Transport, magneto-transport and electron tunneling

studies on disordered superconductors

A Thesis

Submitted to the

Tata Institute of Fundamental Research, Mumbai

for the degree of Doctor of Philosophy

in Physics

by

Madhavi Chand

Department of Condensed Matter Physics and Materials Science Tata Institute of Fundamental Research

Mumbai August, 2012

for Appa

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Table of Contents Declaration ................................................................................................................................ 9

Preamble.................................................................................................................................. 11

Statement of joint work .......................................................................................................... 13

Acknowledgements ................................................................................................................. 15

List of Publications ................................................................................................................. 17

Glossary of Symbols and Abbreviations used in this thesis ................................................... 21

Synopsis ................................................................................................................................... 23

Chapter I: Introduction .......................................................................................................... 49

1.1 Fundamentals of superconductivity and the effects of disorder........................................ 50

1.1.1 Phenomenology of superconductivity ................................................................... 50

1.1.2 London equations and electrodynamics of a superconductor ................................. 51

1.1.2.1 London equations and the magnetic penetration depth ................................... 51

1.1.2.2 Pippards coherence length ............................................................................ 52

1.1.3 BCS: the microscopic theory for conventional superconductors ............................ 53

1.1.3.1 Elementary excitations over the ground state: BCS theory at finite temperatures ................................................................................................. 56

1.1.3.2 Strong coupling superconductors ................................................................... 57

1.1.3.3 Application of BCS theory to disordered superconductors: Andersons theorem .................................................................................................................... 58

1.1.4 The Ginzburg Landau theory of superconductivity ................................................ 59

1.1.4.1 GL theory and fluctuations ............................................................................ 61

1.1.4.1.1 Amplitude fluctuations........................................................................... 61

1.1.4.1.2 Thermal and quantum phase fluctuations ............................................... 63

1.2 Effects of disorder on normal state electronic properties ................................................. 65

1.2.1 Localization of wave functions ............................................................................. 65

1.2.1.1 Weak localization .......................................................................................... 65

1.2.1.2 Anderson localization .................................................................................... 66

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1.2.1.3 Scaling theory of localization........................................................................ 67

1.2.1.4 Localization and superconductivity ............................................................... 69

1.2.2 Coulomb repulsion ............................................................................................... 70

1.2.2.1 Coulomb repulsion and the normal state: resistivity, MR and Hall effect ...... 70

1.2.2.2 Coulomb repulsion and superconductivity .................................................... 71

1.3 Recent studies on disordered superconductors ................................................................ 72

1.3.1 Experimental results ............................................................................................. 72

1.3.2 Theoretical models and numerical studies ............................................................ 75

1.4 Our chosen system: 3-dimensional NbN thin films ......................................................... 79

1.4.1 NbN films: properties and fabrication................................................................... 79

1.4.2 Classification of disorder using the Ioffe Regel parameter kFl ............................... 80

References ........................................................................................................................... 82

Chapter 2: Experimental Details ........................................................................................... 91

2.1 Fabrication of NbN thin films by reactive DC magnetron sputtering .............................. 91

2.1.1 Thin films and epitaxial growth ............................................................................ 91

2.1.2 Sputtering: basic concepts .................................................................................... 91

2.1.3 Fabrication of NbN thin films using reactive DC magnetron sputtering ................ 94

2.2 Techniques used for structural characterization .............................................................. 96

2.2.1 X-ray diffraction .................................................................................................. 97

2.2.2 Four-circle goniometry ......................................................................................... 97

2.2.3 Cross sectional transmission electron microscopy ................................................ 98

2.3 Transport Measurements ...............................................................................................101

2.3.1 Sample patterns, physical measurements, wiring configurations and electrical contacts ...............................................................................................................101

2.3.2 Electronics used for transport measurements .......................................................102

2.3.3 Low temperature measurements ..........................................................................104

2.3.3.1 Basic cryogenic concepts .............................................................................104

2.3.3.2 Continuous flow cryostat .............................................................................105

2.3.3.3 4He VTI in 12T cryostat...............................................................................106

2.3.3.4 3He insert in 12T cryostat ............................................................................109

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2.3.3.5 3He cryostat with sample dipped in liquid .................................................... 110

2.4 Electron tunneling ........................................................................................................ 111

2.4.1 Principle of tunneling measurements ................................................................... 111

2.4.2 Devices used for tunneling measurements ........................................................... 115

2.4.2 Optimization of tunnel junction devices .............................................................. 117

References .......................................................................................................................... 119

Chapter 3: Synthesis and characterization of superconducting NbN thin films ................ 123

3.1 Synthesis and optimization of NbN films ...................................................................... 123

3.1.1 Measurement of critical temperature and resistivity ............................................ 123

3.1.2 Structural characterization of the NbN films ....................................................... 125

3.1.2.1 Characterization by X-ray diffraction .......................................................... 125

3.1.2.2 -scans using 4-circle goniometer................................................................ 126

3.1.2.3 Cross sectional transmission electron microscopy........................................ 126

3.1.3 Analysis of transport and X-ray data to understand the roles of different sputtering parameters and establish a method to control disorder ......................................... 127

3.2 Transport characterization of 3D disordered NbN thin films ......................................... 129

3.2.1 Measurement of critical temperature, resistivity and carrier density; and the calculation of kFl ................................................................................................. 129

3.2.2 Correlation between measured physical quantities .............................................. 134

3.2.3 Measurement of Hc2 and GL ............................................................................... 134

3.2.4 A note on errors in the transport characterization ................................................ 138

3.3 Summary ...................................................................................................................... 139

References .......................................................................................................................... 139

Chapter 4: Normal state properties of disordered NbN films ............................................. 141

4.1 Introduction .................................................................................................................. 141

4.2 Theoretical background ................................................................................................ 142

4.3 Results .......................................................................................................................... 144

4.4 Discussion .................................................................................................................... 147

4.4.1 Understanding the temperature dependence of resistivity .................................... 147

4.4.1.1 Role of electron-phonon interactions ........................................................... 147

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4.4.1.2 Are the samples insulating? .........................................................................148

4.4.1.3 Details of (T) in the metallic regime ..........................................................150

4.4.2 Analysis of the temperature dependence of the Hall coefficient ...........................152

4.5 Summary ......................................................................................................................156

References ..........................................................................................................................156

Chapter 5: Superconducting properties and phase diagram of strongly disordered NbN films close to the metal-insulator transition .........................................................................159

5.1 Introduction ..................................................................................................................159

5.2 Evolution of superconducting properties with disorder: emergence of the pseudogap state ....................................................................................................................................160

5.2.1 Planar tunnel junction measurements...................................................................160

5.2.2 Results from scanning tunneling spectroscopy.....................................................165

5.2.3 Magnetoresistance studies on disordered films ....................................................166

5.3 Establishing the phase diagram .....................................................................................170

5.4 Discussion of the phase diagram ...................................................................................170

5.4.1 Region I ..............................................................................................................171

5.4.2 Region II .............................................................................................................172

5.4.3. Region III...........................................................................................................177

5.5 Quantitative analysis of the tunneling spectra ................................................................179

5.6 Summary ......................................................................................................................183

Appendix 5A: Dependence of Tc on the Coulomb pseudopotential and reduced density of states at Fermi level .....................................................................................................184

Appendix 5B: Quantitative analysis of the decay of MR in region I of the phase diagram ...186

References ..........................................................................................................................188

Conclusions, open questions and future directions ..............................................................193

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Declaration

This thesis is a presentation of my original research work. Wherever contributions of

others are involved, every effort is made to indicate this clearly, with due reference to the

literature, and acknowledgement of collaborative research and discussions.

The work was done under the guidance of Professor Pratap Raychaudhuri, at the Tata

Institute of Fundamental Research, Mumbai.

Madhavi Chand

In my capacity as supervisor of the candidates thesis, I certify that the above statements

are true to the best of my knowledge.

Prof. Pratap Raychaudhuri

Date:

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Preamble

The work presented in my doctoral thesis is an experimental investigation of the

electronic and superconducting properties of disordered superconductors through transport,

magneto-transport, Hall effect and electron tunneling measurements on niobium nitride thin

films.

The effect of disorder on superconductivity has been studied on a variety of systems for

many decades. However, the unprecedented technological development of novel experimental

probes at low temperatures over the past decade has made it possible to access physical

quantities in disordered superconductors, which were not easily accessible before. As I will show

in this thesis, investigation of disordered superconductors using these probes enable us to verify

the validity of a number of theoretical predictions made for systems close to the critical disorder

for destruction of the superconducting state. The reliability of such an experimental scheme

depends on two factors: (i) The ability to synthesize high quality samples with controlled

amounts of disorder and (ii) obtain information from complementary measurements on the same

samples. Towards this goal, in Prof. Pratap Raychaudhuris lab, our strategy was to use a

combination of scanning tunneling spectroscopy, Giaever tunneling spectroscopy, penetration

depth measurement and conventional magneto-transport and Hall effect studies on NbN thin

films with controlled amounts of structural disorder in the form of Nb vacancies in the crystalline

lattice. The experiments have been carried out collectively by four graduate students each

concentrating on a different specialized technique: Mintu Mondal (low frequency electrodynamic

measurements), Anand Kamlapure and Garima Saraswat (low temperature scanning tunneling

spectroscopy) and myself (magneto-transport, Hall effect and Giaever tunneling measurements).

While I will concentrate primarily on the last set of measurements in this thesis, I will use the

insight obtained from these complementary measurements to establish physical points as and

when required.

The thesis is organized as follows: In Chapter 1, I provide a theoretical background of

superconductivity and disordered materials, introduce the field of disordered superconductors

and motivate our experiments. I will also introduce our chosen system of study: homogenously

disordered 3D NbN thin films and the Ioffe-Regel parameter which is used to classify the

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samples. Chapter 2 contains the experimental details of the fabrication of samples by magnetron

sputtering and the measurement techniques including the cryogenics and electronics involved.

This chapter also explains the use of electron tunneling as a tool to study the spectral properties

of superconducting materials. Chapter 3 deals with the standardization, characterization and

classification of sputtered NbN films. In this chapter I discuss properties of the system like

transition temperature, normal state resistivity, number density and upper critical field. Chapter 4

contains the results of Hall effect studies and investigates the unusual normal-state properties of

NbN. In Chapter 5, I will use tunneling and magnetoresistance data to establish a

phenomenological phase diagram for NbN and demarcate three different regimes of disorder. I

will then discuss the physical phenomena that govern the superconducting properties in each of

these regimes.

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Statement of joint work

The experiments reported in this thesis have been carried out in the Department of

Condensed Matter Physics and Materials Science at the Tata Institute of Fundamental Research

under the supervision of Prof. Pratap Raychaudhuri. The results of the major portions of the

work presented in this thesis have already been published in refereed journals.

Most of the experiments discussed in this thesis have been carried out by me. For the sake of

clarity and completeness, I have included some of the data and analyses of experiments

performed by others. These experiments are listed in this statement.

The fabrication and characterization of NbN films, and tunnel junction devices as well as

transport, magneto-transport, Hall effect and planar tunneling measurements, which form the

major part of this thesis, were carried out primarily by me. Some of the above synthesis and

characterization were done jointly with John Jesudasan and Vivas Bagwe and some of the

measurements were performed jointly with Dr. S. P. Chockalingam. All the scanning tunneling

spectroscopy data shown in this thesis were performed by Anand Kamlapure and Garima

Saraswat. Penetration depth measurements, which I have occasionally referred to, were carried

out by Mintu Mondal and Dr. Sanjeev Kumar. Measurements on Be films were carried out in the

group of Prof. P. W. Adams by Dr. Y. M. Xiong at Louisiana State University and the analysis

based on their data was performed by me. Theoretical support for all the analysis presented here

was obtained from Prof. Vikram Tripathi in the Department of Theoretical Physics and Dr. Lara

Benfatto at Sapienza University, Rome, Italy.

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Acknowledgements

Pursuing a PhD at TIFR has been a very rewarding experience and at the same time an

extremely challenging one. I have learnt many valuable lessons while working in this beautiful

institution; I have broadened my horizon and deepened my understanding of physics. To reach

here and complete this degree, I have leaned heavily on the help and support of many people and

I would like to take this opportunity to express my gratitude.

I would like to begin by thanking my thesis advisor Prof. Pratap Raychaudhuri from

whom I have learnt a lot about solid state physics and superconductivity, low temperature

physics and experimental techniques. More importantly, I am grateful for the valuable lessons in

hard work, dedication, perseverance and perfectionism that he always taught by example. Most

of all, I would like to thank him for the faith he had in my abilities as a scientist and for showing

tremendous patience and tolerance as a guide.

I am extremely grateful to my lab-mates John Jesudasan, Vivas Bagwe, S. P.

Chockalingam, Mintu Mondal, Anand Kamlapure, Garima Saraswat, Archana Mishra, Sanjeev

Kumar and Charudatta Galande for all their help in the lab, especially for the innumerable

helium transfers and taking care of the experimental systems when I was not around. I would

also like to thank them for their company and moral support and for making the lab a home away

from home.

I am especially thankful to Subhash Pai and Atul Raut for their assistance with the

fabrication of equipment used in the various experiments. I am also grateful to Bhagyashree

Chalke, Rudheer Bapat, Shashank Purandhare, Nilesh Kulkarni, V. M Chopde, Abdul Kadir,

Masihhur Lashkar and Mahesh Kulkarni for their help with the structural characterization of the

samples discussed here i.e, the SEM, TEM, XRD and 4-circle measurements. I would like to

thank the entire staff of the low temperature facility (LTF) for providing our laboratory with

huge quantities of liquid helium, sometimes even at odd times and on weekends.

I would like to express my gratitude to Prof. Vikram Tripathi for the many explanations

and discussions. I am also grateful to Dr. Lara Benfatto, Dr. Sudhansu Mandal, Prof. Nandini

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Trivedi, Prof. Mohit Randeria, Prof. Mikhail Feigelman and Prof. T. V. Ramakrishnan for useful

and enlightening discussions.

I am grateful to my teachers, especially Dr. Bikram Phookun, colleagues and students at

St. Stephens College who have provided inspiration for me to pursue my studies in physics.

I am extremely grateful to my family who have been extremely supportive and

understanding during these past five years. A special thanks is due to the folks in Bombay whose

homes and help were never more than a phone call away.

I owe a lot of what I am to my parents who encouraged me to be myself and not follow

the crowd. My mother taught me do my level best in anything I undertake, and ensured that Ive

always had the space to follow my dreams. My father, who was an academic at heart, inspired

me to study science and be a thinking human being. This thesis is dedicated to him.

In the end I would like to attempt the impossible; to express my gratitude to my husband

and best friend for always being there, for his reassurance and encouragement and for boosting

my self-confidence whenever it was low. Without his undying love and support, this thesis

would never have come into being.

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List of Publications

In refereed journals, directly related to the material presented here:

Superconducting properties and Hall effect of epitaxial NbN thin films: S. P.

Chockalingam, Madhavi Chand, John Jesudasan, Vikram Tripathi and Pratap

Raychaudhuri, Physical Review B 77 214503 (2008)

Tunneling studies in a homogenously disordered s-wave superconductor: NbN: S. P.

Chockalingam, Madhavi Chand, Anand Kamlapure, John Jesudasan, Archana Mishra,

Vikram Tripathi and Pratap Raychaudhuri, Physical Review B 79 094509 (2009)

Temperature dependence of resistivity and Hall-coefficient in a strongly disordered

metal: NbN: Madhavi Chand, Archana Mishra, Y. M. Xiong, Anand Kamlapure, S. P.

Chockalingam, John Jesudasan, Vivas Bagwe, Mintu Mondal, P. W. Adams, Vikram

Tripathi and Pratap Raychaudhuri, Physical Review B 80 134514 (2009)

Phase diagram and upper critical field of homogeneously disordered epitaxial 3-

dimensional NbN films: Mintu Mondal, Madhavi Chand, Anand Kamlapure, John

Jesudasan, Vivas C. Bagwe, Sanjeev Kumar, Garima Saraswat, Vikram Tripathi and

Pratap Raychaudhuri, Journal of Superconductivity and Novel Magnetism 24 341 (2011)

Phase fluctuations in a strongly disordered s-wave NbN superconductor close to the

metal-insulator transition: Mintu Mondal, Anand Kamlapure, Madhavi Chand, Garima

Saraswat, Sanjeev Kumar, John Jesudasan, Lara Benfatto, Vikram Tripathi and Pratap

Raychaudhuri, Physical Review Letters 106 047001 (2011)

Phase diagram of the strongly disordered s-wave superconductor, NbN, close to the

metal-insulator transition: Madhavi Chand, Garima Saraswat, Anand Kamlapure, Mintu

Mondal, Sanjeev Kumar, John Jesudasan, Vivas Bagwe, Lara Benfatto, Vikram Tripathi

and Pratap Raychaudhuri, Physical Review B 85 014508 (2012)

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In refereed journals, not directly related to the material presented here:

Measurement of magnetic penetration depth and superconducting energy gap in very thin

epitaxial NbN films: Anand Kamlapure, Mintu Mondal, Madhavi Chand, Archana

Mishra, John Jesudasan, Vivas Bagwe, Lara Benfatto, Vikram Tripathi and Pratap

Raychaudhuri, Applied Physics Letters 96 072509 (2010)

Role of the vortex-core energy on the Beresinkii-Kosterlitz-Thouless

transition in thin films of NbN: Mintu Mondal, Sanjeev Kumar, Madhavi Chand, Anand

Kamlapure, Garima Saraswat, Goetz Seibold, Lara Benfatto and Pratap Raychaudhuri,

Physical Review Letters 107 217003 (2011)

As conference proceedings:

Evolution of superconducting properties with disorder in epitaxial NbN films: S. P.

Chockalingam, Madhavi Chand, John Jesudasan, Vikram Tripathi, and Pratap

Raychaudhuri, Journal of Physics: Conference Series 150 052035 (2009)

Fabrication and characterization of epitaxial NbN thin films and tunnel junctions, John

Jesudasan, S. P. Chockalingam, Madhavi Chand, Vivas Bagwe, S. R. Barman and Pratap

Raychaudhuri, Proceedings of the 53rd DAE Solid State Physics Symposium (2008)

Effect of point disorder on superconducting properties of ultrathin epitaxial NbN film,

John Jesudasan,Vivas Bagwe, Mintu Mondal, Madhavi Chand, Archana Mishra, Anand

Kamlapure, S. P. Pai, Pratap Raychaudhuri, Proceedings of the 54th DAE Solid State

Physics Symposium (2009)

Effect of Phase Fluctuations on the Superconducting Properties of Strongly Disordered

3D NbN Thin Films: Madhavi Chand, Mintu Mondal, Anand Kamlapure, Garima

Saraswat, Archana Mishra, John Jesudasan, Vivas C. Bagwe, Sanjeev Kumar, Vikram

Tripathi, Lara Benfatto, and Pratap Raychaudhuri, Journal of Physics: Conference Series

273 012071 (2011)

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Study of Pseudogap State in NbN using Scanning Tunneling Spectroscopy; Madhavi

Chand, Anand Kamlapure, Garima Saraswat, Sanjeev Kumar, John Jesudasan, Mintu

Mondal, Vivas C. Bagwe, Vikram Tripathi and Pratap Raychaudhuri, Proceedings of the

55th DAE Solid State Physics Symposium (2010)

Upper critical field and coherence length of homogenously disordered epitaxial 3-

dimensional NbN films; John Jesudasan, Mintu Mondal, Madhavi Chand, Anand

Kamlapure, Vivas C. Bagwe, Sanjeev Kumar, Garima Saraswat, Vikram Tripathi and

Pratap Raychaudhuri, Proceedings of the 55th DAE Solid State Physics Symposium

(2010)

Phase diagram of a strongly disordered s-wave superconductor, NbN: Madhavi Chand;

to appear in Physics Teacher

Evolution of Kosterlitz-Thouless-Berezinskii (BKT) Transition in Ultra-Thin NbN Films;

Mintu Mondal, Sanjeev Kumar, Madhavi Chand, Anand Kamlapure,Garima Saraswat,

Vivas C. Bagwe, John Jesudasan, Lara Benfatto and Pratap Raychaudhuri, to appear in

Journal of Physics: Conference Series

Effect of disorder and phase diagram of s-wave superconductor, NbN; Anand

Kamlapure, Garima Saraswat, Madhavi Chand, Mintu Mondal, Sanjeev Kumar, John

Jesudasan, Vivas Bagwe, Lara Benfatto, Vikram Tripathi and Pratap Raychaudhuri, to

appear in Journal of Physics: Conference Series

Magnetoresistance studies of homogenously disordered 3-dimensional NbN thin films,

Madhavi Chand, Mintu Mondal, Sanjeev Kumar, Anand Kamlapure, Garima Saraswat, S.

P. Chockalingam, John Jesudasan, Vivas Bagwe, Vikram Tripathi, Lara Benfatto and

Pratap Raychaudhuri, to appear in Journal of Physics: Conference Series

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Glossary of Symbols and Abbreviations used in this thesis

SYMBOLS a lattice constant or characteristic length scale of phase fluctuations e electronic charge Ec mobility edge EF Fermi energy G conductance

=h/2 h is Planck's constant Hc2 upper critical field Hpeak position of MR peak J superfluid stiffness kB Boltzmann constant kF Fermi wave-number kFl Ioffe Regel parameter l mean free path me mass of electron n number density/ electronic carrier density nn number of electrons that remain normal ns superfluid density N(0) density of states at Fermi level R resistance RH Hall coefficient Rsq sheet resistance for a 2-dimensional system t sample thickness T temperature Tc superconducting critical temperature vF Fermi velocity VH Hall voltage

Coulomb pseudopotential coherence length 0 Pippard Coherence length BCS BCS Coherence length GL Ginzburg Landau coherence length resistivity m maximum/peak resistivity n normal state resistivity peak peak resistivity

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electron scattering time penetration depth 0 flux quantum conductivity 0 minimum conductivity D Debye temperature

superconducting energy gap ABBREVIATIONS 2D two dimensions 3D three dimensions AA Altshuler and Aronov AL Aslamazov and Larkin BCS Bardeen, Cooper and Schreiffer DOS density of states GL Ginzburg Landau IVC inner vacuum chamber MIT metal-insulator transition MR magnetoresistance MT Maki and Thompson PID proportional-integral-derivative SIT superconductor-insulator transition STM scanning tunneling microscope STS scanning tunneling spectroscopy TEM transmission electron microscopy VTI variable temperature insert XRD X-ray Diffraction

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Synopsis

I. Introduction

The effect of disorder on superconducting properties has been a topic of active research

for many years. The earliest understanding was due to Andersons theorem1 which predicts that

the superconducting transition temperature (Tc) of a superconductor is not affected by disorder

scattering as long as the system remains a metal. This prediction is based on the spherical

symmetry of the superconducting order parameter as suggested by the Bardeen-Cooper-

Schrieffer (BCS) theory of superconductivity2.

However, it was later realized that this theorem does not hold true in strongly disordered

systems. The Tc does decrease with increasing disorder and sufficiently strong disorder can in

fact suppress superconductivity altogether. Since this disorder driven transition from a

superconducting ground state to a non-superconducting ground state occurs at zero temperature it

is a quantum phase transition (QPT). In 3-dimensional (3D) systems, the transition could be to a

metal or an insulator, while in 2-dimensional (2D) systems, it is to an insulating state as there is

no true metallic state in 2D (Ref. 3). Therefore, the transition is usually referred to in the

literature as the superconductor-insulator transition4 (SIT). In actual experiments, a number of

different parameters can be adjusted to tune the effective disorder. These include atomic scale

disorder in the form of defects and vacancies, granularity, and the effective increase in disorder

by reducing film thickness in 2D samples. A similar QPT can also be observed by the application

of magnetic field.

This field has received renewed attention in recent times because of the observation of

novel phenomena in many strongly disordered superconductors, especially in the vicinity of the

critical disorder for destruction of superconductivity. These include the spontaneous formation of

superconducting islands in physically homogenous films of TiN (Ref. 5) and NbN (Ref. 6),

observation of large magnetoresistance (MR) peaks in thin films of InOx (Ref. 7), TiN (Ref. 8),

amorphous Bi (Ref. 9 ) and NbN (Ref. 6), evidence of Cooper pairing via magnetic flux

quantization in a-Bi (Ref. 10) samples with a non-superconducting ground state, the persistence

of superfluid stiffness11 at temperatures above the mean field transition temperature and the

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existence of a pseudogap in the density of states above Tc (Ref. 12,13,14), similar to that seen

in the High Tc cuprates.

Many of these phenomena are also manifest in numerical studies. A homogenously

disordered 2D system without any physical granularity is seen to spontaneously form domains of

superconducting and non-superconducting regions15,16. This is accompanied by a decoupling of

the energy gap and superfluid stiffness suggesting the existence of a pseudogap state and

persistence of superconducting correlations on the insulating side17 of the SIT as well as a peak18

in the MR.

Understanding these phenomena is a challenging problem because they arise due to a

competition of different many-body effects namely Cooper pairing, Coulomb repulsion and

localization of wave functions19. In addition they may also be governed by universal scaling laws

arising from proximity to a quantum critical point20. It requires us to go beyond mean field

approaches like the BCS theory and perturbative corrections to the same; and look towards

radically different ideas like fluctuations in the phase of the order parameter21, localization of

Cooper pairs22,23 and condensation of vortices24 in the presence of strong disorder.

The system that we have used for this study is a set of 3D homogenously disordered NbN

thin films with different levels of disorder. NbN is a conventional s-wave superconductor with a

relatively high Tc of ~17K. Epitaxial thin films of this material can be grown by reactive DC

magnetron sputtering on nearly lattice matched (~5% mismatch) [1 0 0]-oriented MgO

substrates25. Obtaining films with Tc of 16-17K requires very specific deposition parameters.

Deviations from these optimal conditions results in samples with suppressed Tc and varying

disorder. Thus by adjusting the deposition parameters, we were able to tune the disorder to

obtain samples ranging from Tc~16.8K all the way down to samples in which no

superconducting transition is seen even at 300mK. In addition, these samples are chemically

stable, have good mechanical strength26 and can be thermally recycled from cryogenic to room

temperature without degradation of their superconducting properties.

Most studies of disordered superconductors have concentrated on the 2D limit, with the

tuning parameter being thickness, granularity or magnetic field. Our emphasis has been on the

study of 3D superconducting films as a function of disorder. Our films are homogenously

25

disordered, containing defects on the atomic scale, and have thicknesses t 50nm, which is much

larger than the dirty limit coherence length (2.5dd7nm)27 thereby ensuring that they are in the

3D regime as far as superconducting correlations are concerned.

We have carried out transport, magneto-transport, Hall effect and electron tunneling

experiments to study the evolution of normal state resistivity, number density, density of states

(DOS) and MR. We find that the system evolves from a conventional BCS superconductor to

one with an unusual normal state, a pseudogap above Tc, a superconducting transition governed

by phase fluctuations and a peak in the MR for the samples without a superconducting transition.

I will present these results and use them to establish a phenomenological phase diagram and

highlight the different physical interactions that govern the normal and superconducting states at

varying degrees of disorder.

II. Experimental Details

1. Fabrication and structural characterization of NbN thin films

The films were grown by reactive DC magnetron sputtering a Nb target in an Ar/N2 gas

mixture onto [1 0 0] oriented single crystal MgO substrates. Deposition parameters like substrate

temperature and sputtering gas pressure were kept constant at 600C and 5 mTorr respectively.

The disorder was controlled by varying either one or both of the sputtering power (between 15W

and 250W) and Ar:N2 ratio in the gas mixture (between 90:10 and 30:70). Both decreasing the

sputtering power and increasing the partial pressure of nitrogen reduce the Nb concentration in

the sputtering plasma, thus creating Nb vacancies in the deposited film. It is these vacancies that

form the atomic scale disorder in the system. The time of deposition was adjusted to ensure that

the thickness of each sample was t50nm, which is much larger than 2.57 nm. Therefore

these films are effectively in the 3D limit as far as superconducting correlations are concerned.

The structural properties of the samples were examined through X-ray diffraction -2 scans

using the Cu-K line, -scans using a 4-circle goniometer and cross sectional transmission

electron microscopy (TEM).

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2. Low temperature transport measurements

Resistance (R), MR and Hall effect measurements were carried out using standard AC

and DC four-probe techniques from 285K down to 300mK using a combination of 4He and 3He

cryogenic techniques. The 4He measurements were done in a continuous flow cryostat with the

ability to go down to 2.2K and a home-built conventional 4He variable temperature insert (VTI)

in a 12T magnet operating down to 1.7K. The latter was specially designed to have a stability

better than 10 mK in the temperature range 1.7K - 150K and better than 100 mK in the

temperature range 150K - 285K. This level of stability was essential to obtain reliable Hall effect

data (described later). The 3He measurements were done on a custom built 3He insert in the 12T

cryostat as well as a separate sample-in-liquid 3He cryostat, with a lowest temperature of 350mK

and magnetic field up to 5T.

The AC measurements were carried out using a two-lock-in circuit which allows

independent measurement of the excitation current and induced voltage. For the Hall and MR

measurements, we used the offset and expand functions in the voltage measurement to detect

small changes in voltage over a large background.

The Hall coefficient (RH) was calculated by sweeping the magnetic field (H) from 12T

to -12T, thus subtracting out the resistive contribution. The upper critical field (Hc2) for several

samples was measured from either R-T scans at different H or R-H scans at different T. In all

these measurements, H was perpendicular to the plane of the film.

3. Electron tunneling

Quantum mechanical tunneling provides a useful tool for studying the spectral properties

of superconductors28 . When an electron tunnels across a thin insulating barrier, its energy

remains unchanged and it can therefore be used to carry spectroscopic information. We have

studied normal metal-insulator-superconductor (NIS) tunneling which is used to measure the

single particle density of states for the following reason: In a typical superconductor, the

superconducting energy gap feature extends over a few milli-electron-volts around the Fermi

energy, whereas the Fermi energy for a typical metal is in electron-volts. Therefore, in the

vicinity of EF, the DOS of the normal metal can be considered constant. At each value of bias

27

voltage, the tunneling current will depend on Ns(E) and Nn(E) i.e. the densities of states of the

superconducting and normal metal electrodes respectively as well as the Fermi-Dirac distribution

function f(E). The differential conductance G(V) which is proportional to the superconducting

DOS is obtained from the tunneling equation:

( ) ( ) ( ) ( ) ( ){ }

dEeVEfEfeVENENdV

d

dV

dIVG

ns

V

The tunneling experiments were carried out using NbN/native oxide/Ag planar tunnel

junctions down to 300mK. The geometry of the tunnel junction devices was designed so that

current (I)-voltage (V) and R-T could both be measured on the same sample without having to

take it out of the cryostat29. The I-V curves were then differentiated numerically to obtain the

differential conductance.

III. Characterization of NbN Samples

In order to classify the NbN samples and have a quantitative characterization of disorder,

we use the Ioffe Regel parameter kFl (which is a measure of the mean free path in units of de-

Broglie wavelength). This quantity decreases with increasing disorder. At kFl ~1, the Bloch state

is completely destroyed 30 and the system becomes insulating. The kFl values are extracted

experimentally from RH and resistivity () measurements and calculated from the free electron

formula: ( ) ( )[ ]{ } ( )[ ]3/53/13/22 2852853 eKKRlk HF h= . The usage of free electron formulae here is justified by the very short mean free path even in the samples with relatively high Tc. The

significant disorder scattering in each of the samples is expected to smear out any fine structure

in the electronic density of states leaving an almost free electron like structure.

I will demonstrate in a later chapter that strong el-el interactions are present in disordered

NbN. Since ne

RH1

= is truly valid only in the absence of el-el interactions, we have calculated kFl

at the highest temperature of our measurements i.e. 285K, where the interactions are believed to

be small31. The set of films described here range from the moderately disordered (kFl ~10) to

very strongly disordered, with kFl ~0.4.

28

We begin with the structural characterization of the NbN films. The films have a NaCl

structure and grow epitaxially on MgO substrates oriented along the [1 0 0] direction. Figure

S1(a) shows the -2 scans for NbN films of varying disorder, deposited with the same gas ratio

but different sputtering powers. The [2 0 0] MgO peak is ~43 and the [2 0 0] NbN peak is

observed ~ 41 We find that the optimal NbN film, i.e. the one with the highest Tc is at a

structural phase boundary between NbN and Nb2N. Films for which the sputtering plasma has a

greater Nb concentration than the optimal one form a different phase, Nb2N which is non-

superconducting.

An example of this is the sample

deposited at 250W shown in Figure S1(a). This is

evident from the fact that it does not show the

NbN peak, but instead shows a peak ~38.6,

which corresponds to the [1 0 1] peak of Nb2N.

Although many of our samples showed a small

peak corresponding to the Nb2N impurity phase,

on comparing the area under the peaks we

estimate that the volume fraction of the impurity

was ~ 0.5% for these films. Only such films, with

a significant NbN peak and very small (or absent)

Nb2N peak were considered in further experiments. The lattice constants of the different films

Figure S2: -scans of samples with kFl~4 and 9 showing epitaxy of the films.

0 60 120 180 240 300 360

kFl~9

Inte

nsity (

arb

. u

nits)

(degrees)

kFl~4

Figure S1 (a) X-ray diffraction spectra for NbN films deposited with a Ar:N2 ratio of 80:20 and sputtering powers 40W, 150W, 200W and 250W. (b) Lattice constants derived from the X-ray spectra for all samples deposited with the same Ar:N2 ratio and different sputtering power.

40 80 120 160 200

4.39

4.40

4.41

4.42

4.43

38 39 40 41 42 43 44 45

(b)

a (

)

Sputtering Power (W)

(200)

MgO

*

*

NbN-40W NbN-150W

NbN-200W

NbN-250W

Lo

g In

ten

sit

y (

arb

un

its

)

2 (degrees)

(200)

NbN

(a)

29

were calculated from the position of the [2 0 0] NbN peak. These were found to vary only by a

small amount [Figure S1(b)].

The epitaxial nature of the films was verified by taking -scans around the [1 1 1]

direction using a 4-circle goniometer. Representative scans for samples with kFl ~4 and 9

showing the peaks separated by 90 are presented in Figure S2. Further verification of the

epitaxial growth is seen from cross sectional TEM images of samples with kFl~3 and kFl~9 in

which the lattice planes are clearly seen to be parallel to the corresponding ones in the MgO

substrate (Figure S3(a) and (b)). The inset in Figure S3(b) shows the high resolution (HR) TEM

image with atomic scale resolution for the ordered NbN film.

Figure S3 (a) and (b) Cross sectional TEM images of NbN films with kFl~3 (a) and 9 (b) showing the lattice planes both in the NbN films and the MgO substrates. A high resolution TEM image is shown for the latter in the inset of 3(b).

From the preceding structural characterization we can see that the NbN thin films are

homogenously disordered and there is no evidence of any physical granularity.

We now proceed to the transport characterization of our films. Figure S4(a) shows the

resistivity vs. temperature for samples with different disorder in the temperature range 300mK to

300K. An enlarged picture of the low-temperature region is shown in the inset. The Tc is

extracted from these curves by recording the temperature at which the resistance reaches 1% of

MgO MgO (b) (b)

NbN kFl~3

kFl~9

NbN

(a)

30

its normal state value just above the transition. In Figure S4(b), Tc is shown as a function of

disorder (kFl). We have observed samples with Tc ranging from ~17K down to samples which do

not show a transition even at 300mK. A number of explanations are afoot for this strong decrease

of Tc with disorder, like Coulomb effects driven by loss of screening and phase fluctuations in

the superconducting order parameter. At present I will not elaborate on these and defer our

understanding of this decrease in Tc to the section V. In Figure S4(a) we see that all except the

least disordered sample have a negative temperature coefficient of resistance that becomes more

and more pronounced as disorder in increased. This is also evident from Figure S4(c) where the

resistivity at the peak (for superconducting samples) and at 300mK (for non-superconducting

samples) as well as the resistivity at 285K have been plotted in the log scale as functions of kFl.

This figure clearly shows that the disparity between low temperature and room temperature

resistivities gets magnified with increasing disorder. Hall effect measurements have been carried

out for samples of different disorder and the number density values so obtained are also plotted

in Figure S4(c) on the right hand axis. The number density for the least disordered sample comes

out to be ~2.63X1029 electrons/m3, which is close to the theoretical estimate32 of 2.39X1029

electrons/m3. This estimate has been arrived at by counting the number of electrons per Nb atom

and assuming that the nitrogen electrons do not contribute to electrical conduction. The

calculation assumes perfect stoichiometry, which suggests that our least disordered sample is

very close to stoichiometric and the disordered samples have varying degree of Nb vacancy.

50 100 150 200 250 3000.1

1

10

100

1000

10000

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10

1

10

100

1000

10000

0 2 4 6 8 10 12 14 16 18 200.1

1

10

100

1000

10000

10.1

(

m

)

T (K)

kFl~0.42

(a) (b) (c)

Tc (K

)

kFl

(

m

)

kFl

0

5

10

15

20

25

n (

X 1

028m

-3)

(

m

)

T (K)

Figure S4: (a) vs. T for films with kFl~0.42, 0.49, 0.82, 1.23, 1.58, 1.68, 1.96, 3.27, 3.65, 4.98, 5.5, 8.02, 8.13, 8.82 amd 10.12. The inset shows an expanded plot of the low temperature region. (b) Tc vs. kFl for all the films and (c) Peak (black squares) and room temperature (blue triangles) values of resistivity as well as number density (purple stars) as functions of kFl.

31

The upper critical field (Hc2) and coherence length were also measured for the different

samples. For the less disordered samples, for which the upper critical fields were not directly

accessible at 12T, they were measured through -T measurements at different fields and then

calculated from the dirty limit relation 33 cTT

c

ccdT

dHTH

=

= 22 693.0)0( where dHc2/dT was

calculated at the point where the resistance becomes 90% of its normal state value.

10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

6 8 10 12 14 160

2

4

6

0 2 4 6 8 100

5

10

15

20

25

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16

kFl~8.7

(

m

)

T(K)

B=0

B=5.8T

(a)

(d)(c)

Hc2(T

) (T

)

T (K)

Hc2 (

0)

(T)

kFl

4

5

6

7

8

(

nm

)

(b) B=11T

(

m

)

T(K)

B=0

kFl~2.2

Figure S5: (a) and (b) vs. T at different magnetic fields for samples with kFl~8.7 and 2.2 respectively. (c) Hc2(T) vs. T for films with kFl~ 8.7, 7.09, 5.5, 4.6, 4.38, 3.91, 3.5, 2.85 and 1.5. (d) Hc2(0) and for all the films.

Representative plots of -T measurements at different fields are shown in Figures S5(a)

and S5(b). In S5(b), the Hc2(0) is directly read off the graph as the field for which the

superconducting downturn is no longer present. The dHc2/dT lines close to Tc for all the samples

that are measured using the dirty limit relation are shown in Figure S5(c). The Ginzburg-Landau

coherence length is then extracted from the upper critical field using)0(2 2

0

c

GLH

= . Both Hc2

and GL have been plotted as functions of kFl in Figure S5(d). The initial increase in Hc2 with

32

disorder (5.5dkFld10) seems counter-intuitive, but can be understood in the following way: In

the low disorder regime, Andersons theorem applies to some extent and the Tc and (the

superconducting energy gap) are not significantly affected. Therefore, in this regime, the Pippard

coherence length, BCS (=vF/), where vF is the Fermi velocity, does not change either.

However, GL depends on both BCS and the mean free path l ( lBCSGL = ), the latter of which

decreases with disorder even in the low disorder regime, resulting in a net decrease in GL. At

higher disorder, starts to decrease, thereby increasing BCS. The competition between BCS and l

results in a net increase of GL and decrease in Hc2.

Finally, I make some observations on the sources of error that affect the basic quantities

that we measure. The single largest source of error comes from the measurement of thickness of

the samples (using a stylus profilometer), which affects RH and measurements. This error is

~15%. In addition the values also depend on the widths and lengths of the patterned devices.

However, since all the samples are deposited using a shadow mask, there will be no relative error

and we estimate the absolute error to be not more than 1-2%, which is the tolerance of our mask.

In comparison to these mechanical quantities, the electronic quantities (V and I) are measured to

within a few parts per million. Therefore the errors in and RH are estimated to be ~15% and in

kFl ( RH1/3/ thickness-2/3) ~10%. The temperature measurements are accurate to within 0.1%.

In the measurements from 1.8K to 300K, using the 4He VTI, the sample and sensor are both in

contact with the same copper block, and also the experiment is carried out in a He atmosphere,

ensuring that thermal gradients are negligible. In the 3He insert, where the sample is in vacuum

there are two sensors, one near the 3He-pot and the other close to the sample. These two sensors

are a few centimeters apart and can have a gradient of up to 100mK. However, the distance

between the sample and the sensor closest to it is much smaller, therefore we do not expect a

significant error in the estimation of temperature.

IV. Understanding the normal state of disordered NbN

A number of factors control the transport properties of a metal. In most metals, resistance

increases with temperature primarily due to phonon-scattering and follows Mattheissens rule34.

In the weakly disordered regime, the role of disorder is to increase the temperature independent

33

part of the resistance. However, strong disorder can localize states at the band edges and can

drive the system into an insulating state30, even for a partially filled band, when the all the states

up to Fermi level get localized. This is known as the Anderson metal-insulator transition (MIT)

and corresponds to kFl~1. The scaling theory of localization3 studies the effect of disorder

scattering right down to the MIT and describes the eigenstates of a system of size (2L)d (where d

is the dimensionality) in terms of the eigenstates of the corresponding systems of size Ld. i.e. it is

used to study how the properties of a system scale with the system size. It predicts that the metal-

insulator transition is continuous in 3D without any minimum conductivity and predicts power

law dependence of conductivity on temperature.

Another phenomenon that affects the transport properties is el-el Coulomb repulsion,

which becomes important in disordered systems because of the effective loss of screening in the

diffusive transport regime. Coulomb repulsions have also been shown to change the sign of

d/dT. In the weak scattering regime in 3D, the Altshuler-Aronov35 theory for el-el correlation

predicts a conductivity of the form ,

DTk

Fe

B

hh)2

33

4(2

3.1

4

1 ~2

2

= .. (1)

where D =

is the diffusivity, and F

~is a Fermi liquid parameter such that

F~

23

34

is of the order of unity.

Since the temperature dependence of resistivity cannot be unambiguously attributed to

either disorder scattering or e-e repulsion, one can study Hall effect as a function of temperature

to distinguish between the two. Shapiro and Abrahams36 have applied the scaling theory of

localization to the Hall effect and they predict that for a 3D disordered system without e-e

interactions, RH will retain its metallic character down to the metal-insulator transition i.e. it will

be temperature independent. On the other hand, according to the A-A theory, xy is unaffected

by temperature 37 and therefore RH=xy/xx2 varies twice as fast with temperature as i.e.

=

2H

H

R

R (Ref 38).

34

In this chapter I will present a detailed study of resistivity and Hall effect studies to

understand the effect of these different phenomena on the normal state properties of disordered

NbN films.

3 6 9 12 15 18

0

1

2

3

4

5

6

7

8

0 30 60 90 120 150 1800

1

2

3

4

5

6

7

8

180 210 240 270 300

0.736

0.738

0.740

0.742

0.744

0 1 2 3 40

200

400

600

800

1000

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(d)

1.68

5.50

kFl

(

10

5

-1 m

-1)

T1/2

(K1/2

)

4.98

3.65

3.27

(c)

1.68

(b)

3.27

3.654.98

(

10

5

-1 m

-1)

T (K)

kFl

5.50

(

m

)

T (K)

kFl ~ 8.82

~0.002 m

kFl~0.82

kFl~0.49

kFl~0.42

(

-1)

T (K)

kFl~10.12

(a)

(

m

)

T (K)

kFl~8.82

Figure S6: (a) vs. T for samples with kFl 10.12 and 8.82. The inset shows an expanded view for the sample with kFl ~8.82 at high temperatures, vs. T for samples with wide range of disorder. (c) vs. T for the most disordred samples at low temperature. (d) vs. T1/2 for the same samples as in (b)

We begin by looking at the role of phonon scattering. Figure S6(a) shows -T for samples

with kFl~ 10.12 and 8.82 up to 300K. The positive slope seen for the sample with kFl ~ 10.12

indicates that this sample obeys Matthiessens rule and we can attribute the temperature

dependence largely to phonon scattering. In the next sample (kFl~8.82), only a small region at

higher temperature has a positive slope (see expanded view of -T for the film with kFl~ 8.82

shown in the inset) which can be attributed to the el-ph scattering. This results in a resistivity

increase of 0.002 m in the interval 240 K to 300 K. At lower temperatures, this sample too

has a negative temperature coefficient of resistance. Therefore the contribution of el-ph

scattering to the overall (T) is less than 2% for this sample. For the more disordered samples

35

where we do not see any positive slope at all in the -T curves, this contribution would be

negligible compared to impurity scattering.

To explore the reasons for the negative d/dT for all the films with kFl 8.13, we turn

our attention to the precise temperature dependence of for the disordered samples. For the

more disordered samples, varies linearly with T from 40 K to 150 K (Figure S6(b)). The films

with kFl>1 show an upward deviation from the linear T behaviour at temperatures below ~40 K,

but for the samples with kFl

36

In Figure S8(a)-(b) we plot ( )

( )

=

KR

KRTR

R

R

H

HH

H

H

285285)(

vs.

=

)285(

)285()(K

KT

for

all NbN samples with 1 kFl 8.13. All the curves are linear within error bars of our

measurements and the slope,

=

dR

Rd

H

H has a universal value 0.680.1. The

variation in is shown as a function of kFl in the inset of Figure S8(a), which emphasizes the fact

that there is no monotonic trend in as a function of kFl. The observed value of =0.68 differs

significantly from the predictions from localization (=0) as well as AA theory of interactions

(=2).

The first argument to support being different from 2 is the following: Since we expect

both localization and interaction effects to be present, could in principle take any intermediate

value between 0 and 2. However, this scenario can be ruled out for two reasons. First, since for

0 2 4 6 8 10 12-16.48

-14.12

-11.77

-9.42

-7.06

-4.71

-2.35

0.00

0 2 4 6 8 10 12

-3.23

-2.69

-2.15

-1.61

-1.08

-0.54

0.00

0 50 100 150 200 250 3000.0

0.4

0.8

1.2

1.6

2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

0.3

0.6

0.9

1.2

B (T)

(a) kFl ~ 3.27

12K 25K

50K 75K 100K

150K 195K

240K 285K

xy (

10

-4

m

)

B (T)

(b)

20K 40K

70K 105K

145K 180K 232K

285K

xy (

10

-4

m

)

kFl ~ 8.82

(c)

RH (

10

-10m

3C

-1)

T (K)

RH (

10

-10m

3C

-1)

(d)

(

m

)

T (K)

RH (

10

-10m

3C

-1)

Figure S7: (a) and (b) xy vs. applied field at different temperatures. (c) RH vs. T for samples with kFl ~1.68, 3.27, 3.65, 4.98, 5.5, 8.01, 8.13, 8.82 and 10.12. (d) and RH as functions of temperature for the sample with kFl~3.27, superposed to illustrate the linear relationship between them.

37

samples with larger kFl, el-el interactions should be

more predominant than localization effects, a

systematic deviation of towards 2 should have

been observed with increasing kFl. Secondly, in 3D

the temperature dependence of due to

localization and el-el interactions are different.

Since RH is affected only by el-el interactions, the

linear relation between (T) and RH is not expected

over a large temperature range.

Another possibility is that our films are not

in the weak scattering regime (kFl>>1) so we

should not consider AA theory at all for the more

disordered samples. However, since the level of

disorder in the films shown in Fig. 7 spans a large

range of kFl~3.27-8.13, one would have expected

to be asymptotic to the theoretical value of 2 with

increasing kFl. Such a systematic change was

actually observed in a 2D electron gas 39 in Si

inversion layers where a gradual increase of

towards the theoretical value of 2 was observed as

the sheet resistance Rsq0. No such systematic

variation is observed in our data.

It can also be argued that a match is not

expected as the AA calculation is strictly valid for

the limit H0. The importance of this limit is also

illustrated in Ref. 39. However, taking this limit would not alter our results since for our samples

xy is linear over the entire magnetic field range (Figures S7(a)-(b)).

To check if the observed behavior is specific to NbN, similar measurements were also

performed at Louisiana State University on a 2nm thick Be film with resistance, Rsq3.61 k40.

Figure S8: (a) and (b) RH/RH vs. //// for NbN films with kFl~1.68, 3.27, 3.65 and 4.98 and 5.5, 8.01 and 8.13 respectively. The inset in (a) shows , the slope of the plots as a function of kFl (c) RH/RH vs. Rsq/Rsq for a 2nm thick Be film.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.4

0.8

1.2

1.6

2.0

2.4

2 4 6 80.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

kFl

1.68

3.27

3.65

4.98

R

H/R

H

/

(a) NbN

kFl

Be(c)

R

H/R

H

Rsq

/Rsq

= 0.69

kFl~2

NbN

kFl

5.50

8.01

8.13

(b)

R

H/R

H

/

38

While for this thickness the film is not superconducting the thickness is much smaller that the

coherence length (~20nm) of its superconducting thick counterpart. This film is thus in the 2D

limit with kFl~2. The Rsq(T) and RH(T) measured in this film is summarized in Figure S8(c).

While this film displays a logarithmic temperature dependence of Rsq typical of a disordered 2-D

system, the slope, 69.0

=

sq

sq

H

H

R

Rd

R

Rd , is strikingly similar to the value observed in

disordered NbN. Since at present we are able to compare with only one such Be film, it would be

interesting to investigate whether the value of in the 2D Be films remains robust as a function

of disorder (Rsq), similar to NbN.

The theoretical value of 2 has been observed in 2D electron gas in Si inversion

layers41 in the limit of large sheet resistance and at intermediate magnetic fields (~0.1-0.5 T).

However, decreases from 2 for both very low fields as well as higher field values. The former

is due to the antilocalization effect of magnetic field even at relatively small fields (where the

localization effects get suppressed and increases from zero) and the latter is presumed to be

attributed to breakdown of the low field limit where this calculation is valid. The dominance of

localization effects is also observed in 3D disordered In2O3 films42 where RH measured at very

low magnetic field was reported to be temperature independent. While the very low limit of RH is

below our experimental resolution, we do not observe any non-linearity in xy vs. H at fields

above ~ 0.2 T.

On the other hand, Hall effect measurements on uncompensated Si:As samples43 in the

metallic regime showed that ranges from 0.4 to 0.6 for different samples. This is speculated to

be due to the anomalous Hall effect44 driven by spin-orbit interaction45. For the various reasons

listed above, it is evident that these Hall effect data cannot be understood on the basis of existing

theories and further theoretical considerations may be needed to establish the value of in a

disordered metallic system.

39

V. Tunneling studies, pseudogap, phase fluctuations and phase diagram

To understand the superconducting properties of disordered NbN, we begin by studying

the electronic DOS through tunneling measurements. Figures S9(a), (d) show the conductance

spectra observed in tunneling measurements for samples of Tc ~14.9 and 7.7K at temperatures

ranging from 2.2K to Tc. These spectra are fitted to the tunneling equation after subtracting a

linear background to symmetrize the data. The density of states used for fitting is the BCS

density of states( )

=

22Re)(

iE

iEEN s where is the superconducting energy gap and

(=1/) is the inverse of the lifetime of the quasiparticle excitations but phenomenologically

takes into account all non-thermal sources of broadening. The symmetrized data, along with the

BCS fits are shown in Figures S9(b) and (e). The best fit values of and are then plotted as a

function of temperature along with the theoretical BCS curve for (T) in Figures S9(c) and (f) .

The resistances vs. temperature curves of the underlying films are also shown on the

same graphs. In Figure S9(c) we see that the temperature dependence of closely follows the

Figure S9: (a) and (d): Raw dI/dV curves for tunnel junctions with kFl~7.9 and 3.5. (b) and (e): Symmetrized and fitted conductance curves for the same two junctions as in (a) and (d) respectrively. (c) and (f) and as function of temperature, with the R-T measurement on the underlying films shown alongside on the right hand axes.

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

1

2

3

4

5

2 4 6 8 10 12 14 16 18 200.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

-3 0 30.0

0.2

0.4

0.6

0.8

1.0

1.2

-6 -3 0 3 60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

(

m

)

T(K)

,

(m

eV

)

BCS

(f)

(c)

, (

me

V)

T(K)

(

m

)

BCS

-10 -5 0 5 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.9K

2.8K

3.6K

4.4K

5.2K

6K

6.8K

7.5K

8.8K

dI/d

V (

-1)

V (mV)

(d)

kFl ~ 3.5

kFl ~ 3.5

(e)

1.9K

2.8K

3.6K

4.4K

5.2K

6K

7.5K

8.1K

dI/

dV

(

-1)

V (mV)

-15 -10 -5 0 5 10 150.0

0.5

1.0

1.5

kFl ~ 7.9

(a)

1.9K

3K

4.5K

6K

7.5k

9K

10.5K

12K

13.5K

14.4K

15K

dI/

dV

(

-1)

V (mV)

(b)k

Fl ~ 7.9

3K

4.5K

6K

7.5K

9K

10.5K

12K

13.5K

15K

dI/d

V (

-1

)

V (mV)

40

BCS curve and goes to zero at Tc. However, the sample with Tc ~7.7K shows a significant

deviation from the BCS curve and does not seem to go towards zero close to Tc. At the same

time, the value of increases with temperature and becomes comparable to close to Tc. If we

extrapolate (T) beyond Tc (dashed line in Figure S9(f)), we can assume that the gap in the DOS

will persist in the normal state. Such a gap above Tc is known as a pseudogap and is usually

seen in the underdoped high Tc cuprates. Unfortunately, it was not possible to directly measure

the tunneling DOS above Tc using these planar tunnel junction devices because of the low

resistance of our tunnel junctions which get overpowered by the resistance of the NbN film in the

normal state. However, this problem was overcome by using a scanning tunneling microscope

(STM) as the tunnel barrier in that case is of the order of hundreds of megaohms. Scanning

tunneling spectroscopy (STS) was carried out on the NbN samples by my lab-mates Anand

Kamlapure and Garima Saraswat.

Figure S10: Temperature evolution of the DOS for samples with Tc: (a) 11.9K; (b) 4.1K; (c) 2K and (d)

41

Figure S10 shows the temperature evolution of the tunneling DOS in the form of a colour

plot for samples of Tc ~ 11.9K, 4.11K, 2K and one that shows an incomplete transition even at

300mK. The spectra shown here are obtained by taking the average of 32 spectra measured along

a 150nm line. In each of the plots, the x-axis is temperature and the y-axis is the bias voltage.

The color scale shows the normalized conductance values. These conductance values have been

arrived at after subtracting a V-shaped background from the conductance curves which is due to

AA type corrections to the DOS that persist up to high bias and the highest temperatures at

which STS has been performed here. In the lower part of each panel, the R-T for the same sample

is also shown. In Figure S10(a), for the sample with Tc 11.9K, we see that the superconducting

gap feature vanishes at Tc whereas for the other, more disordered, samples it is seen to persist

well above Tc. This persisting gap is called the pseudogap and the temperature at which it finally

vanishes is known as T*.

Figures S11(a)-(d) show the resistivity vs. perpendicular magnetic field for strongly

disordered samples with kFl~0.42, 0.49, 0.82 and 1.23. The samples with kFlTc. T* is apparently independent of disorder in this regime. In

region III, superconductivity is suppressed altogether and the samples do not show any

superconducting transition at all. In addition, a peak is observed in the MR which vanishes close

to T*.

42

We now discuss each of these regimes in some detail and try to understand which

physical interactions are dominant in each of them. The superconducting order parameter is

represented as a complex number ei, where is the magnitude of the binding energy of the

Cooper pair and is the phase of the macroscopic wave function which must be constant

throughout the sample in order for it to be in the zero resistance state. Destruction of the

superconducting state, therefore, can have two main causes: reduction of due to weakening of

the pairing interaction, and phase decoherence between different parts of the sample without

suppression of . Within the BCS theory, and Tc are coupled together by the relation

0 1 2 3 4 5 6 7 8 9 10 11 125000

10000

15000

20000

25000

0 2 4 6 8 10 122000

3000

4000

5000

6000

7000

0 2 4 6 8 10 12

2000

2500

3000

3500

4000

0 2 4 6 8 10 120

10

20

30

40

50

60

70

0 2 4 6 8 10

-150

-100

-50

0

50

0 2 4 6 8 10

-90

-60

-30

0

30

60

300mK

500mK

750mK

1K

1.25K

1.6K

2K

kFl~0.42

(

m)

0H (T)

(b)

300mK

500mK

750mK

1.1K

2K

(

m)

0H (T)

kFl~0.49

(c)300mK

600mK

900mK

1.2K

2K

kFl~0.82

(

m)

0H (T)

(d)

300mK

500mK

1K

2K

3K

(

m

)

0H (T)

kFl~1.23

0H

c2~7.55T

(f)(e)

2K 2.5K 3.5K

4.5K 5K 5.5K

6K 8K

(

)

(0)

(

m)

0H (T)

kFl~0.82

kFl~0.49

2K 2.5K 3K

3.5K 4K 4.5K

5K 5.5K 6K

(

)(

0) (

m)

0H (T)

Figure S11: as a function of magnetic field at different temperatures for four strongly disordered NbN thin films with kFl~(a) ~0.42, (b) ~0.49, (c) ~0.82 and (d) ~1.23. The samples with kFl

43

2/kBTc=constant and the transition

temperature is when goes to zero.

Here phase fluctuations have no role to

play in the superconducting transition.

In region I of the phase diagram,

Tc reduces with increasing disorder, but

the superconducting gap vanishes at Tc

in accordance with BCS theory. We

believe that this decrease in Tc can be

attributed to two effects. The first is

electron-electron (el-el) Coulomb

repulsion. Due to disorder scattering,

the electrons are in the diffusive

regime and electron-screening is less

efficient. This effectively weakens the

attractive Cooper pairing, reducing

(Ref. 46 , 47 ). The second effect

arises because with increasing disorder, not only do the band edges get localized, but also the

bandwidth increases, thus reducing the DOS at the Fermi level (EF) (Ref. 48). Both these effects

can be included in a modified BCS relation49:

=

*)0(1

expVN

T Dc where D~Debye

temperature, V is the attractive electron-phonon potential, N(0) is the DOS at EF and * is a

Coulomb pseudopotential. Although the current theoretical development is not sufficient to

calculate * and attempt a quantitative fit, the combination of enhanced * and suppressed N(0)

qualitatively explains our data.

As disorder is increased and we enter regime II, an additional energy scale i.e. the

pseudogap temperature T* becomes relevant. In this region, Tc continues to decrease

monotonically but T* does not vary significantly. These two temperatures can be associated with

two energy scales: The strength of the pairing interaction and the phase stiffness of the

Figure S12: Phase diagram of strongly disordered NbN, showing Tc and T

* as functions of kFl. Tc is obtained from transport measurements while T* is the crossover temperature at which the low bias feature disappears from the observed tunneling conductance. The samples with kFl

44

superfluid J. In conventional BCS

superconductors, J>> and therefore

is not relevant for the

superconducting transition. In order

to compare these two energy scales

for our system, we estimate J using

the relation: *)4/()( 2 manJ sh=

where a is the distance over which

the phase fluctuates, ns is the

superfluid density and m* is the

effective mass of the electron. ns is

obtained from low temperature

penetration depth measurements12

and a is taken to be ~ (Ref. 27). The values of J thus obtained are plotted in temperature units

against Tc in Figure S13. A dotted line separates regions I and II in this figure. We can see that

in region I, J>>Tc, and therefore the superconducting transition will be governed by the

temperature at which (T) becomes zero. On crossing over to region II the superfluid density is

reduced due to disorder scattering and J~kBTc. This renders the system susceptible to both

quantum and thermal phase fluctuations. It has also been observed that the ns values for samples

in regime II deviate from the dirty limit BCS predictions both at zero and finite temperatures, but

these deviations can be accounted for by the presence of dissipation and phase fluctuations12.

This certifies the assertion that phase fluctuations are the dominant phenomenon in this regime.

The STM studies also observe that the strongly disordered samples show a spontaneous

inhomogeneity in the DOS6 as predicted in the literature15,18. Therefore in this regime, the

superconducting state is destroyed due to phase fluctuations between the superconducting islands

while phase incoherent Cooper pairs continue to persist up to T*.

In region III, there is no superconducting ground state, but superconducting correlations

are expected to persist as T* is finite at the boundary between regions II and III. We therefore

believe that in this regime, the system comprises of superconducting islands that are unable to

Figure S13: Superfluid stiffness (J/kB) and penetration depth ((T0)) for NbN films with different Tc. The solid line corresponds to J/kB=Tc. Regions I and II corresponding to the phase diagram are segregated by the dashed vertical line.

3 6 9 12 15 181

10

100

II

J/

k B (

K)

Tc (K)

I

0

2000

4000

6000

(0

) (n

m)

45

form a zero resistance ground state

due to quantum phase fluctuations.

The MR peak vanishes close to T*

(Figures S11(e) and (f)) and is

therefore attributed to

superconducting correlations.

The peak can be understood

on the basis of the superconducting

islands scenario as follows16: at low

fields, MR is positive as the

superconducting paths shrink due to

magnetic field. Beyond a certain

field, the superconducting paths become small enough that the chosen conduction path now goes

through the normal regions. The gradual decrease in MR at higher fields is because of the slowly

increasing normal regions as the superconducting islands shrink further. The peak field (Hpeak) is

then interpreted as the field at which the superconducting correlations are almost destroyed and

is expected to evolve from Hc2 for the superconducting samples. To verify this, we plot Hc2(0 or

300mK) and Hpeak(300mK) as functions of kFl in Figure S14. For the strongly disordered

samples, Hc2 values are taken as the field at which the resistance reaches 90% of its normal state

value and for the more ordered samples whose Hc2 is not directly accessible up to 12 T, we use

the dirty limit formula as in Chapter 3. We observe that Hc2 decreases with disorder and

smoothly evolves into Hpeak for kFl1) and Hp (for kFl

46

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47

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48

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49

Chapter I: Introduction

Superconductivity is a phenomenon that is extremely resilient to disorder. Experimentally

it is seen that many superconducting materials continue to show superconductivity with

reasonably high critical temperatures even when they are significantly disordered. Indeed, many

of the common superconducting materials used for making superconducting magnets are alloys

like Nb-Sn and Nb-Ti (Ref. 1). Even the high-Tc superconductors like cuprates and pnictides are

doped, disordered materials. The earliest theoretical understanding was due to Anderson2 who

said that as long as the disorder is caused by non- magnetic impurities*3 it does not significantly

affect the superconducting critical temperature. However, this turned out to be true only at low

disorder and is in fact invalid for many experimental systems 4 , 5 , 6 , 7 , 8 . At strong disorder,

electronic and superconducting properties undergo drastic changes, to the extent that both

metallicity and superconductivity can be destroyed completely by sufficiently strong disorder9,10.

In recent times, a number of novel phenomena11,12,13,14,15,16,17,18,19,20,21 have been observed in

disordered superconductors