Resistivity and the vortex solid-to-liquid transition in high...

Resistivity and the vortex solid-to-liquid transition

in high-temperature superconductors

BEATRIZ ESPINOSA ARRONTE

Doctoral Thesis

Stockholm, Sweden 2006

Typeset in LATEX.

Cover picture: YBa2Cu3O7−δ single crystal. Electrical contacts are applied withsilver paint for measurements of c-axis resistance. The dimensions of the crystalare 660 × 525 × 14 µm.

TRITA-ICT/MAP AVH Report 2007:2ISSN 1653-7610ISRN KTH/ICT-MAP/AVH-2007:2-SEISBN 978-91-7178-545-9

Fasta Tillståndets FysikMikroelektronik och Tillämpad Fysik

Kungliga Tekniska HögskolanElectrum 229

SE-164 40 Kista

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av Teknologie doktorsexamen fredagen den19 januari 2007 i F3, Kungl Tekniska Högskolan, Lindstedtsvägen 26, Stockholm.

c© Beatriz Espinosa Arronte, November 2006

Tryck: Universitetsservice US AB

Abstract

In high-temperature superconductors a large region of the magnetic phase diagram isoccupied by a vortex phase that displays a number of exciting phenomena. At low tem-peratures, vortices form a truly superconducting solid phase which at high temperaturesturns into a dissipative vortex liquid. The character of the transition between these twophases depends on the amount and type of disorder present in the system. For weak pointdisorder the vortex solid-to-liquid transition is a first-order melting. In the presence ofstrong point disorder the solid is thought to be a vortex-glass and the transition into theliquid is instead of second order. When the disorder is correlated, like twin boundariesor artificially introduced columnar defects, the transition is also second order, but hasessentially different properties. In this work, the transition between the solid and liquidphases of the vortex state has been studied by resistive transport measurements in mainlyYBa2Cu2O7−δ (YBCO) single crystals with different types of disorder.

The vortex-glass transition has been investigated in an extended model for the vortex-liquid resistivity close to the transition that takes into account both the temperatureand magnetic field dependence of the transition line. The resistivity of samples withdifferent properties was measured with various contact configurations at several magneticfields and analyzed within this model. For each sample, attempts were made to scalethe transition curves to one curve according to a suitable scaling variable predicted bythe model. Good scaling was found in a number of different situations. The influence ofincreasing anisotropy and angular dependence of the magnetic field in the model were alsoconsidered.

The vortex solid-to-liquid transition was also studied in heavy-ion irradiated YBCOsingle crystals. The ions create columnar defects in the sample that act as correlateddisorder. A magnetic field was applied at a tilt angle with respect to the direction ofthe columns. At the transition the resistance disappears as a power law with differentexponents in the three orthogonal directions considered. This provides evidence for a newtype of critical behavior with fully anisotropic critical scaling properties not previouslyfound in any physical system.

The effect on the vortex solid-to-liquid transition of high magnetic fields applied par-allel to the superconducting layers of underdoped YBCO single crystals was also studied.Some novel features were observed: a sharp kink appearing close to Tc at high magneticfields and a triple dip in the angular dependence of the resistivity close to B ‖ ab in someregions of the phase diagram.

Keywords: high-temperature superconductors, vortex dynamics, vortex solid-to-liquidtransition, critical scaling, glass transition, Bose glass, heavy-ion irradiation, B ‖ ab,YBa2Cu2O7−δ , Tl2Ba2CaCu2O8+δ, oxygen deficiency.

Sammanfattning

I högtemperatursupraledare består en stor del av det magnetiska fasdiagrammet av envortexfas som uppvisar ett flertal spännande fenomen. Vid låga temperaturer bildar vor-texarna en fast vortexfas utan elektriskt motstånd. Vid högre temperatur övergår dennafas till en dissipativ vortexvätska. Egenskaperna hos denna fasövergång beror på oord-ningen i form av defekter. Vid svag punktoordning är fasomvandlingen mellan det fastaoch flytande vortextillståndet en första ordningens smältövergång. Vid stark punktoord-ning anses den fasta fasen vara ett vortexglas och övergången till vortexvätskan är iställetav andra ordningen. När oordningen är korrelerad, som för tvillinggränser eller artifici-ellt skapade kolumndefekter, är övergången också av andra ordningen men med väsentligtannorlunda egenskaper. I detta arbete har övergången mellan det fasta och det flytandevortextillståndet studerats med resistiva transportmätningar i framförallt enkristaller avYBa2Cu2O7−δ (YBCO) med olika typer av oordning.

Vortexglasövergången har undersökts i en utvidgad modell för resistansen i vortexväts-kan nära fasövergången där hänsyn tas till såväl temperatur- som fältberoendet. Resistan-sen hos prover med olika egenskaper mättes i varierande magnetfält och i flera kontakt-konfigurationer och analyserades inom denna modell. Övergångskurvorna skalades till enkurva med en skalningsvariabel som givits av modellen. God skalning uppnåddes i fleraolika fall. Effekten av ökande anisotropi och vinkelberoendet i modellen undersöktes också.

Vortexövergången mellan det fasta och det flytande vortextillståndet undersöktes äveni enkristaller av YBCO bestrålade med tunga joner. Jonerna skapade kolumndefekter somfungerar som korrelerad oordning. Vinkeln mellan pålagt magnetfält och dessa kolumnde-fekter varierades. Vid fasövergången avtar resistansen som en potenslag med olika expo-nenter i de tre undersökta ortogonala riktningarna. Detta ger experimentell belägg för enny typ av kritiskt beteende med fullständigt anisotropa kritiska skalningsegenskaper.

Egenskaparna hos på vortexövergången mellan fast och flytande fas vid höga mag-netfält parallella med de supraledande lagren hos underdopade YBCO enkristaller under-söktes också. Några nya effekter observerades: en skarp knyck uppstod nära Tc vid högamagnetfält och en tredubbel dipp i den vinkelberoende resistiviteten nära B ‖ ab i någraregioner av fasdiagrammet.

Nyckelord: Högtemperatursupraledare, vortexdynamik, fasövergången i vortexfasen, glasö-vergång, Bose glas, jonbestrålning, B ‖ ab, YBa2Cu2O7−δ, Tl2Ba2CaCu2O8+δ, underdo-pade prover.

Resumen

En los superconductores de alta temperatura una amplia región del diagrama de fasesestá ocupada por una fase de vórtices. Conocida como estado mixto, exhibe un grannúmero de fenómenos interesantes. A bajas temperaturas los vórtices forman una fasesólida en la que la resistencia electrica es cero. A temperaturas más elevadas esta fasesólida se transforma en un líquido de vórtices. Las propiedades de esta transicion de fasedependen del grado y el tipo de desorden existente en el sistema. Cuando el desorden esdébil y puntual la transición entre el sólido y el líquido de vórtices es de primer orden. En elcaso de desorden fuerte y puntual el solido es un vidrio de vórtices y la transición a la faselíquida es de segundo orden. Cuando el desorden es correlacionado, como maclas o defectoscolumnares inducidos artificialmente, la transición también es de segundo orden, pero conpropiedades esencialmente distintas. En esta tesis se investiga la transición entre las fasessólida y líquida del estado mixto mediante medidas de transporte eléctrico principalmenteen monocristales de YBa2Cu2O7−δ (YBCO) con distintos tipos de desorden.

La transición del vidrio de vórtices a la fase líquida se estudia en un modelo ampliadode la resistividad del líquido de vórtices cercana a la transición. Este modelo tiene en cuentaque la línea de transición sólido-liquido depende tanto del campo magnetico como de latemperatura. La resistividad de muestras con distintas propiedades, medida en distintoscampos magneticos y con contactos en varias configuraciones, se analiza con este modelo.Se emplea con éxito un análisis de escala de las curvas de resistencia con una variable deescala proporcionada por el modelo. Se investiga también la influencia en el modelo de laanisotropía y del ángulo entre el campo magnetico y el eje c del cristal.

La transición entre la fase sólida y líquida del estado mixto se estudia también enmonocristales de YBCO irradiados con iones pesados. Los iones crean defectos columnaresque actuan como desorden correlacionado. El campo magnético se aplica inclinado conrespecto a la dirección de los defectos. Al acercarse a la transición la resistencia desaparecesiguiendo una ley de potencia con distintos exponentes críticos en cada una de las tresdireciones ortogonales consideradas. Este hecho muestra un nuevo tipo de comportamientocrítico con propiedades de escala totalmente anisótropas.

También se estudia el efecto en la transición sólido-liquido de vórtices de altos cam-pos magneticos paralelos a los planos superconductores de monocristales de YBCO condeficiencia de oxígeno. Se observan algunos fenomenos nuevos. El primero es un cambiobrusco cerca de Tc en las curvas de resistencia en función de la temperatura cuando elcampo magnetico es alto. El segundo es una depresión triple cerca de B ‖ ab en las curvasde resistencia en función del angulo, observada en algunas regiones del diagrama de fase.

Palabras clave: superconductores de alta temperatura, dinámica de vórtices, transicio-nes de fase en el estado mixto, vidrio de vórtices, vidrio de Bose, irradiación por iones,B ‖ ab, YBa2Cu2O7−δ, Tl2Ba2CaCu2O8+δ, deficiencia de oxígeno.

Preface

The results presented in this thesis are based on my research as a PhD studentwithin the Solid State Physics group (FTF), now belonging to the Departmentof Microelectronics and Applied Physics (MAP) at the School of Information andCommunication Technology of the Royal Institute of Technology (KTH). The workpresented here was carried out between 2002-2006, mostly in the FTF lab at KTH,with the exception of the work on the solid-to-liquid transition at high in-planefields, carried out at Argonne National Laboratory (Argonne, IL) and at the Na-tional High Magnetic Field Laboratory in Tallahassee, FL.

This thesis consists of two parts. In the first one, a general introduction tosuperconductivity is presented, as well as the theoretical background upon whichthe work is based. A short description of the experimental techniques and summaryof results are also included. The second part consists of the publications.

List of publications

The following papers are included in the thesis:

I. Scaling of the vortex-liquid c-axis resistivity in underdoped YBa2Cu2O7−δ

B. Espinosa-Arronte, Ö. Rapp, and M. Andersson.Physica C 408-410, 583 (2004).

II. Scaling of the vortex-liquid resistivity in high-Tc superconductors

B. Espinosa-Arronte, and M. Andersson.Phys. Rev. B 71, 024507 (2005).

III. Scaling of the B-dependent resistivity for different orientations in Fe doped

YBa2Cu2O7−δ

B. Espinosa-Arronte, M. Djupmyr and M. Andersson.Physica C 423, 69 (2005).

IV. Fully anisotropic superconducting transition in ion irradiated YBa2Cu3O7−δ

with a tilted magnetic field

B. Espinosa-Arronte, M. Andersson, C. J. van der Beek, M. Nikolaou,J. Lidmar and M. Wallin.Manuscript submitted for publication.

The author has also contributed to the following publication, which is not in-cluded in this thesis:

V. Sphalerite-Chalcopyrite Polymorfism in Semimetallic ZnSnSb2

A. Tengå, F. J. García-García, A. S. Mikhaylushkin, B. Espinosa-Arronte,M. Andersson, and U. Häussermann.Chem. Mater. 17, 6080 (2005).

Comments on my participation

In Paper I, I prepared the sample, performed the measurements, analyzed the dataand wrote the paper.

In Paper II the data was obtained from a previous work by Björn Lundqvist. Iperformed the data analysis and wrote the paper.

The work for Paper III was done in close collaboration with the diploma workerMärit Djupmyr. I supervised her work in sample preparation and data analysis.Measurements were carried out together and I wrote the paper.

The original idea for the work in Paper IV was proposed by Mats Wallin andJack Lidmar. I grew the crystals, prepared the electrical contacts and performedthe measurements. Kees van der Beek was responsible for the irradiation of thecrystals with heavy ions. The data analysis was carried out by myself and MariosNikolaou. I coordinated the contributions of all coauthors to the writing of thepaper, that were based on a first draft written by Magnus Andersson and modifiedby myself.

ACKNOWLEDGMENTS ix

Acknowledgments

There is so many people that have helped me over the years that the task of thank-ing them all without forgetting anyone seems more difficult than writing the thesis.

I want to start by thanking my supervisor, Magnus Andersson. He gave methe opportunity to join FTF and has encouraged me all the way through. Healways had time to listen to me, to answer my questions, to come to the lab when Iencountered any trouble. His guidance and suggestions have been essential for thiswork. I also appreciated all the fast and good feedback on the thesis manuscript.Thanks!

I also want to thank Östen Rapp. I have learned a lot from our discussions andhis sharp comments on the manuscripts have been extremely helpful.

Therese Björnängen taught me how to do everything in the lab: growing crys-tals, making contacts, measuring. She also saved me a number of sleepless nightswith her wonderful going home checklist. And I had the pleasure to share officewith her for over two years and become her non-identical twin. We had a lot offun, the only drawback is that I missed her at work after she left.

I want to thank everyone involved in the experiments on irradiated YBCO. MatsWallin and Jack Lidmar for proposing the experiments, helping with the writingof the manuscript and for all the discussions we have had. Marios Nikolaou forthe incredible work he did with the data analysis. Kees van der Beek, not just forhelping with the irradiation and the writing of the manuscript, but for having thepatience to answer all my questions on what’s going on with the vortices, both inDresden and over the summer. It gave me a much better insight in the problem.

I also want to thank everyone at Argonne. Starting with Wai Kwok for welcom-ing me in his group. Ulrich Welp spent hours and hours in the lab helping me withthe experiments, he struggled with me through sample U9 ordeal and he alwaysfound time to answer my questions. He was also excellent company in Tallahasseeand so that the 13 hours trip there didn’t feel all that long. So thanks. I also wantto thank Ruobing Xie for showing me how to do things in the lab and being sofriendly. And Andreas Glatz for showing me around in Chicago.

Andreas Rydh deserves a special thanks. I started when he was just about toleave towards Argonne, but he still found the time to show me the art of contacting.After that we kept in touch and he finally made possible my visit to the US. I cannotbe more grateful for his help with Labview in Tallahassee. It was really great toknow that things were going to work out. And we also had a good time in Florida.And he even found the time to look at the thesis manuscript and give me somegood comments. So thanks for so many things.

I also want to thank the Vlasko-Vlasov family, Vitalii, Galina and Katya, forhaving me at their place during my stay in Chicago. We had a good time andGalina took extremely good care of me: finding a hot meal when arriving late atnight from the lab is something I cannot thank enough.

I want to thank as well some of the people in Kista for making life so muchnicer at work: Shun-Hui, Reza, Märit, Yuri, Sören, Mats, Stefano, Matteo, Roland,Yeganeh, Audrey, Maciej, Rina. Also our secretary Zandra Lundberg, for all thehelp. Special thanks deserves Rickard Fors, my room mate for the past two years,for showing me all the useless things that make life way more fun.

I don’t forget the money: financial support from the Foundation for StrategicResearch (SSF) under the OXIDE program is gratefully acknowledged.

And of course there are also a lot of people that have helped me through outsidework. My friends, specially Maria and Silvia, for having the patience to listen toall my problems, relevant or not, throughout the years.

Javi, Nacho, Isa and Ana, who are my brothers and sisters, for keeping thedistance short and cheering for me during the writing process. Gracias nenes.

Albin, for so many things that I would need a whole thesis to write them down.But they all sum up in the care and love that I needed to make it happen. Tack.

And above all I want to thank my father, for making our education a priorityand for teaching us to keep balance in life. Y esto además lo escribo en español.Sobre todo quiero darle las gracias a mi padre, por poner siempre nuestra educaciónpor delante y por enseñarnos a vivir sin perder el equilibrio.

Beatriz Espinosa Arronte

Stockholm, November 2006

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Comments on my participation . . . . . . . . . . . . . . . . . . . . . . . . viii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Contents xi

1 Introduction 1

2 The superconducting state 5

2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Theories of superconductivity . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 London theory . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . . 82.2.3 BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Type II superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 High-Tc superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Vortex matter in high-Tc superconductors 15

3.1 Vortex motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.1 Flux flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.2 Flux creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.3 Thermally assisted flux flow . . . . . . . . . . . . . . . . . . . 18

3.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 Vortex liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Vortex solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Effects of the layered structure . . . . . . . . . . . . . . . . . 23

xii CONTENTS

3.2.4 The melting transition . . . . . . . . . . . . . . . . . . . . . . 243.3 The vortex-glass transition . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 FFH scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 The vortex molasses scenario . . . . . . . . . . . . . . . . . . 28

3.4 The Bose-glass transition . . . . . . . . . . . . . . . . . . . . . . . . 28

4 The solid-to-liquid transition 33

4.1 Extended model of the vortex-glass transition . . . . . . . . . . . . . 334.1.1 The vortex glass line . . . . . . . . . . . . . . . . . . . . . . . 354.1.2 Scaling in the extended model: Experimental results . . . . . 36

4.2 The transition in samples with columnar defects . . . . . . . . . . . 384.2.1 The effect of tilting the field . . . . . . . . . . . . . . . . . . . 394.2.2 The effect of tilting the field: experimental results . . . . . . 414.2.3 The effect of tilting the field: discussion . . . . . . . . . . . . 44

4.3 In-plane magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . 454.3.1 The solid-to-liquid transition for in-plane fields . . . . . . . . 474.3.2 The transition at high in-plane fields: experimental results . . 49

5 Experimental overview 53

5.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.1 Growth of YBCO single crystals . . . . . . . . . . . . . . . . 535.1.2 Oxygenation . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.3 Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.4 Contact preparation . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.1 Temperature control . . . . . . . . . . . . . . . . . . . . . . . 595.2.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Summary, conclusions and future work 65

6.1 Scaling of the vortex-liquid resistivity in the extended model . . . . 656.2 Anisotropic scaling in heavy-ion irradiated YBCO . . . . . . . . . . 666.3 Measurements at high in-plane magnetic fields . . . . . . . . . . . . 666.4 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . 66

Bibliography 71

Papers 85

Chapter 1

Introduction

Despite the fact that superconductivity has been known for almost a century, theinterest in superconducting materials and the mechanisms behind such a remark-able phenomenon is still enormous. A material with zero electrical resistance cantransport currents without any power losses, so there is a large number of potentialapplications. However, the development has been restrained by the low tempera-tures at which materials become superconducting. Despite that, some applicationsare already a reality. Superconductors allow us to build strong magnets of reducedsize, which can be found, for example, in magnetic levitating trains or in medicalapplications such as magnetic resonance imaging.

Even though the most characteristic properties of superconductors, like the dis-appearance of electrical resistance or the expulsion of magnetic fields, might seemrather exotic, superconductivity is a widespread phenomenon. Most elements inthe periodic table become superconducting at low temperatures, even though someof them achieve superconductivity only under high pressures. Superconductivityis also found in many more or less complex compounds and the family of super-conducting materials is continuously growing. Some of the new incorporations areO2 under pressure [1], B-doped diamond [2] or the most recent multiwalled carbonnanotubes [3], or B-doped Si [4].

Superconductivity has attracted a large attention over the years. In fact sev-eral Nobel prizes have been awarded for discoveries within this subject, the mostrecent in 2003 to A. A. Abrikosov and V. L. Ginzburg. In 1987 J. G. Bednorzand K. A. Müller received their prize for the discovery of high-temperature su-perconductivity in a cuprate, the La-Cu-O system. During the twenty years thathave passed since then an impressive amount of work has been devoted to trying tounderstand the properties of this new kind of superconductors. Although a lot ofprogress has been made, there are still a number of issues that remain controversial.One of them is the microscopic theory to explain superconductivity in this kind ofmaterials, where very little is known. A second one is the generic phase diagramcommon to all cuprate superconductors, that considers the temperature versus the

2 CHAPTER 1. INTRODUCTION

Figure 1.1. Schematic mean-field phase diagram of a high-temperature supercon-ductor. At fields below Hc1 there the magnetic field is expelled (Meissner phase).Between Hc1 and Hc2 magnetic fields penetrate in the form of vortices. At low tem-peratures the vortex state is a truly superconducting solid that at high temperaturesmelts into a dissipative vortex liquid.

concentration of charge carriers. Superconductivity appears only in limited area ofthis phase diagram and when the concentration of charge carriers is low, cupratesare instead insulators with antiferromagnetic order. The properties of the otherregions are still under discussion, but some proposals include a zone of charge den-sity fluctuations (“stripes”) and a “pseudogap” regime, where electrons might startto form pairs. A general overview about the phase diagram of the cuprates can befound in a short review by Orenstein and Millis [5].

High-temperature superconductors (HTSC) are interesting not only from theprospect of possible applications, but also as a good system to experimentally studyphase transitions. Over a large region of the magnetic field versus temperaturephase diagram, shown in Fig. 1.1, magnetic flux penetrates in the form of vortices.At low temperatures they are organized in a solid that is truly superconducting.But in HTSC vortices are rather soft and thermal fluctuations are large, so at highenough temperatures the vortex solid melts into a vortex liquid, which dissipatesenergy. The properties of the liquid are a third example of a still not well understoodfeature of high-temperature superconductors. The transition between the vortexsolid and the vortex liquid depends on the strength, type and amount of disorderpresent in the system. So if the type and amount of defects can be controlled,different types of phase transitions can be probed. The problems addressed in

this thesis concern the properties of the vortex liquid and the vortex solid-to-liquidtransition. The goal has been both to test some previously proposed models and toinvestigate novel scenarios. To achieve that resistive transport measurements havebeen used. This kind of experiments study the motion of vortices under appliedelectrical currents and magnetic fields.

This thesis is outlined as follows. Chapter 2 is a general introduction to super-conductivity. The main properties of superconductors, together with the theoriesdeveloped to explain them and the basic characteristics of type II superconductorsare shortly reviewed. A more extensive introduction to superconductivity can befound in the books by Tinkham [6] or Orlando and Delin [7]. In chapter 3 thephysics of vortices is shortly presented. Detailed attention is paid to the propertiesof the transition between the vortex solid and the liquid and to the effect that dif-ferent types of disorder have on it. An extensive overview about vortices in HTSCcan be found in the review by Blatter et al. [8]. Chapter 4 is devoted to describethe three main problems considered in this thesis. In each case the necessary back-ground is provided, together with a short description of the results obtained in ourexperiments. Chapter 5 describes the experimental techniques used in this work, aswell as the procedures for data analysis. Finally, chapter 6 summarizes the resultspresented on the appended papers and gives an outlook for a possible continuationof the work. The publications originated from this work are appended in the end.

Chapter 2

The superconducting state

2.1 Basic properties

The most renowned property of superconductors is their ability to carry an electricalcurrent without any losses. This occurs because in the superconducting state theirdc resistivity is zero. The property of perfect conductivity was discovered in 1911 byH. Kammerlingh Onnes in Leiden. He had managed to liquefy helium three yearsbefore, which allowed him to reach stable low temperatures. Together with hiscoworkers he studied the electrical resistance of pure metals. In a thin Hg capillarythey observed that “between 4.21 K and 4.19 K the resistance diminished veryrapidly and disappeared at 4.19 K” [9–11]. The temperature at which the drop inresistance takes place is known as the critical temperature, Tc. During the followingyears the Leiden group was the only group capable of liquefying helium and achievethe low temperatures required to study superconductivity. They discovered thatthe superconducting state is limited not only by a critical temperature, but also bya critical magnetic field Hc and a critical current density jc.

Perfect conductivity is not the only remarkable property present in supercon-ductors. In 1933 W. Meissner and R. Ochsenfeld observed that at temperaturesbelow Tc the magnetic field inside a superconducting sample was zero even whenthe cooling took place in the presence of an applied magnetic field [12]. The prop-erty of perfect diamagnetism is not a direct consequence of perfect conductivity.Maxwell equations predict that the magnetic field inside a perfect conductor shouldbe time independent, implying that if there is magnetic flux inside a perfect con-ductor above Tc it will remain when the sample is cooled to a temperature below Tc,as illustrated in Fig. 2.1. The expulsion of magnetic flux out of a superconductoris known as Meissner effect, and it takes place through the appearance of surfacecurrents in a thin layer of the material1. These currents create a magnetic fieldthat cancels the external field in the interior of the superconductor.

1The thickness of this layer is determined by a temperature dependent parameter, the pene-tration depth λ.

6 CHAPTER 2. THE SUPERCONDUCTING STATE

Figure 2.1. In the presence of an external field H < Hc, a superconductor expelsthe magnetic flux when cooled below Tc, while in a perfect conductor the flux wouldremain the same.

The critical temperature of superconductors is closely related to their isotopicmass. In 1950, both E. Maxwell and C. A. Reynolds with coworkers noticed that thecritical temperature of mercury varied with the isotopic mass M in such a way thatMαTc = constant [13, 14], where α is close to 0.5, but depends on the material [15].The isotope effect was the first indication that some interaction between the latticeand the electrons was involved in the superconducting phenomenon, as a propertyof the lattice like the isotopic mass was affecting an electronic property like thecritical temperature. This was a fundamental discovery for the development of themicroscopic theory of superconductivity, which is based on the pairing of electronsthrough phonons.

Another important property common to most superconductors is the existenceof an energy gap in the density of states. The gap was directly observed by I. Giaeverin a tunneling experiment in 1960 [16, 17]. It has a width 2∆ of the order of kBTc

and it is centered around the Fermi energy εF . A direct consequence of the gapis the peculiar behavior of the heat capacity. As the temperature is decreased asudden jump to a value above that of the normal metal is observed near Tc, followedby a low temperature behavior of the form exp(−∆/kBT ) [18].

In 1962 B. D. Josephson predicted that electron pairs could tunnel between twosuperconductors separated by a thin insulating layer [19]. The consequence is thatcurrents can flow in this kind of junctions even at zero voltages. This is known as dc

Josephson effect and was confirmed experimentally in 1963 by P. W. Anderson andJ. M. Rowell [20]. There is also an ac Josephson effect : when a dc voltage is appliedto the junctions the electron pairs start oscillating with a frequency proportionalto the applied voltage. Since it is possible to measure frequencies very accurately,this effect has been used to set a voltage standard. There are many applicationsof both effects. They are sensitive to magnetic fields, so Josephson junctions areused in very sensitive SQUID (Superconducting QUantum Interference Device)magnetometers. These devices are widely employed to measure magnetic signals inmedical research. Josephson junctions are also a fundamental part in the develop-ment of a superconducting quantum bit [21–23].

2.2. THEORIES OF SUPERCONDUCTIVITY 7

2.2 Theories of superconductivity

2.2.1 London theory

In 1935 the brothers Fritz and Heinz London proposed a phenomenological theorythat could explain some of the properties of superconductors [24]. The first equationin their theory is based on Ohm’s law and accounts for perfect conductivity bypostulating an infinite scattering time for the charge carriers. It relates the localelectric field E to the supercurrent density j in the form:

E =∂

∂t(Λj) , (2.1)

where Λ = µ0λ2L = m∗/ns(q

∗)2. Here m∗, ns and q∗ are the mass, density andcharge of the superconducting electrons and λL is the London penetration depth.The second London equation postulates that the supercurrent density j is propor-tional to the vector potential A,

j = − 1

µ0λ2L

A . (2.2)

Taking into account that the local induction B is given by B = ∇×A, an alternativeway of writing Eq. (2.2) is,

∇× j = − 1

µ0λ2L

B . (2.3)

Combining Eq. (2.3) with Maxwell’s equation ∇× B = µ0j we obtain

∇2B = B/λ2L . (2.4)

A uniform magnetic field B0 is not a solution to this equation unless B0 is identically0. The only solution allowed is an exponential decay with λL as the length overwhich the field penetrates inside the superconductor. In this way, Eq. (2.4) accountsfor the Meissner effect.

London theory is a local theory, that is, considers fields, currents, etc. at a cer-tain point r. In 1953 A. B. Pippard proposed a non-local generalization in whichEq. (2.2) was replaced by an equation where the supercurrent was related to anaverage of the vector potential over a region around r [25]. The size of this region isgiven by the coherence length ξ, which is a measure of the minimum distance overwhich the superconducting electron concentration ns can change change apprecia-bly. For a pure superconductor, the coherence length ξ0 can be estimated from theuncertainty principle to be

ξ0 ≈ ~vF /kBTc , (2.5)

where vF is the Fermi velocity. In general, the coherence length is determined bytwo lengths, the mean free path between two scattering events l and the coherencelength for a pure superconductor ξ0. In fact, ξ−1 ≈ l−1 + ξ−1

0 , so ξ is given roughlyby whichever length l or ξ0 is smaller.

2.2.2 Ginzburg-Landau theory

In 1950, V. L. Ginzburg and L. D. Landau proposed a new phenomenological theoryto explain superconductivity that contained London theory and could account fora density of superconducting electrons ns varying in space [26]. In their model,the superelectrons are described by a wave function Ψ(r) =

√ns(r) eiϕ(r), such

that ns ∝ |Ψ(r)|2. They introduced this wave function as the order parameterin Landau’s theory of second-order transitions and wrote Gibbs free energy G asa series expansion in powers of Ψ and ∇Ψ. By minimizing G, they obtained aSchrödinger-like equation with a nonlinear term,

αΨ(r) + β |Ψ(r)|2 Ψ(r) +1

i∇− q∗A

Ψ(r) = 0 . (2.6)

α and β are the expansion coefficients, which are related to Hc and ns, and A isthe vector potential. Equation (2.6), called Ginzburg-Landau equation, also definesthe coherence length ξ(T ), a characteristic length for the variation of the orderparameter |Ψ|2. This quantity is temperature dependent,2 and therefore differentfrom Pippard’s coherence length, which was a constant. Only for temperatures wellbelow Tc, ξ(T ) ≈ ξ0 in pure superconductors. From the minimization of G, theyalso obtained an equation for the supercurrent,

j =q∗

m∗Re

i∇− q∗A

. (2.7)

The spatial variation of the vector potential A is given by the penetration depthλ(T ), which has the same temperature dependence as ξ close to Tc and is re-lated to London’s penetration depth λL. The ratio between λ and ξ, κ = λ/ξ,is the Ginzburg-Landau parameter. Its value determines the magnetic behaviorof the material, and it is very useful for distinguishing between different types ofsuperconductors. Ginzburg-Landau theory has however a strong limitation. Theequations are derived from a series expansion around the critical temperature and,therefore, are only valid close to Tc.

2.2.3 BCS theory

Both London and Ginzburg-Landau theories are phenomenological theories. Theycan explain many characteristic features of superconductivity, but do not provideany information about its origin or nature. It was not until 1957 that Bardeen,Cooper, and Schrieffer developed a microscopic theory of superconductivity, knownas the BCS theory [27, 28]. BCS theory is based on the formation of so called Cooperpairs. In 1956 L. N. Cooper had studied the interaction of a pair of electrons abovea frozen Fermi sphere [29]. This was six years after the discovery of the isotope

2The temperature dependence of ξ is ξ(T ) ∝ (1 − T/Tc)−1/2 for temperatures close to Tc.

2.2. THEORIES OF SUPERCONDUCTIVITY 9

Figure 2.2. Schematic picture of the electron-phonon interaction. Electron 1creates a polarized region in the lattice as it travels through the solid. In thisdeformed part of the lattice there will be an accumulation of positive charge thatin turn attracts electron 2.

effect, which pointed toward the electron-phonon interaction as a key to understandsuperconductivity. So besides the screened Coulomb repulsion, Cooper consideredan attractive interaction via phonons, showing that the ground state in this casewas a bound state. As long as there is a net attractive interaction, no matter howweak, the formation of a pair would be energetically favorable for the system. Buthow does this attractive interaction appear? The physical idea is illustrated inFig. 2.2. When the first electron travels through the solid it attracts the positiveions of the lattice and creates a polarization in the system. The second electron isin turn attracted by the positive region created by the first electron. In this way,electrons are bound to each other. With this idea as a starting point, Bardeen,Cooper and Schrieffer went one step further and in their ground state all electronswere paired. The pair wave functions were spin singlets and orbital s states, formedby electrons with opposite spins and wave vectors. This ground state is separatedfrom the excited states by an energy gap 2∆. As long as the thermal energy isless than the gap, the resistivity will be zero, i.e the electrons will not have enoughenergy to collide with the lattice. In fact, collisions occur, but they just lead toan exchange of partners, so the correlated motion that creates zero resistivity isconserved. The size of the pairs is given by the coherence length ξ, which can beup to a few µm in some conventional superconductors. This length is much largerthan the spacing between electrons, so the pairs strongly overlap.

In 1959, L. P. Gor’kov showed that Ginzburg-Landau equations can be derivedfrom BCS theory near Tc [30]. This was a major boost for Ginzburg-Landau theory.The charge q∗ and the mass m∗ that appeared in Ginzburg-Landau equations turnedout to be twice the electronic charge and the electron mass respectively, and thedensity of superconducting charge particles ns was half of the density of normalelectrons. These facts relate to the physical meaning of Ψ(r) as the wave functionof the Cooper pairs.

Meissner

Normal

Meissner

Normal

Vortex

Type I Type II

Figure 2.3. Left: Schematic H − T phase diagram for a type I superconductor.Right: Schematic H − T mean-field phase diagram for a type II superconductor.Between the critical fields Hc1 and Hc2 the system is in the vortex state, wherepartial penetration of magnetic flux occurs.

2.3 Type II superconductors

Depending on the value of the Ginzburg-Landau parameter, superconductors canbe divided into two different types. The classical pure superconductors like lead ortin, with λ < ξ and therefore κ < 1, are known as type I superconductors. In thiskind of materials the Meissner effect is present at low fields and temperatures. AsH and T are increased above their critical values, the system undergoes a first-ordertransition into the normal state and superconductivity is destroyed. The schematicphase diagram for type I superconductors is shown in the left panel of Fig. 2.3.

In 1957, A. A. Abrikosov investigated the effect of a large κ in the Ginzburg-Landau theory, reversing the inequality λ < ξ typical of type I superconductors [31].He found a completely different magnetic behavior for materials with κ > 1/

which he called type II. A Meissner phase is also present in these materials belowa certain lower critical field Hc1. However, instead of a discontinuous transitionto the normal state, at Hc1 the magnetic flux starts to penetrate in the form offlux tubes surrounded by screening currents.3 Each of these vortices carries a fluxquantum of value

φ0 =h

2e= 2.0679 · 10−15 Vs. (2.8)

The response of the flux density and the superelectron concentration in a vortexis shown in Fig. 2.4. In the core, the superconducting order parameter |Ψ|2 goesto zero. The size of the core is thus determined by the coherence length, ξ, whichis the characteristic length for the variation of the order parameter. As the field

3The length scale around the vortex core at which these currents circulate is given by thepenetration depth λ, which determines the electromagnetic response of the superconductor.

2.4. HIGH-TC SUPERCONDUCTORS 11

Figure 2.4. Flux density and density of superconducting electrons in a vortex.The electromagnetic response is determined by the penetration length λ, while thevariation of the order parameter |Ψ|2 ∝ ns depends on the coherence length ξ.

is increased, the density of flux lines also increases until the cores overlap. Thisoccurs at the upper critical field Hc2. This field is much higher than the criticalfield Hc of type I superconductors, as the energy cost to maintain the field out ofthe superconductor is less due to the partial flux penetration. This fact makes thistype of materials more suitable for high field applications.

2.4 High-Tc superconductors

In 1986 J. G. Bednorz and K. A. Müller discovered the Ba-La-Cu-O system [32].The highest critical temperature of this material, around 30 K, was not much higherthan the highest Tc known at the moment, around 23 K for Nb-Ge films [33],but it involved a significant breakthrough. To find superconductivity in an other-wise poorly conducting ceramic opened the path for studying the superconductingproperties of cuprates. Just one year later, superconductivity was discovered inYBa2Cu2O7−δ (YBCO) with Tc around 93 K [34]. This provided an importantpractical advantage, as liquid N2 could now be used as a coolant. Mercury basedcuprates hold the highest Tc achieved at the moment, above 130 K [35]. The highestTc reported so far, measured at atmospheric pressure, is 138 K in Tl doped sinteredHgBa2Ca2Cu3O8+δ [36]. If high pressures are applied, this temperature can beincreased to over 160 K [37]. However, the effect of pressure on Tc is not the samein all cuprates, and in YBCO, Tc decreases with increasing pressure [38]. Althoughmost high-temperature superconductors are cuprates, there is a number of non-cuprate HTSCs: MgB2, with Tc around 40 K and a remarkable simple structure fora HTSC [39], Ba1−xKxBiO3, with a cubic perovskite structure and Tc ≈ 30 K [40],or alkali doped fullerenes like KxC60 (Tc ≈ 18 K) [41] or RbxC60 (Tc ≈ 30 K) [42].

High-temperature superconductors are not well described by BCS theory. Themechanism of superconductivity and the symmetry of the order parameter in HTSCsis in fact one of the most controversial issues in superconductivity. The general be-lief is that the pairing is d-wave [43–45], but there are some authors that suggestthat it might actually be s-wave [46, 47].

Figure 2.5. The crystal structure of YBa2Cu2O7−δ . Lattice parameters area ≈ 3.82 Å, b ≈ 3.89 Å and c ≈ 11.68 Å for δ = 0.07 (Jorgensen et al. [48]). Atomsizes are not to scale.

All cuprates have similar crystal structures: they are perovskite-like with oxygendefects. The conduction takes place in CuO2 sheets parallel to the crystal ab plane.In the primitive cell of YBCO, shown in Fig. 2.5, there are two such conductionplanes separated by an Y plane. In between these conducting blocks, there are twoBaO planes and an additional layer with CuO chains along the b direction. Theseact as charge reservoirs, controlling the electron density in the planes.

2.4.1 Anisotropy

The particular structure of cuprates make their electrical properties highly anisotropic.The resistivity ρ in the c-axis can be up to 104 times larger than in the a or b direc-tions. Ginzburg-Landau theory can account for anisotropy if the effective mass isconsidered as a tensor instead of a scalar. This tensor is diagonal and normalizedas (mambmc)

1/3 = m,

(mij) =

ma 0 00 mb 00 0 mc

. (2.9)

In high-temperature superconductors the effective mass along the c-axis, mc, ismuch larger than along the a or b directions, ma and mb, so the anisotropy isalmost uniaxial. It is therefore convenient to define the mass anisotropy parameteras

mab, (2.10)

2.4. HIGH-TC SUPERCONDUCTORS 13

Compound Highest Tc (K) γYBa2Cu2O7−δ YBCO 93 7-8

Bi2Sr2CaCu2O8+δ BSCCO 110 55 - 200 [51–53]Tl2Ba2CaCu2O8+δ Tl-2212 130 20 - 300 [54–56]

Table 2.1. Highest transition temperatures and anisotropy of some cuprates.

with mab = (mamb)1/2. γ is most often determined through magnetic torque mea-

surements 4. In cuprates, both the value of γ and the maximum Tc that can be ob-serve depend on the exact stoichiometry. For YBCO, γ goes from 7-8 for optimallydoped samples (δ = 0.07) to larger than 30 for samples with δ = 0.5 (Tc around50 K) [50]. Other cuprates, like Bi2Sr2CaCu2O8+δ (BSCCO) or Tl2Ba2CaCu2O8+δ

(Tl-2212), are more anisotropic, but reported values of γ are highly scattered. Theanisotropy and highest Tc of some of the most common cuprates are summarizedin Table 2.1.

The effective mass enters London equations through Λ = m/ns(q∗)2, so in an

anisotropic theory Λ is also a tensor. Since Λ = µ0λ2L, the penetration depth will

also take different values in different directions. Taking into account Eq. (2.10) itis easy to show that λc = γλab. The coherence length however obeys the inverserelation, ξc = γ−1ξab. This follows from introducing Eq. (2.10) in the definition ofξ given in Ginzburg-Landau theory, ξ2 ≡ −~/2mα, where α is one of the expansioncoefficients of the free energy. As a consequence, the upper and lower critical fields,that depend on λ and ξ, will also be different in the ab planes and the c-axis.

In general, the physical features of high-temperature superconductors are stud-ied with the magnetic field applied along either the c-axis or the ab-plane. Butin some cases it can be interesting to study what happens at intermediate angles.A way to solve the anisotropic problem would be to introduce the effective masstensor given by Eq. (2.9) into Ginzburg-Landau (GL) equations and redo the cal-culations made in the isotropic case. A different approach was proposed by Blatter,Geshkenbein and Larkin (BGL) [57]. Their main idea was to map the anisotropicproblem to a corresponding isotropic one at an initial level of the GL equations anduse the scaling rules obtained in this way to generalize the isotropic results to theanisotropic situation, as shown schematically in Fig. 2.6. To get an isotropic GLGibbs free energy they rescaled the coordinates, vector potential and magnetic field

r = (x, y, z/γ) (2.11)

A = (Ax, Ay, γAz) (2.12)

H = (γHx, γHy, Hz) . (2.13)

In these expressions, the tilde distinguishes isotropic quantities. By minimizingthe free energy obtained in this way, they found a scaling rule to map an isotropic

4When the external magnetic field does not point in a principal crystal direction, the magne-tization M has a nonzero component perpendicular to the field H. This results in a mechanicaltorque τ = µ0M × H that can be related to the masses along the ab and c directions [49].

Isotropic Anisotropic

scaling

approach

scaling

Basic Equations(GL or London)

Characteristic

Quantity

Figure 2.6. Schematic illustration of the BGL scaling approach and the tradi-tional way of solving the anisotropic problem. In the scaling approach anisotropicquantities can be obtained directly from the isotropic ones using the scaling rulesobtained at the basic equations level. Figure from Blatter et al. [57].

quantity Q into the anisotropic Q,

Q(θ, H, T, ξ, λ, γ) = sQQ(εθH, γT, ξ, λ) . (2.14)

Here sQ = 1/γ for volume, energy, temperature and action and sQ = 1/εθ formagnetic field. εθ is given by

ε2θ =

γ2sin2 θ + cos2 θ , (2.15)

where θ is the angle between the applied magnetic field and the c-axis.

Chapter 3

Vortex matter in high-Tc

superconductors

High-temperature superconductors are extreme type II materials: they have shortcoherence lengths and long penetrations depths, which results in large values of theGinzburg-Landau parameter κ. A direct consequence of a large κ is that the mixedstate extends over a large part of the H −T phase diagram. This is clear if we lookat the expressions for the lower and upper critical fields. Magnetic flux will notstart to penetrate in the superconductor until the Gibbs free energy of the systemwith one vortex becomes smaller than the Gibbs energy in the pure Meissner state.This occurs at fields just above the lower critical field Hc1,

Hc1 =φ0

4πµ0λ2ln

ξ. (3.1)

A long penetration depth λ leads to a low value of Hc1 and therefore the Meissnerstate in high-Tc materials is confined to small fields. At the upper critical field Hc2

the density of vortices is so high that the cores begin to overlap. From Ginzburg-Landau theory Hc2 is found to be

Hc2 =φ0

2πµ0ξ2. (3.2)

So a short coherence length ξ leads to a value of Hc2 much larger than in conven-tional superconductors. The vortex state occupies thus a large part of the phasediagram of high-Tc materials.

In a clean sample vortices are not arranged in a random manner. The super-currents that circulate around them create repulsive forces that tend to keep themapart, pushing them out of the material. However, at fields above Hc1 the energy islowest in the presence of vortices, so a force will appear to keep the flux density con-stant by preventing the vortices from leaving the sample. The balance between thesetwo forces leads to an equilibrium configuration where the vortices are arranged in

16 CHAPTER 3. VORTEX MATTER IN HIGH-TC SUPERCONDUCTORS

Figure 3.1. Bitter decoration image of the vortex lattice produced by a 52 Gmagnetic field in YBa2Cu2O7−δ at 4.2 K. Reproduced from Gammel et al. [59].

a triangular lattice, with a separation a = (2φ0/√

3B)1/2. This lattice is charac-terized by a shear modulus, c66 ∼ 1/λ2, and a tilt modulus, c44 ∼ (1 + k2λ2)−1,where λ is the penetration depth and k is the wave vector. The vortex lattice isalso known as Abrikosov lattice, due to the discovery by Alexei Abrikosov in 1957of a solution to the linearized Ginzburg-Landau equation that predicted a lattice ofparallel flux lines [31]. The first experimental observation of the Abrikosov latticewas an image taken by Essmann and Träuble [58] in 1967 using Bitter decoration1

on Pb-4at%In. Shortly after the discovery of YBCO, Gammel et al. [59] used thesame technique to observe the vortex lattice in this material. One of the imagesthey obtained is shown in Fig. 3.1. Several other techniques have also been succes-fully used to image vortices, among others scanning tunnelling microscopy [60] ormagnetic force microscopy [61]. Vortex dynamics have been directly studied, forexample, with Lorentz microscopy [62] and magneto-optical techniques [63].

3.1 Vortex motion

When a current density j is applied to a superconductor in the mixed state, aLorentzlike force will appear, setting the vortices in motion. The force exerted perunit length on each vortex is

fL = j × Φ0 , (3.3)

1In this technique, small ferromagnetic particles, produced by evaporating a magnetic materialin an inert gas atmosphere, settle on the regions of the surface of the superconductor where themagnetic field is stronger (in this case on top of the end of the vortex lines). The pattern is thenexamined with an electron microscope.

3.1. VORTEX MOTION 17

where Φ0 = φ0z, assuming that the vortices are parallel to the z axis. The Lorentzforce per unit volume over the entire sample is FL = j×B, where B is the mean fluxdensity. A viscous drag force opposes the vortex flow, giving a constant velocity v

of the vortices. The viscous drag force per unit length on a vortex is

fη = −ηv, (3.4)

where η is the viscous drag coefficient. The motion of vortices generates an electricfield E = B×v parallel to j, which causes power dissipation in the system. In thisway, perfect conductivity is lost.

Another force is thus necessary in order to keep the vortices from moving evenwhen FL 6= 0 and in that way recover perfect conductivity. In real materials, thisextra force comes from the static disorder present in the system. For a vortex it isenergetically convenient to be located in a defect, as the order parameter is alreadydepressed there and it is not necessary to create a whole normal core. However,a random distribution of point pinning centers will not be effective to pin a rigidvortex lattice, since the sum of the individual pinning forces from each pinning sitewould add up to zero. In 1979, Larkin and Ovchinnikov [64] studied this problemin their theory of collective pinning. The pinning force scales with volume only as√

V , while the Lorentz force scales with V , so pinning is effective only over smallvolumes of the lattice. Flux bundles, in which the vortices remain correlated, arethus formed. The volume of the bundles, Vc, is determined by the competitionbetween the pinning energy and the elastic energy of the lattice, that allows thevortices to adapt to the pinning landscape.

The property of dissipation-free current flow is recovered when vortices arepinned by disorder. However, if the current density is increased above the criti-cal value jc, the Lorentz force will overcome the pinning force and the vortices willstart moving again. This depinning critical current density jc is bounded by thecurrent at which the Cooper pairs are destroyed, known as the depairing currentdensity j0, so jc<j0. Pinning is most effective at low temperatures, where thermalfluctuations are not as important, so at higher temperatures the flux lines can bedepinned at current densities smaller than jc.

3.1.1 Flux flow

At current densities larger than jc vortices will not be influenced by the pinningbarriers. Their motion in the presence of a driving force will only be affected by theviscous drag. The mean velocity of the system will be determined by the equilibriumbetween the Lorentz force and the drag, so v = (j− jc)×Φ0/η. Assuming (j − jc) ⊥Φ0, the flux flow resistivity is given by

ρff =E

η. (3.5)

Around 1965, J. Bardeen and M. J. Stephen developed a model to explain howdissipation occurs in the flux flow regime [65]. Their starting point was a simple

model of the vortex based on a calculation by Caroli et al. [66]: a normal core,of radius approximately equal to the coherence length ξ, around which the super-currents circulate. They found two equally important contributions to dissipation:currents inside the normal core and normal currents in the transition region outsidethe core. They obtained a flux flow resistivity:

ρff = ρnB

Hc2, (3.6)

where ρn is the resistivity in the normal state. This relation was experimentallyobserved already in 1965 by Kim et al. [67] by resistance measurements in type IIsuperconductors.

3.1.2 Flux creep

At high enough temperatures the thermal energy may be sufficient to allow fluxlines to jump from one pinning center to another, even if the current density is notlarger than jc. Flux bundles jump between adjacent pinning points in a processknown as flux creep, explained in the theory developed by P. W. Anderson andY. B. Kim [68–70]. The jump rate is given by:

R ∝ e−U0/kBT (3.7)

where U0 is the energy of the pinning barrier. If no current is applied, the proba-bility of jumping is the same in all directions and no creep is observed. A currentwill induce a flux density gradient favoring the jumps in the direction in which theflux density decreases, as shown schematically in Fig. 3.2. The electric field causedby the flux creep is [71]

E(j) ∝ e−U0/kBT sinh

). (3.8)

3.1.3 Thermally assisted flux flow

Even when the applied current is small, some type II superconductors show anohmic response at low temperatures in the presence of a magnetic field. In thiscase, the pinning barriers are large compared to the thermal energy, but still finite,and the system acts as a very viscous liquid. Kes et al. [72] proposed the thermallyassisted flux flow (TAFF) model to describe the physics of this regime. The ohmicresponse can be obtained from Eq. (3.8), as for small currents is possible to linearizesinh(x) ∽ x, obtaining a resistivity

ρ = ρ0 e−U0/kBT , (3.9)

where ρ0 and the activation energy U0 are field and temperature dependent. Thegeneral form suggested for the dependence of U0 near Tc is

U0 ∝ (1 − t)qB−β , (3.10)

3.2. PHASE DIAGRAM 19

U (x) =j 0

Figure 3.2. Flux bundles jump between adjacent pinning centers. When no cur-rent is applied, the probability of jumping is the same in all directions. An electricalcurrent induces a flux density gradient that favors the jumps in one direction. Inthis schematic view, the free energy is shown as a function of the position of thebundle.

where t is a reduced temperature, t = T/Tc. The exponent q has been reported tobe around ∼ 1.5 [73–75], but q ≈ 1 has also been suggested [76]. This latter value isin agreement with the theory of a pinned vortex liquid developed by Geshkenbeinet al. [77] and Vinokur et al. [78], in which the barriers for plastic motion had thetemperature and field dependence of Eq. (3.10), with q = 1 and β = 0.5. Theexponent β has been found to depend on the anisotropy of the material [79] andpossibly on the magnetic field [80]. Reported values lie around 0.5 for BSCCO [76,81] and scattered in the range ∼ 0.6− 1 for YBCO [75, 79, 82, 83]. At low currentdensities U0 is current independent, while at high currents follows U0 ∝ ln(jc/j) [84].

According to Eq. (3.9) the resistivity should obey ln ρ ∝ 1/T . In an Arrheniusplot, of ln ρ vs. 1/T , the TAFF regime corresponds to the linear low resistancepart. Such a plot is shown in Fig. 3.3. At higher temperatures and resistances, theflux flow regime is entered.

3.2 Phase diagram

In the previous section we have seen how vortex dynamics are determined by thecompetition of several energies. The system gains energy due to the Lorentz forcethat sets the vortices in motion. If this energy is smaller than the pinning energy,the vortices will be trapped by the defects present in the system. But for thepinning to be effective, the vortex lattice has to adapt to the pinning landscape atthe cost of some elastic energy. Thermal energy also plays an important role: athigh enough temperatures, the pinning barriers become irrelevant compared to thethermal energy (flux flow regime), and thermal activation is also what makes thecreep and TAFF phenomena possible. The competition between elastic, pinning and

1.1 1.2 1.3 1.4

10−3

10−2

10−1

100 / T (K−1)

ρ / ρ

Figure 3.3. Arrhenius plot of normalized resistance for the YBCO single crystalused in Paper III, in magnetic fields B ‖ c 0, 0.5, 1, 2, 4, 6, 9 and 12 T (left toright). The dashed line marks the crossover from flux flow at high temperatures toTAFF at low resistivities.

thermal energies shapes the phase diagram of high-temperature superconductors.The importance of thermal fluctuations is measured by the Ginzburg number Gi,

(γkBTc

4πµ0H2c ξ3

, (3.11)

where γ is the mass anisotropy parameter, ξab(0) is the coherence length along theab plane and Hc is the thermodynamical critical field, Hc = φ0/(2

√2πµ0λξ),. The

large anisotropy parameter γ, short ξ and high Tc characteristic of high-temperaturesuperconductors yield a large Gi and very soft vortex matter, giving rise to a veryrich phase diagram. In these materials, thermal fluctuations are important enoughto overcome the elastic energy of the vortex lattice in a large part of the phasediagram, melting the lattice into a liquid. The mixed state is then separated intoa solid at low temperatures and a liquid at high temperatures. At low fields theshear modulus of the lattice, c66, vanishes exponentially causing melting and aliquid phase also at low fields. The phase diagram for high-Tc superconductors isshown schematically in Fig. 3.4. The properties of the solid depend on the type andamount of disorder, which also determine the nature of the solid-to-liquid transition.

3.2.1 Vortex liquid

At high enough temperatures or magnetic fields the flux line lattice melts into aliquid. In the vortex liquid phase the correlation between vortices is lost and theymove around under the influence of thermal energy and Lorentz forces, causingdissipation. Due to the lack of correlation between vortices, pinning in the liquid

Figure 3.4. Schematic phase diagram of a superconductor with κ >> 1. TheMeissner phase is suppressed to low fields. The mixed state occupies a large partof the diagram and is divided into a solid phase at low temperatures and a liquidphase at high temperatures. A liquid phase is also predicted at low fields, but itand at low fields.

state is not as effective as in the solid. Thus the liquid always shows a non-zeroresistivity, even for small current densities. At temperatures just below the meltingtransition the lattice becomes softer and a peak appears in jc. Soft vortex mattercan adapt better to the pinning landscape, explaining the appearance of the peak.The first observation of this effect in high-temperature superconductors was madeby Kwok et al. [85]. Just above the melting temperature the TAFF regime isentered. In this region pinning is still effective and the liquid is very viscous, witha resistivity smaller than in the flux flow regime. Nelson and Seung [86] suggestedthat the reduced resistivity could be due to entanglement in the liquid. Marchettiand Nelson [87] proposed a hydrodynamic theory where the liquid is describedthrough a flux liquid viscosity, which is large due to the entanglement. The flowof the viscous flux liquid is impeded by the effect of a few “strong” pinning centerswhose effect is transmitted over large distances, causing the observed decrease inresistivity in the TAFF regime. Finally, at high enough temperatures the flux flowregime is reached, where the liquid is unpinned. The current dependence of theresistivity in the different regimes is schematically shown in Fig. 3.5.

In high-temperature superconductors the magnetic relaxation processes are quickover a large region of the phase diagram below Hc2, leading to a reversible mag-netization. This region is limited at low temperatures by the irreversibility line2.The first to observe this behavior of the magnetization were Müller et al. [88] in

2Below the irreversibility line, the zero-field cooled and the field cooled magnetization curvesare different. In the latter case some flux lines get trapped and M 6= 0 even when H = 0.

log ρ

Liquid

CreepT

Figure 3.5. Schematic plot of resistivity ρ vs. current density j for a cleansuperconductor. At high temperatures the pinning barriers are much smaller thanthermal energy and the system is in the flux flow regime (FF). As T decreasesand the system gets closer to the solid phase, the pinning barriers become largecompared to the thermal energy (TAFF regime). In the solid the resistivity comesfrom creep motion of the vortices.

1987 in the Ba-La-Cu-O system. They interpreted their observation as evidence ofa superconducting glass state. Yeshurun and Malozemoff [73] and Tinkham [74],described the irreversibility line as a depinning line, using a thermally-activatedflux-creep model. This interpretation has been supported by several works [89–91].However, the origin of the irreversibility line has been a controversial issue, andsome authors attribute it to the phase transition from the solid to the liquid, eithermelting [35, 92] or second-order glass transition [93].

3.2.2 Vortex solid

The properties of the vortex solid are determined by the competition between thepinning energy and the elastic energy that keeps the lattice together. Vorticeswill try to accommodate to the defect landscape while maintaining the lattice.In an ideal material with no pinning centers the solid state would be a perfectAbrikosov lattice. If the density of defects present in the material is not very high,the distortions of the lattice will not be very important and the solid is a quasi-lattice or Bragg glass [94, 95]. In this phase the translational long-range order ispreserved and Bragg diffraction peaks are observed [96]. When the temperaturereaches the melting temperature Tm, the system undergoes a first-order transitioninto the liquid state.

If the pinning centers are strong or if there is a large amount of them, the pinningenergy will prevail and the solid can no longer be regarded as a perfect lattice. In thepresence of uncorrelated disorder, like oxygen vacancies or impurities, the proposedmodel for the solid state is a vortex glass [93]. This phase is expected to be truly

Pancake

vortices

Josephson

vortices

Figure 3.6. In layered superconductors, pancake vortices in the CuO2 planes arecoupled by Josephson vortices that run parallel to the planes. Left: magnetic fieldapplied along c-axis. Right: tilted magnetic field.

superconducting: for small currents (j → 0) the pinning barriers become infinite,preventing any kind of creep motion and making ρ → 0. The transition betweenthe glass and the liquid is second order, instead of the first-order melting observedfor clean samples.

The situation in the presence of correlated disorder, like the twin boundaries inYBCO or the columnar defects caused by heavy-ion irradiation, is slightly different.Vortices are pinned by the columnar defects, causing a strong angular dependenceand anisotropy not observed in the vortex glass. When the magnetic field is parallelor almost parallel to the defects, the solid is a Bose glass. This system has beenstudied in detail by Nelson and Vinokur [97, 98] and Larkin and Vinokur [99]. Theinfluence on the solid-to-liquid transition of the tilt angle between the magneticfield and the columnar defects is the subject of Paper IV (see Chapter 4).

3.2.3 Effects of the layered structure

In highly layered materials the coherence length along the c-axis, d, can becomeshorter than the separation between the superconducting CuO2 planes and thenvortex cores will be confined to the conduction planes, forming what is known aspancake vortices, see Fig. 3.6. The coupling between pancake vortices in differentlayers occurs via Josephson [100] and/or magnetic [101, 102] interactions. If thiscoupling is not too weak, like in optimally doped YBCO, the system can be re-garded as anisotropic three-dimensional (3D). However, in systems like BSCCO,the coupling is weaker and the system can be considered as a stack of decoupledtwo dimensional (2D) layers. The physics of such a system is described by theLawrence-Doniach model [103], that reduces to the anisotropic Ginzburg-Landautheory in the limit of long wavelength of the modulation of the order parameterperpendicular to the layers. The crossover between the 3D and quasi-2D behavioroccurs at a temperature Tcr = Tc[1 − 2(ξc(0)/d)2], which for BSCCO is very closeto Tc, while for YBCO lies quite far below Tc [104]. Glazman and Koshelev [105]studied the influence of thermal fluctuations in layered superconductors and pro-

posed a phase diagram in which there is a crossover field Bcr that separates theregions of 3D and 2D melting. The crossover field is

Bcr ∝ φ0

(dγ)2, (3.12)

where γ is the anisotropy as defined in Eq. (2.10) and d is the spacing betweenthe CuO2 layers [100, 106]. Experimentally, changes in the field dependence of thecritical current density jc of DyBa2Cu3O7/Y1−xPrxBa2Cu3O7 superlattices [107],the appearance of a two step transition in Tl-2212 thin films [108] as well as changesin the irreversibility line [109] have been attributed to this crossover.

3.2.4 The melting transition

In clean samples, where the solid phase is a Bragg glass, the transition between thesolid and the liquid is expected to be a first-order melting transition. This occurs attemperatures and fields such that the thermal energy is equal to the elastic energyof the lattice. A simple model to locate the melting line is based on the Lindemanncriterion: ⟨

th= c2

La2 . (3.13)

Here⟨u2

is the mean-squared amplitude of the thermal fluctuations and a is thedistance between the flux lines in the lattice. What Eq. (3.13) states is that themelting occurs at the temperature at which the displacement of the flux lines dueto the thermal fluctuations equals some fraction of the lattice parameter a. TheLindemann number cL is usually around 0.1-0.3. To determine the melting line itis then necessary to calculate

. This was done by Houghton et al. [110] forisotropic materials or anisotropic materials with H ‖ c. They obtained an implicitequation for the melting line that de Andrade, Jr. and de Lima [111] simplified fortemperatures close to Tc, obtaining an explicit form,

Hm(T ) ≈ βthHc2(0)c4L

(1 − T/Tc

. (3.14)

The numerical constant βth ≈ 5.6 and Gi is the Ginzburg number defined inEq. (3.11). The temperature dependence of the melting line is usually approxi-mated by a simple power law, (1 − T/Tc)

2. Far away from Tc, it is necessary totake into account quantum fluctuations besides thermal fluctuations. This can bemade by combining

and⟨u2

in equal weights and applying Lindemann

criterion [112]. Here⟨u2

is the mean-squared amplitude of the quantum fluctua-tions. The melting line obtained in this way is displaced toward lower temperaturesand does not follow a simple power law,

bm(T ) =4θ2

(1 +√

1 + 4SθTc/T )2, (3.15)

3.3. THE VORTEX-GLASS TRANSITION 25

θ = c2L

)1/2 (Tc

T− 1

S = q + c2L

Here, bm(T ) = Bm(T )/Hc2(T ) and q ≈ 2.4ν/kF ξ, with kF the Fermi wave vectorand ν a fitting parameter. Blatter et al. [8] compared Eq. (3.15) to the simplepower law that has been experimentally observed in YBCO, Hm ∝ (1 − T/Tc)

n,with n ≈ 1.35−1.45. They found that the agreement of the power law and Eq. (3.15)with the data was equally good. For BSCCO, due to the high anisotropy thermalfluctuations are more important than quantum fluctuations (see the definition of Gi,Eq. (3.11)), and the best description of the data is instead provided by Eq. (3.14).

There is plenty of experimental evidence of the melting transition and its first-order character. Schilling et al. [113] were the first to observe the latent heatthat is associated to a first-order transition, L = T∆S. They obtained a valueof ∽ 0.45kBT per vortex per superconducting layer in untwinned YBCO. As thedensity of vortices is different in the liquid than in the solid phase a jump in themagnetization ∆M and the local field ∆B = µ0∆M should be expected at thetransition. This was observed in BSCCO single crystals by Pastoriza et al. [109]and Zeldov et al. [114] respectively. Even though resistivity is not a thermodynamicproperty, transport measurements also show signs of the melting transition. Asharp step in the resistivity is observed at the melting temperature [115, 116].Welp et al. [117] showed that the location of the melting step coincides with thelocation of the magnetization jump by measuring both in the same YBCO singlecrystal. Continuous phase transitions are reversible, so a hysteretical behavior ofthe resistivity would reveal a first-order transition. This was observed by Safaret al. [118] but it was later shown that the hysteresis was not related to the latentheat [119]. The melting has been observed directly by magneto-optical techniquesby Soibel et al. [120].

3.3 The vortex-glass transition

The long range order characteristic of the vortex lattice is unstable in the presence ofuncorrelated disorder [64]. A model proposed to describe this situation is the vortex-glass model, in which the vortex solid is expected to be a truly superconductingphase with infinite dc conductivity. The solid-to-liquid transition changes in nature,becoming a continuous second-order transition.

3.3.1 FFH scaling

Fisher, Fisher and Huse (FFH) [106] developed a scaling theory to explain theglass-to-liquid transition. At a continuous transition the correlation length,

ξ ∽ |T − Tg|−ν (3.16)

and relaxation timeτ ∽ ξz (3.17)

will diverge as the critical temperature is approached. In the case of the vortex-glass, the critical temperature is the glass temperature Tg. ν and z are the static anddynamic critical exponents, which are constants within the vortex-glass universalityclass. Taking into account Eqs. (3.16) and (3.17), dimensional analysis can be usedto establish a scaling ansatz that will help us study the I-V characteristics of thesystem. We want to know how E and j scale. The vector potential A scales as aninverse length L−1. Since E ∝ ∂A/∂t, the electric field should scale as L−1T−1, sothe appropriate scaling combination should be Eξz+1

g . The same reasoning can beapplied to the current density j ∝ ∂f/∂A, where f is the free energy density. f hasdimensions L−d, where d is the dimensionality of the system, so the correspondingscaling combination for the current density should be jξd−1. The scaling ansatz isthus

Eξz+1 ∝ E±(jξd−1), (3.18)

where E± is an appropriate scaling function for temperatures above (+) and below(-) the transition temperature Tg. The shape of E± is unknown, but some infor-mation can be extracted by looking at the behavior expected for the I-V curves.For temperatures above Tg and large distances (or small currents) the response isexpected to be ohmic. So in this case E+(x → 0) ∽ x, which gives

E ∝ j ξd−2−z . (3.19)

Taking into account Eq. (3.16) we obtain an ohmic resistivity that follows

ρ(T ) ∝ |T − Tg|ν(z+2−d) (3.20)

Experimentally, it is possible to obtain Tg and the combined exponent s = ν(z +2 − d) by calculating the inverse derivative of the resistivity,

(d ln ρ

s(T − Tg) . (3.21)

For temperatures below Tg and small currents, a glassy behavior is expected, withE−(x → 0) ∽ exp(−a/xµ) and therefore with I-V characteristics described by

E ∝ e−c(jc/j)µ

, (3.22)

3.3. THE VORTEX-GLASS TRANSITION 27

Figure 3.7. Scaled I − V curves obtained by Koch et al. [122] on an epitaxialYBa2Cu2O7−δ thin film. In this figure 119 isotherms collapsed into two curvescorresponding to temperatures above and below Tg . This was the first experimentalevidence of vortex-glass behavior. Figure taken from Koch et al. [122].

where µ is the glass exponent. At the glass transition temperature Tg the vortex-glass coherence length ξ diverges. To cancel out the dependence on ξ in Eq. (3.18),we must have E±(x → ∞) ∽ xα with α = (z+1)/(d−1). Right at the transition, andalso for high currents both above and below the transition, the I-V characteristicsfollow thus a power law,

E ∝ j(z+1)/(d−1) . (3.23)

The crossover between the high current and low currents regime is given by jx ∝|T − Tg|ν(d−1).

The scaling functions and the critical exponents should be universal, so a goodway to experimentally test the vortex-glass model and calculate the exponentsis to plot appropriately scaled I-V curves. Looking at Eq. (3.18), and assumingd = 3, a scaled current density, j = j ξ2 = j/|T − Tg|2ν , and a scaled electric fieldE = E ξz+1 = E/|T −Tg|ν(z+1) can be defined. E/j vs. j curves taken at differentfields and temperatures should collapse into two curves, one for T > Tg and one forT < Tg. This kind of scaling analysis was performed by Koch et al. [121] on theresistive data from an epitaxial film of YBCO, finding excellent agreement with themodel. Their result is shown in Fig. 3.7. They obtained z = 4.8± 0.2, ν = 1.7 andthus s = 6.5, in agreement with the expected values for a vortex-glass transition in3D (z > 4 and ν ∽ 1−2). Afterward, vortex-glass transitions have been observed inseveral high-temperature superconductors: optimally doped YBCO [123], BSCCO

single crystals [124], Tl-2212 thin films [108] or oxygen-deficient YBCO [125]. Thereis some variation in the critical exponents reported for different materials, partlydue to the difficulties in the experimental determination, as very low resistivitiesare probed. A comparison of exponents for different YBCO samples can be foundfor example in Roberts et al. [126] and Hou et al. [127].

3.3.2 The vortex molasses scenario

The vortex molasses scenario is an alternative model to the vortex glass for thesolid phase in disordered superconductors. The idea, proposed by Reichhardt et al.

[128], is that at the transition from the solid to the liquid vortices freeze like thewindow glass. In this kind of transition only a diverging relaxation time is observed,as the dynamics are frozen so rapidly that the divergence of the coherence lengthbecomes unobservable. The resistivity is then expected to decay exponentiallyat Tg, ρ ∝ exp[−1/(T − Tg)], instead of the ρ ∝ (T − Tg)

s behavior expectedfor the vortex glass. Reichhardt et al. [128] observed this behavior in numericalsimulations, but experimentally it is difficult to distinguish between both models,as the exponent s in the vortex glass is quite large.

3.4 The Bose-glass transition

The vortex-glass model was proposed for cases in which the disorder was uncorre-lated, like oxygen vacancies or point defects. When the disorder is correlated, liketwin boundaries in YBCO or columnar defects created by heavy-ion irradiation,the solid is described by a Bose glass. The solid-to-liquid transition in supercon-ductors with columnar defects and fields applied along the direction of the columnswas studied in detail by Nelson and Vinokur [97, 98]. They established an anal-ogy between the vortex system and the problem of localization of two-dimensionalquantum-mechanical bosons [129]. From this mapping they proposed a phase dia-gram in which the low temperature phase is a Bose glass separated by a sharp phasetransition from a liquid of entangled vortices at high temperatures. The tempera-ture at which the transition takes place is the Bose-glass temperature TBG. In thevortex-glass phase wandering of vortex lines was enhanced in order to adapt betterto the pinning landscape. In contrast, in the Bose glass vortices are localized inthe columnar defects. Some segments of the vortex lines might however be excitedout of the columns when an electrical current is applied transverse to the appliedmagnetic field. The shape and size of these loops are determined from the balanceof the elastic energy of the vortex lines, εlr

2/z, the pinning energy UP z and theenergy gained due to the Lorentz force, φ0jzr [98]. Here ε is the vortex line energy,UP is the pinning energy per unit length, z is the length of the portion of the vortexthat gets depinned and r is the transverse size of the loop, see Fig. 3.8. From thebalance of the pinning energy and the energy due to the Lorentz force it is found

3.4. THE BOSE-GLASS TRANSITION 29

Figure 3.8. When a transverse current is applied, the vortex lines can be excitedout of the columnar defects. At high currents the loops are small and do not reachneighboring defects (a). At lower current densities the excitations grow and vortexlines get pinned by adjacent columns (b).

that the size of the loops decreases with increasing current density,

jφ0(3.24)

For low enough currents the excited vortex segment might reach a neighboringdefect, forming a double kink. An schematic illustration of vortex structure in theBose-glass phase is shown in Fig. 3.8.

The shape of the Bose-glass line, that separates the Bose-glass phase from theliquid in the phase diagram, depends on the magnetic field. Larkin and Vinokur [99]studied this dependence and proposed the following picture. At low fields, wherethe distance between vortices is much larger than the penetration depth (dilutelimit), the Bose-glass line follows

BBG ∝(

Tc − T

, (3.25)

where Tc is the zero field transition temperature. At higher fields, the interactionbetween vortices should be taken into account. In this case the shape of the linedepends on the amount of disorder. At low temperatures it lies slightly above themelting line of a pristine sample. At higher temperatures disorder dominates thetransition and the line follows either BBG ∝ (Tc − T )6 or BBG ∝ (Tc − T )3. Thefirst expression is valid for temperatures above a characteristic temperature Tdp atwhich the size of the transversal thermal fluctuations is similar to the coherencelength, while the latter is valid at T < Tdp. A schematic phase diagram is shownin Fig. 3.9.

Figure 3.9. Schematic phase diagram for superconductors with a high concentra-tion of columnar defects and fields applied parallel to the columns. The white lineis the melting line for a non irradiated sample. At fields around the matching fieldBφ the Bose-glass line deviates from the melting line.

A scaling analysis similar to the vortex-glass scaling described in Section 3.3.1can be performed for the Bose glass [97, 98]. It is assumed that the columnar defectsare parallel to the c-axis of the sample and that the magnetic field is applied in thedirection of the defects. The main difference with respect to the vortex-glass scalingis that the Bose glass is described by two correlation lengths instead of one. Thescaling is thus anisotropic, in contrast to the fully isotropic vortex-glass scaling.The correlation lengths are related to the size of the vortex loops. Vortex linesmay wander within a distance given by the localization length ξ⊥, which definesthe correlation length perpendicular to the applied magnetic field. It diverges atthe Bose-glass temperature TBG:

ξ⊥(T ) ∽ |T − TBG|−ν⊥ , (3.26)

with ν⊥ ≥ 1. There is also a correlation length parallel to the magnetic field, ξ‖,that diverges at TBG with a different exponent:

ξ‖(T ) ∽ |T − TBG|−ν‖ , (3.27)

where ν‖ = 2ν⊥ ≡ 2ν′. The relaxation time of the fluctuations is given by τ ∼ ξz′

⊥ .Using dimensional analysis and following the same procedure as for the vortex-glass,an appropriate scaling ansatz can be found for the Bose-glass transition:

Eξz′+1⊥ ∝ F±(jξ⊥ξ‖) . (3.28)

As for the vortex-glass case, F± is a scaling function, of unknown shape, for tem-peratures above (+) and below (-) the transition temperature TBG. At low currentsthe resistivity is expected to be linear, so F+(x → 0) ∼ x. Using the expressions for

3.4. THE BOSE-GLASS TRANSITION 31

ξ⊥ and ξ‖ given by Eqs. (3.26) and (3.27), it can be found that the ohmic resistivityin the liquid state follows

ρ ∼ |T − TBG|ν′(z′−2)

. (3.29)

This result is very similar to the power law found for the vortex-glass scaling,i. e. Eq. (3.20), where the combined critical exponent was ν(z − 1), assuming thedimensionality of the system was d = 3. Numerical simulations predict ν′ ≈ 1.0and a value of z′ between 4.6 and 6 [130, 131], which yields an s′ = ν′(z′ − 2)between 2.6 and 4. The experimental determination of the exponents is difficultand the dispersion in the reported values is large. For BSCCO it was found thats′ ≈ 9 [132, 133], while in Tl-2212 it was reported s′ ≈ 4.3 [134]. In YBCO,consistent values were found in untwinned single crystals with columnar defects,s′ ≈ 2.4 [135], and in twinned single crystals, s′ ≈ 2.8 [136], even though previousmeasurements gave smaller s′ [137–139]. This large scattering in the values makesvery difficult to distinguish between the Bose glass and the vortex glass from thecritical exponents.

Chapter 4

The solid-to-liquid transition

In Chapter 3 it was discussed how the amount and type of disorder present in asuperconductor change the character of the vortex solid-to-liquid transition. Threeproblems regarding this transition are considered here in more detail, all of themdirectly related to my research activity. The first one is the applicability of anextended model of the vortex-glass transition for various conditions. The secondproblem considers samples with correlated disorder in the form of columnar defects.Resistivity measurements in the three spatial directions are studied to understandthe effect on the transition of the tilt angle between the columns and an appliedmagnetic field. Finally, the transition is studied in strongly anisotropic YBCOin the presence of strong magnetic fields applied parallel to the superconductingplanes.

4.1 Extended model of the vortex-glass transition

In the vortex glass scaling theory of Fisher, Fisher and Huse presented in Sec-tion 3.3.1, the ohmic resistivity above Tg vanishes following a power law as theglass-transition temperature is approached,

ρ = ρ0

∣∣∣∣T

Tg− 1

∣∣∣∣s

. (4.1)

The transition is thus driven by a temperature difference T − Tg. However, in asuperconductor relevant energy scales like the condensation energy or the pinningenergy change also with magnetic field. Rydh et al. [79] proposed a generalizationof the FFH vortex-glass scaling theory that takes this dependence into account.In this extended model the driving force of the transition is instead the differencekBT − U0(B, T ) between the thermal energy kBT and an effective pinning energyU0. Eq. (4.1) is then replaced by

ρ = ρn

∣∣∣∣kBT

U0(B, T )− 1

∣∣∣∣s

. (4.2)

34 CHAPTER 4. THE SOLID-TO-LIQUID TRANSITION

10−2

10−1

10−2

10−1

T(Tc−T

c−T)−1

ρ/ρ n

Figure 4.1. Normalized resistivity versus scaled temperature according to Eq. (4.4)for the YBCO single crystal studied in Paper III. Curves for fields 0.5, 1, 2, 4, 6, 9and 12 T applied parallel to the c-axis are shown. The exponent s can be determinedfrom the slopes in the linear region of these curves.

The glass-to-liquid transition occurs when the two energy scales are equal U0(B, Tg) =kBTg. The explicit field and temperature dependence of U0 is not well established,but good scaling of the resistivity of disordered high-temperature superconductorshas been found using an effective pinning energy of the form

U0 = kBTc1 − T/Tc

(B/B0)β= UB(1 − T/Tc) . (4.3)

Here both B0 and β are temperature and field independent constants [79]. InvertingEq. (4.2) it is possible to obtain U0 directly from the measured resistivity. Thiswas done by Andersson et al. [140] for optimally doped single crystals of YBCO.They also fitted their data to a power law according to the TAFF model discussedin Section 3.1.3, obtaining the activation energy for flux motion. They found thatboth energies had the same magnetic field dependence, justifying the form for U0

proposed in Eq. (4.3), which is similar to the suggested expression for the activationenergy in the TAFF model, i. e. Eq. (3.10).

When U0, given by Eq. (4.3), is introduced into Eq. (4.2) a useful scaling formfor the resistivity is obtained

ρ = ρn

∣∣∣∣T (Tc − Tg)

Tg(Tc − T )− 1

∣∣∣∣s

. (4.4)

Here the magnetic field dependence of ρ enters implicitly through Tg. According toEq. (4.4), the resistive curves for different fields should scale into one when ρ/ρn

is plotted versus the scaling variable T (Tc − Tg)/Tg(Tc − T ) − 1. An example ofsuch a plot is shown in Fig. 4.1. The combined exponent s, which is the same asin the vortex-glass model of Fisher, Fisher and Huse, can be obtained from the

4.1. EXTENDED MODEL OF THE VORTEX-GLASS TRANSITION 35

20 40 60 80 1000

Tg (K)

Figure 4.2. Glass line of the Tl-2212 thin film studied in Paper II. The solid lineis a fit to Eq. (4.6) with B0 = 1.5 T and β = 0.55. The dashed line is a fit to thepower law given by Eq. (4.5) with n ≈ 1.3.

slopes of the linear part of these curves. The similarity between this scaling vari-able and the one predicted by the Ginzburg-Landau Coulomb-gas model for the2D Kosterlitz-Thouless transition is remarkable: they have the same form, with Tg

replaced by the Kosterlitz-Thouless temperature TK [142]. This type of Coulomb-gas scaling has been used to describe the resistance of BSCCO crystals [142],YBa2Cu2O7−δ/PrBa2Cu3O7−δ superlattices [143] and Bi2+ySr2−x−yLaxCuO6+δ

thin films [144]. These are strongly anisotropic materials, in which the couplingof the CuO layers is very weak, and they can be considered two dimensional, whilein YBCO the coupling of the layers is stronger, and can be described by a three di-mensional anisotropic model. Therefore the Ginzburg-Landau Coulomb-glass scal-ing and the extended model of the glass scaling considered here address differentphysical problems, so the similarity between the scaling variables should at themoment be considered merely formal.

4.1.1 The vortex glass line

The glass line, that separates the glass from the liquid phase in the phase diagram,has been empirically described by a simple power law of the form

Bg ∝ (1 − T/Tc)n , (4.5)

where n ≈ 1.5 [127, 145]. However, as shown in Fig. 4.2, this power law does notaccount for the data at low temperatures and high fields for the most anisotropicsamples. The deviations set in at a temperature close to Tc/2 and a field propor-tional to 1/γ2 [127, 146]. These deviations have been related to the field inducedcrossover from a three-dimensional (3D) to a quasi-two dimensional (2D) regime ofthe vortex fluctuations discussed in Section 3.2.3.

10−1

10−4

10−3

10−2

10−1

T(Tc−T

c−T)−1

ρ/ρ n

Figure 4.3. Normalized c-axis resistivity versus scaled temperature according toEq. (4.4) for the YBCO single crystal studied in Paper I. Curves for fields 1, 2, 4,6, 8, 10 and 12 T applied parallel to the c-axis are shown.

An alternative expression for the glass line can be obtained from the extendedscaling theory presented above when U0(B, T ) (given by Eq. (4.3)) is considered atthe glass temperature Tg,

Bg(T ) = B0

(1 − T/Tc

. (4.6)

The field B0 roughly agrees with the field at which the deviations from the power-law appear, as expected from the fact that Eq. (4.6) implies B0 ≡ Bg(Tc/2). A fitof the experimental data to this line is shown as a solid line in Fig. 4.2.

4.1.2 Scaling in the extended model: Experimental results

The development of the extended model was based on measurements of in-planeresistivity in optimally doped YBCO, for which excellent scaling was found usingEq.(4.4) [79, 140]. In Paper I it was shown that the model can also be applied tomeasurements of c-axis resistivity. Figure 4.3 shows that good scaling is also foundin this case.

In the first works on the extended model some underdoped YBCO sampleswere studied, finding a somewhat worse scaling than in the case of optimally dopedYBCO. The influence of anisotropy on the model was explored in Paper II, wheresamples with different anisotropies were considered. In all cases good scaling wasfound only at fields below a certain field B0, related to B0 in the glass line givenby Eq. (4.6). The value of B0 is consistent with the crossover field considered in

4.1. EXTENDED MODEL OF THE VORTEX-GLASS TRANSITION 37

10−2

10−1

10−3

10−2

10−1

T(Tc−T

c−T)−1

Figure 4.4. Modified temperature versus normalized resistance for the Tl-2212thin film studied in Paper II. Good scaling is found for magnetic fields 0 6 B 6

1.5 T. For this sample, B0 ≈ 1.5 T. Deviations from the scaling are observed alreadyat 3 T and become more important for increasing fields.

Section 3.2.3, Bcr ∝ φ0/(dγ)2, at which the vortex fluctuations go from a three-dimensional (3D) to a quasi-two-dimensional (2D) regime. The fact that the scalingdeteriorates for fields B > B0 could then be taken as further evidence for a fieldinduced crossover at B0. However, the existence of a sharp dimensional crossover atB0 is still an open issue. The combined exponent s depends on the dimensionalityof the system, and therefore some changes in s would be expected at a dimensionalcrossover. We did not find any essential change in the exponent above and belowB0 for the Tl-2212 film and only a slight decrease, within the uncertainty of ourdetermination, for the YBCO crystals. The fact that, as shown in Fig. 4.2, the glassline in Eq. (4.6) describes fields below and above B0 is also opposed to the ideaof a sharp dimensional crossover at B0. We believe that a gradual transition intoa quasi-2D situation takes place instead. This idea is supported by the fact thatthe deviations from the scaling become more important with increasing fields. Thefact that scaling is found for low fields indicates that vortex lines are still presentin the liquid state in this region. As the field is increased, the correlation withineach vortex line becomes weaker while the interaction between pancake vorticesincreases, leading to a quasi-2D situation at high fields. The results for the Tl-2212thin film studied in Paper II are shown in Fig. 4.4.

In all the cases considered above, the magnetic field was applied along the c-axisof the samples. The effect of tilting the field with respect to the c-axis was studiedin Paper III. The angular dependence of the magnetic field can be introduced in twofundamental parameters of the model: B0 in the glass line given by Eq. (4.6) and

0 20 40 60 80

θ (deg)

Figure 4.5. Angular dependence of the glass transition temperature Tg for fields1, 2, 4, 6, 9 and 12 T (from top to bottom) for the YBCO single crystal measuredin Paper III. Open circles are experimental data. Solid lines are given by Eq. (4.8).There is good agreement between the experimental data and the predicted Tg(θ).Pinning by twin boundaries leads to a cusp in Tg(θ) at θ = 0 [116]. To compensatefor this effect Tg(0) − 0.5 K is also shown (filled circles).

Tg. The scaling approach proposed by Blatter, Geshkenbein and Larkin (BGL) [57],described in Section 2.4.1, can be used for this purpose, giving:

B0(θ) = ε−1θ B0(0) (4.7)

Tg(θ) =ε−β

1 +Tg(0)

Tc(ε−β

θ − 1)Tg(0) . (4.8)

Here ε2θ = γ−2 sin2 θ + cos2 θ, where θ is the angle with respect to the c-axis. In

Paper III it was found that the experimentally obtained B0 and Tg for a slightlyunderdoped YBCO single crystal were in very good agreement with the predictionsfrom Eq. (4.8). The result for Tg is shown in Fig. 4.5.

4.2 The transition in samples with columnar defects

In previous sections the solid-to-liquid transition was considered in cases in whichuncorrelated disorder was the main source of pinning. But correlated disorder canappear naturally in a sample, in the form of twin boundaries or screw dislocations,changing the properties of the transition. Pinning by this type of disorder is veryefficient and it is thus technologically interesting to be able to control the amountand direction of the defects. Columnar defects are a type of correlated disorderthat can be artificially produced in a sample by bombarding the material with

4.2. THE TRANSITION IN SAMPLES WITH COLUMNAR DEFECTS 39

heavy ions. The ions traverse the crystal, leaving behind linear tracks that arenot superconducting. The result of irradiation is thus a random array of columnardefects with a diameter that depends on the type of ions, but it is usually around50-70 Å [149]. The concentration of columns in the sample can be controlled byvarying the irradiation dose, nd, which is usually measured in ions cm−2. Thefield at which the density of vortices is equal to nd is known as matching field,Bφ ≡ ndφ0, and it is an important parameter for characterizing irradiated samples.Civale et al. [150] showed that in samples with this kind of defects pinning is greatlyenhanced and the critical current density is thus increased. They also showed thatpinning is most effective when the magnetic field is aligned along the columnardefects. In this case the vortices are trapped within the columns in the Bose-glassphase. This was directly observed by Tonomura et al. [151] in Bi-2212 thin filmsusing Lorentz microscopy.

4.2.1 The effect of tilting the field

The vortex glass and the Bose glass respond very differently to a tilt of the appliedmagnetic field. In anisotropic materials the vortex-glass temperature Tg increases asthe angle θ between the applied magnetic field and the c-axis increases, see Fig. 4.5.For the Bose glass there is instead a cusp at θ = 0 1. The Bose glass is stableto small tilts, that is, the vortex lines remain locked within the columnar defectseven when the applied field is not parallel to the columns. This phenomenon isknown as transverse Meissner effect, since B remains parallel to the columns evenwhen the transverse component of the applied magnetic field H⊥ 6= 0. Smith et al.

[152] found experimental evidence of this effect from measurements of transversemagnetization. This so called lock-in phase is stable up to a critical tilt angleθL, which for YBCO is θL ≈ 2. A schematic illustration of this phase is shownin Fig. 4.6(a).

When the magnetic field is tilted with an angle larger than θL there is a kinkedvortex structure in the solid. Different segments of the vortex lines are trappedin different defects and connected by weakly pinned kinks, forming a staircasestructure. This is illustrated in Fig. 4.6(b). In this phase there is an increase of theresistivity in comparison to the lock-in phase, since the kinks can move along thecolumnar defects. The kinked phase is expected to persist up to the accommodationangle θa, which is determined by the competition between the line energy of thevortex, εl, and the pinning energy per unit length, UP , where

εl ≈ ε0γ−2 lnκ (4.9)

UP ≈ ε0(c0/2ξab)2 . (4.10)

Here ε0 is an energy scale ε0 = φ20/4πµ0λ

2ab, c0 is the radius of the columnar

defects and ξab and λab are the in-plane coherence length and penetration depth

1In this discussion it is assumed that the columnar defects are parallel to the crystallographicc-axis.

Figure 4.6. When the applied magnetic field is tilted with respect to the columns,three directions can be defined: parallel to H⊥ (x), perpendicular to H and thecolumns (y) and parallel to the columns (z). The columnar tracks are along the thecrystallographic c-axis. (a) For angles θ < θL the vortex lines are locked withinthe columns. (b) For angles θL < θ < θa vortices accommodate to the columnardefects by forming a staircase structure. (c) At angles θ > θa the kinks disappearand vortices take the direction parallel to the applied field.

respectively. Equation (4.10) is valid for c0 ≪√

2ξab. The accommodation angle isthus,

tan θa ≈(

≈ γc0

1√lnκ

(1 − T

, (4.11)

where the temperature dependence of the coherence length has been introduced,ξab(T ) = ξ0(1 − T/Tc)

−1/2, with Tc being the transition temperature at H = 0.This expression is valid for low applied magnetic fields, for which the interactionbetween vortices and columnar defects is more important than the vortex-vortexinteraction. The field and temperature dependence of θa were studied in detail byPomar et al. [153] from resistance measurements in BSCCO. Experimentally, θa

is usually defined as the angle at which the angular dependent resistivity in theliquid state has a maximum [134, 153, 154], as shown in Fig. 4.7. Finally, whenthe angle between the applied field and the columns is increased to values aboveθa, it is believed that the kinks disappear and the vortex lines take a direction thatis on average parallel to the applied field, with some deviations due to thermalfluctuations. The vortices intersect thus the columns over a length correspondingto the vortex core radius, see Fig. 4.6(c).

Vestergren et al. [156] applied the same scaling procedure used for both the vor-tex glass and the Bose glass to the transverse Meissner phase. Due to the presenceof a transverse applied field three inequivalent spatial directions can be defined:parallel to the transverse component of the field, x, perpendicular to the field andthe columns, y, and parallel to the direction of the columns, z. The geometry ofthis situation is shown in Fig. 4.6(a). The correlation length is expected to divergewith different exponents in each of these directions, ξy ∼ |T − TtM |−ν , ξx ∼ ξχ

and ξz ∼ ξζ . Here TtM is the temperature at which the transition between thetransverse Meissner phase and the liquid takes place. The scaling is thus expectedto be fully anisotropic. The predicted critical exponents were χ = 2 and ζ = 3, and

-80 -60 -40 -20 0 20 40 60 800

θ (o)

Figure 4.7. Angular dependent resistivity for the YBCO sample studied in Pa-per IV. The matching field was Bφ = 2 T and in this figure the resistivity wasmeasured with an applied magnetic field µ0H = 1 T at a temperature T = 90.2 K.For θ = 0 the vortices are locked within the columnar defects and there is a min-imum in the resistivity. Dotted lines show the value of the accommodation angleθa, defined as the maximum of ρ(θ).

from Monte Carlo simulations ν was expected to be ν = 0.68 ± 0.06. Performinga dimensional analysis similar to the analysis for the vortex and Bose glasses, thescaling ansatz Eiξiξ

z ∝ Ei(jξxξyξz/ξi) is found, with i = x, y, z. The resistivitywill diverge following a power law as TtM is approached, with different exponentsin each direction,

ρx ∼ ξ(2−z) ∼ |T − TtM |ν(z−2) (4.12)

ρy ∼ ξ(4−z) ∼ |T − TtM |ν(z−4) (4.13)

ρz ∼ ξ−z ∼ |T − TtM |νz. (4.14)

4.2.2 The effect of tilting the field: experimental results

In the transition from the transverse Meissner phase to the vortex liquid the re-sistivity was predicted to disappear with different critical exponents in the threedirections defined. The presence of columnar defects and a tilted applied field wasthought to make the three spatial directions nonequivalent, allowing for differentcritical properties in each direction. These symmetry breaking elements are alsopresent at higher angles, so fully anisotropic scaling might also be found in thissituation. This idea was tested experimentally in Paper IV, where the resistivityof an irradiated YBCO single crystal was measured in the x, y and z directions

described above. Measurements were made at several angles and fields, and thecombined critical exponents were obtained assuming the resistivity vanishes follow-ing a power law with a different exponent in each direction:

ρi = ρ0i

∣∣∣∣T

Ttr− 1

∣∣∣∣si

, (4.15)

Here Ttr is the temperature at which the solid-to-liquid transition takes place. Theexponents obtained as a function of field are shown in Fig. 4.8. Different exponentsare observed in each direction at fields comparable to the matching field, Bφ = 2 T.At higher fields the interaction between the vortices and the columns becomes lessrelevant compared to the vortex-vortex interactions and thus the columns are notas effective pinning centers. The exponents as a function of angle are shown inFig. 4.9. At angles up to 15 all si’s are approximately the same. It is importantto note that the transition to the liquid in this regime is not from a transverseMeissner phase. As mentioned above, the lock-in phase was stable only at anglessmaller than θL, which in YBCO is expected to be around 2. This small values ofθ were never reached in these experiments. So the solid in this region should consistof vortex segments pinned to different columns forming a staircase structure. Atangles above θ ≈ 25 a clear difference between the exponents is observed. Thisangle is consistent with the accommodation angle obtained experimentally from theangular dependence of the resistivity, see Fig. 4.7, which in turn is in agreement withwhat Eq. (4.11) predicts for the temperatures considered. A solid phase with nokinks and with vortices aligned approximately parallel to the applied magnetic fieldwould thus be expected here. The fact that fully anisotropic scaling is found in thisregime indicates, however, that the correlated nature of the disorder is still playinga fundamental role in the transition to the liquid state. A possibility is that theeffective accommodation angle is in fact larger than the experimentally defined θa.If the anisotropy of YBCO is taken into account, the effective transverse field shouldbe Heff

⊥ = H⊥/γ, according to the BGL scaling rules presented in Chapter 2. Theeffective angle would thus be smaller and the kinked regime would persist for highervalues of H⊥.

The idea that correlated disorder still plays and important role in the regimewhere the different exponents are observed is also supported by the shape of thephase diagram shown in Fig. 4.10, obtained by plotting the experimental transitiontemperatures. The anisotropy of YBCO was taken into account by using a mag-netic field scaled according to the rules proposed by Blatter et al. [57], described inSection 2.4.1. The transition temperatures corresponding to fields where no differ-ence in the exponents was observed were fitted to a vortex-glass line of the shapeBg ∝ (1 − T/Tc)

n. At fields comparable to the matching field the transition liesabove this glass line. This behavior is similar to that of the Bose-glass line shownin Fig. 3.9, thus indicating that the columns still act as correlated pinning centersabove the experimentally defined θa.

0 2 4 6 8 10 122

µ0H (T)

Figure 4.8. Power-law exponents si for the disappearance of resistivity in the solid-to-liquid transition as a function of applied magnetic field µ0H for the irradiatedYCBO single crystal studied in Paper IV. At fields comparable to the matchingfield, Bφ = 2 T different exponents are observed. Dotted lines indicate the averagevalues of si in this region.

0 15 30 45 60 75 902

θ (o)

µ0H = 2 T

Figure 4.9. Power-law exponents si for the disappearance of resistivity in thesolid-to-liquid transition as a function of angle θ for the irradiated YCBO singlecrystal studied in Paper IV. Dotted lines indicate the average values of si in theinterval in which different exponents are observed.

0.80 0.85 0.90 0.95 1.000

8 θ = 45o, same si

θ = 45o, different si

µ 0H = 2 T

s2 θ +

sin2 θ

)1/2 (

Hg (T)

Figure 4.10. Phase diagram for the sample studied in Paper IV. Filled symbols( ,•) correspond to measurements performed at θ = 45 and several fields, whileopen symbols () correspond to a constant applied field, µ0H = 2 T, and severalangles.

4.2.3 The effect of tilting the field: discussion

The same scaling procedure followed for the transition from the transverse Meissnerphase to the liquid can be used in this case. The correlation lengths are also assumedto diverge at different rates in each direction, ξy ∼ |T − Ttr|−ν , ξx ∼ ξχ and ξz ∼ ξζ .However the assumption of χ = 2 and ζ = 3 is valid only for the transverse Meissnertransition and should be dropped in this case. The correlation time again divergesas τ ∼ ξz. The electric field and current density should be considered in eachdirection, so Ei ∝ ∂Ai/∂t and ji ∝ ∂f/∂Ai, with Ai ∼ ξ−1

i and f ∼ (ξxξyξz)−1,

with i = x, y, z. The resistivity is ρi = ∂Ei/∂ji and follows

ρx ∼ |T − Ttr|ν(z−1+χ−ζ)

ρy ∼ |T − Ttr|ν(z+1−χ−ζ) (4.16)

ρz ∼ |T − Ttr|ν(z−1−χ+ζ) .

The values of ν, z, χ and ζ are unknown, but can be calculated from the experimen-tally obtained combined exponents and the fact that ζ − χ − 1 = 0. The resultingcritical exponents are different of those predicted for the transverse Meissner tran-sition, indicating that the transition is of a different nature.

The vortex solid-to-liquid transition in the cases studied above was continuous,with physical quantities like the correlation length or the resistivity diverging as

4.3. IN-PLANE MAGNETIC FIELDS 45

a power law as the transition temperature was approached. Phase transitions indifferent physical systems, not necessarily superconductors, might share the samecritical exponents. In that case it is said that they belong to the same universalityclass, that depends on features like dimensionality or symmetry and not on under-lying microscopic properties. Thus the vortex glass and the Bose glass belong todifferent universality classes. The presence of columnar defects in the latter case isa symmetry breaking element not present in the vortex glass. The columns intro-duce uniaxial anisotropy that leads to a different exponent for the divergence of thecorrelation length in the direction of the columns. A natural question is if there areany physical systems with different critical exponents in all three spatial directions.An extra symmetry breaking element should be introduced in the Bose-glass systemto obtain fully anisotropic scaling of physical quantities. The results presented hereindicate that a tilt in the applied magnetic field can be this element.

4.3 In-plane magnetic fields

Cuprates are strongly layered materials: superconducting CuO2 planes are sepa-rated by non-superconductive layers. The superconducting order parameter is thusstrongly modulated along the c-axis. As described in Section 3.2.3, depending onthe size of the coherence length along the c-axis compared to the separation betweenthe layers the system can be considered 3D continuous anisotropic or quasi-2D. Inthe latter case the discreteness due to the layers plays an important role. Due tothe modulation of the order parameter, when a magnetic field is applied parallelto the planes the free energy of the superconductor is minimized if the vortex linesare placed between the superconducting layers, where the order parameter is al-ready depressed. If the system can be considered quasi-2D the vortex cores will beconfined in the space between two layers, as shown schematically in Fig. 4.11. Thelayered structure acts thus as natural source of pinning [157]. Vortices will remainlocked within the planes even when the magnetic field is tilted away from the planes,as long as the tilt angle is smaller than a certain critical angle θc [158]. This lock-inprocess is somehow similar to the lock-in by columnar defects described in previoussections. At a larger tilt angle, the ground state can be described by a crossingof Josephson vortex lattice and the lattice of stacks of pancake vortices [159]. Forfields aligned with the planes, the vortices in the solid state form a triangular lat-tice. There are two stable possibilities for such a lattice, one just rotated by π/6with respect to the other [8]. Campbell et al. [160] showed that the equilibriumconfiguration is the one depicted in Fig. 4.11. The distance l between the vorticesin the direction normal to the planes in this configuration is given by:

(2φ0√3γB

, (4.17)

where γ is the mass anisotropy as defined in Eq. (2.10), φ0 is the flux quantumand B = µ0H is the applied magnetic field. The energy of the vortex lattice is

Figure 4.11. Vortex lattice in a layered superconductor in a strong magnetic fieldapplied parallel to the layers. In strongly anisotropic materials each vortex core,here represented by filled circles, will be confined between two superconductinglayers. In this example the lattice is commensurate with the layers, with a periodl = 2d along the c-axis. Arrows indicate the direction of the currents. Figure takenfrom Bulaevskii and Clem [161].

minimized when l is commensurate with the distance between the layers, d, thatis, when l = nd with n = 1, 2, 3, .... To accommodate the vortices in the non-superconducting layers, the distance l increases with the applied magnetic field.Transitions between different commensurate states could explain the oscillatorybehavior, with a period proportional to B−1/2, observed in the magnetic momentand the resistivity of YBCO single crystals with different oxygen contents [162–164].

An intermediate smectic phase between the solid and the liquid has been pro-posed in the phase diagram of high-temperature superconductors when magneticfields are applied parallel to the superconducting planes [165]. This phase would bedisordered within the planes and periodic in the direction perpendicular to them.Several experimental observations have been taken as evidence for the existence ofa smectic phase: the resistive tail in YBCO single crystals [166], the suppression ofthe step in the resistivity related to the first-order melting of the vortex lattice [167]or the presence of a peak in the temperature dependence of the amplitude of theoscillations in magnetization due to commensurability effects [163]. These effectsdisappear even for small misalignments of the magnetic field with respect to theplane, which indicates that the they are intimately related to the layered struc-ture of the superconductor. Signs of intrinsic pinning by the layers are the sharpmaximum in the critical current density [168] and the dip in the resistance [169]found for B ‖ ab. The angular dependence of the resistance close to B ‖ ab for anunderdoped YBCO single crystal is shown in Fig. 4.12.

When the current and the field are applied along the ab planes and perpendicularto each other, the Lorentz force of the vortices, FL = j × B, is directed along thec-axis of the sample. In that direction vortex motion is hampered by the presence ofthe superconducting layers, explaining the dip observed in the angular dependenceof the resistance. However, at high current densities Chaparala et al. [170] observed

88 89 90 91 920.00

θ (deg)

Figure 4.12. Angular dependent resistance in the region close to B ‖ ab of anunderdoped YBCO single crystal (Tc = 59 K). The data was taken at T = 53.7 Kwith an applied magnetic field µ0H = 8 T and a current I = 200 µA. When thefield is parallel to the planes, θ = 90, a dip related to the intrinsic pinning by thelayers is observed.

a maximum in R(θ) in several high-temperature superconductors, instead of theexpected dip. They characterized the peak by fitting the data to a background termof the form R ∼ |sin θ|α and a Lorentzian peak. The combination of both termscould explain the peak with a central dip that Iye et al. [171, 172] had observedbefore. The presence of the peak was explained as follows. Due to fluctuations, theJosephson vortices that are being pushed towards the superconducting layers bythe Lorentz force may form vortex-antivortex pairs in the planes. These pancakevortices will move in opposite directions, but their motion will not be restrainedby the presence of other pancake vortices, as for fields aligned exactly parallel tothe planes there are no pancake vortices in the layers. In this way dissipationis increased. High currents enhance the nucleation process, explaining why thisphenomenon was not observed at low currents. The same kind of peak is observedwhen the current is applied along the c-axis [173, 174], despite the fact that in thiscase the Lorentz force is directed parallel to the planes and the influence of thelayered structure should not be as important. Lundqvist et al. [175] studied thec-axis resistance of an underdoped YBCO single crystal and observed that the peakturned again into a dip as the temperature was lowered.

4.3.1 The solid-to-liquid transition for in-plane fields

The properties of the transition between the solid and the liquid when the magneticfield is applied exactly aligned with the superconducting layers depend strongly onthe strength of the applied field. As mentioned in Chapter 3, at a certain magnetic

Figure 4.13. Phase diagram proposed by Hu and Tachiki [176] from Monte Carlosimulations. Above the crossover field Bcr there is an intermediate 2D Kosterlitz-Thouless (KT) phase between the liquid and a three dimensional phase with longrange order (LRO). Below Bcr the transition would be first order. Figure takenfrom Hu and Tachiki [176]. Some features not relevant to the discussion here arenot displayed.

field, given by

Bcr =φ0

3γd2, (4.18)

a crossover between a 3D and a quasi-2D behavior of the system is expected. At thisfield the distance between vortices along the c-axis, l, is comparable to the separa-tion between the superconducting layers, d. From Monte Carlo simulations Hu andTachiki [176] suggested the phase diagram presented in Fig. 4.13. For fields belowBcr there is 3D long range order in the solid and the transition to the liquid at Tm

is expected to be a first-order melting. Above Bcr a 2D Kosterlitz-Thouless (KT)phase is expected between the 3D solid and the liquid at intermediate temperatures.In this region dissipation occurs through the motion of nucleated vortex-antivortexpairs in the superconducting layers. The transitions between the 3D solid and theKT phase (at Tx) and between the KT phase and the liquid (at TKT ) are bothexpected to be continuous.

Measurements of in-plane resistance at fields below the expected Bcr in YBCOwith different oxygen content revealed that the solid-to-liquid transition becomesalmost field independent at a certain temperature that decreases with increasedanisotropy [164, 175, 177]. In all cases oscillations are observed in the B(T ) curvedue to commensurability effects. These oscillations are schematically representedin the phase diagram of Fig. 4.13. Lundqvist et al. [175] suggested that the onset ofthe vertical transition could be related to a field induced decoupling of the planes,while Gordeev et al. [164] connected it to the smectic phase discussed above. Anidentical vertical transition was observed as well in the c-axis resistance of an un-

42 44 46 48 50 52 54 56 58 600.0

Figure 4.14. Normalized c-axis resistance, measured at a 2 µA current, of anunderdoped YBCO single crystal for magnetic fields (from right to left) µ0H =0, 2, 4, 6, 8, 10, 15, 18, 21, 25, 30 and 34 T accurately aligned parallel to the supercon-ducting planes.

derdoped YBCO single crystal [178]. This is a puzzling fact, as both the smecticand decoupling models are based on an effective intrinsic pinning of vortices by thelayered structure, which is not as relevant for c-axis currents.

4.3.2 The transition at high in-plane fields: experimental results

In the experiments mentioned above the magnetic fields considered were below theexpected crossover value Bcr. In an attempt to investigate the transition at fieldsabove Bcr the c-axis resistance of two underdoped single crystals of YBCO, withtransition temperatures Tc ≈ 60 K, was studied in Paper V (possible TRITA-FYS).The anisotropy parameter for this sample is expected to be γ ≈ 29, which givesa crossover field Bcr ≈ 15, according to Eq. (4.18). Some novel features wereobserved. One of them appears in the temperature dependence of the resistanceat fields accurately aligned with the superconducting planes, shown in Fig. 4.14.At the highest fields measured there is a sharp kink close to Tc. At lower fieldsthe kink is located further down in the transition, in agreement with previousmeasurements of c-axis resistance by Charalambous et al. [115]. They observed astep at a resistivity level ρn/12, below which the resistance was non-ohmic, andsuggested that it could be related to a melting transition. A similar step wasobserved by Lundqvist et al. [178] in a sample with the same characteristics as theones considered here. In Fig. 4.14, the transition between the high field and lowfield behavior is smooth. The resistance level at which the kink is located increases

30 35 40 45 50 55 600.0

1o from ab

B || ab

Figure 4.15. c-axis resistance, measured at a 2 µA current, of an underdopedYBCO single crystal at a magnetic field µ0H = 34 T. The sharp kink observed whenthe field is accurately aligned parallel to the superconducting planes disappearsalready at small misalignments, increasing the width of the transition.

gradually with magnetic field from R/Rn ≈ 0.1 at B = 4 T to R/Rn ≈ 1 at thehighest fields. The kink also becomes sharper as the magnetic field is increased.The resistance at temperatures below the kink is current dependent. The presenceof the kink should be in some way related to the layered structure of the material,as it disappears already at small misalignments from the orientation parallel to theplanes, making the transition broader, as shown in Fig. 4.15.

Another novel feature is found in the angular dependence of the resistivity insome regions of the phase diagram. At angles close to B ‖ ab two peaks form insidethe characteristic dip, giving rise to a triple dip feature, as shown in Fig. 4.16. Theresistance inside the dip was non-ohmic and the height of the peaks increased withthe applied current. Figure 4.17 shows the region in the phase diagram where thetriple dip was found.

88 89 90 91 920.00

θ (deg)

T = 46 K

Figure 4.16. Angular dependence of the c-axis resistance, measured with a 50 µAcurrent, of an underdoped YBCO single crystal at a temperature T = 46 K andmagnetic fields (from bottom to top) µ0H = 18, 19, 20, 21 and 23 T. Two peaks areobserved inside the dip at 19 T, that create a triple dip. The peaks increase inheight with magnetic field, but a certain field the side dips disappear.

25 30 35 40 45 50 550

T (K)Figure 4.17. Location of the triple dip feature in the phase diagram. Filled circlesshow the highest temperature at which the triple dip feature was found at eachmeasured field.

Chapter 5

Experimental overview

5.1 Sample preparation

In order to obtain meaningful results it is necessary to produce high quality sampleswith low impurity concentration. In this thesis mainly single crystals of YBCOwere studied. The growth, oxygenation and contact processes are described in thissection. In Paper II a Tl-2212 thin film was investigated among three YBCO singlecrystals. The film had previously been studied in Ref. [179] and therefore not grownspecifically for this work. The fabrication is described in Ref. [180].

5.1.1 Growth of YBCO single crystals

The YBCO single crystals employed in this work were grown using the flux flowmethod [181], in which the crystals grow in an eutectic melt of 28 at% BaO and72 at% CuO known as flux. A detailed review of the growth process can be found inLundqvist [182, 183]. In order to obtain high quality crystals, high purity (99.999%)dry powders of Y2O3, BaCO3 and CuO are carefully ground in an agar mortar.The choice of BaCO3 instead of BaO is based on the higher purity and stabilityagainst moisture of the former. The powder mixture is fixed to obtain 15 wt%YBCO and 85 wt% flux. Each 6 g batch contains 0.154 g Y2O3, 3.241 g CuOand 3.346 g BaCO3. Point disorder can be introduced by using less pure BaCO3

powders (99.x%) and by introducing small amounts of Fe2O3. Using 3 mg of Fe2O3

and reducing the amount of CuO by 4 mg, 1% of the Cu was substituted by Fe.The powders are then placed in an yttria stabilized zirconia (YSZ) crucible,

with a 19 mm inner diameter and a 29 mm height. YSZ is suitable for this processdue to its high melting point, that prevents any mixing of the crucible materialwith the melt. The melting process occurs in a tube furnace. First the BaCO3

is decalcinated to BaO at 900C. The temperature is then raised to 1000C toallow for the melting of the flux. After 10 hours, the crucible is displaced insidethe furnace in order to create a temperature gradient of approximately 1C/mmover the melt. The temperature is then slowly decreased. At some point the flux

54 CHAPTER 5. EXPERIMENTAL OVERVIEW

Figure 5.1. Critical temperature Tc as a function of oxygen deficiency δ forYBa2Cu2O7−δ (from Jorgensen et al. [48]). A two plateau behavior is observed.The error bars are related to the transition width.

moves to the colder side of the crucible, leaving behind the YBa2Cu2O7−δ crystalsthat have grown out of the melt. After cooling to room temperature the crystalscan easily be removed from the bottom of the crucible. Typical dimensions are0.5 × 0.5 × 0.02 mm, with the short distance along the c-axis.

5.1.2 Oxygenation

The superconducting properties of YBa2Cu2O7−δ depend on the oxygen contentand distribution. The relation between the critical temperature Tc and the oxy-gen deficiency δ has been studied in detail by Jorgensen et al. [48]. As shown inFig. 5.1, the Tc(δ) curve is characterized by two plateaus: one for 0 < δ < 0.15,corresponding to Tc around 90 K, and the other for 0.3 < δ < 0.5, correspondingto Tc around 60 K. Even though most of the oxygen is taken from the CuO chainswhen the oxygen content is reduced, there is also some redistribution and someoxygen can occupy empty sites in the chain plane. This causes a transition fromorthorhombic to tetragonal crystal structures at δ ≈ 0.65 and Tc goes to zero. If theoxygen content is further reduced, the system enters an antiferromagnetic phase, asshown in the electronic phase diagram in Fig. 5.2. Doping occurs through exchangebetween the superconducting planes and the chains and therefore, when the oxygencontent is changed in the chains, the doping of the superconductor also changes.The oxygen deficiency δ is also related to the anisotropy γ, as the removal of theoxygen from the chain sites decreases the coupling between the CuO2 layers causing

5.1. SAMPLE PREPARATION 55T

Hole doping (arb units)

Normal

Strangemetal

Figure 5.2. Schematic electronic phase diagram of a hole-doped high-Tc super-conductor.

an increase in γ. The relation between γ and δ has been studied for example byChien et al. [50] and Janossy et al. [184].

The crystals resulting from the growth process have an inhomogeneous oxygendistribution and are oxygen-deficient, which causes broad transitions and low Tc.In order to improve these properties, crystals need to be annealed. The final oxygendeficiency is determined by both annealing temperature and oxygen partial pressure[48, 185]. Optimally doped crystals, with Tc ≈ 92 K, are obtained by annealingin an oxygen atmosphere at temperatures around 400C-450C for 5 to 10 days.Long annealing times are required due to the slow diffusion along the c-axis andin order to minimize the concentration gradients, which is necessary to obtain anhomogeneous oxygen distribution. To obtain crystals with lower oxygen contentand therefore lower Tc, higher annealing temperatures or lower oxygen pressuresare required. Annealing in air gives a very stable oxygen partial pressure, but hightemperatures and the presence of CO2 might deteriorate the surfaces, and thereforeannealing times must be shorter. To avoid this problem an argon-oxygen mixturecan be used to reduce the oxygen partial pressure, which allows for lower annealingtemperatures. Table 5.1 summarizes the typical annealing conditions to obtainYBCO single crystals with different oxygen contents.

Around 95% of the samples obtained in every batch of optimally doped samplesare heavily twinned, that is, the a and b axes alternate. In underdoped crystalsuntwinned samples are very rare. The twin boundaries, that are along the <110>directions, can be observed with a polarized light microscope as dark lines, seeFig. 5.3. Twins act as sources of correlated pinning and therefore in the experimentswith columnar defects of Paper IV it was specially important to employ untwinnedsamples to assure that the observed effects are due to the columns.

Atmosphere P(O2)% T (C) Expected Tc (K) γO2 100 400 - 450 ≈ 92 7-8

Ar - O2 2.2 450 ≈ 80 ≈ 15Ar - O2 1 450 ≈ 70 ≈ 22Ar - O2 0.2 450 ≈ 60 ≈ 29

Air - 510 (3 days) ≈ 80 ≈ 15

Table 5.1. Typical annealing conditions to obtain YBCO single crystals withdifferent anisotropies. Typical annealing times are around 1 week.

5.1.3 Irradiation

The irradiation with heavy ions of the samples used in Paper IV was performed atGrand Accélérateur National d´Ions Lourds (GANIL), Caen, France. The incidentbeam was parallel to the crystallographic c-axis of the crystals and had an energyof 1 GeV. The Pb ions produce amorphous tracks with a diameter of around 7 nmthat extend through the whole thickness of the sample. The irradiation dose formost of the samples used in Paper IV was 1×1011 ions cm−2, which corresponds toa matching field Bφ = 2.0 T. Some test measurements were made also with samplesirradiated at half that dose, giving a matching field of Bφ = 1.0 T.

5.1.4 Contact preparation

Once the crystals have been annealed they are cut into a suitable shape. If thesurface deteriorates during annealing it is recommendable to polish the crystalwith thin sand paper. When the crystal is ready for contacting, it is placed on asapphire substrate. Electrical contacts are then painted on the crystal with silverepoxy (EPO-TEK H20E) or silver paste (DuPont 5504). Two contact configurationshave been employed in this study. For measurements of in-plane currents, currentcontacts are made by covering the short edges of the crystal with paint, providingan homogeneous current distribution. Potential contacts are made by applyingsmall strips or dots on the crystal and attaching gold wires. If silver paste isused, it is also possible to extend the fibers of paint from the pad of paint in thesubstrate onto the crystal surface, avoiding the use of gold wires. Examples ofsamples contacted in this way are shown in Fig. 5.5. For Paper I and the studies athigh in-plane fields, where the current was applied along the c-axis, the horse shoeshape configuration shown in Fig. 5.4 was used. This configuration provides a fairlyhomogeneous current distribution along the c-axis. To avoid any in-plane currentcomponents in the measurements of c-axis resistance the ideal contact configurationwould be to completely cover the ab surfaces of the crystal for the current contactsand measure voltages with contacts placed along the edge of the crystal. However,this requires thick crystals, which are rarely found. Fortunately, it has been shownthat the horse shoe shaped configuration gives the same results [178], which allowsfor the use of thinner crystals. Gold wires of diameter 12.5 µm are attached to thecontacts.

5.1. SAMPLE PREPARATION 57

Figure 5.3. Typical twinned YBa2Cu2O7−δ single crystal before contacting ob-served in polarized light. Black lines are twin boundaries, which appear due toalternation of a and b axes. Photo courtesy of Lundqvist [183].

Figure 5.4. Sketch of the contact configuration used for measurements with cur-rent along the c-axis.

Once the contacts are painted heat treatment is required to obtain low contactresistances. To avoid changes in the oxygen content, curing is made in the sameconditions of partial oxygen pressure and temperature as used for annealing. Curingtimes span from around a minute for air and high temperatures to a few hours for anO2 atmosphere and 400C if silver paste is used (epoxy requires somewhat shortertimes). Before curing in the furnace, the samples are heat treated for a few minuteson a plate at 150C to remove possible solvents. Finally, the crystals are coveredwith a protective paraffin layer. Detailed descriptions of the contact procedure fordifferent configurations can be found in Rydh [186] and Björnängen [187].

The method described above provides stable contacts with low contact resis-tances. For samples cured in flowing O2 or in an argon-oxygen mixture contactresistances below 1 Ω can be obtained. When curing in air at high temperatures,the deterioration of the surfaces makes the contact process more difficult and con-tact resistances are around 3 to 5 Ω. Low contact resistances are essential to avoidheating and noise.

Figure 5.5. Scanning electron microscope pictures of contact configurations usedfor in-plane resistivity measurements. The samples are YBa2Cu2O7−δ single crys-tals.

5.2. MEASUREMENTS 59

5.2 Measurements

5.2.1 Temperature control

The measurements for Papers I-IV were made in a flowing 4He gas cryostat fromOxford Instruments equipped with a NbTi/Nb3Sn 12 T superconducting magnet.The system, shown in Fig. 5.6, has been described in great detail by Andersson [188].The magnet is in a liquid He bath at 4.2 K during operation. For measurements attemperatures above 4.2 K the cooling takes place by pumping cold 4He gas fromthe He bath through the sample cell, where the sample holder is placed. Sampleholders are equipped with a heater, allowing temperature control between 4.2 Kand 300 K. All measurements for this work were performed in this regime. If theHe bath is pumped, the temperature can be decreased to the lambda point of 4He(Tλ = 2.17 K) and magnetic fields up to 13.5 T can be achieved. The measurementsfor the studies of in-plane magnetic fields were performed partly in a 3He cryostatwith a 8 T superconducting magnet at Argonne National Laboratory and partlyin a cryostat placed inside a 35 T resistive magnet at the National High MagneticField Laboratory in Tallahassee, FL.

The measurements for Papers I-IV were performed in the rotatable sampleholder schematically shown in Fig. 5.7. The sample holder can be rotated by360 a step motor and a gear drive, which allows for measurements at arbitraryangles between the magnetic field and the sample. An accuracy down to 0.01 canbe achieved, which makes it possible to align the field parallel to the ab-planes ifneeded. The sample holder has a carbon-glass thermometer and a heater windingfor temperature regulation. Carbon-glass thermometers are not extremely sensi-tive to magnetic fields, but still they should be field calibrated and correctionsshould be taken into account. Temperature errors up to 1 K can be found at hightemperatures and fields.

5.2.2 Instrumentation

The measurements in Papers I-IV were performed with a standard four-probe DCtechnique. The samples were mounted in the sample holder and connected by thincopper wires attached by electrically conducting silver paint. Samples were cooledin magnetic field and resistivity was measured with increasing temperatures. Thisprovides a better temperature control and therefore less noise. The temperaturewas controlled by a LakeShore 340 temperature regulator. The current sources,Keithley 220 or Keithley 2400, can provide currents up to a 100 mA. The typicalcurrents used in this work were 50 µA or 100 µA. To compensate for thermopowereffects the current was supplied alternating two opposite directions (+,-,+). Volt-age measurements were made with high quality multimeters (Hewlett Packard 3458,Schlumberger 7081). The voltage was preamplified by a picovoltmeter (EM Elec-tronics, model P13), obtaining a resolution of ∽1 nV.

to He recoverysystem

to pump

Cu thermal link

Radiation shields

Liquid He bath

Liquid N2 bath

Outer vacuum chamber

Inner vacuum chamber

Radiation shield

Superconducting magnet

Sample cell

He inlet tube

Needle valve

Figure 5.6. Schematic illustration of the flowing gas cryostat used in the measure-ments in Papers I-IV. Figure courtesy of Lundqvist [183].

5.3. DATA ANALYSIS 61

Carbon-glassthermometer

Heater windingWire drum

Worm gear

Sample position(rotatable)

Sample holder

Gearwheels

Sample cell

He gas inlet

Heater winding(for preheatingthe flowinghelium gas)

Figure 5.7. Schematic view of the sample cell in the rotatable sample holder.Figure courtesy of Lundqvist [183].

In the experiments at Argonne and Tallahassee an AC technique was used in-stead. The voltage response to the AC current sent to the sample, with frequencyof around 30 Hz, is measured with a lock-in amplifier.

5.3 Data analysis

The data in Papers I-III was analyzed within the extended model for the vortex-liquid resistivity described in Section 4.1. According to the model, when normalizedresistivity is plotted versus the modified temperature T (Tc−Tg)/Tg(Tc−T )−1, allcurves should scale into one. Several parameters enter the modified temperature.In this analysis, Tc was defined at every resistivity level as the temperature at whichthe zero-field curve has the same resistivity, in order to compensate for the width ofthe zero-field transition. The normal state resistivity ρn was used to normalize theresistivity. The definition of ρn does not affect the scaling result, and it was definedin each case depending on the available data. For the c-axis resistivity measuredin Paper I, ρn was defined as the maximum resistivity at each field. For the otherYBCO samples, for which the in-plane resistivity was measured, ρn was defined byextrapolating the resistivity data in the normal state, as shown in Fig. 5.8.

50 60 70 80 900

ρ/ρ 29

0 50 100 150 200 .

Figure 5.8. Definition of the parameter ρn. Left: c-axis resistivity of a slightlyunderdoped YBCO single crystal (Paper I). Right: in-plane resistivity of an un-derdoped YBCO single crystal (sample 2 in Paper II).

The extraction of the glass temperature Tg from the data is more complicated.One way is to calculate the inverse of the logarithmic derivative of the resistivity,

(∂ ln ρ

s(T − Tg)

Tc − T

Tc − Tg. (5.1)

The only difference between (5.1) and its equivalent in the traditional FFH scalingis a factor A(T ) = (Tc −T )/(Tc −Tg), that close to Tg is nearly one. Therefore thetraditional expression, (

∂ ln ρ

s(T − Tg) , (5.2)

can be used. This power-law expression is characteristic of the resistivity for anysecond-order phase transition, where the correlation length and time diverge aspower laws as the transition is approached. Therefore an expression similar toEq. (5.2) was also used to determine the vortex solid-to-liquid transition tempera-ture in the experiments on heavy-ion irradiated YBCO single crystals described inPaper IV. The low resistance data can be fitted to Eq. (5.2) to obtain the exponents of the power law of the resistivity. It is important to choose the fitting data asclose as possible to the transition and in a consistent manner. In our experimentsonly data below 5% of the normal state resistance was considered. However, thederivative introduces a large noise in the data and therefore the the scattering inthe value of s obtained in this way can be rather large. In Paper IV a direct fit toa power law was also performed, giving consistent results. The transition temper-ature can be extracted from by extrapolating (d ln ρ/dT )−1 to zero resistivity, asshown in Fig. 5.9.

5.3. DATA ANALYSIS 63

87.5 88.0 88.5 89.0 89.50.0

(dlnρ

sx = 4.5

θ = 35o

µ0H = 2 T

Figure 5.9. The vortex solid-to-liquid temperature Tc can be found by extrapo-lating (∂ lnρ/∂T )−1 to zero resistivity (dashed line). The combined exponent s canbe obtained as the inverse of the slope .

Chapter 6

Summary, conclusions and future

6.1 Scaling of the vortex-liquid resistivity in the extended

All the previous work based in the extended model of the glass transition had beenbased on measurements of in-plane resistivity [79, 140]. In Paper I, c-axis resistivityis investigated in a slightly underdoped (Tc ≈ 82 K) YBa2Cu2O7−δ single crystal.The resistive curves at all fields studied successfully scale into one curve whenplotted according to Eq.(4.4).

In the work presented by Andersson et al. [140] excellent scaling was found foroptimally doped YBCO single crystals. Some underdoped YBCO samples werealso studied, finding a somewhat worse scaling. The problem of scaling within theextended model for samples with higher anisotropy was considered in Paper II. Thein-plane resistivity of a Tl2Ba2CaCu2O8+δ thin film and four YBCO single crystalswith different oxygen contents was studied. The fact that good scaling was foundin all cases only at low fields, below a certain value B0, suggests that there could bea dimensional crossover from a regime of 3D to one of quasi-2D vortex fluctuationsat B0. However, the combined exponent s of the power law at which the resistivitydisappears in the vortex-glass model was approximately the same above and belowB0. The exponent s depends on the dimensionality of the system, and therefore achange in s would be expected at a dimensional crossover. Also against the idea ofa sharp crossover is the fact that the glass line in Eq. (4.6) describes fields belowand above B0. The transition into a quasi-2D situation should thus be gradual, assuggested by the fact that the deviations from the scaling become more importantwith increasing fields.

In Papers I and II and also in previous works [79, 140], the extended model ofthe glass transition was studied with fields applied along the c-axis. In Paper IIIthe angular dependence of the magnetic field is introduced into the model. The

66 CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK

in-plane resistivity of a slightly underdoped YBCO single crystal was studied atdifferent orientations with respect to the c-axis. Good agreement is found betweenthe Tg(θ) and B0(θ) extracted from the data and the expressions expected from theextended model when the scaling approach proposed by Blatter, Geshkenbein andLarkin (BGL) (described in Section 2.4.1) is introduced. All the transition curves,measured at several fields and orientations, scaled into one curve.

6.2 Anisotropic scaling in heavy-ion irradiated YBCO

Evidence for phase transition with a new type of critical scaling was found in Pa-per IV from measurements performed in heavy-ion irradiated YBCO single crystalswith magnetic field applied at a tilt angle with respect to the columns. The re-sistivity was measured in three orthogonal directions: parallel to the transversecomponent of the field (ρx), perpendicular to the transverse component of the fieldand the columns (ρy) and parallel to the columnar defects (ρz). The power-lawexponents at which the resistivity disappears in each direction were found to bedifferent in a certain angular range. Among all phase transitions known in physicsup to now, critical exponents are either the same in all directions, like in the vortex-glass transition, or different in one direction and the same in the plane perpendicularto it, like in the Bose-glass transition. This result thus suggests a completely newtype of critical behavior with critical exponents that are different in all three spatialdirections.

6.3 Measurements at high in-plane magnetic fields

Two novel features were observed in the c-axis resistance of underdoped YBCOsingle crystals at high magnetic fields applied parallel to the superconducting planes.At high fields there is a sharp kink in the resistance curves close to Tc, that makesthe superconducting transition very narrow. The kink disappears when the field ismisaligned with respect to the layers, indicating that the phenomenon is related tothe layered structure of YBCO. The second feature observed is the appearance of atriple dip in the angular dependence of the resistance close to B ‖ ab. This featureis only present in a certain region of the B − T diagram. So far explanations forthese effects are lacking.

6.4 Conclusions and future work

The goal with this thesis has been to try to gain some better understanding of theH − T phase diagram of high-temperature superconductors. The work has beencentered around the transition from the vortex liquid to the vortex solid phase.Some of the results obtained confirmed previously proposed models. In that way,Papers I-III provide more evidence to support the coherent description of the vortexliquid resistivity given by the extended model of the glass transition.

6.4. CONCLUSIONS AND FUTURE WORK 67

In contrast, Paper IV investigates a type of phase transition that, to the bestof my knowledge, had never been experimentally studied before. The results so farsuggest that there is anisotropic critical scaling, but there are many open questionsand experiments that could be carried out in the future to shed some light onthis problem. One of them is a systematic study of the I-V characteristics inall the three directions considered. Would it be possible to scale them in a waysimilar to the vortex or Bose glass scaling (see Fig. 3.7)? During the experimentsfor the work on this thesis a few I-V curves were measured, finding a somewhatodd behavior that should be further investigated. It would also be interestingto study the angular dependence of the transition temperatures at fields abovethe matching field. Would there be any effect when the effective magnetic field,i. e. the field rescaled by the anisotropy parameter, equals the matching field?Would the transition temperatures also lie above the melting line for non irradiatedsamples? Would the same results be obtained if the columns were not parallel tothe crystallographic c-axis of the sample? What happens if the defects are splayed?

The experiments on the transition at high in-plane fields also opened up nu-merous questions. The next logical step would be to repeat the experiments forin-plane resistances, to investigate if the new features observed here also appear inthat case. No conclusive results were obtained from our experiments regarding thephase diagram proposed in the calculations by Hu and Tachiki [176] (see Fig. 4.13).It is not clear how the intermediate Kosterlitz-Thouless phase at B > Bcr can beprobed. Besides, the prefactor in the expression for the crossover field Bcr is notwell established, and therefore it cannot be determined if the fields used in thisstudy were above or below this value. However, it would still be interesting toinvestigate how the kink feature observed here develops at higher fields. To studythe behavior of the resistance curves inside each of the three dips observed close toB ‖ ab could also provide some helpful information to elucidate the origin of thetriple dip in the angular dependent resistance.

Abbreviations

3D Three dimensional

2D Two dimensional

3D Three dimensionalBCS Bardeen–Cooper–Schrieffer superconductivity theory

BGL Blatter, Geshkenbein and Larkin scaling theory

BSCCO Bi2Sr2CaCu2O8+δ

FFH Fisher, Fisher and Huse glass scaling theory

GL Ginzburg-Landau

HTSC High Temperature Superconductor

SEM Scanning Electron Microscope

SQUID Superconducting QUantum Interference Device

TAFF Thermally assisted flux-flow

Tl-2212 Tl2Ba2CaCu2O8+δ

YBCO YBa2Cu2O7−δ

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84 BIBLIOGRAPHY

[186] A. Rydh, ‘Vortex properties from resistive transport measurements on extreme type-II superconductors’, Ph.D. thesis, TRITA-FYS 5275, Royal Institute of Technology,Stockholm, SE-100 44, Sweden (2002).

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high temperature superconductors’, Ph.D. thesis, TRITA-FYS 5172, Royal Insti-tute of Technology, Stockholm, SE-100 44, Sweden (1992).

Papers

Paper I

Scaling of the vortex-liquid c-axis resistivity in underdoped YBa2Cu2O7−δ

B. Espinosa-Arronte, Ö. Rapp, and M. Andersson.Physica C 408-410, 583 (2004).

Paper II

Scaling of the vortex-liquid resistivity in high-Tc superconductors

B. Espinosa-Arronte, and M. Andersson.Phys. Rev. B 71, 024507 (2005).

Paper III

Scaling of the B-dependent resistivity for different orientations in Fe dopedYBa2Cu2O7−δ

B. Espinosa-Arronte, M. Djupmyr, and M. Andersson.Physica C 423, 69 (2005).

Paper IV

Fully anisotropic superconducting transition in ion irradiatedYBa2Cu3O7−δ with a tilted magnetic field

B. Espinosa-Arronte, M. Andersson, C. J. van der Beek,M. Nikolaou, J. Lidmar, and M. Wallin.Manuscript submitted for publication.

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