Dobramysl, M. Pleimling, and U. C. Tauber, “Dynamicalregimes of vortex flow in type-II superconductors with parallel twin boundaries,” Eur. Phys.J. B, vol. 91, p. 294, 2018.

We devote this chapter to the exploration of dynamics of driven magnetic flux lines indisordered type-II superconductors in the presence of twin boundaries oriented parallel to thedirection of the applied magnetic field, using a three-dimensional elastic line model simulatedwith Langevin molecular dynamics. The lines are driven perpendicular to the planes tomodel the effect of an electric current applied parallel to the planes and perpendicular tothe magnetic field. A study of the long-time non-equilibrium steady states for varyingsample orientation and thickness reveals a rich collection of dynamical regimes spanningthe depinning crossover region that separates the pinned and moving-lattice states of vortexmatter. We observe the emergence of a preferred direction for the ordering of the Abrikosovlattice in the free-flowing vortex regime due to asymmetric pinning by the planar defects.We have performed novel direct measurements of flux line excitations such as half-loops anddouble kinks to aid the characterization of the topologically rich flux flow profile.

Planar defects are commonly found in the form of twin boundaries in high-Tc cuprates suchas (doped) YBa2Cu3O7−x (YBCO) and La2CuO4+δ. Twin boundaries are formed in thesematerials as they undergo a tetragonal to orthorhombic structural phase transition duringthe oxidative cooling phase of synthesis [149,150]. Twin boundaries tend to occur naturallyas a mosaic of twins from one of two orthogonal families [9,10]. Samples containing a singlefamily of twin planes are fabricated artificially [11–13]. The work in this paper pertains tothe latter. In the case of a single family of twin planes, the pinning effect of twin boundarieson flux lines is highly anisotropic [12,151–159], i.e., it strongly depends on the angle between

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the magnetic field and the twin planes (that are both oriented along the crystallographic caxis). Early experiments exploring this variation of pinning strength with field orientationyielded contradictory results [160, 161], with later experiments [151] conclusively showingthat pinning is strongest when the field is parallel to the twin planes and the current isflowing in the ab plane parallel to the twins thus exerting a Lorentz force on the flux linesperpendicular to the planar defects, confirming the results first presented by Kwok andcollaborators [160, 162]. Experiments on flux-boundary pinning can be broadly classifiedinto two types – those that measure electrical transport properties of the system such asresistivity and critical depinning current, and those where the flux lines are directly imagedvia techniques like small angle neutron scattering [163] and scanning tunneling microscopy[164].

In transport experiments, planar defects are seen acting as strong pinning centers by theirinfluence on the linear resistivity of a sample near the melting point of the Abrikosov latticeinto a flux liquid. The monotonic increase of resistivity with temperature observed in theabsence of disorder is interrupted by a drop near the lattice melting transition in the presenceof material defects due to the pinning of vortices by the disorder. This drop is more pro-nounced for spatially correlated disorder such as columnar or planar defects owing to theirsuperior pinning properties [160,165]. Experiments measuring the critical depinning currentdensity Jc in systems with planar defects also confirm the strong flux-boundary pinninghypothesis, with a sharp maximum in Jc observed as a function of temperature just belowthe melting point of the Abrikosov lattice in a phenomenon known as the peak effect [166].Real-time imaging experiments of flux lines driven perpendicular to a single family of twinplanes also show strong pinning of magnetic vortices at the twin boundaries [167–169]. How-ever, relatively recent experiments utilizing scanning superconducting quantum interferencedevice microscopy to probe vortex motion near twin boundaries in pnictide superconductorsshow that vortices avoid pinning to twin boundaries in these materials owing to enhancedsuperfluid density near the boundaries. They instead prefer to move parallel to them. Thisflux-boundary repulsion is offered as a possible explanation for the enhanced critical currentsobserved in twinned superconductors [170].

Numerical studies of vortex behavior in the presence of planar defects range from solving thefull time-dependent Ginzburg-Landau equations [171–174] to more approximate descriptions[175–179] of vortices in two-dimensional thin-film and three-dimensional bulk samples asstructureless point- or string-like objects that are studied with either Monte Carlo simulationsor Langevin dynamics methods. The experimentally detected anisotropy of pinning andtransport has been observed in numerical simulations of twinned superconductors [180,181]with thermal fluctuations being enhanced and vortex motion facilitated within defect planes.Reichhardt et al. identified three phases of flux flow in London-Langevin studies of drivenvortices subject to planar pinning, viz. guided plastic flow at low drives characterized bypartially-ordered vortices, highly disordered plastic flow at intermediate drives, and elasticflow at high drives with the vortices reordering into a lattice in this phase [176]. This isin agreement with earlier results of Crabtree et al. [182] which were obtained by solving

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the time-dependent Ginzburg-Landau equations on a discrete grid. Recent analytical worksargue that for randomly placed parallel planar defects, the flux line lattice displays a novelplanar glass phase with an exponential decay of long-range translational order as opposedto the algebraic decay seen in the case of the Bragg glass obtained in the absence of planardefects [183–185]. In their analytical studies of the low-temperature dynamics of magneticflux lines in type-II superconductors, Marchetti and Vinokur find that flux lines driventransverse to a family of parallel twin planes do so in a manner analogous to the motion ofone-dimensional charge carriers in a disordered semiconductor induced by an electric field.They discuss various linear and non-linear transport mechanisms for vortex motion that areassociated with different flux line excitations in the system [68,186].

In this present work, we use an elastic line description of vortices in a three-dimensionalsample modeled to mimic the behavior of flux vortices in the mixed phase of YBCO (Chapter3). The elastic lines are mutually repulsive and are subject to a horizontal drive representingthe Lorentz force exerted by an external current. The sample contains two planar defectsperpendicular to the drive direction as well as many randomly distributed point-like pinningsites that represent point disorder such as those produced by oxygen vacancies (Figure 6.1).The dynamics of this model are simulated by numerically solving overdamped Langevinequations that account for the fast degrees of freedom in the system as stochastic forcingthat is subject to certain physical constraints. This particular implementation of the elasticline model was previously used by Dobramysl et al. [41] to study relaxation and agingphenomena of flux lines in the presence of point-like and columnar disorder. Since then,it has been employed to investigate relaxation dynamics of vortex lines following magneticfield, temperature and drive quenches [42, 43, 187], as well as the pinning time statistics forflux lines in disordered environments [188]. We have extended this work to here addressthe dynamics of vortices driven parallel to the x axis, and perpendicular to two parallelplanar defects that are placed either a short distance (16 pinning center radii b0) apartor a large distance (160b0) apart. The system is periodic in the x direction and thereforethe planar defect pair configuration employed here is comparable to a long YBCO samplecontaining evenly spaced pairs of parallel twin boundaries. We observe the long-time steady-state behavior of this system of flux lines for two sample orientations (aspect ratios) andseveral sample thicknesses L, i.e., system extensions along the magnetic field or z direction.These observations involve measuring several physical attributes and occurrence statistics fordifferent flux line excitations as well as static and dynamic visualizations of the system undera range of conditions and from a number of (both two- and three-dimensional) perspectives.

The characterization of the depinning process, by which magnetic vortices subject to planarpinning transition from the pinned to the moving lattice state, has been greatly enhancedby direct measurements of the unique vortex excitations that emerge from planar defect-induced elastic deformations of these vortices. These measurements are made possible bythe full three-dimensional specification of our simulated model coupled with the structuralsimplicity of the infinitesimally thin elastic lines that represent the vortices. The steady-state results pertaining to the depinning region reveal a rich assortment of drive regimes

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Figure 6.1: Simulation snapshot of flux lines (red) driven along the x axis inthe presence of two planar defects (blue) oriented perpendicular to the directionof drive and many randomly positioned point defects (blue). Copyright (2018) byThe European Physical Journal.

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that culminate in the dynamical freezing of the vortices into a moving hexagonal lattice.Upon increasing the driving force, distinct crossover regimes are encountered between thefully pinned glassy phase and the freely flowing ordered lattice, depending on the orientationof the system. In these intermediate current regimes, the vortices remain partially pinnedto the planar defects; flux transport in these regimes is mediated through vortex half-loop,single-kink, and double-kink excitations. The quantitative analysis of vortex excitationscomplement these results by providing us with insight into the types of structures thatfacilitate the realization of different depinning regimes.

6.1 Depinning Drive Regimes and Preferred Ordering

We configure the horizontal xy plane of the system in one of two equal-area, orthogonalorientations: (A) 16/

√3λab × 8λab or (B) 8λab × 16/

√3λab. Orientation B is a 90 rotation

of orientation A about the z axis. The two orientations produce markedly different flux flowprofiles that are discussed in detail below. We begin with results for the case where theplanar defects are placed closely together. In both orientations, A and B, we see a pinnedregime at the beginning of, and a moving-lattice regime at the end of the driving-forcerange under consideration. For orientation A, where the system is more extended alongthe x than the y direction, we find three intermediate regimes (liquid, partially-ordered,and smectic) that span the depinning crossover region connecting the two extremal regimes(Figure 6.2) [189], while for orientation B (simulation domain longer in the y direction),there exists only one intermediate liquid / smectic regime. In this section we discuss themechanisms underlying the development of the differing crossover flow profiles in the twoorientations. In our simulations with rather few interacting vortex lines, the boundariesbetween these distinct drive regimes are not well defined; the crossover regions separatingthem have non-zero widths and their sizes are not uniform (Figure 6.2). These steady-state results are supplemented by simulation snapshots of the system under different driveconditions, which provide visual evidence for the defining structural configurations that theflux lines assume in different regimes.

6.1.1 The Pinned Regime

At low driving currents, the first dynamical steady-state region encountered in the systemis the pinned regime. Observed in both orientations, the pinned regime is characterizedby zero mean velocity v (Figure 6.3a/e) and a sizable fraction of pinned line elements fp(Figure 6.3b/f). The degree of local hexatic order h is relatively low (∼ 0.2 for L = 250b0)as seen in Figure 6.2a/c. In this regime, a proportion of the flux lines are trapped in the firstplanar defect (the one with the lower x coordinate) while the remainder are held stationaryat a fixed distance behind them by the long-range inter-vortex repulsions. This is visible inFigure 6.4a1 and 6.4a2. Although the snapshots in these figures are taken for the system

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Figure 6.2: Steady-state (a, c) local hexatic order parameter h, and (b, d) radiusof gyration rg (units of b0) for interacting flux lines in the presence of two closelyplaced planar defects, in a sample of thickness L = 250b0. The top two figures(a, b) display results for the system in orientation A (simulation domain longer inthe x direction along the drive) and the bottom two (c, d) do so for orientation B(simulation domain more extended along the y direction). Vertical gray bars areused to indicate the crossover regions that separate consecutive drive regimes. Theregimes are labeled with the acronyms P : pinned, L: liquid, O : partially-ordered,S : smectic, M : moving lattice, and L/S : liquid/smectic. Note that these data arereplicated in Figure 6.3. Here and in the following figures, only error bars largerthan the symbol sizes are shown. Copyright (2018) by The European PhysicalJournal.

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Figure 6.3: Steady-state (a, e) mean vortex velocity v (units of b0/t0); (b, f)fraction of pinned line elements fp; (c, g) local hexatic order parameter h; and(d, h) radius of gyration rg (units of b0) as a function of drive Fd (units of ε0)for interacting flux lines in the presence of two closely placed planar defects, insamples of varying sample thickness L (units of b0). The figures on the left (a, b,c, d) display results for the system in orientation A and those on the right (e, f,g, h) do so for orientation B. Vertical gray bars are used to indicate the crossoverregions that separate consecutive drive regimes for L = 250b0. Copyright (2018) byThe European Physical Journal.

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Figure 6.4: Simulation snapshots of a system with 16 interacting flux lines in asample with orientation A and thickness L = 150b0, in the presence of two closelyplaced planar defects, projected onto the (1) xy plane for a top view, and (2)xz plane for a side view, in respectively the (a) pinned (Fd = 0.06ε0), (b) liquid(Fd = 0.11ε0), (c) partially-ordered (Fd = 0.14ε0), (d) smectic (Fd = 0.16ε0), and(e) moving-lattice (Fd = 0.22ε0) drive regimes. The green dotted lines represent thevortices, the black vertical lines indicate planar defects. The red dotted sectionsmark those portions of the flux lines that are trapped by planar defects. Thepurple circles and lines in (c1) indicate a central vortex and its nearest neighborsforming a hexagon-like structure. The drive Fd is oriented in the positive x (right)direction. The system boundary lengths in the x and y directions are 314b0 and272b0, respectively. The full videos from which these snapshots have been takencan be viewed at https://figshare.com/s/8a1e4bf34f463f988ebd. Copyright(2018) by The European Physical Journal.

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in orientation A, they are qualitatively faithful representations of vortex behavior in thepinned regime for orientation B as well. Due to the vertically correlated configuration of theplanar defects (along the magnetic field or z direction), the flux lines trapped within themare nearly perfectly straight and therefore display a low radius of gyration (Figure 6.3d/h).Lateral fluctuations for the unpinned vortices are suppressed as well on account of boththeir intrinsic elastic line tension and the repulsive caging induced by the lines trapped inthe defect planes positioned in front of them. This results in the low overall gyration radiuswe observe in the steady state in the pinned regime. Note that the fraction of pinned lineelements fp actually grows monotonically with drive (Figure 6.3b/f): higher driving forcesinduce larger pinning fractions since an increase in drive shrinks the distance between thefree-standing caged vortices and those trapped in the first defect plane, thereby increasing thelikelihood of a free-standing vortex being pulled into that extended defect. Correspondingly,the influence of sample thickness L on all the measured quantities is negligible, with thesteady-state curves for different L appearing identical within our statistical errors, since theflux lines are virtually motionless barring thermal fluctuations in the pinned state. Theinfluence of the flux line length only becomes appreciable once the vortices start moving, asdescribed below.

6.1.2 The Liquid Regime

Increasing the drive further, we exit the pinned regime and begin entering the liquid regimefor orientation A and the liquid/smectic regime for orientation B. We explain the differen-tiating factors between these regimes in the next subsection on the partially-ordered andsmectic regimes. During the crossover from the pinned to the liquid(/smectic) regime, thedrive is sufficiently strong for the vortices to detach from the defect planes. In the course ofdepinning, the flux lines suffer large distortions (Figure 6.4b1 and Figure 6.4b2) that resultin a sharp increase of their mean gyration radius (Figure 6.2b/d). The degree of orientationaldisorder is maximized as the local hexatic order parameter h approaches zero (Figure 6.2a/c)and remains suppressed for the extent of the regime. The incipient vortex motion naturallyresults in the mean vortex velocity assuming non-zero values (Figure 6.3a/e). Both gyra-tion radius and mean vortex velocity increase monotonically with drive, while the pinningfraction diminishes (Figure 6.3b/f) as the driving force is increased to enter deeper into thedepinning region.

The crossover from the pinned into the liquid regime occurs at lower drive values for thickersamples (greater L) as evidenced in all observables shown in Figure 6.3. This is becauselonger vortex lines comprise a larger number of segments along their trajectory which canpotentially be set free from the defect plane holding them, by the applied drive with assistancefrom thermal fluctuations. This increases the probability of neighboring line elements tobreak free as they are elastically coupled to the first detached element, inducing a cascadingeffect whereby the entire line is pulled free from the defect plane. For any given drive inthis regime, longer flux lines display a larger gyration radius (Figure 6.3d/h), since they are

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capable of incorporating larger distortions as they are pulled away from the defect planes atdifferent locations along their contour. The opposite trend holds true for the pinning fractionfp (Figure 6.3b/f), with shorter lines being more likely to remain trapped by planar defects.The propensity to be partially depinned also results in longer lines moving faster on average(Figure 6.3a/e) as their motion is less impeded by the disorder.

6.1.3 The Partially-Ordered and Smectic Regimes

As the drive is increased beyond the liquid(/smectic) regime, we see differences develop inthe steady-state behavior for different system orientations. For orientation A, the systembegins to develop local hexatic order (Figure 6.2a), which marks the onset of the partially-ordered regime, the second of the three intermediate regimes in the depinning region. In oursimulations, the sixteen flux lines arrange themselves into eight horizontal pairs flowing alongthe positive x direction as seen in Figure 6.4c1. The pairs are approximately equally spacedin the y direction. In this regime, we observe the formation of hexagon-like structures in thevicinity of the planar defects. Figure 6.4c1 illustrates that in the xy plane, each approximatehexagon consists of a central vortex trapped at a planar defect surrounded by six nearest-neighbor flux lines. Two of the adjacent vortices are located in the row above the centralone, while two are in the row below. The fifth and sixth nearest neighbors are symmetricallysituated in the second rows above and below the central vortex, respectively. Each pair ofconsecutive neighbors subtends an angle of ≈ 60 at the central vortex. The organizationof the lines into this hexagonal lattice-like configuration results in each line experiencingenhanced repulsive caging by its neighbors giving rise to comparatively straighter flux lines(Figure 6.4c2). This is evident in the reduced gyration radius rg (Figure 6.2b) in this region.The stronger external forcing propels the flux lines faster through the defect planes, as seenin the rising drive-velocity (or current-voltage) curves (Figure 6.3a) for all sample thicknessesL. The pinning fraction continues to decline with drive, albeit at a slower rate than in theprevious regime (Figure 6.3b).

The partially-ordered regime seen for orientation A exists in a rather small driving forceinterval. As the drive is increased further, the modest gains in local hexatic order h inthe partially-ordered regime are rapidly lost again (Figure 6.3c), as the more disorderedsmectic regime is reached. The eight horizontal vortex pairs of the partially-ordered regimegive way to four distinct horizontal flux flow channels that are akin to dynamic smecticordering. Each channel consists of four flux lines in our simulations; the channels are equallyspaced along the y axis. This geometric arrangement is characterized by a larger typicalinter-vortex distance. This essentially gives the flux lines more wiggle room and subjectsthem to weaker repulsive caging by neighboring lines as compared to the partially-orderedregime. This is evident visually from the extended nature of the flux lines along the xaxis as seen in Figure 6.4d1, and quantitatively from the enhanced radius of gyration rg(Figure 6.3d). Mean vortex velocity and pinning fraction continue to monotonically in- anddecrease, respectively, within the entire smectic region, with the four-row smectic eventually

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crystallizing into a four-row moving lattice.

We explain this rather peculiar sequence of orientational-order transitions for system orien-tation A, i.e., disorder-to-order (liquid to partially-ordered), followed by order-to-disorder(partially-ordered to smectic), followed again by disorder-to-order (smectic to moving lat-tice), via the following mechanism: The partially-ordered regime consists of eight rows andfour columns of flux lines, with each row containing two vortices and each column comprisingfour (Figure 6.4c1). The geometry of this configuration is similar to that of the ultimate de-pinned moving lattice, which consists of the flux lines sorted into four rows and eight columns(Figure 6.4e1). The partially-ordered eight-row pseudo-lattice is actually an approximationof the highly ordered four-row moving lattice rotated by 90 about the z axis. In the absenceof defects, the flux lines in orientation A will naturally form a four-row moving lattice sincethis configuration is more energetically favorable in this orientation of the system (containinga longer x boundary). However, strong anisotropic pinning in one direction favors a vortexlattice configuration that maximizes the number of line segments being pinned at any giventime. This translates to the planar defects along the y direction favoring a vortex lattice withthe maximum number of vortices along the y direction (i.e. per column), which happensto be the eight-row, four-column rotated lattice. The vortex configuration favored by theanisotropic planar pinning is thus distinct from, and competing against the flux line struc-ture that arises naturally in this system orientation. An approximation of the former (theeight-row partially-ordered configuration) is energetically favorable for a brief drive intervaljust as the vortex system begins to depin into the liquid regime, while the effective pinningstrength is not yet overpowered by the drive. Yet at elevated driving forces, the influenceof the planar defects predictably weakens, resulting in the re-shuffling of the vortices intothe four-row smectic that is a precursor to the natural ordering of flux lines favored by thesystem orientation A.

Further evidence for preferred vortex lattice ordering by the planar disorder is found in thesimulation results for orientation B. Here, the system boundary length in the y directionis greater than that in the x direction, and consequently, the natural configuration for thevortex lattice is the eight-row, four-column version, which is orthogonal to the expectednatural configuration for orientation A. Note that for this orientation, the default vortexlattice configuration in the absence of defects also happens to be the arrangement preferredby the planar defects parallel to the y axis, according to our hypothesis of maximal vortexpinning. Unlike in the case of orientation A, in orientation B there are no competing vortexlattice configurations. This is borne out by the fact that for orientation B, there is norearrangement of the vortices at higher drives, when the pinning effectiveness of the defectsis diminished. The vortices in the liquid state begin forming an eight-row smectic by the endof the liquid/smectic regime (similar to that in Figure 6.4c1), and continue to monotonicallycrystallize into a highly ordered (Figure 6.3g) eight-row hexagonal lattice. This is supportedby the absence of the local maximum in hexatic order h (Figure 6.2c) seen for orientation Athat signifies the onset and cessation of the partially-ordered regime, as well as the absenceof the corresponding local minimum in gyration radius (Figure 6.2d).

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The partially-ordered regime is not observed in thin samples with L ≤ 50b0, and is markedlysuppressed for lines shorter than 150b0 (Figure 6.3c). For shorter flux lines, the systemdirectly transitions from the liquid to the moving-lattice regime, forming a smectic alongthe way, but completely bypasses the partially-ordered regime. As in the case of orientationB, this claim is supported by the missing first local maximum in the h curve for L = 50b0

(Figure 6.3c), and the absence of the local minimum in the rg curve (Figure 6.3d). Thepartially-ordered regime likely constitutes a metastable state that (for any system orienta-tion) is inaccessible to short and stiff vortices which can be approximated as one-dimensionalobjects moving in a two-dimensional domain. This is supported by the significant similaritiesin the steady-state profiles of h and rg at low L for different system orientations (Figure 6.3).Sample thickness and hence vortex line length L plays a crucial role in allowing the system toaccess this metastable region, which can be explained as follows. In a 3D system, a spatiallyextended one-dimensional flux line is subject to pinning by a two dimensional planar defect,while the analogy for this in a 2D system is a zero-dimensional flux point being subjected topinning by a one-dimensional line defect. In the 2D system, the flux point depins from theline defect as a whole, when the drive is of sufficient strength. Contrarily in a 3D system, theextended flux line depins in a staggered fashion, with regimes where some portions of the lineare pinned by the defect plane while simultaneously other portions are depinned, allowing theextended vortices with additional degrees of freedom to access the unstable partially orderedregime albeit for only a brief drive interval. Essentially, it is the quenched disorder of theplanar defects that may be responsible for the appearance of the partially-ordered regime.This hypothesis could be tested in a future study by performing simulations in which thepinning strength is varied, which should shift the value of L at which the partially-orderedregime disappears.

The peculiar flux flow profile observed for orientation A is a finite-size effect – a consequenceof the particular choice of system boundary lengths in our small system of 16 vortices. Anexperimental system would be one to several orders of magnitude larger in both extension(in the x and y directions) and the number of vortices. The vortex lattice orientation insuch a system would most likely not depend on the ratio of boundary lengths (as is the casein our simulations), and in the absence of strong anisotropic pinning, several orientations ofthe vortex lattice would be equally likely. However, the results of our study for orientationA, especially when compared to those for orientation B, strongly suggest that the presenceof strong anisotropic pinning, such as that due to parallel planar defects, should breakrotational symmetry and prefer flux line arrangments that would maximize the number ofvortices encountered by the planar defects during flux flow.

6.1.4 The moving-lattice regime

The final drive regime for both orientations (A and B) is the moving-lattice regime. Thedriving current is sufficiently strong that the pinning to the attractive defects becomes neg-ligible, and the lines are once again almost perfectly straight as in the low-drive pinned

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regime (Figure 6.4e2). The flux flow channels along the x direction that had formed in thesmectic (or liquid/smectic) regime persist into the moving-lattice regime. With the desta-bilizing influence of disorder effectively removed, the flux lines arrange themselves withinthese channels to form a moving hexagonal Abrikosov lattice (Figure 6.4e1), with the localhexatic order h approaching unity (Figure 6.3c/g). The moving lattice marks the completionof the dynamic freezing process. The mean velocity of the lines shows a linear dependenceon drive strength, indicating that the system has entered an Ohmic regime with linear I-Vcharacteristics (Figure 6.3a/e). The pinning fraction (Figure 6.3b/f) plateaus a little abovezero (at fp ≈ 0.04) as does the gyration radius (Figure 6.3d), as the lines move practicallyfreely through the system without the pinning and roughening caused by the defects. Therole of sample thickness or vortex line length becomes negligible, with the h, rg, v, and fpcurves for different L practically coinciding for the extent of the entire moving-lattice regime(Figure 6.3).

6.2 Flux Line Excitations

In order to better understand the flux line structures that characterize the distinct regimesobserved in our studies of vortex matter subject to planar defects, we performed directmeasurements of flux line excitations, viz. half-loops, single kinks, and double kinks (Section3.5), which appear in our system due the interactions of the flux lines with the closely placedattractive defect planes.

The two local gyration radius maxima (Figure 6.3d) that mark the liquid and smectic driveregimes for orientation A coincide exactly with corresponding peaks in the single-kink (Fig-ure 3.1b) steady-state curves (Figure 6.5b). In the case of orientation B, the steady-stategyration radius (Figure 6.3h) and single-kink population (Figure 6.5e) move practically to-gether with non-zero drive, indicating that the two quantities are strongly correlated withregard to their evolution with applied driving force, in either system orientation. For alldrive strengths, we notice a positive correlation between the number of single kinks andvortex length (or sample thickness) L: longer flux lines afford considerably more possiblelocations along their contours for kinks to form. This correlation between line length andnumber of steady-state excitations holds true for the other line structures (half-loops anddouble kinks) under consideration as well (Figure 6.5).

Vortex half-loops (Figure 3.1a) occur in the system with a frequency comparable to that ofsingle kinks. They peak at the end of the smectic regime (Figure 6.5a) for orientation A,and within the crossover region separating the liquid/smectic and moving lattice regimes fororientation B. In contrast to single kinks, the formation of half-loops requires the flux lines tobe relatively straight. We thus observe a steady increase in the number of half-loops beyondthe pinned regime as the lines that start out distorted in the liquid regime steadily straightenout with increasing drive until the end of the smectic regime, where the number of half-loops

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Figure 6.5: Steady-state number of (a, d) half-loops, (b, e) single kinks, and (c,f) double kinks as a function of drive Fd (units of ε0) for interacting flux lines in thepresence of two closely placed planar defects, in samples of varying sample thicknessL (units of b0). The figures on the left (a, b, c) display results for the system inorientation A and those on the right (d, e, f) do so for orientation B. Verticalgray bars are used to indicate the crossover regions that separate consecutive driveregimes for L = 250b0. Copyright (2018) by The European Physical Journal.

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acquires its maximum. Beyond this regime, the pinning influence of the planar defects startsto wane in comparison to the relatively high driving force; hence we observe the number ofhalf-loops monotonically decline with drive. It is worth noting that of the three types ofexcitations under study, we found the half-loops to be the most resilient structures in thesystem, with their population being significantly above zero (∼ 0.2 . . . 0.5) in the moving-lattice state, even though single kinks (Figure 6.5b) and double kinks (Figure 6.5c) practicallycease to appear in the system well before the onset of this regime. This is consistent withthe theoretical findings of Marchetti and Vinokur that for sufficiently large current, half-loopconfigurations of transverse width smaller than the average separation between the planesbecome the dominant excitations [68,186].

Double kinks (Figure 3.1c) occur far less frequently (Figure 6.5c/f) than single kinks or half-loops as they require a flux line to assume a spatial structure of relatively higher complexity.The dominant double-kink peak occurs at the end of the liquid regime for orientation A(liquid/smectic for orientation B), at a noticeably higher drive value than that correspondingto the major single-kink peak, which is observed in the center of the liquid(/smectic) regime:As the flux lines evolve from their most distorted configurations in the middle of the liquidregime to straighter shapes at higher drives, the formation of double kinks is facilitated asthese vortices start to loop back on themselves and reattach to the first defect plane.

These different vortex line excitations mediate thermally activated flux transport with non-linear force-velocity or current-voltage characteristics, as worked out analytically by Marchettiand Vinokur [68, 186] for dilute vortex arrays. The linear transport regime in our systemmay be characterized by either a rigid flow of flux lines or motion facilitated by double kinksdepending on the length of the sample. The non-linear regime is dominated by vortex singlekinks and half-loops. There exists a characteristic current scale JL ∼ 1/L that separatesthe regions of linear and non-linear current-voltage response in the (L, J) plane. Here, Lis again the sample thickness in the direction of the magnetic field, and J is the externalelectric current that exerts a Lorentz driving force Fd ∼ J in the direction perpendicularto the defect planes (the x direction); JL denotes the characteristic current scale, at whichflux transport in a sample of thickness L crosses over from a linear to a non-linear regime.Since J ∼ Fd, it follows that FL ∼ 1/L where FL is the crossover drive corresponding to thecharacteristic current JL.

In order to numerically obtain the crossover boundary curve that separates the regions oflinear and non-linear transport, we have identified the critical drive strength FL when thesteady-state number of half-loops in the system (with orientation A) starts assuming non-zero values. We repeated this process for 21 systems of varying sample thickness, evenlyspaced between L = 50b0 and 250b0 (Figure 6.6a). We then plotted the resulting FL valuesagainst 1/L and successfully fit a straight line to the data points (Figure 6.6b), therebyconfirming the analytical prediction. Note that our best linear fit has a non-zero y intercept0.04, implying that for infinitely long flux lines (L → ∞) in our system, a non-zero finitedrive is required for them to form half-loops. This is likely a consequence of the discretizationof the flux lines along the z direction in our model, which results in a minimum line tension

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Figure 6.6: (a) Steady-state number of half-loops as a function of drive Fd (unitsof ε0) for interacting flux lines in samples with thicknesses varying from L = 50 toL = 250 (units of b0) in steps of 10b0. (b) Crossover drive strength FL separating theregions of linear and non-linear response, as a function of inverse sample thickness1/L. The onset points in the half-loop curves for different L in (a) are the FL valuesused in (b). Copyright (2018) by The European Physical Journal.

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and thus depinning energy necessary to detach a single vortex element.

6.3 Widely Spaced Defect Planes

Upon increasing the distance between the planar defects to 50% of the system length, therichness and variety of the observed phenomenology in the depinning regimes encounteredfor closely placed planes for system orientation A is diminished significantly. The flux linesof all lengths under consideration, except for the longest (L = 250b0), are then too short toallow for the formation of single-kink (Figure 6.7b) or double-kink excitations. For shortersamples (L < 150b0), the gyration radius valley that marks the partially-ordered regimedisappears (Figure 6.7a). Consequently, the system is characterized by a flux flow profileresembling samples with columnar defects, i.e., containing a single maximum in the steady-state gyration radius curve that marks the transition of the system directly from the liquidto the smectic regime in the depinning region. For all sample thicknesses considered in thisstudy, the only flux line excitations to appear in any appreciable quantity are half-loops(Figure 6.7c), since they require merely one defect plane to form.

The defect spacing likely sets the characteristic length scale in the system (as evidenced bythe maximum gyration radius being ∼ 16b0 (Fig. 6.3a), which is practically the same as thedefect spacing used), but only up to a critical value. Below the critical spacing, the pinningstrength may be strong enough to overcome the elastic contraction of the flux lines, allowingthe lines to span the distance between the defect planes. Above the critical spacing, theelastic force would dominate, deterring the simultaneous pinning of the lines by both defectsand suppressing the associated signatures as seen in Fig. 6.7. The effect would also non-trivially depend on line length and temperature, both of which facilitate flux line wandering.This would be a useful and interesting dependence to study quantitatively in a future study.

6.4 Summary

We have utilized Langevin molecular dynamics simulations to examine a system of drivenflux lines in the presence of two planar defects aligned parallel to the magnetic field andperpendicular to the direction of drive. We have probed the steady-state drive dynamicsof the system for two horizontal orientations and several sample thicknesses. For closelyplaced defect planes, we have observed characteristic flux flow regimes that range froma fully pinned stationary configuration at the lowest drive strengths to a perfectly orderedmoving lattice at the highest drive strengths. In addition there appear intermediate crossoverregimes with vortex matter in different stages of disorder that however manifestly dependon the orientation of the simulation domain. The depinning region in the flux flow profile isbroad and displays non-trivial vortex structures. We have characterized these structures by

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Figure 6.7: Steady-state (a) radius of gyration (units of b0), (b) number ofsingle kinks, and (c) number of half-loops as a function of drive Fd (units of ε0)for interacting flux lines in the presence of two widely spaced planar defects, insamples of varying sample thickness L (units of b0). Copyright (2018) by TheEuropean Physical Journal.

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analyzing the unique spatial flux line configurations and excitations that appear during anddominate the different drive regimes. These methods supplement measurements of essentialglobal average observables such as local hexatic order and mean gyration radius in thesystem. The analysis is also aided by rich visualizations of vortices in the various regimesfrom different perspectives.

The steady-state results and simulation snapshots for the two orthogonal system orientationsindicate that strong anisotropic pinning due to parallel planar defects results in a preferredorientation of the vortex lattice that maximizes the number of flux lines encountered by thecorrelated defects as the former are driven across the sample by an external current. Thisis evidenced by the tendency of the vortices to arrange into the defect-preferred orientationeven when this orientation is orthogonal to the natural lattice orientation for the specificboundary conditions of the system, as is the case for orientation A.

Quantitative measurements of the flux line excitation populations were utilized to detect theboundaries separating distinct crossover regimes for linear and non-linear current-voltageresponse in the (L, J) or (L, Fd) plane. By identifying the drive strength FL correspondingto the emergence of half-loops in the system for each sample thickness L, we have numericallyconfirmed that the critical drive strength FL needed to push the system from a linear to anon-linear transport regime shows a 1/L dependence, as analytically predicted by Marchettiand Vinokur [68].

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Chapter 7

Conclusions

This doctoral work can be broken up into three major studies on the behavior of magneticvortex lines in type-II superconductors driven through disordered media – a system far fromequilibrium that has intrigued physicists for over 70 years. Each of these studies has led tointeresting original findings regarding the static and dynamic properties of flux lines underdifferent physical conditions; findings that have added at least a tiny (hopefully net-positive)amount to the body of knowledge on type-II superconductors.

Our first study focused on the relaxation phenomena of flux lines subject to random pointdisorder following drive down quenches both within the high-drive, moving-lattice, non-equilibrium steady state, and from the moving state to the low-drive, pinned, glassy state.When we suddenly lowered (quenched) the driving force (drive) from a higher to a lower valuewithin the moving regime, we saw fast relaxation of flux lines evidenced by the exponentialdecay of one-time observables such as radius of gyration and two-time observables such asheight autocorrelations. The time translation invariance of curves for the latter quantityfor different waiting times definitively ruled out any possibility of aging for intra-moving-regime quenches. On the other hand, quenches deep into the zero-transport glassy pinnedregime gave us flux lines that relaxed in a much slower fashion, with height autocorrelationsdecaying algebraically with time and displaying clear breaking of time translation invariance.The height autocorrelation curves were found to be amenable to dynamical scaling, with themaster curve showing a power-law dependence on t/s. Thus we observed simple aging forflux lines following quenches into the pinned regime, albeit with non-universal aging scalingexponents that are not independent of microscopic details such as temperature.

Having explored the moving and pinned regimes, we decided to explore next the depinningcrossover region that separates the two regimes. We obtained steady-state results for radiusof gyration and vortex velocity for flux lines over a range of temperatures, and by applyingfinite-temperature scaling techniques, were able to confirm that vortex depinning constitutesa continuous second-order phase transition at zero temperature, as has been shown in earlierstudies. We directly computed scaling exponents for this critical phenomenon including

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the growth exponent ν, the thermal exponent δ, and the order parameter exponent β, thelatter turning out to be in good agreement with recent experiments. We also investigatedaging near the critical point by quenching the drive for flux lines in a high-drive movinglattice regime to a value in the depinning regime, and analyzing height autocorrelations fordifferent waiting times post quench. As in the case of quenches into the pinned regime, weobserved dynamical scaling and simple aging of the height autocorrelations but with onekey difference – the aging scaling exponents (b and λC/z) were found to be consistent acrossa range of different temperatures, i.e., we found critical aging characterized by universalexponents independent of the microscopics of the system. Using hyperscaling relations, wewere additionally able to compute the roughness exponent ζ, the dynamic exponent z, andthe autocorrelation exponent λC for both non-interacting and interacting flux lines. It wouldbe an interesting future undertaking to directly compute the dynamic exponent z via thetime decay analysis of height autocorrelations following quenches to different drive valuesnear the critical point, and comparing this to the z value obtained from the hyperscalingrelations.

Besides our two comprehensive studies on flux lines driven through point defects, we havealso studied planar defects. We studied the steady-state flux flow profile for two orthogonalhorizontal configurations of the system, with one yielding a far richer collection of driveregimes compared to the other. Further analysis revealed the richer configuration to be afinite size effect due to the low number of vortices (sixteen) in our simulations. More im-portantly though, this finite size effect revealed a preferred ordering of the Abrikosov vortexlattice by the planar defects on account of the strong spatial anisotropy in their pinningaction. In this study, we directly measured for the first time (to the best of our knowledge),flux-line excitations such as half-loops, single kinks and double kinks for a variety of samplethicknesses, that emerge as a consequence of the peculiar pinning mechanism of planar de-fects, and that do not occur in the case of point defects. We were able to use the quantitativemeasurements of these excitations to compute the boundary curve separating the regions oflinear and non-linear current-voltage response in the (sample thickness)-(current density)plane. The numerical values we computed for the boundary curve were found to be in agree-ment with the functional form predicted by analytical studies. We found that the richness ofthe flux flow profile diminished with decreasing sample thickness, as the system evolved froma three-dimensional bulk sample towards a two-dimensional thin film. A good future lineof investigation would be to quantitatively determine where this transition between effectivedimensionalities occurs, as well as the influence of pinning strength on this transition point.

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